Dielectric Analysis Of Pharmaceutical Systems (Pharmaceutical Science Series)

Dielectric Analysis of Pharmaceutical Systems

Dielectric Analysis of Pharmaceutical Systems Duncan Q.M.Craig Centre for Materials Science, School of Pharmacy, University of London

UK Taylor & Francis Ltd, 4 John St., London WC1N 2ET USA Taylor & Francis Inc., 1900 Frost Road, Suite 101, Bristol, PA 19007 This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledges’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Copyright © Taylor & Francis 1995 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-203-30257-5 Master e-book ISBN

ISBN 0-203-34468-5 (Adobe eReader Format) ISBN 0-13-210279-X (Print Edition) Library of Congress Cataloging in Publication Data are available

Dedicated to Susan

“Everything should be made as simple as possible, but not simpler” Albert Einstein, quoted in Readers Digest, Oct. 1977

Contents

1. 2. 3. 4. 5. 6. 7. 8.

Preface Acknowledgments List of symbols Principles of dielectric spectroscopy Methods of dielectric measurement Dielectric analysis of solutions The analysis of colloids and suspensions Dielectric analysis of solids The analysis of polymeric systems Dielectric analysis of biological systems Conclusions References Subject index

viii x xii 1 37 48 87 130 154 188 220 224 237

Preface The characterisation of the materials and tissues involved in the delivery of drugs to the body is of vital importance for the development of effective medicines. There is therefore a need to explore the use of new analytical methods in order to gain a greater insight into the structure and behaviour of drugs and medicines. This text describes the theory and uses of dielectric spectroscopy, a technique which is well established in the fields of physics, polymer science and colloid science but which has only recently been introduced to the pharmaceutical sciences. As will hopefully become clear to the reader, the technique may yield information on both dosage forms and biological tissues which is of considerable use in the development of medicines. At present, neither the theory nor the applications of dielectric analysis are widely known within the pharmaceutical sciences, hence this book is intended to redress that balance. Furthermore, it is hoped that the book will be of interest to others working in the field of dielectric spectroscopy, as it describes a novel application for the technique. It should be emphasised that this text takes a slightly different approach to others written on the subject of dielectric analysis, over and above the novelty of outlining pharmaceutical applications. A large proportion of pharmaceutical scientists have not had an extensive training in mathematics, hence while a sound basic description of the theory is absolutely necessary in order to interpret the data, it is hoped that this has been explained in such a way as to be accessible to a wide readership without loss of scientific rigour. Similarly, an outline of the pharmaceutical aspects of the various dosage forms has been included for the non-pharmaceutical scientist. It should also be emphasised that the field of dielectric analysis is very large indeed and no single text can cover all the aspects of the studies that have been performed, even in a relatively focused text such as this one. What is intended, however, is that the reader should become aware of the type of information that may be obtained using the technique in any of the areas under discussion and where more information on any of these subjects may be found. In this way, it is hoped that this book will aid the growing recognition of dielectric analysis as a useful and interesting novel means of pharmaceutical analysis.

Acknowledgments A number of individuals made positive contributions to the writing of this book and I would like to particularly thank the following: Professor Robert Hill, Department of Physics and Mathematics, Kings College, University of London, and Professor J.M.Newton, School of Pharmacy, University of London, for helpful comments and criticism. The staff of the Library, School of Pharmacy, for help during the many literature searches. The various authors who were kind enough to send reprints of their work. Dr Susan Barker and Dr Kevin Taylor (and the staff of the ‘office’) and the research group for persistent encouragement.

List of symbols a= c= e= g= h= i= k= l= n= r= u= v= x= A= B= C= E= ET= F= F0= G= ∆G= ∆H= I= J= L= N= N0 = P= Q,q= R= T= T0= V= X= Xc= X2= W=

radius of a sphere speed of light charge on an electron correlation factor Planck’s constant square root of −1 Boltzmann’s constant, rate constant length electrons/unit volume, refractive index, correlation factor for relaxation time/temperature relationship, porosity distance mobility of ions velocity of electron movement distance, mole fraction area, ampere, constant in WLF equation constant in WLF equation capacitance field strength, induced emf, activation energy transition energy from ground to excited state force, ‘inner field’ of molecule, formation factor resonance frequency conductance free energy enthalpy current current density inductance number of molecules per unit volume Avogadro’s constant polarisation charge resistance, gas constant temperature melting point voltage, molar volume mole fraction reactance solubility energy, power

Z= = = ∆= = 0= r= s=

impedance, ionic charge polarisability phase angle, dielectric decrement, solubility parameter propagation constant permittivity permittivity of free space relative permittivity static permittivity = resistivity = charge density, conductivity, specific surface area = relaxation time = mean residence time in hydration shell r susceptibility, crystallinity = = frequency MW=Maxwell-Wagner crossover frequency

1 Principles of dielectric spectroscopy 1.1 DIELECTRIC SPECTROSCOPY AS AN ANALYTICAL TECHNIQUE Dielectric spectroscopy involves the study of the response of a material to an applied electric field. By appropriate interpretation of the data, it is possible to obtain structural information on a range of samples using this technique. While the use of dielectric spectroscopy has previously been largely confined to the field of physics, the generality of dielectric behaviour has led to the technique being used in more diverse fields such as colloid science, polymer science and, more recently, the pharmaceutical sciences. There are two reasons why one would perform dielectric studies. Firstly, the data will give information on the electrical properties of the sample. This is of interest from a purely theoretical viewpoint but also has practical application in the electronics industry, particularly in the development of semiconductor devices and in the characterisation of insulators. Secondly, the technique can be used as an analytical tool whereby the dielectric data is related to other properties such as changes in crystal structure or gel morphology. It is this second application on which subsequent discussions will be focused. As the majority of this book will be dedicated to giving examples of established or potential uses of dielectric spectroscopy within the pharmaceutical sciences, it would be inappropriate to discuss these uses of the technique at length here. However, some general comments on the method may be helpful at this early stage. The term ‘pharmaceutical sciences’ covers an extremely wide range of disciplines including molecular biology, materials science, physiology etc. This wide range of topics reflects the fact that pharmaceutical science includes any discipline which is relevant to the development of drugs and medicines. Consequently, the problems faced by pharmaceutical scientists are so diverse that it is necessary to have a similarly wide range of techniques available with which to study the various systems under investigation. There will therefore always be a need for further analytical methods to be introduced into the field. As will be shown later, most pharmaceutical systems may be described as dielectrics, which for present purposes may be defined as materials which contain dipoles. In principle, therefore, the majority of such materials may be studied using the technique. The use of the information obtained may be broadly divided into two categories. Firstly, dielectric data may be used as a fingerprint with which to compare samples prepared under different conditions; this therefore has implications for the use of dielectric spectroscopy as a quality control tool. Secondly, each spectrum may be interpreted in terms of the structure and behaviour of the sample, therefore leading to more specific

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information on the sample under study. Both approaches are useful and obviously require different levels of understanding regarding the theory behind the technique. It is also useful to consider the type of information that may be obtained from the spectra. Techniques can be very broadly divided into those which examine molecular structure (e.g. IR, NMR) and those which examine the physical arrangement and behaviour of molecules within structures (e.g. rheological measurements, DSC). Dielectric spectroscopy tends towards the latter category, although information on molecular structure may also be gained. As will be demonstrated, the technique is also useful for systems with complex physical structures, in which the number of different chemical components present is so high that many techniques are rendered inapplicable. As with any technique, there are associated advantages and disadvantages. The wide applicability of the approach to systems of pharmaceutical interest and the usefulness of the information obtained will be outlined in subsequent chapters. In addition, the sample preparation technique is generally very simple. For example, low frequency measurements may be made via the application of two electrodes to the sample, either by attachment or immersion. Samples with a range of sizes and shapes may therefore be studied; solid compacts, powders, gels or liquids may be easily measured. Furthermore, in most cases the technique is non-invasive, as the voltages used are small. Finally, the method and conditions of measurement may be varied. For example, the sample may be examined under a range of temperatures, humidities, pressures etc., allowing direct investigation of the system in conditions which would preclude the use of most other techniques. The principal disadvantages of the technique with respect to pharmaceutical uses are firstly that not all samples may be usefully analysed, a fault which is common to all analytical methods. For example, a sample may give such a low response that measurement is outside the range of the instrument. Alternatively, the response of a component of interest may be swamped by that of a less interesting constituent. The second disadvantage lies with the general inaccessibility of the dielectrics literature to pharmaceutical scientists. This has arisen largely for historical reasons, as most of the dielectric literature has been written on the (hitherto) reasonable assumption that any reader interested in the subject will already have a prior knowledge of dielectrics (or at least physics). In order to attempt to alleviate this problem, a detailed theoretical chapter is given in the present text which, it is hoped, will enable the reader not only to understand the arguments used in subsequent chapters but also to extract relevant information from the dielectric literature. It should be emphasised, however, that even this relatively lengthy introduction will only give a broad outline of the principles involved and is not a comprehensive treatise on all dielectric theories that may be encountered. A number of texts may be referred to for further details regarding the technique (e.g. McCrum et al., 1967; Daniel, 1967; Grant et al., 1978; Blythe, 1979; Jonscher, 1983). Furthermore, while broad agreement exists regarding the general principles of the technique, controversy regarding the ‘fine detail’ interpretation of dielectric data remains. This should in many ways act as encouragement to pharmaceutical scientists, as by providing data on a range of systems it is possible to contribute to the debate by providing experimental evidence to validate or refute different approaches.

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The final disadvantage of the technique is that there is not as great a precedent for the use of the technique within the pharmaceutical literature as exists for other techniques such as IR, NMR etc., hence there is still a great deal to be learnt regarding the capabilities of the technique. It is intended that this text will go some way to correcting this by firstly bringing together the work that has been performed on pharmaceutical systems to date and also discussing work which, while not falling under the direct classification of ‘pharmaceutical’, is nevertheless of direct relevance to the field.

1.2 PRINCIPLES OF ELECTRICITY 1.2.1 Electrostatics Electrostatics is defined as the study of electric charges at rest. This subject provides a good introduction into the more complex concepts of the behaviour of materials containing moving charges. The first quantitative theory of electrostatics can be attributed to Coulomb, who stated that the force between two point charges is directly proportional to the product of the charges divided by the square of their distance apart, or

(1.1) where F is the force between two point charges Q1 and Q2 separated by a distance r. This equation was later developed to include the proportionality constant

(1.2) where is the permittivity of the medium in which the charges lie, a concept which will be developed later. When the medium between the two charges is a vacuum, then the permittivity is denoted by 0, known as the permittivity of free space. This has a constant value of 8.85×10−12 Fm−1 where F (farad) is the unit of capacitance (dealt with later). The permittivity of air at standard temperature and pressure is 1.0005 0, hence the two values are usually taken as being equal. If one accepts the concept of forces acting on an electric charge, then it follows that regions of space exist whereby any charge introduced into that region will experience a force. There therefore exists a field of force, the field strength being denoted by E, given by

(1.3) where Q is the magnitude of the point charge introduced into the field and F is the force

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acting on that charge. If one now considers a point charge Q introduced into a field generated by a charge Q0 at a point distance r, then by Coulomb’s law (1.2)

As E is the force per unit charge (1.3)

(1.4) hence E is independent of the magnitude of Q, the charge that has been introduced into the field. Instead, E depends on Q0, r and . It should be noted that E is a vectorial quantity, as the field will have a direction as well as a magnitude. Another useful concept is that of charge density ( ), representing the charge per unit area of a conductor. If one considers a sphere of radius r having a charge Q0 uniformly distributed across the surface, then that conductor will have a charge per unit surface area given by

(1.5) hence by (1.4)

(1.6) at the surface of the conductor. Continuing from the concept of a charge experiencing a force, one may now consider the movement of a charge from A to B in such a direction that the movement opposes that of the field.

As the field will tend to force a charge to move in the opposite direction to that shown above, then work must be performed to allow movement from A to B. An analogous situation is the movement of a point mass against gravity. In moving that charge to B, the charge now has stored energy which will be lost when the charge returns from B to A.The change in energy that occurs when a charge is moved from one point to another is therefore dependent on the magnitude of both the charge and the field. If one then extends the argument to the movement of a charge from infinity (or at least a position where there is negligible field) to a specific position in a field, the energy required (W) is related to the magnitude of the charge (Q) by a parameter known as the potential. This quantity may be given in JC–1 (where C is the charge in coulombs), as it is a measure of the energy required to move a charge within a field. However, it is more usually given in

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volts (V), where

(1.7) Potential is in many ways a more useful quantity than field strength, firstly because it is a scalar quantity, hence easier to calculate than the vectorial E, and secondly because it is almost invariably more useful to deal in terms of energy rather than force. If one now considers a charge Q in a field of strength E, generated by a point charge Q0 at a distance x, the force on that charge is EQ (from (1.3)). If the charge is moved a distance x (where x is small compared to x) against E, then the work ( W) performed by the system is given by

thus

(1.8) This is a differential equation which describes the work involved in moving a charge Q a finite distance. If one then wishes to know the work involved in moving a charge from an infinite distance to a distance r from Q0, (1.8) is integrated such that

(1.9) As V=W/O (1.7) then

(1.10) In practice, one is usually concerned with the difference in potential between two points in a field, rather than between a given point and infinity. This is known as potential difference (or p.d.) and is again given in terms of JC−1. Therefore, if the p.d. between the points of a field is 10 V, 10 J of work are performed in moving one coulomb of charge between the two. Another useful concept is the electronvolt, which is in fact a measure of energy and not voltage. This is the work performed in moving an electron between two points in a field, between which is a p.d. of 1 V. The charge on the electron is constant at

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1.6×10−19 C, hence the work involved is 1.6×10−19 J. As the potential difference represents the work performed in moving a charge between two points in a field, it follows that the field and voltage must be numerically related. If a given field (E) is considered uniform, then

(1.11) hence E is equal to the voltage (V) across the distance d. This can be most easily envisaged by considering two parallel plates between which is a uniform field E (Figure 1.1). If a field has a value of 2 Vm−1 and the plates have a separation distance of 3 m, then the p.d. at points a, b and c will be 6 V, 4 V and 2 V respectively. The lines denoting regions where the voltage is equal are known as equipotentials.

Figure 1.1: Diagrammatic representation of potential difference in a uniform field If one now imagines a pair of parallel plates between which is placed an insulating material (for the purposes of the present argument we will define an insulator as a material which does not conduct electricity), we can see that charge is effectively stored on the plates, as shown in Figure 1.2. The amount of charge is related to the potential difference between the plates as, if the p.d. is increased, the charge will also increase, i.e.

The constant of proportionality between the two is the capacitance (C) which may be regarded as the charge-storing ability of a particular system. The unit of capacitance is the farad (F) when the charge is in coulombs and the p.d. is in volts.

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Figure 1.2: Diagrammatic representation of a capacitor within a circuit practice, very large, hence the micro (µF), nano (nF) or pico (pF) farad This unit is, in may be used. One may therefore express the capacitance as

(1.12) It should be noted that the conventional diagrammatic method of representing the battery of a circuit is by using two parallel lines of unequal length, while a capacitor is represented by two parallel lines of equal length. The battery generates a current between the terminals which results in a potential difference across the circuit. Electrons flow from the negative terminal of the battery onto plate Y while also flowing from plate X to the positive terminal of the battery. This process will continue, with charge accumulating on the plates until the potential difference between the electrodes is equal to that of the battery, when charge flow will stop. If one then disconnects the battery and then reconnects the circuit so as to bypass that battery, the charge stored on Y will flow around to X, thus neutralising the charge on the plates. This is known as the discharge of the capacitor and will be dealt with later. The relationship between capacitance and field strength may be seen by considering the charge density at the electrodes, given by Q/A. From (1.6)

Combination with (1.11) gives

thus

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(1.13) The above equation (1.13) is important as it indicates the factors which determine the value of the capacitance of a particular sample. The importance of the area A is simply that the larger the area of the electrodes, the greater the quantity of charge that may be stored on them per unit volt. This is not always strictly true, as edge effects cause nonlinearity in the relationship. However, for the present purposes it is sufficient to consider (1.13) to be valid in this respect The role of the inter-electrode distance (d) may be considered in terms of the potential difference between the electrodes. A capacitor will store charge until the potentials across the cell is the same as that of the battery. Examination of (1.13) shows that for a given p.d., the smaller the distance between the electrodes the greater the charge that may be stored, hence the greater the capacitance. The intrinsic parameter determining the capacitance for any particular material is the permittivity , which will be discussed in the next section. It is also helpful to consider the theoretical zero of potential (i.e. infinity). In practice, one uses the potential of the earth (itself a conductor) as zero. By Gauss’s law, the potential of a conducting sphere is given by

(1.14) where a is the radius of the sphere, hence using (1.12)

(1.15) This shows that the capacitance of a sphere will increase with the radius. When a charged system of small capacitance and an uncharged system of large capacitance are connected in parallel, there is charge flow between the two until the potentials are the same, i.e.

therefore

If one imagines the earth as being a spherical capacitor, consideration of (1.15) shows that the capacitance of the earth is very high, as the radius (a) will be large. In fact, the value is approximately 700 mF (which also gives an indication as to how large a quantity

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the farad is). If C1 is the capacitance of the earth, then the ratio of C1 to C2 will be large, hence Q1/Q2 will be similarly large. This has two implications. Firstly, it means that a charged capacitor, when in contact with the earth, will effectively lose charge. This is the principle of ‘earthing’. Secondly, if charge will effectively always flow to the earth, then it is reasonable to use the potential of the earth as being zero potential, as it will effectively be lower than (or equal to, after earthing) the potential of any system likely to be found in practice. The permittivity of a material is a property which describes the charge storing ability of that substance, irrespective of sample dimensions. A related quantity with which permittivity is often confused is the relative permittivity r, otherwise known as the dielectric constant. This is a useful parameter as the values are more manageable than the absolute values of permittivity. Relative permittivity is given by

(1.16) where C is the capacitance of the sample, C0 is the capacitance of a vacuum in a cell of identical dimensions and 0 is the permittivity of free space. Equation (1.13) therefore becomes

(1.17) It should be noted that permittivity (like conductivity) is a material property, while capacitance (like conductance) is specific to a sample of given dimensions. Table 1.1 gives some typical values of r. The value given for water is for pure water, although in practice it is virtually impossible to remove all the dissolved impurities, hence experimental values tend to vary from those given in the table below.

Table 1.1—Typical dielectric constants of common materials at 20°C Material Dielectric constant Paraffin wax 2 Carbon tetrachloride 2.2 Benzene 2.3 Oleic acid 2.5 Olive oil 3.1 Chloroform 4.8 Glass 5–10 Acetone 21.4 Ethanol 25.7 Methanol 33.7 Glycerin 43.0 Water 80.4

Dielectric analysis of pharmaceutical systems

Hydrogen fluoride Hydrogen cyanide

10

83.6 116

Examination of Table 1.1 shows that materials which contain polar molecules have a high dielectric constant and vice versa for non-polar materials. Again, this concept will be discussed in greater detail later, but for the purposes of the present argument it is useful to consider the response of materials to an electric field in a qualitative sense. When a field is applied across a sample, the dipoles within that sample will react in such a way as to oppose the charge on the plates of the capacitor, a process known as polarisation. A material which is ‘polar’ is readily polarised and will therefore be able to neutralise a greater quantity of charge on the plates. The greater the propensity to polarise, the greater the charge that may be placed (or stored) on the plates for a given potential difference, hence the permittivity of polar materials is greater than that of nonpolar materials. In practice, capacitors may be used in combination within a circuit. The two types of combination are known as series and parallel. A typical parallel circuit is shown in Figure 1.3a. In this case, the applied potential difference across each capacitor is the same. By definition, therefore, the charge across each must vary if the capacitances are different, thus, from (1.12)

and likewise for the other two. The total value of the charge across all three capacitors (QT) is

hence

(1.18) In the case of series circuits, the charge flows across the various elements in sequence, as shown in Figure 1.3b. While the total voltage across the three capacitors must, at equilibrium, equal the voltage of the battery, the potential across each one will not be the same. This can be visualised by considering electrons flowing to plate Y, causing a buildup of charge on that plate which must be neutralised by charge on plate X. However, the presence of capacitors C2 and C3 means that the charge on plate X must also be equal in magnitude (but not sign) to that on the adjacent plate and so on. Therefore, unlike the previous example, the charge on all three must be equal. If the charge is equal across different capacitors then the voltage must vary, thus

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Figure 1.3: Diagrammatic representation of a circuit containing two capacitors a) in parallel and b) in series

hence

thus

(1.19) In summary, therefore, for parallel circuits the p.d. is the same across each element but

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the charge varies with capacitance. The total capacitance is the sum of the individual values. For series circuits, the charge is the same on each capacitor and the total capacitance is the inverse sum of the inverse individual values. 1.2.2 Moving charges The previous section dealt with situations whereby the charges were at rest. However, in most practical situations the charges will be moving, hence the system will not be in a state of static equilibrium. In the first instance, it is useful to consider the situation whereby this movement is unidirectional. The concept of moving charges can best be understood by considering a metal. With no applied potential, electrons move from atom to atom in a random manner, thus while there is movement of charge in the system, there is no net movement in any one direction at a macroscopic level. On applying a potential difference across the ends of the metal, e.g. by connecting a battery, the establishment of the field results in the electrons experiencing a unidirectional force. The electrons then move in the direction of the field via defects in the structure at an average velocity. The rate at which charge moves through a material is the current (denoted by I) and the unit of current is the ampere (A). This unidirectional movement of charge is known as direct current (d.c.). If one imagines electrons drifting through a material of cross-sectional area A, the current is given by

(1.20) where n is the number of electrons per unit volume, v is the average velocity and e is the charge on each electron (1.6×10−19C). Examination of the dimensions of the parameters in (1.20) shows that the current may also be expressed as C sec−1. The average rate at which electrons travel through the material will be dependent on the applied potential difference. The constant of proportionality between the voltage and current is the resistance R (given in ohms, symbol ), thus

(1.21) which is known as Ohm’s law. The resistance is therefore a measure of the difficulty with which charges move through the system. It should also be noted that resistance is also associated with energy loss through the system due to the dissipation of heat. This energy loss is caused by the collision of electrons with atoms or other electrons during the drift through the conductor. This is the principle of the lightbulb, whereby the heat generated by the passage of charge through the resistor will be converted to light energy by the use of an appropriate material as a resistor. Connecting a bulb to a battery will therefore cause electrons to flow round the circuit from one terminal to the other such that the resistor (bulb) is heated and emits visible light. The energy lost by such a system per second is given by

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(1.22) where W is known as the power (given in watts, symbol W). This type of energy loss is known as Joule heating and will be considered in greater depth later. In summary, it may be seen that when a current flows through a material under a potential difference, that material will have a finite resistance and, as a consequence, heat will be lost from the system. It should also be noted that the bulb will only be lit when a current is flowing through the circuit, rather than when the system is at electrostatic equilibrium. A further unit that may be used in place of the resistance is the conductance (G), where

(1.23) The units of conductance are siemens (S) or, more commonly, mho. Ohm showed in his earliest experiments that the resistance of a piece of wire was proportional to its length (L) and inversely proportional to the cross-sectional area (A), i.e.

thus

(1.24) where is the resistivity of that material (units m). There is an equivalence between the resistivity of resistors and the permittivity of capacitors, as both represent the relationship between the extrinsic parameter (capacitance or resistance) and the dimensions of the material, i.e. they are both fundamental quantities of that substance. Further useful parameters are the current density J and the conductivity, (note that the same symbol is used to denote both charge density and conductivity). Conductivity is the inverse of resistivity, thus

hence

(1.25) The current density J is therefore the current per unit cross-sectional area. Resistors may be built up into combination circuits. In the case of parallel resistors, the

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voltage will be the same across each but the currents will differ, the total being the sum of the individual values. The total resistance will therefore be inversely additive. It also follows from this argument that the current will flow through the resistor with the smallest value. This principle is known as ‘following the path of least resistance’. Similarly, for series resistors, the current flowing through each will be the same but the voltage will vary. Both the voltages and the resistances will be additive in this case. The relationship between capacitance and resistance may be visualised by considering that current will flow through a resistor, as, by definition, a material with a resistance also has a conductance. A capacitor, however, will not allow the direct passage of electrons or charges through that material, but will instead act to neutralise that charge by polarisation, whereby a polarisable entity within the sample will reorientate. If one therefore attaches a battery to a system whereby a resistor and capacitor are arranged in series (Figure 1.4), the charge will flow through the resistor with a current (I) of V/R and (from (1.20)) a velocity of I/nae. However, that flow of charge will effectively stop when it reaches the plates of the capacitor in order to establish a potential difference which corresponds to that across the terminals of the battery. Similarly, electrons will flow off the other plate of the capacitor to the battery. The system will therefore reach equilibrium and no current will flow after the plates have stored the maximum amount of electrons, hence a connected lightbulb would flicker momentarily while the plates of the capacitor were becoming charged but would remain unlit thereafter.

Figure 1.4: Diagrammatic representation of a series RC circuit It is also important to consider the way in which a capacitor discharges. Using Figure 1.4 as an example, if one were to disconnect the battery and reconnect the capacitor as shown by the dashed line in the diagram, the potential difference causing the plates to be charged would be lost. The plates would therefore discharge in order to reestablish equilibrium conditions. However, this discharge would not be instantaneous, in the same way that charge flow is not instantaneous (i.e. I is never infinitely high). In fact, the decay of charge Q against time is exponential. This can be shown as follows. The initial charge on the plates is given by

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(1.12) The current flow off the plates is

(1.26) The negative sign shows that Q decreases with t. Combining (1.12) and (1.26) with (1.21) gives

hence

where Q0 is the initial charge. This gives

thus

(1.27) where e is the exponent of natural logarithms. This is the equation of an exponential decay curve. The value RC will have the units seconds, as

and

RC is known as the relaxation time or the time constant of the system. Note that the resistance of the circuit determines the relaxation time, as well as the value of the capacitance, as a capacitor can neither become charged or discharge without the presence of a material with finite conductivity (and hence resistance) through which charge may flow. Two further concepts should be briefly covered at this point. Firstly, the principle of electromotive force. Any battery will have a small internal resistance, for example the resistance of the electrolyte solution in a voltaic pile. This resistance acts in series with the circuit and hence will not be seen when the battery is connected to a resistor of much

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larger value. However, it will act to reduce the voltage across a battery connected to a smaller resistor. The battery is therefore considered to have an electromotive force (or e.m.f., measured in volts) and an internal resistance. When the value of the connected resistor is large, the e.m.f. and measured voltage will be the same. When the resistance is low, the measured voltage will be less than the e.m.f.. Secondly, it is useful to examine one aspect of magnetic field theory. Any conductor carrying a moving charge will generate a magnetic field at right-angles to the movement of charge (hence the concern regarding the effects of living underneath overhead power lines). Similarly, physically moving a conductor through a magnetic field or holding a conductor in a fluctuating magnetic field will generate an electric charge. This phenomenon is known as electromagnetic induction. While this process has many practical applications, it may be also be troublesome during dielectric measurements, as it leads to the establishment of induced (eddy) currents which may be measured along with the response of the sample. It is therefore necessary to earth the equipment adequately in order to obtain reliable results. 1.2.3 Alternating current In the previous section, systems whereby the charge drift velocity was unidirectional (d.c.) were discussed. In this section, systems whereby the direction of the drift velocity alternates with time will be considered. The simplest and most important category of alternating current (a.c.) is the sinusoidal wave. This may be described in terms of applied voltage via

(1.28) where V0 is the maximum current and is the angular frequency (i.e. number of radians per second, where one complete cycle consists of 2 radians). It should be noted that when describing the frequency of a field, the value is usually given in hertz (i.e. cycles per second). A sinusoidal wave form is shown in Figure 1.5a; other forms are shown in Figures 1.5b and 1.5c. However, all cyclical wave forms may be considered to be summations of a number of perfect sinusoidal waves, hence it is only necessary to consider this relatively simple wave form in detail. The value of the current or voltage in an alternating system may be conveniently described in terms of the root mean square value. Taking the current as an example, the root mean square (Irms) is given by

(1.29) The mean value of sin 2 t over a complete cycle is 1/2, hence

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Figure 1.5: Diagrammatic representation of alternative wave forms: a) sinusoidal wave, b) step function wave, c) irregular wave form

(1.30) It is now useful to consider the relationship between current and voltage in an alternating field. The voltage will be given by

(1.28) The charge on the capacitor will be equal to

(1.12) The current I within the circuit will therefore be given by

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As

then

(1.31) The above shows that the current and voltage will be 90° out of phase. Using (1.31), we can also write that at the peak current (i.e. cos ( t)=1), then

hence

(1.32) Equation (1.32) predicts that the presence of a capacitor in an a.c. circuit will cause opposition to a.c. flow, as in a d.c. circuit V/I=R (1.21). The quantity 1/ C is known as the reactance of the circuit and is given by XC. The important difference between resistance and reactance is that the presence of a resistance implies the dissipation of heat, while the presence of a reactance does not. This will be explained in greater detail shortly. If one again considers a simple RC series combination, across which an alternating field is applied, then the applied current will result in different voltages across the two components: the voltage across the resistor will be a function of the resistance, while that across the capacitor will be due to the reactance. In order to calculate the total voltage across the system one must therefore take into account the phase differences across the components as well as the magnitudes of the individual voltages. Consequently, the two voltages may be represented by the phase diagram shown in Figure 1.6. The resulting voltage VT will therefore be given by

(1.33) VR=IR and VC=IXC where XC is the reactance, hence

(1.34) The quantity (R2+XC2)1/2 is known as the impedance (Z) of the circuit and can be

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regarded as a measure of the total opposition to a.c. current. It can be seen from (1.34) that it has both resistive and reactive components, and like both parameters is measured in ohms. The impedance may also be described by

(1.35) The impedance is therefore similar to the resistance in d.c. circuits, but differs in that it also takes into account the opposition to flow caused by the presence of the capacitor in a.c. circuits (i.e. the reactance). Were the capacitor absent, the overall voltage would be the same as the voltage through the resistor and would be exactly in phase with the current, thus Z=R and energy would be lost via Joule heating, i.e. W=IR2. Were the resistor absent, the voltage and current would be exactly 90° out of phase and Z=XC, with no energy lost as heat.

Figure 1.6: Vector diagram illustrating the relationship between voltages across a resistor (VR) and capacitor (VC) in a series circuit

As stated in section 1.2.2, a circuit will, in practice, always have a resistance (or conductivity), as otherwise charge could not flow on and off the plates of the capacitor. A further important quantity is the phase angle. From the above arguments (leading to Figure 1.6), it can be seen that the voltages across the capacitor and the resistor are 90° out of phase. The phase behaviour of the resultant voltage will therefore be somewhere between these two extremes. The angle between the voltage and current is known as the phase angle and is usually designated by (sometimes ). A more commonly used quantity is the tan value, where

(1.36) This quantity is generally used rather than the other characteristic parameters of the circuit.

value itself as it is more readily related to

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The above discussion shows that a material may exhibit an impedance to charge movement (and hence, by definition, a conductivity) in addition to the conductivity of the other elements within a circuit. The former process is known as a.c. conductivity and is distinct from d.c. conductivity in that the latter involves a continuous movement of charge, while a.c. conductivity involves charge movement between localised sites. This distinction will be explained in greater detail in a later section. However, it is important to emphasise the difference between these two at an early stage as it is a concept that will be referred to frequently throughout the book. As shown above, not only must one consider the magnitudes of the voltage across elements in a circuit but also the phase angle between the two. In practice, it is most convenient to use complex variables for this purpose. These are numbers which include the exponent i (sometimes written as j), this being the square root of –1. Complex numbers take the form

where a is the complex number, b is the real component and c is the imaginary component Complex terms have two mathematical properties which are of special interest Firstly, i2 will equal –1, which means that by multiplying two terms involving i together one obtains a real number. Secondly, and more importantly for the present argument, the relationship

means that vectorial quantities may be expressed as complex numbers, with one quantity being ‘real’ (cos(a)) and the other ‘imaginary’ (i sin(a)). Using this notation, the voltages shown in Figure 1.6 may be expressed as

(1.37) The complex variable effectively acts as a marker to separate the in-phase and out-ofphase components, as the real and imaginary components will inevitably remain distinct. This means that the expressions may undergo various manipulations, but the two phase components will be easily distinguished by the presence or absence of the i term. Two points should be made regarding notation. Firstly, the notation I( ) signifies the current at frequency and does not imply that the current should be multiplied by frequency. The same notation applies to the voltage, capacitance, conductance etc. at a frequency . This notation is standard and hence will be used here. However, it is appreciated that this may lead to confusion, as a bracketed term may also be intended for inclusion in the given equation. Secondly, complex numbers are often denoted with an asterisk, e.g. I*. While this is not standard mathematical notation, this format is often used in the dielectric literature and hence will be used here.

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1.3 THE RESPONSE OF DIELECTRICS TO ELECTRIC FIELDS 1.3.1 Conductors, semiconductors, insulators and dielectrics It is necessary to consider what is meant by the terms conductor, semiconductor and insulator, as the expressions are often used loosely in the literature and can thus lead to confusion. The term conductor applies to materials which have a high value of conductivity, the most important example being metals. The high conductivity of metals is due to the electrons existing in a ‘free’ state within the system, often referred to as a ‘sea’ of electrons, hence charge flow is relatively uninhibited. While each metal has a finite value of resistance, depending on the material and the density of imperfections, this is small compared to most other materials. Semiconductors are materials which have a small number of free charges, this number increasing as the temperature is raised. This is in contrast to metallic conductors, whereby the excess charge is generated by the battery and travels through the material. There are two types of semiconductor, intrinsic and extrinsic, examples of intrinsic semiconductors being pure silicon and germanium. When extremely pure, these materials show little conductivity at low temperatures. On heating to room temperature, the conductances increase as a greater proportion of electrons gain sufficient energy to break free from the valence band. However, as the electron moves away in the lattice, the original atom is left with an overall positive charge, resulting in a ‘hole’. The positive charge is neutralised by a second valence band electron from a neighbouring atom, hence the ‘hole’ is effectively mobile. In this way, negative charge is considered to flow through the system in one direction and positive charge in the opposite direction. Extrinsic semiconductors contain small amounts of ‘impurities’ within the pure (intrinsic) semiconductor lattice, a process known as doping. For example, if one dopes silicon (which is quadrivalent) with phosphorus (which is pentavalent), a covalent bond will form between these atoms, leaving a ‘spare’ phosphorus electron. The presence of this free electron increases the conductance of the material as a whole. This is known as an n-type semiconductor, due to the presence of the excess free electron. If one incorporates a trivalent material such as boron, the atom can only form a covalent bond with three out of the four surrounding silicon atoms. The ‘missing’ electron acts as a ‘hole’ and can accept electrons from neighbouring silicon atoms, thereby creating more holes and hence propagating the charge flow within the material. This is known as a ptype semiconductor. These semiconductors are used extensively in the electronics industry, as by incorporating known amounts of impurities it is possible to accurately control the conduction properties of the materials. Insulators are considered to be materials which do not exhibit d.c. conductivity, i.e. there are no free charges within the system, hence the use of insulating materials (usually plastics) to coat electrical wires and cables. A dielectric may be defined as a material which contains dipoles, either permanent or induced. A dipole is an entity that has a charge separation but which maintains overall neutrality. The dipole moment µ is given by

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(1.38) where q is the charge magnitude and L the separation distance. For example, H-Cl is a dipolar molecule due to the dissimilarity of the electron distributions round the H and Cl atoms. In this case, the dipole is permanent, i.e. it will be present in the absence of any electric field. When placed between two electrodes, therefore, the assembly of dipoles will act as a capacitor, i.e. the molecules will allow charge to be stored on the plates but will not show d.c. conductivity, as the two atoms are covalently bound and hence will not travel between the plates as individual ions. However, when dissolved in water the covalent bonds are broken and the resulting electrolyte solution will act as a conductor, with H+ and Cl− carrying charge from one plate to the other. In this latter case, therefore, the system will exhibit d.c. conductance. Alternatively, molecules such as CCl4 have only a small dipole under standard conditions but will show greater dipolar activity when subjected to an applied field due to distortion of charges within the molecule. Dipoles may therefore be described as permanent (such as HCl) or induced by an electric field. It should be noted that this classification differs from the traditional categories of dipoledipole (Keesom), dipole-induced-dipole (Debye) and induced-dipole-induced-dipole (London) in that these three terms generally refer to dipoles induced by the presence of other molecules rather than the application of an external field. 1.3.2 Polarisation of dielectrics It is now appropriate to consider the concept of polarisation mentioned in section 1.2.1 in more detail. Polarisation arises from a finite displacement of charges in an electric field and is distinct from conduction, which arises from a finite average velocity of charges in an electric field. This returns to the concepts of finite and continuous movement of charge, as reflected by the two components of the impedance Z (section 1.2.3). For the present purposes we will consider polarisation in a material in a static field. It is also assumed that the material shows no d.c. conductivity. Mathematically, polarisation is the dipole moment per unit volume of a sample which is also equal to the charge per unit area on the plates of the adjoining electrodes. The magnitude of the polarisation therefore incorporates both induced and permanent dipoles. The mechanisms by which polarisation occurs may be summarised as follows. Firstly, there is molecular polarisation, which involves the distortion of the charge balance within the atoms of the material due to the presence of an electric field and is principally due to electronic polarisation, whereby the electron cloud surrounding a nucleus becomes distorted. Secondly, there is orientational polarisation, whereby dipoles are considered to physically rotate so as to align themselves with the applied field. This mechanism is applicable to both permanent and induced dipoles. The relative predominance of molecular polarisation and orientational polarisation depends largely on the nature of the material. In gases, reorientational polarisation will often be prevalent, while in solids molecular polarisation may predominate due to the rigidity of the material preventing physical reorientation of dipoles. A charge-hopping mechanism may also be involved

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(Mott and Davis, 1979), which may be regarded as an intermediate case between polarisation as discussed above and d.c. conduction. Charges may move between specific localised sites within a material under the influence of an external electric field, thus the charges are neither bound (as in polarisation in the conventional sense) but nor are they truly free. The significance of this mechanism will be considered in greater detail in subsequent sections. Having discussed what is meant by polarisation, it is helpful to examine the relationship between the polarisation (P) and the field (E) using non-polar gases as a simple model system to begin with. At relatively low field strengths there is a linear relationship between the two parameters. This proportionality may be expressed in terms of the polarisability, . In the simplest case of induced dipoles, the moment µi is given by

(1.39) In this case is also known as the molecular (or atomic) polarisability. The field experienced by the dipole is given by EL, which is the local field rather than the applied field. This distinction is drawn because the local field will be the vectorial sum of the applied field and the fields generated by the presence of the surrounding charges (i.e. the other dipoles). The question then arises as to how the local field may related to the applied field. One of the earliest approaches involves the general relationship between induced polarisation (Pi) and the applied field strength

(1.40) where r is the relative permittivity and 0 is the permittivity of free space (Bleaney and Bleaney, 1965). Combination of (1.39) and (1.40) lead to the Clausius-Mossetti equation

(1.41) where N is the number of polarisable molecules per unit volume. This equation has been shown to be valid for a number of gases up to high pressures (Bleaney and Bleaney, 1965). However, it should be appreciated that this relationship is limited in its application, as it may only be used for gases with no permanent dipole with any reliability. Nevertheless, it is useful because all gases will undergo induced polarisation to some extent, hence the Clausius-Mossetti equation may be used to quantify the induced component of the polarisation process. The situation becomes slightly more complex for polar gases, i.e. gases in which a permanent dipole is present. In the absence of a field, these dipoles are randomly orientated and hence will have no net polarisation. On applying a field, however, the dipoles will reorientate in the direction of that field, hence there will be an additional

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component to the polarisation (Pp), given by

(1.42) where µ is the permanent dipole moment, N0 is Avogadro’s constant, V is the molar volume, k is the Boltzmann constant and F is the magnitude of the ‘inner’ field, i.e. the field experienced by the sample, as opposed to the field applied to the material. The value of F is given by

(1.43) Equations (1.42) and (1.43) may be combined to give Debye’s equation (Debye, 1929; 1945)

(1.44) The Debye equation therefore combines contributions to the response made by the electronic and reorientational polarisation mechanisms. However, as with the ClausiusMossetti equation, (1.44) is of limited use when applied to liquids and is even less applicable for solid systems. A more sophisticated model was proposed by Onsager (1936), resulting in the relationship

(1.45) where n is the refractive index. This equation will not be discussed in detail, except to say that it is superior to the Debye model in that it can predict the behaviour of a wider variety of liquid samples. However, like the Debye model, it assumes that the contributions made by neighbouring dipoles are seen purely in terms of the average local field. In fact, neighbouring dipoles will be exerting directional forces on each other, hence not only will there be effects on the local field but there will also be coupling effects, particularly in condensed samples. Kirkwood (1936) introduced a ‘correlation parameter’ (g) into the analysis which was later refined to

(1.46)

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These approaches therefore demonstrate how attempts have been made to correlate the polarisation (and hence permittivity or dielectric constant) to the molecular structure and behaviour of the sample. As will be discussed in later chapters, dielectric constants have been used extensively as a means of characterising pharmaceutical liquids. The above discussion shows how more information could be obtained on these systems than is obtained using the more empirical approach commonly in use within the field. 1.3.3 The dielectric response in the frequency domain

Figure 1.7: Summary of the types of response that may be encountered over a range of frequencies Dielectric analysis usually involves applying a field of fixed or varying frequency to a sample and measuring the response. As the frequency of the field changes, different mechanisms of polarisation will predominate. It is the analysis of these mechanisms that provides the basis of dielectric spectroscopy. Dielectric phenomena may be measured over a frequency range of 10–5 Hz to 1012 Hz, which is clearly a much wider spectral window than is found with most other spectroscopic techniques. As will be discussed in the following chapters, different responses will be seen at different frequencies. A summary is given in Figure 1.7. It should be emphasised that these ranges are very approximate and are given merely to give an idea of the types of responses that may be seen. In addition, there is still debate regarding the interpretation of dielectric spectra, hence Figure 1.7 should be regarded only as a rough guide. It was shown in section 1.2.2 that when an external p.d. is removed from a charged capacitor, that capacitor will discharge over a period of time, depending on the capacitance itself and the resistance in the discharge circuit. In an alternating system, charge movement will change direction in order to ‘keep up’ with the fluctuations in the field when that field changes direction. As this realignment will inevitably be non-

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instantaneous, the response will take place over a period of time. The time dependent response function G(t) may be transformed to a frequency dependent function, G( ). Time and frequency domain data may be interchanged using a Fourier transform. This expresses a function B(t) in terms of B( ) via

(1.47) where F denotes a Fourier transform between time limits and – . It is not necessary for the purposes of this text to go into further details of this function or its derivation, except to stress one important point. The inclusion of i in the above expression means that any transformed term will be complex, i.e. it will possess real and imaginary components. This therefore indicates that any resulting expressions will take into account the phase behaviour of the response of a sample. If the Fourier transform is applied to polarisation phenomena, one obtains an expression for P( ) for a single relaxation process such that

(1.48) where is the susceptibility of the sample. This parameter is complex and may therefore be expressed in terms of its real and imaginary components, i.e.

(1.49) The susceptibility is related to the permittivity, which may also be expressed in terms of the real and imaginary components, i.e.

(1.50) The difference between the susceptibility and permittivity is that the term ( ) refers to the sum of all the permittivities between infinity and the frequency of interest, while ( ) refers to the permittivity at that specific frequency. The real part of the permittivity therefore includes the permittivity of free space, which is necessarily real as there can be no dielectric loss in a vacuum, i.e.

(1.51) This is an important distinction because permittivity and susceptibility are both frequently used in the dielectrics literature.

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A further important consideration is the physical significance of the real and imaginary parts of the susceptibilities and permittivities. The simplest method of visualising this concept is to remember that when voltage and current are in phase, then heat is lost via Joule heating (see section 1.3.2), while no energy is lost when the two are out of phase (hence all the energy put in by the field is stored within the system). The real component of the permittivity therefore relates to the energy stored (i.e. the capacitive properties of the system, involving neutralisation of applied charge) and the imaginary component is related to the energy lost as heat (i.e. the a.c. conductance process), hence the imaginary permittivity is often referred to as the loss component. 1.3.4 The Debye model of dielectric relaxation If a unidirectional field is applied to a sample, any dipoles present will reorientate with the field, and return to their original state (relax) at a characteristic rate when that field is removed. When an alternating field is applied to a dielectric sample, the response of that sample will vary with the frequency used. This dependence of the response on frequency is known as dispersion. The study of this frequency dependence forms the basis of many of the studies in the dielectrics field. There are in fact two processes which may be associated with frequency dependence. Firstly, there are regular oscillations of a part of a system at a definite frequency, an example being the vibration of intramolecular bonds. These oscillations will absorb energy over a narrow range of applied frequencies and such phenomena may be classified as resonance responses. The frequencies at which these phenomena are observed tend to be higher than those of interest here and hence will not be discussed further. Secondly, there are relaxation processes. These processes are essentially cooperative, as relaxation phenomena involve the damping of the response of dipoles to an electric field, this damping being due to the inertia of the dipoles and the structure of the surrounding environment. Relaxation processes are therefore distinct from resonance processes, which essentially reflect the behaviour of individual atoms or molecules. Relaxation processes are most easily envisaged by considering the behaviour of a sample containing mobile dipoles which is being subjected to an oscillating electric field of increasing frequency. In the absence of the field, the dipoles will experience random motion due to the thermal energy in the system and no ordering will be present. At low frequencies, the applied field will result in changes in the mean position of the dipoles as the direction of the field changes, although those dipoles will still be oscillating around that mean. At very high frequencies, the changes in field direction are so rapid that the dipoles are unable to reorientate with that field because of their inertia and viscous damping, hence the total polarisation of the system falls. However, at a characteristic frequency ( p) between these two extremes the efficiency of the reorientation process is at a maximum, as the rate of change in direction of the applied field matches the relaxation time of the dipoles. Those dipoles will therefore undergo maximum reorientation, but the random oscillations superimposed on that system will be at a minimum. The rate at which a dipole relaxes will be dependent on the nature of that dipole and the environment in which it is situated, hence dielectric studies may yield information concerning both the nature of the relaxing species and the structure

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surrounding that dipole. It is convenient to study relaxation processes by measuring the real and imaginary components of the permittivity over a range of frequencies. As will be shown in subsequent chapters, these relaxation processes are a function of the structure of the sample, hence they may be used as a means of characterising materials. The frequency dependence of the reorientation process was the subject of extensive work by Debye (1945) and is summarised as follows. The real and imaginary (loss) components of the susceptibility for a reorientating dipole may be derived from (1.47) and are given by

(1.52) where (0) is the static susceptibility (i.e. the susceptibility as the frequency tends to zero). The complex susceptibility can be expressed in terms of the real and imaginary (loss) components (see (1.44)) via

(1.53) and

(1.54) In terms of permittivity, these equations may be given by

(1.55) from which

(1.56) and

(1.57)

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where n is the refractive index (equivalent to 1/2 at optical frequencies) and S is the static field permittivity (i.e. the permittivity at zero frequency). is the relaxation time of the system and is a characteristic of the material, as it gives the time constant of the dipolar reorientation process. The concept of a relaxation time for a dipole is similar to that described in section 1.2.2 for an RC circuit. These equations have several implications. Firstly, the frequency dependence of any (ideal) dielectric material can be defined in terms of only two intrinsic variables, (0) and . Secondly, the equations predict the relationship between ’ and " shown in Figure 1.8. It can be seen that the imaginary part of the susceptibility shows a peak at a characteristic frequency, known as p or the peak loss frequency. This corresponds to the frequency of maximum energy absorption described earlier. The breadth of the peak at half the height is constant at 1.144 decades (log cycles) of frequency and the logarithmic slopes of susceptibility against frequency above and below the loss peak frequency are 1 and −1 respectively. The loss peak frequency is related to the relaxation time by

Figure 1.8: Idealised Debye model of relaxation

(1.58) The real part of the susceptibility remains effectively constant at low frequencies but decreases with a logarithmic slope of –2. This is the classic form of the dielectric response and has formed the basis of much of the subsequent theory in this field. The peak loss frequency may be regarded as the frequency at which the rate of power absorption by the system is at a maximum, as discussed previously. Indeed, Debye (1945) derived a simple expression for the relaxation time of a sphere of radius a in a medium of viscosity

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(1.59) indicating a direct relationship between the relaxation time and the size of the relaxing species. This is a useful relationship and has a number of pharmaceutical applications. For example, it may be used to characterise the size and shape of molecules in solution or colloidal particles such as DNA molecules in an aqueous medium, as will be discussed later. The relaxation time therefore contains information regarding the system, as shall be seen in subsequent chapters. 1.3.5 Modifications to the Debye theory In practice, the Debye response is seldom seen, as systems invariably contain more than one relaxing species, hence the behaviour is likely to be complicated by interactions between these components. A common deviation from the ideal response is power law behaviour above and below the peak frequency (i.e. the magnitudes of the logarithmic slopes of the real and imaginary susceptibilities become fractional). Moreover, the response of many liquids and almost all solids may assume a variety of forms which often bear little or no resemblance to the Debye response. Therefore, while the usefulness of the Debye model as a basis cannot be overemphasised, the use of only two parameters ( and (0)) is inadequate to describe the dielectric behaviour of a number of systems. It is therefore helpful to consider the developments to the original Debye theory that have been proposed over subsequent years. The first modification to the Debye theory was that of Cole and Cole (1941), who introduced the concept of susceptibility functions to correct for non-Debye behaviour. These are essentially correction factors, which, when inserted into the Debye equations (1.53 and 1.54), may yield a better fit to the observed results than may be obtained with the Debye equation alone. The Cole-Cole modification is of the form

(1.60) where is the permittivity at infinite frequency. This equation can then be applied most readily to what is known as a Cole-Cole plot, as shown for the ideal (Debye) case in Figure 1.9. This involves plotting the imaginary against the real permittivity, which gives a symmetrical half-circle for a perfect Debye response. In practice, the symmetry shown in Figure 1.9 is seldom seen, hence the authors suggested a shape factor a to account for non-ideal behaviour. The theoretical basis behind this function is the assumption that the deviation from the ideal model is due to there being a particular range of relaxation times within the sample, with the peak loss frequency corresponding to the geometric average of this range. In terms of the standard Debye plot, this would appear as a broadening of the width of the loss peak. As with almost all dielectric interpretations, there is a certain

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amount of controversy regarding the validity of this approach on theoretical grounds. Hill and Jonscher (1983) have argued that it is unlikely that a range of relaxation frequencies would explain the considerable width of loss peaks seen for many system and consider relaxation rate distributions to be less important than is commonly thought. Whatever the theoretical arguments that may be submitted regarding this model, the Cole-Cole correction does not in fact fit a large number of real systems. A further correction was suggested by Davidson and Cole (1951) to account for the non-Debye behaviour, given by

Figure 1.9: Cole-Cole plot corresponding to the ideal Debye curve (single relaxation time) and to a sample showing deviation described using Cole-Cole equation (1.60)

(1.61) Again, however, many responses are not described by (1.61) and so a further modification was suggested by Havriliak and Negami (1966) which essentially combines the above two, i.e.

(1.62) There are several further modifications that have been described, most relying on the use of two power law indices as it is arguable that a single susceptibility function is insufficient in most cases. These will not be discussed here, with the exception of the many-body model outlined by Dissado and Hill (1979) and Jonscher (1983). It is worth discussing these theories in more detail, as their approach represents an attempt to interpret the deviations from the Debye theories in a non-empirical manner.

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Jonscher (1975a) has argued that power law behaviour is so common in dielectric behaviour that there must be a universal mechanism involved in virtually all dielectric materials. He suggested that the presence of many-body interactions, rather than a distribution of relaxation times, was a more likely explanation for the observed behaviour. While other workers had acknowledged the possibility of interactions influencing the response, these had been essentially single dipole models whereby a relaxing entity is considered to behave in a Debye-like manner with surrounding dipoles exerting an influence on the ‘ideal’ dipole. Jonscher suggested that the flaw in this argument lies in the fact that the dipole will itself be influencing the local environment and not just the other way round. This early work led Dissado and Hill (1979) to develop a quantum mechanical approach (the Dissado-Hill theory) which makes certain predictions of real dielectric behaviour. The essential features of the Dissado-Hill theory are firstly that materials are assumed to be composed of clusters. These are spatial groups within a sample that show cooperative behaviour during the relaxation process. The relaxation behaviour of these clusters will affect the overall shape of the response, as well as the absolute values at any particular frequency. The authors suggested that two power law exponents were required to fit a given set of data. These two exponents (n and m) are considered to refer to the degree of cooperation within a cluster and between separate clusters, respectively, with 0
(1.63) where 0≤m, n≤1, and 2F1(, ; ;) is the Gaussian hypergeometric function, which is essentially an averaging function. The significance of the Dissado-Hill theory will be explained in conjunction with individual examples of applications of the model in subsequent chapters. The advantage of this approach compared to those described in the previous section is that, being non-empirical, it is considerably easier to predict the dielectric behaviour of a range of systems, particularly those which apparently show considerable deviation from the Debye response. It is in this respect that the real importance of the approach lies, as it is theoretically possible to predict any dielectric response in terms of four exponents: , (0), n and m. This then means that apparently diverse responses of, for example, molten liquids and their respective solids may now be usefully compared and insights gained into the changes that occur structurally on solidification. 1.3.6 Alternating and direct current conductivity It is helpful to emphasise the difference between a.c. and d.c. conductivity. It was shown in section 1.3.3 that under an a.c. field, the presence of a capacitance will in itself result in a finite conductivity. This is known as an a.c. conductivity, as it signifies the

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movement of fixed charges between specific sites. D.c. conductivity, however, involves the movement of free charges at a steady velocity. It is not possible for any instrument to distinguish between these two types of conductivity; that task is left to the interpretive skills of the operator. An important instance under which the Debye theory may be modified is when a d.c. conductivity is present within the system. In this case, the d.c. response will be seen in addition to (and may swamp) the a.c. conductivity, particularly at low frequencies (kilohertz and below). The most pernicious aspect of this problem is that the d.c. conductance is frequency independent, as it is equivalent to a resistor being present in parallel to the dielectric, with the result that the loss slope will be –1. This has the same appearance as the loss slope above p for an ideal Debye response. It is therefore very important to consider the possibility of either mechanism being present when examining dielectric data showing constant conductivity. A further point that is relevant to low frequency work is that the charge hopping mechanism described by Jonscher and Hill (1975) will also result in a conductance which may be seen in addition to the reorientational polarisation. However, it is comparatively easy to distinguish this response from the above two, as charge hopping is characterised by an corresponding increase in capacitance, as will be shown in subsequent examples.

1.4 PRESENTATION OF DIELECTRIC FUNCTIONS As can be seen from the above discussion, there are several methods of presenting essentially the same data which may lead to confusion on examining the literature. It is therefore helpful to survey some of the methods used to present dielectric data. It is usual to present the frequency on a logarithmic scale, as the width of the range covered would usually preclude the use of linear plots. The use of a logarithmic frequency scale has therefore been assumed in the subsequent discussions. 1.4.1 Permittivity plots One of the simplest methods (at least theoretically) is to present data as permittivity against frequency, usually on a log/log scale. This method of presenting data has several advantages. It involves the use of intrinsic parameters which have no dependence on external variables such as electrode size, hence it is a theoretically sound approach in this respect. Susceptibility plots are not used in practice, as it is considerably more difficult to calculate this parameter from experimental capacitance data (see section 1.4.3). Permittivity data may be presented in a Cole-Cole plot, as shown in Figure 1.9. While there are some theoretical objections to this method (see section 1.4.4), this technique is frequently used and at least represents a fairly standard method of presenting dielectric data. 1.4.2 Tan

values

The data may also be expressed in terms of the tan

value, which is given by

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(1.64) It should be remembered that in calculating tan values, one is substituting two pieces of information for one, as it is a ratio between two absolute values. The tan method is often used in temperature domain studies, with a peak in the value at a particular temperature being ascribed to a phase change or glass transition phenomena. 1.4.3 Capacitance and dielectric loss This method involves the use of the extrinsic parameters capacitance and dielectric loss (G/ , where G is the conductivity), via

(1.65) and

(1.66) There are several points to be made concerning this method. Firstly, the capacitance measured above is the real part of the complex capacitance (see section 1.3.3), while the dielectric loss is the imaginary part of the capacitance. The dielectric loss G/ contains a term (G) which is the a.c. conductance of the sample. However, as discussed in section 1.4.4, the presence of a d.c. conductivity may complicate the analysis. This method of presenting dielectric data is advantageous in that respect, as in order to account for the d.c. element one simply has to modify (1.65) to

(1.67) The method of using capacitance and loss has the advantage that the cell dimensions are included in the data, i.e. one is measuring the capacitance and loss for the whole cell. However, it therefore has the disadvantage that one is not measuring an intrinsic property of the sample, hence it is often difficult to compare the absolute values of different sets of data unless the cell capacitance is known. 1.4.4 Admittance and impedance Admittance and (more usually) impedance are also commonly used. The impedance may

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be regarded as the overall resistance of a series RC circuit to an a.c. current and is a complex number, with

(1.68) (see section 1.2.3) where R refers to the a.c. loss. Admittance is the reciprocal of impedance and is often used to describe parallel RC circuits. The advantages and disadvantages of using capacitance and loss also tend to apply to these two complex parameters and the various sets of data may be interchanged without any further information being required. 1.4.5 Circuit analysis of the dielectric response

Figure 1.10: Circuit analysis of dielectric functions, showing typical circuit diagrams and corresponding plots of log susceptibility against frequency (after Jonscher, 1983)

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A further technique that is often used to characterise the response of dielectric materials is to compare the sample to a resistor/capacitor circuit which would give an identical response. This technique will be discussed in greater depth with reference to individual examples. A summary of such circuit elements is given in Figure 1.10.

1.5 TYPES OF DIELECTRIC ANALYSIS As can be seen from the above discussion, the dielectric behaviour of a sample will be characteristic of the structure of that material and hence has actual and potential uses as an analytical technique. These will be discussed in subsequent sections. However, one point should be emphasised. The term ‘dielectric analysis’ covers a variety of methods, including conductivity measurements and dielectric constant measurements (both of which are familiar tools in the pharmaceutical sciences), step function measurements, time domain measurements, temperature measurements using a single frequency and, of course, frequency domain measurements as discussed above. The bulk of this text will be concerned with the last form of dielectric analysis which, by virtue of the type of response obtained, can be considered to be a spectroscopic technique. However, reference will frequently be made to some of the other techniques mentioned above in order to put the spectroscopic data in context. Furthermore, it should also be emphasised that there are different approaches within the broad category of frequency domain measurements, particularly in terms of the frequency range under study.

2 Methods of dielectric measurement 2.1 INTRODUCTION

Figure 2.1: Summary of the various types of frequency domain measuring techniques (reproduced from Blythe (1979) with permission of the Cambridge University Press) One of the distinguishing features of dielectric relaxation is that phenomena of interest may occur over an extremely wide frequency range. This range, from below 10−4 Hz to 1012 Hz, cannot be covered by any single instrument, hence a variety of techniques have been developed in order to allow accurate measurement in sections of this range. The basic principles of these methods will be described here; for more detailed information, a number of texts are available (e.g. Grant et al., 1978; Blythe, 1979). Most of the studies described here refer to frequency domain measurements. At the low frequency end of the spectrum, bridge techniques are commonly used, which will be discussed in the next section. At higher frequencies, transmission methods are used, which include techniques involving coaxial lines and waveguides. Coaxial lines are used between approximately 50 MHz and 12 GHz, while waveguides are used above this up to 100 GHz. Methods are available for even higher frequency measurements, but they will not be covered here as they are of little relevance to systems of pharmaceutical interest. A summary of the general methods used at each frequency range is given in Figure 2.1. Measurements may also be made in the time domain, whereby a pulsed voltage is applied to the sample and the response measured as a function of time. It is possible to convert

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this data into the frequency domain, often covering a wide frequency range. Finally, temperature domain methods will be considered, which are used extensively for the measurement of polymeric samples. It should be stressed that a wide range of equipment is available for making dielectric measurements and this chapter is not intended as a review of individual models. Instead, the basic concepts behind the measurements will be discussed and interested readers are advised to contact manufacturers directly for further details of individual sets of equipment.

2.2 LOW FREQUENCY METHODS 2.2.1 Principles of bridge measurements Bridge measurements are the most commonly used method of determining the capacitance and loss of dielectrics in the low frequency range. The approach involves measuring the electrical properties of a sample with reference to known resistors and capacitors, most instruments being based on the principle of the Wheatstone bridge. This apparatus (and various similar systems) involves an arrangement of four circuits such that on applying an a.c. signal the four are electrically balanced. One of the four is the sample and the other three have known impedances. An example (the Schering bridge) is shown in Figure 2.2, where the sample is in arm 4. The bridge is at balance (i.e. detector D shows no output) when

(2.1) The bridge is first balanced with the sample cell out (switch S open). The switch is then closed and the cell capacitance CX and resistance RX are compensated for by adjusting C4 and C1 respectively. By varying the frequency of measurement, the values of capacitance and resistance of the sample may be measured at a number of individual frequencies and hence a spectrum may be obtained. In all cases, care must be taken in setting up the bridge to ensure that there are no stray impedances between the circuits, which may be achieved by effective shielding of the components from one another. Other types of bridge are available, including the transformer ratio arm bridge, which uses inductive ratio arms to compare a sample with standard circuits (Starr, 1932; Cole and Gross, 1949). At frequencies below approximately 10–2 Hz, problems may arise due to the long period of the signal making measurements extremely slow, while at frequencies above approximately 106 Hz the stray impedances mentioned above may become appreciable. Furthermore, the sample cell and associated leads may exhibit self-inductance which will be seen as an additional response in this frequency range. While cells which may be suitable for use with bridge circuits up to 100 MHz have been described (Jordan and Grant, 1970), it is generally considered advisable to use alternative techniques such as resonance methods in the megahertz (MHz) region and upwards. For more details on bridge methods, a number of texts are available, including Hill et al. (1969), Grant et al.

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(1978) and Blythe (1979), amongst others.

Figure 2.2: Diagrammatic representation of a Sobering bridge (reproduced from Blythe (1979) with permission of the Cambridge University Press) 2.2.2 Cell design for bridge measurements For low frequency measurements, the cells used may be coaxial or may comprise two circular cross-sectional electrodes (Lovell and Cole, 1959; Pauly and Schwan, 1966; Payne and Theoodorou, 1972), but are more usually parallel plate systems. These involve the insertion of a thin disc of solid material between parallel plates or immersion of those plates in a liquid. While this method has the considerable attraction of simplicity, it does suffer some drawbacks. Firstly, the electric field between the electrodes may exhibit fringing, whereby non-linear field effects occur at the edges of the electrode systems, as shown in Figure 2.3. This may be corrected for or, alternatively, may be overcome by using a guard electrode, whereby the the guard is held at the same potential as the measuring electrodes but is not connected to them. As a result, the field in the guarded area is uniform. Furthermore, problems arising due to conductivity on the surface of hygroscopic samples as a result of water adsorption may be overcome using this method. Other problems include the presence of stray capacitances, for example between the lead and the opposite electrode, which may be avoided by appropriate shielding. A further difficulty is that of the contact between the sample and the electrodes. If an air gap exists between the two, then this will act as a further series capacitance. This is a particular problem for solid samples, especially in the very low frequency region (<1 Hz). This may be corrected for if the size of the air gap is known, but it is generally preferable to ensure good contact between the two surfaces. Methods include the application of pressure to the sample via the electrodes, which is somewhat crude but may often be satisfactory. Other methods include using soft metal electrodes such as metal foil which may be pressed onto the sample, or alternatively metals may be vacuum deposited onto the surface of the sample. If possible, the sample may be solidified within the cell when, for example, solid polymers are being examined, as this tends to overcome electrode contact problems. We have also used a silver-loaded conducting resin which may be applied to either side of a solid sample and left to cure with short wires immersed in the

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resin which then act as leads (Bauer-Brandl et al., 1992). This method gives good reproducible results, with the proviso that the resin should not interact with the sample.

Figure 2.3: Electric field pattern in parallel-plate electrode systems: a) unguarded b) with one plate guarded 2.2.3 The problem of electrode polarisation A persistent problem, and source of debate, has been that of electrode polarisation. For very conductive solutions, a layer of ions will form adjacent to the electrodes (Oncley, 1942; Schwan and Maczuk, 1960; Mandel, 1965; Rosen et al., 1969). This will alter the charge distribution within the system and results in a marked rise in capacitance as the frequency is lowered. There is some debate as to whether electrode layers are a nuisance to be avoided or an integral part of the dielectric response, from which useful information may be obtained. As will be outlined in subsequent chapters, there is growing evidence that in many cases, such electrode layers are themselves of interest. However, for the purposes of this discussion, it will be assumed that, as in the case of highly conductive solutions, electrode polarisation is an unwanted effect which should be removed or corrected for. One of the simplest methods of reducing electrode polarisation is to roughen the electrodes, thereby increasing their surface area. A popular method of achieving this is to coat the electrodes with platinum black (Schwan, 1963; Takashima, 1963), although care must be taken to ensure that the coating material does not interact with the sample. There are also methods of correcting for electrode polarisation. Grant et al. (1978) have described the electrode layer as being equivalent to an impedance in series with the test liquid and have discussed various approaches that are based on this premise. One method of overcoming the problem is to measure a series of electrolyte solutions such as KCl to find the concentration which gives an equivalent response to that of the test solution, from which it is possible to calculate the contribution of ions to the low frequency response (Takashima, 1966). Alternatively, a cell with variable electrode spacing may be used (Broadhurst and Bur, 1965; Young and Grant, 1968; Rosen et al., 1969; van der Touw and Mandel, 1971). Schwan and Maczuk (1960) suggested that when the interelectrode distance is small with respect to the electrode area, the polarisation effects will be large as the electrode layer will effectively constitute a relatively large component of the sample, while at large electrode spacings, stray capacitances tend to dominate the response.

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A further method of dealing with this phenomenon is to use a four-terminal bridge in which two electrodes supply the alternating field and the other two (probe) electrodes sample the field. Theoretically, the probe electrodes should not accumulate any ions and hence the effect of electrode polarisation should be eliminated (Ferris, 1963; Berberian and Cole, 1969). In practice such systems are not straightforward to use (Grant et al., 1978), particularly for very conductive solutions. In conclusion, therefore, there are methods available by which electrode polarisation may be reduced or corrected for. A further approach which has been outlined by Hill and Pickup (1985) is to measure the response down to sufficiently low frequencies such that the effect may be seen in full, in which case interpretation and, if necessary, correction, becomes considerably easier. In any case, the phenomenon of the establishment of electrode layers must be appreciated and understood in order to allow accurate interpretation of low frequency responses. This will be discussed in more detail in subsequent chapters.

2.3 RESONANCE METHODS In the region of 105 Hz to 108 Hz, it is possible to use resonance methods, whereby the sample becomes part of a resonance circuit. The original approach was developed by Hartshorn and Ward (1936) and the basic circuit is shown in Figure 2.4. The sample is represented by the parallel RC circuit (subscripted x) and connected to a coil of fixed inductance L. When a current is passed through the coil, the change in direction of the charge flow will generate a magnetic field within that system. If the magnitude of the current is altered, the change in the magnetic field will itself generate an e.m.f. within the circuit. The inductance is given by

(2.2) where E is the induced e.m.f. and dI/dt is the rate of change of current. When a coil of inductance L is included in a circuit in parallel, the p.d. across the circuit will be at a maximum at a particular frequency ( 0). This frequency is given by

(2.3) At this frequency, the circuit is said to be in resonance. In the apparatus shown in Figure 2.4, an alternating signal is passed via a loosely coupled oscillator and the frequency tuned to resonance, as indicated by the voltmeter. By comparing the resonance behaviour with and without the sample, it is possible to calculate the real and imaginary permittivities of that material. By using a range of resonance coils, it is possible to obtain readings over a range of frequencies. This method

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is less frequently used than transmission methods and hence will not be discussed in more detail here.

Figure 2.4: The basic electrical circuit of the Hartshorn and Ward (1936) resonance method of dielectric measurements (reproduced from Blythe (1979) with permission of the Cambridge University Press) 2.4 WAVE TRANSMISSION METHODS As the frequency of the field increases, the wavelength of the signal decreases and becomes of the same order of magnitude as the dimensions of the sample. Applied fields therefore vary from place to place within the sample which necessitates interpretation of the response in terms of Maxwell’s wave equations. Full details of this approach have been given by Grant et al. (1978). In terms of measuring techniques, coaxial lines are used up to the low gigahertz (GHz) region, while waveguides which are used at higher frequencies. 2.4.1 Coaxial lines Coaxial lines are used for frequencies ranging between approximately 50 MHz to 12 GHz (Grant et al., 1978). The coaxial line cell consists of a hollow metal cylinder containing a concentric inner conductor. At the end of the line is a flat metal plate which acts as a short circuit, as shown in Figure 2.5. An electromagnetic wave passes through the liquid sample in the line and is reflected by the short circuit. This reflected wave will interfere with the transmitted wave, resulting in a standing wave being set up in the liquid. This phenomenon arises due to transmitted wave being combined with its reflection. For example, a cord tied to a solid object at one end and moved from side to side by the operator at the other will, at certain frequencies, form standing waves, whereby a wave pattern will be generated which oscillates but does not travel along the cord. The same principle occurs in coaxial lines. The characteristics of the standing wave are related to the permittivity of the sample; if at a point x=0 the field is E0 and at a point x=d the field is E, then in the absence of any reflections the

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relationship between the two is given by

Figure 2.5: A simplified coaxial line cell (reproduced from Grant et al. (1978) with permission of the Oxford University Press)

(2.4) where

is the complex propagation constant, given by

(2.5) where is the attenuation coefficient (a measure of the decrease in field per unit length, indicating the potential between the inner and outer tubes) and is the phase constant, which is related to the wavelength of the electromagnetic wave in the medium ( M) by

(2.6) These basic principles may be applied to coaxial lines and hence by measuring the propagation characteristics of the standing wave, the complex permittivity of the sample

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may be calculated via

(2.7) where A is the wavelength in free space. The above complex equation may be expanded into the real and imaginary components of permittivity, given by

(2.8) and

(2.9) There are now a number of open-ended coaxial probes available which comprise a coaxial line ending abruptly at the tip. This probe is simply placed in contact with the material under examination, hence it is relatively simple to use for the majority of applications. 2.4.2 Waveguides Above approximately 12 GHz, it is not possible to propagate electromagnetic radiation in coaxial lines in a sufficiently controlled manner to allow useful measurements to be made, necessitating the use of waveguides. These are essentially hollow metal tubes, through which electromagnetic radiation may pass. In many respects, therefore, they may be considered to be similar to a coaxial line but without the inner conductor. Consequently, it is not possible to consider the system in terms of potentials between the inner and outer tubes. Coaxial lines involve propagation of radiation by what is known as the transverse electromagnetic mode (TEM), in which the wave form has electrical and magnetic components perpendicular to the direction of propagation, i.e. the field is between the inner and outer conductors while the wave form travels along, rather than between the conductors. The TEM mode does not exist in waveguides but instead one of a number of higher modes dominates. Maxwell’s theories demonstrated that in addition to the TEM mode, two further series of modes may exist, namely the H or transverse electrical (TE) and the E or transverse magnetic (TM) modes. With H modes, the electric field component is perpendicular to the direction of propagation, while for E modes the magnetic component is perpendicular to the propagation. This is in contrast to the TEM mode, in which both components are perpendicular to the propagation. Depending on the characteristics of the waveguide, one of a number of H or E modes

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other than the TEM mode may be detected. However, these higher modes tend to have a very narrow frequency range, hence waveguides are often used only for one-point determinations or at best for a very limited frequency range. From a pharmaceutical viewpoint, these systems are of relevance in terms of measuring the relaxation of free water, which occurs in the gigahertz region. However, pharmaceutical scientists are more likely to be interested in the relaxation of bound water, which generally occurs within the frequency range measurable by coaxial lines.

2.5 ALTERNATIVE APPROACHES TO FREQUENCY DOMAIN MEASUREMENTS In addition to the techniques mentioned above, a number of additional approaches to dielectric measurement have been described which, while not necessarily being of direct use pharmaceutically at present, nevertheless represent interesting developments within the dielectrics field. For example, it is possible to use electrode systems whereby there is no direct contact with the sample. For example, Hewlett-Packard have developed a ‘free space’ measuring system whereby remote antennae focus microwave energy onto the sample. While this system has the obvious advantage of not requiring a test fixture, measurements are only possible in the gigahertz region and the sample needs to be a large, flat thin material. Yu (1993) has described a method whereby a coil is wound round a sample in place of a conventional parallel plate electrode system. The author suggested that this system may be used for the measurement of polar liquids, although at present the data obtained is not as accurate as that obtained by conventional methods. These methods therefore may currently have limited applicability as far as pharmaceutical systems are concerned, although it is possible that developments of this type may be of use in the future.

2.6 TIME DOMAIN MEASUREMENTS The techniques outlined so far have been concerned with measuring the properties of dielectric materials by analysing their response to an alternating field, i.e. by measuring those samples in the frequency domain. In addition, it is also possible to measure those samples in the time domain. In many ways, time domain measurements may be simpler to perform, as they involve simply the application of some sort of pulsed signal to the sample rather than an oscillating one. The time domain response may then be converted to the frequency domain by Fourier transformation, as mentioned in section 1.3.3. This techniques is known as time domain spectroscopy (TDS) and has been used as an alternative to low frequency measurements (Davidson et al., 1951), allowing analysis down to frequencies down to 10–4 Hz without necessitating the use of specialised equipment required for frequency domain measurements in this range. In order to provide a time scale which is equivalent to a signal in the audiofrequency range, measurements of signal growth or decay from several hours to milliseconds have been performed. A greater problem lies with higher frequency measurements, as pulse signals and detection

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mechanisms with a sensitivity of picoseconds are required (Oliver, 1964; FellnerFeldegg, 1969; Loeb et al, 1971). For low frequency measurements, a step voltage of 1 to 500 V is applied across a sample using a guarded electrode system (or equivalent) and the resulting charging current is measured via the potential across a standard resistor in the circuit. The transient polarisation currents tend to be small (approximately 10–12 A), hence the standard resistor has a correspondingly high value (109 to 1012 ). For high frequency measurements, coaxial lines or waveguides are used. A more detailed description of TDS has been given by Grant et al. (1978) and will not be entered into here, other than to mention that TDS may hold considerable potential as an alternative means of assessing the frequency dependent behaviour of samples.

2.7 TEMPERATURE DOMAIN MEASUREMENTS In addition to measurements made with respect to frequency or time, it is also possible to measure the response of materials over a range of temperatures. Naturally, this may be achieved by measuring over a range of frequencies at a number of temperatures, but it is also possible to scan a range of temperatures at a single frequency, thereby using dielectric analysis as a thermoanalytical technique. Indeed, the term dielectric thermal analysis (DETA) has been used to describe these techniques, which are widely used in the polymer field. Measurements over a range of temperatures allow activation energies to be obtained, thereby allowing discrimination between different processes. A number of methods have been described for measuring the temperature dependent dielectric behaviour of solids (especially polymers) and a brief description is given here. For more details, the interested reader is referred to Chapter 6 of this text or any of a number of books dealing specifically with the dielectric response of polymeric samples (e.g. Hedvig, 1977). For most polymeric samples, low frequency measurements are considered to be more useful than high frequency data, although for cryogenic transitions and studies concerning lattice vibrations high frequency measurements may be desirable. Most low frequency measurements are made using bridges, particularly the transformer ratio arm bridge mentioned earlier. For higher frequency measurements (above 107 Hz) resonance methods may be used, while from approximately 108 Hz to 1011 Hz coaxial lines and waveguides are used. A further, related technique is dielectric depolarisation spectroscopy. The method is also known as thermostimulated current depolarisation spectroscopy (or any one of several variations on this nomenclature) and involves measuring the short-circuit current during heating of a sample following polarisation in a constant d.c. field above a transition temperature. The sample is heated at a constant rate to the polarisation temperature under a field of approximately 10 kV/cm, maintained at this temperature for a specified time, then cooled to –150°C. The external field is then removed and an electrometer applied to measure the short-circuit current while the sample is again heated. At transition temperatures, a peak in current is recorded. The temperatures at which these peaks occur, the areas under the peaks and the dependence on thermal history all give a

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new dimension to measuring polymer transition phenomena. This technique is just beginning to be be used for pharmaceutically relevant samples and it is likely that more work will be conducted in this field in the future.

3 Dielectric analysis of solutions 3.1 INTRODUCTION In this chapter, the dielectric properties of solutions, particularly aqueous solutions, will be discussed. A knowledge of solution behaviour is of considerable pharmaceutical importance, in terms of both the formulation of medicaments and also the understanding of the distribution of drugs in tissues and membranes throughout the body. In particular, the aqueous solubility of a new chemical entity is one of the first physico-chemical properties to be measured during the development of a medicine, as it is considered to be a fundamental factor in the choice of dosage form and formulation. Liquid dosage forms are usually prepared for parenteral medicines or for oral administration to the elderly and the very young. If a solution, rather than a suspension, is to be administered, then the choice of liquid vehicle will be dictated by the stability of the drug in that vehicle, the bioavailability and toxicity profile of the vehicle and the solubility of the drug within the vehicle. However, the importance of solubility extends to all dosage forms, as it is almost invariably necessary for a drug to be in aqueous solution before it may cross biological membranes, hence the drug must to some extent be hydrophilic. However, as biological membranes are composed largely of lipid, the drug must be relatively lipophilic in order for partitioning to take place. This contradiction leads to perhaps the most important problem in the formulation of medicines, that of facilitating the dissolution of the drug within an aqueous environment such as the gastrointestinal tract while also allowing drug partitioning into lipid membranes to occur. Clearly, a knowledge of the solution behaviour of the drug is of considerable importance in achieving this objective. Furthermore, the solution behaviour, and in particular oil/water partitioning, may be directly related to the biological activity of certain drugs, hence the potency and distribution behaviour of untested compounds may be estimated by measurement of appropriate physico-chemical properties. A knowledge of drug solubility and dissolution is not, however, the only relevance of solution chemistry to the pharmaceutical sciences. Most reactions occur in solution, hence the stability of drugs and interactions between drugs and other substances must be considered. In this chapter, therefore, the discussion will not be limited solely to systems in which one or more constituents are used pharmaceutically. Instead, the broader question of how one may understand and predict the behaviour of a substance in solution and how dielectric analysis has contributed to such discussions will be outlined. Indeed, a solution is considered to be ‘pharmaceutical’ solely due to the biological relevance of one or more constituents, the physico-chemical considerations being the same as for any other system. This chapter has been structured to provide firstly a theoretical background to

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dielectric analysis of liquids in general. This is in itself an extremely large topic and hence the information given here is intended only as an introduction rather than a comprehensive review. However, emphasis has deliberately been placed on the effect of solutes on water structuring, as this is an area which has arguably been somewhat neglected within the pharmaceutical sciences despite the clear possibility that such structuring may have a profound effect on the solution behaviour of drugs, particularly in terms of solubility. The existing uses of dielectric analysis within the pharmaceutical sciences will then be outlined. As these studies have concentrated almost entirely on the use of dielectric constants rather than the examination of the frequency dependent response, this chapter will provide a discussion of the further possibilities afforded by using dielectric spectroscopy for these systems. The discussion will be limited to the solution behaviour of relatively small molecules, as the responses of larger molecules such as proteins will be covered elsewhere.

3.2 THE DIELECTRIC RESPONSE OF WATER An understanding of the properties of water is of importance in a wide range of scientific fields. Similarly, the electrical properties of water are of interest, both as an aid to understanding the behaviour of aqueous solutions and as a means of probing the structure of water itself. Indeed, as will be shown, dielectric studies have made a significant contribution to the debate regarding the structuring of water, although the issue is far from resolved and studies are still continuing. 3.2.1 The structure of water One difficulty in assessing the dielectric and other properties of solutions lies in the lack of understanding concerning the nature of liquids in general. More specifically, there is debate as to whether they should be treated as compressed gases or disordered solids. The volume change on melting a solid is much smaller than the volume change caused by vaporisation of the corresponding liquid. Furthermore, heats of vaporisation are usually much greater than heats of fusion for the same substance. Both these observations therefore indicate that liquids are more equivalent to solids than they are to gases. However, the negligible shear strength of liquids compared to that of solids indicates similarities between liquids and gases. The problem of how to consider liquids is even more complex when considering water, in which hydrogen bonding results in a greater degree of structuring than in non-polar liquids. This hydrogen bonding accounts for many of the unusual properties of water, including the low vapour pressure (and hence comparatively high boiling point), the larger molar volume in the liquid than in the solid state, the high specific heat capacity, the density maximum at 4°C and the high dielectric constant (static permittivity). In the H2O molecule itself, the covalent binding of the oxygen atom to two hydrogens results in the two covalent bonds and the two remaining lone pair electrons forming a near-tetrahedral shape with an angle of 104.5° in water and 109.4° in ice. While the molecules in a sample of water will be in continuous motion, a configuration

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will exist at any one instant that may be regarded as being a ‘structure’. Early work on this structure by Bernal and Fowler (1933) indicated that X-ray diffraction patterns of water and ice were remarkably similar, the sharp pattern for ice having become ‘blurred’ on melting. From this it was proposed that water consists of a ‘broken down’ ice structure. The ways in which a such a structure may form is the subject of some controversy. One model is that the lattice structure of ice becomes effectively disrupted but not entirely broken down, with a decrease in unit cell volume. The resulting liquid would therefore be effectively homogeneous with a negligible number of truly ‘free’ water molecules, hence it is arguable that a substance corresponding to such a model would not constitute a liquid at all. A modification was suggested by Pople (1951) based on the idea that water undergoes a considerable degree of bond bending, thus allowing the structure to assume liquid properties due to the flexibility, rather than simply the rupture of the hydrogen bonds. An alternative model considers the structure of ice to collapse on melting, leaving small regions of ice-like order (known as icebergs) within the system. These icebergs are considered to be continuously breaking down and reforming, with a sufficiently high proportion of broken bonds present to allow the structure to assume liquid properties (which is in itself an advantage over the previous model). The iceberg model allows predictions to be made regarding the behaviour of water which has been tested experimentally, both in terms of the dielectric properties (Haggis et al, 1952) and the thermodynamic properties (Nemethy and Schetraga, 1962, 1964). Such experimental data has led to refinements being made to the original approach, notably the ‘flickering cluster’ model whereby bond breaking is cooperative, rather than random (Frank and Wen, 1957). This cluster model has been disputed by Grant et al. (1978), although evidence for cooperative relaxation processes in a number of systems has been presented (e.g. Dissado and Hill, 1979). The two broad approaches (near-homogeneous, flexible structure or broken ‘iceberg’ structure) have been described as ‘uniformist’ and ‘mixture’ respectively. The area of water structuring has been the subject of controversies which have in the past spilled over into the public domain. In the 1960s, the concept of poly water was popular amongst many workers, a model whereby water molecules were thought to effectively polymerise to form long chains. In fact the ‘pure’ polywater that was isolated by repeated distillation of water was merely a concentrate of trace impurities arising from the distillation process itself. A more recent controversy regarding water structure concerns the studies of Benveniste (Davenas et al., 1988), who reported antibody activity to antigen solutions which had been diluted sufficiently so as to remove any antigen molecules. These findings appeared to validate the claims of homeopathic physicians who suggest that water structuring can lead to a therapeutic effect. This resulted in a somewhat acrimonious dispute within the scientific community, fuelled by Nature taking the unusual step of publishing an editorial disclaimer in the same issue as that containing the paper in question. At the time of writing, the debate has still not been resolved to the satisfaction of all parties. While the structure of liquid water is still a matter of debate, the structure of ice is relatively well understood. The oxygen atoms in ice are surrounded tetrahedrally by four other oxygens via hydrogen bonds. A number of different crystal forms of ice have been

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described in addition to that described above, although these other forms may only be manufactured under high pressures and are not found under normal circumstances. The density of water compared to ice is a function of the bonding between the water molecules in the two phases. In the solid, a tetrahedral structure is formed, while in water, these bonds are either broken or bent (depending on which model one assumes to be correct), hence the size of the unit cell is reduced and the density consequently increased. 3.2.2 Spectral features of the water response The dielectric relaxation behaviour of water is characterised by a single relaxation peak at a frequency of 17 GHz at 20°C, corresponding to a relaxation time of 9.3 ps (Grant et al., 1978). A representation of the response is given in Figure 3.1. The response between 35 and a few hundred gigahertz is represented as a dashed line as there is less information available in this frequency range due to difficulties in obtaining precise measurements, although more recent studies have produced high precision data in this region (Mattar and Buckmaster, 1990). The dielectric behaviour of water has been reviewed by Bussey (1967) and Kaatze and Giese (1980).

Figure 3.1: The dielectric response of water at 20°C (reproduced from Grant et al. (1978) by permission of the Oxford University Press) The response is typical for a material which undergoes reorientational polarisation, as discussed in section 1.3.4, but is unusual in that the dispersion demonstrates almost ideal (Debye) behaviour. Given the complexity of the structure of water, this similarity between theory and real behaviour is remarkable. The Cole-Cole parameter has been measured as lying between 0.01 and 0.02 over a temperature range of 0°–60°C (Hasted, 1973), with a slightly higher value of 0.025–0.045 being found near 4°C (Grant et al., 1978). These values are comparatively small and reflect the near-Debye behaviour of water. This behaviour presents interpretative difficulties regarding the Cole-Cole model, as one would expect a relatively wide range of relaxation times for water, given that H2O molecules exist in a range of structural environments (Walrafen, 1968).

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The relaxation time of water ( ) decreases with increasing temperature in an approximately exponential manner (Grant et al., 1978). The value of at 0°C is 18 ps (18x10−12 s), falling to 4 ps at 60°C. The relationship between and temperature (T) may be described by the equation

(3.1) where H is the activation enthalpy, R is the gas constant and n=0, 0.5 and 1 for an Arrhenius, Bauer and Eyring relationship respectively. In fact, Grant et al. (1978) were unable to obtain a linear relationship between log and 1/T using a range of n values (n=0, 0.5 and 1), although they were able to estimate the activation energy for the three cases as being 18.3 kJ/mol, 16.3 kJ/mol and 15.3 kJ/mol respectively. These measurements therefore provide approximate values for the energy of the relaxation process, the significance of which will be demonstrated shortly. The lower frequency response (<1 GHz) shows negligible loss, while the real part of the permittivity remains independent of frequency, corresponding to the static (relative) permittivity or dielectric constant ( s). At frequencies much greater than that corresponding to the loss peak, the real part of the permittivity decreases to a constant value (the infinite permittivity, ). From the discussion in section 1.2.2, it may be assumed that is due to electronic polarisation, thus dipolar reorientation is not involved. The (extrapolated) data shown in Figure 3.1 corresponds to a value of approximately 4.5 for (at 20°C), while the square of the refractive index (which is equivalent to the permittivity at optical frequencies) is only 1.7. There must therefore be a further process or processes occurring between these two frequency ranges. Zafar et al. (1973) have suggested the existence of a second dipolar relaxation process in the far infrared frequency range in order to explain the drop in , although the mechanism of this high frequency process has not been fully established. 3.2.3 Relationship between dielectric behaviour and structural properties of water The similarities in X-ray data reported for water and ice (Bernal and Fowler, 1933) are to some extent reflected by their dielectric properties. In particular, ice has a static permittivity of 92 at 0°C and a value of of 3.1, compared to 88.2 and 4.3 respectively for water at the same temperature (Hasted, 1973). As with water, the response of ice may be reasonably described by the Debye plot. However, the relaxation times for water and ice show large differences. The value of for water at 0°C is 18 ps while that for ice is 2×10−5 s, hence ice shows a loss peak in the kilohertz region. Furthermore, the activation energy for of ice is 55 kJ/mol, compared to approximately 18 kJ/mol found for water (Grant et al., 1978). The observation that values are of the same order of magnitude is not particularly surprising, as these values reflect the behaviour of the electrons within individual molecules. The observation that the static permittivities are also similar is of greater interest as this parameter is a reflection of the structure of the sample as well as the nature of the constituent molecules, hence the results indicate that, under a low frequency electric field, the dipolar structures adopted by water and ice are similar. The

Dielectric analysis of solutions

53

data has been interpreted in terms of both the uniformist and mixture approaches by Pople (1951) and Haggis et al. (1952) respectively, thus the results do not help to vindicate one or other approach. However, the data does support the hypothesis that water and ice are structurally similar. The significance of the differences in relaxation time and activation energy is best appreciated by considering the mechanisms involved in either process. The diffusion coefficient and viscosity of water show similar activation energies to that of the relaxation time (i.e. approximately 18 kJ/mol), implying that all three have a common mechanism. The relationship between relaxation time and viscosity for a single dipole may be described by

(1.59) hence one would expect a material with a higher viscosity or rigidity to have a higher relaxation time than one with a lower viscosity. The above relationship is based on the assumption that relaxation of a molecule is similar to a sphere of radius a relaxing in a viscous fluid, with k corresponding to the Boltzmann constant, the viscosity and T the absolute temperature. Plots of .T against for water yield a linear relationship (Grant et al., 1978), with the calculated value of 0.14 nm for the radius a, which corresponds to half the separation between two molecules in liquid water. The differences in the relaxation behaviour of water and ice has been exploited in the study of hailstones, with a view to characterising the radar backscattering observed under certain weather conditions (Chylek et al., 1991).

3.3 AQUEOUS AND NON-AQUEOUS SOLUTIONS As outlined in the introduction, a knowledge of the properties of drugs in solution is of importance not only in terms of formulation design but also in terms of understanding the behaviour of drugs in the body. In this section, the basic principles of how solutions behave in an electric field will be outlined before going on to discuss specific pharmaceutical examples of the use of dielectric analysis in this area. In order to avoid confusion between absolute and relative permittivities, the symbol refers to relative permittivity for the remainder of the chapter.

Dielectric analysis of pharmaceutical systems

54

3.3.1 Aqueous solutions 3.3.1.1 Solutions of electrolytes Electrolytes may be defined as substances which dissociate into two or more ionic species in solution. They may be further classified into strong electrolytes (such as NaCl), which may be considered to dissociate completely into their respective ionic species, and weak electrolytes, for which dissociation is only partial. An understanding of electrolyte behaviour is essential within virtually all branches of the pharmaceutical sciences, not only in terms of the behaviour of drugs and excipients but also the mechanisms by which many biological processes occur, examples being the transmission of nerve impulses and the control of urinary excretion. The measurement of the static permittivity of electrolyte solutions is in itself a matter of some difficulty, as the presence of the additional ionic substances causes a high dielectric loss due to d.c. conductivity (as discussed in section 1.3.4). Consequently, the value of tan ″ ( ′) will be so high that most bridge instruments will not be capable of measuring the proportionately smaller real part of the permittivity. It is easier to measure the static dielectric constant at higher frequencies (i.e. in the megahertz to gigahertz region) where the conductivity effect is proportionately smaller. Indeed, until higher frequency equipment was developed, studies on the effects of electrolyte concentration on the static permittivity gave largely inconsistent results. However, even at higher frequencies these measurements are not entirely straightforward as the relaxation peak of water and the still significant d.c. conductivity of the solutions need to be taken into account. Given these difficulties, much of the original work on the static permittivity was theoretical. Debye and Huckel (1923) predicted that the static permittivity of strong electrolytes will increase with the square root of concentration, although this has been found to be applicable only at very low concentrations. In fact, the static permittivity decreases significantly at concentrations in the region of 0.1–1 mol/1, as shown for KC1 and LiCl in Figure 3.2 (Collie et al., 1948). The relationship between concentration and static permittivity is approximately linear at the lower limit of concentration and is given by

(3.2) where SS and SW are the static permittivities of the sample and water respectively, is a negative quantity known as the dielectric decrement, which relates the permittivity lowering of the solute to the concentration c of that solute. A summary of decrement values is given in Table 3.1. At higher concentrations, however, the relationship between the static permittivity and concentration becomes non-linear. The physical basis of the permittivity lowering is considered to involve the polar ions orientating the water molecules around them, thus reducing the ability of those water

Dielectric analysis of solutions

55

molecules to reorientate on the application of an electric field. The dielectric decrement therefore gives an idea of the structure of the hydration sheath around the ions.

Figure 3.2: Concentration dependence of static dielectric constants of KCl ( ) and LiCl ( ) (reproduced from Collie et al. (1948) with permission of the American Institute of Physics) The hydration of such ions is sometimes expressed in terms of the hydration number, which is effectively the number of water molecule attached to an ion as it moves through the solution. Hydration numbers may be calculated from a number of sources, including viscosity measurements, activity coefficients, entropies and ionic mobilities as well as from the static permittivity via the dielectric decrement value. The hydration number found this way in fact refers to the number undergoing ‘irrotational binding’, i.e. the number of water molecules that are bound sufficiently strongly so as to prevent reorientation, and hence polarisation, of the water molecules in question. The number of bound water molecules (nirr) may be calculated from the dielectric decrement (Haggis et al., 1952). The relationship is given by

(3.3) where V and Vw are the molar volumes of the solute and water respectively,

is the

Dielectric analysis of pharmaceutical systems

56

static permittivity of water and and are the infinite frequency dielectric constants of the bound and free water. Using crystallographic data or molecular modelling to estimate V, the value of nirr is commonly of the order of 6 for a monovalent ion. It is therefore possible to predict the static permittivity of solutions using dielectric decrement values and from this to calculate the hydration of the solute in question. Further refinements to this approach have been given by Hasted (1973).

Table 3.1—Dielectric decrement values of a range of ions (Hasted, 1973) Ion Dielectric decrement (molar) + –17 H –11 Li+ + –8 Na + –8 K –7 Rb+ 2+ –24 Mg 2+ –22 Ba –35 La3+ – –5 F – –3 Cl –7 I– − –13 OH 2− –7 SO 4

Table 3.2—Relaxation time decrements for a range of ions (after Hasted, 1973) Ion decrease (ps/mol) + +1.3 H –1.0 Li+ + –1.3 Na + –1.3 K –1.7 Rb+ 2+ –1.3 Mg 2+ –3.0 Ba –5.0 La3+ − –1.3 F − –1.3 Cl –5.0 I− − –0.7 OH

Dielectric analysis of solutions

SO42–

57

–3.7

Figure 3.3: Effect of potassium fluoride concentration on the relaxation time of water (reproduced from Hasted (1973) with permission of Chapman and Hall Ltd.) The relaxation behaviour of electrolyte solutions may also be measured in the high frequency region where the contribution of the d.c. process is small. The effect of added electrolyte tends to be a reduction in relaxation time (i.e. an increase in the frequency at which the loss peak occurs). Moreover, the decrease in relaxation time is linear with concentration at low electrolyte levels, hence relaxation time decrements may be used in the same manner as static permittivity decrements. Examples are given in Table 3.2. It should be noted that the presence of hydrogen ions actually cause an increase in the relaxation time. The mechanism behind the decrease in relaxation time has been suggested as a structure breaking effect of the ions (Haggis et al., 1952; Hasted and Roderick, 1958). This model predicts that the addition of the ions causes a larger number of free water molecules to be present due to hydrogen bond breakage during hydration. Haggis et al. (1952) suggested that the extent of water molecule reorientation is related to the proportion of water molecules that are present in a free or semi-free state, whereby the number of bonded neighbours is less than the four predicted for a completely hydrogen bonded water sample. The presence of the ions therefore causes an increase in the proportion of less-bonded water molecules in the bulk liquid, i.e. a reduction in the total number of water-water bonds, increasing the number of molecules which are sufficiently ‘free’ to allow rotation. This results in a faster (smaller) relaxation time.

Dielectric analysis of pharmaceutical systems

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At higher electrolyte concentrations, the above analysis is inapplicable as the relaxation time increases and may exceed that of pure water, as shown for potassium fluoride in Figure 3.3. Pottel and Lossen (1967) and Giese et al. (1970) have attempted to explain this behaviour by considering there to be an exchange between water molecules in the bulk which may undergo reorientation and water molecules in the hydration sheath surrounding the ions, given by a mean residence time, r, which denotes the mean time spent in the hydration layer. The relaxation time of the solution compared to the relaxation time of water ( w) is then given by

(3.4) where c is the concentration and nh is the number of hydration molecules per ion. This model has not been fully accepted but has the advantage that it does predict a decrease in at low concentrations but a rise at higher concentrations. The half-widths of the loss curves of electrolyte solutions are generally greater than that predicted by the Debye model, i.e. greater than water alone. This has been attributed to a spread in relaxation times (e.g. Harris and O’Konski, 1956; Giese et al., 1970), although this interpretation is by no means fully established, as discussed in section 1.3.4. Overall, therefore, dielectric analysis of aqueous electrolyte solutions may yield information on the hydration state of the ions in question and the effect of those ions on the structuring of the surrounding water. 3.3.1.2 Aqueous solutions of non-electrolytes The treatment of non-electrolyte solutions found in the literature is different to that of electrolytes in an important respect, namely that the electrolyte systems have been considered to consist of simple ions (as shown by the examples given in the previous section), while non-electrolytes are treated as more complex molecules. There is no dipolar response for the ions, hence the above treatment concentrates on the effect of the electrolytes on the response of the solvent. In the case of non-electrolytes, however, the solute molecules may exhibit a dielectric response in their own right which must be considered alongside that of the solvent. Clearly, this differentiation is for convenience only but it does allow analysis of these systems in a somewhat simplified manner. It should also be pointed out that the majority of texts on the basic theory use solutes which are liquid at room temperature. The more pharmaceutically relevant systems of solid solutes will be discussed in subsequent sections. The static permittivity of aqueous non-electrolyte solutions may be related to the permittivities of the individual components using the treatment of Oster and Kirkwood (1946). The dielectric decrement is calculated using the equation

Dielectric analysis of solutions

59

(3.5) where s2 and s1 are the static permittivities of the solute and solvent (water) respectively and V2 is the mean molar volume of the solute. The decrement is calculated in essentially the same manner as that described in equation (3.2). This approach provides a reasonable correlation between experimental and theoretical values of the decrement, as shown in Table 3.3. It is therefore possible to estimate the dielectric constant of a solution from the permittivity and molar volume of the solute.

Figure 3.4: The dependence of the Kirkwood correlation factor on solute concentration (mole fraction) for isopropylwater systems (reproduced from Hasted (1973) with permission of Chapman and Hall Ltd.) It is also possible to assess the degree of association between the solute and solvent by using a modification to the Debye theory of relaxation which considers the deviation from ideal behaviour to be due to interactions (correlations) between relaxing species. The degree of correlation is given by the Kirkwood correlation factor (g). By using this approach, it is possible to estimate the effect of the addition of a solute on the cooperative behaviour of the solute.

Table 3.3—Comparison of experimental and calculated values of the molar dielectric decrement of a number of solutes (Hasted, 1973) – (calculated) – (experimental) Methyl alcohol 1.79 1.4 3.11 2.6 Ethyl alcohol 4.22 4.0 n-propyl alcohol t-butyl alcohol 6.26 6.3 Acetone 4.13 3.2

Dielectric analysis of pharmaceutical systems

Diethyl ether Glycol Aniline Methyl acetate Pyridine Acetonitrile Nitromethane

7.57 2.04 6.40 5.65 5.27 2.08 2.11

60

7.1 1.8 7.6 5 4.2 1.7 2.0

The Kirkwood correlation factor may be calculated via

(3.6) where Ps is the polarisation of the solution (calculated from the static permittivity) and the refractive index of the solution, k is the Boltzmann constant, x1,2 is the mole fraction of the solvent and solute respectively, is the dipole moment of the two species and N0 is Avogadro’s constant. More details of this and other approaches have been given by Hasted (1973) and will be discussed in the context of pharmaceutical systems in the next section. The point to emphasise here is that the correlation factor may give an idea of the degree of structuring in aqueous solutions and the effect that the addition of various solutes has upon this structuring, as the correlation factor will indicate the cooperative behaviour between the molecules. For example, acetone decreases the value of g, indicating structure-breaking properties, while ethanol causes a slight increase at high concentrations in water, indicating structure formation. Figure 3.4 shows the effect of isopropyl alcohol on the value of g, indicating that at approximately equimolar concentrations structure breakage is at a maximum. This could therefore be a highly useful approach to understanding how solutes effect aqueous solvents which could in turn lead to a greater understanding of the factors determining solubility. While the study of the static permittivity of non-aqueous solutes is of use, it is also of interest to have a knowledge of the relaxation behaviour of such systems. Typical data are shown in Table 3.4. Wessling et al. (1991) studied the relaxation behaviour of a range of PEG binary systems, including ethanol, methanol, toluene and water. The authors showed that the magnitude of the response varied linearly with concentration of solute for the non-aqueous systems. On addition of water, peaks corresponding to ‘free’ and ‘bound’ water were observed, the magnitudes of which varied with water content. A maximum in the lower frequency (‘bound’ water) relaxation peak magnitude was noted at a concentration corresponding to two water molecules per PEG repeat unit This ratio is in good agreement with IR and NMR measurements for the level of water binding to PEG molecules, thus indicating that analysis of relaxation behaviour may be of use in measuring not only the stoichiometry of molecular interactions in solution but also, from the relaxation times, the strength of that interaction.

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3.3.2 Non-aqueous systems 3.3.2.1 Dielectric behaviour of non-aqueous liquids While the behaviour of aqueous systems is of considerable pharmaceutical interest, nonaqueous liquids are also of importance, both in terms of formulation and drug distribution within the body. The most extensively studied non-aqueous liquids are water-soluble alcohols and other relatively polar substances. Liquids such as xylene and carbon tetrachloride do not show measurable relaxation behaviour and are used as solvents for studies on more polar molecules, as will be discussed shortly.

Table 3.4—Dielectric data for a range of non-aqueous solutes in water (Hasted, 1973) Solute c(mol/l) ×10−11 S S Pyrazine 2-Methylpyrazine

2,6-Dimethyl-pyrazine Quinoxalone

2-Methylquin-oxalone Pyridine

0.5 1.0 1.5 0.5 1.0 2.0 4.0 1.0 1.5 0.5 1.0 2.0 4.0 1.0 1.5 0.5 1.0 2.0 4.0

75.0 71.8 68.5 74.8 70.8 63.4 48.4 70.0 65.1 73.6 68.9 58.7 40.5 67.2 61.6 76.9 74.3 70.6 62.3

0.88 0.94 0.99 0.92 1.00 1.19 1.83 1.05 1.22 0.91 0.97 1.22 2.11 1.05 1.26 0.90 0.99 1.20 1.77

The static dielectric constants of a range of non-aqueous liquids are given in Table 3.5. These liquids will, to a greater or lesser extent, show some form of self-association which may be described using the Kirkwood correlation factor, as described in section 3.3.1. The dielectric behaviour of aliphatic alcohols has been studied by Oster and Kirkwood (1943) using this approach, whereby the correlation factor g is used to assess the degree of self-association. This analysis has been extended to other liquids and representative values are given below. In particular, the Kirkwood factor may be associated with the degree of hydrogen bonding within the various solvent systems. The value increases with

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aliphatic chain length for straight chain aliphatic alcohols, although for tertiary alcohols the value is low, possibly due to steric inhibition of hydrogen bond formation.

Table 3.5—Dielectric parameters associated with a range of non-aqueous liquids (Hasted, 1973) T(°C) g S n-Amyl alcohol n-Butyl alcohol n-Propyl alcohol Ethyl alcohol Methyl alcohol s–Butyl alcohol t-Butyl alcohol Isobutyl alcohol Benzyl alcohol Hydrogen cyanide Hydrogen fluoride Hydrogen peroxide Ammonia Diethyl ether Acetone Nitrobenzene Ethyl bromide Pyridine Benzonitrile

20 20 20 20 20

15.8 18.0 19.5 24.6 32.8

20 0 0 15 20 20 20 20 20 25

116 83.6 91 17.8 4.4 21.5 36.1 9.4 12.5 25.2

3.43 3.21 3.07 3.04 2.94 2.83 2.38 3.36 2.08 4.1 3.1 2.8 1.3 1.7 1.1 1.1 1.1 0.9 0.8

The relevance of cooperative behaviour in the relaxation of liquids is now considerably more established and cooperative models for a number of non-aqueous liquids such as methylene chloride (Vij et al., 1991a), acetone (Vij et al., 1991b) and methyl chloride/fluoride (Gershel, 1987) have been described (Vij et al., 1991a). Primary aliphatic alcohols show three relaxation peaks; one low frequency Debye—like response which increases in relaxation time with chain length and two higher frequency processes, one of which may be ascribed to rotation of hydroxyl groups around C–O bonds and the other to rotation of the monomer. The low frequency response has been ascribed to the breaking of hydrogen bonds followed by rotation of the monomers, whereby the former is the rate determining step. A further approach that may prove to be of considerable pharmaceutical interest in the future is the measurement of the relationship between dielectric constant and field strength (E). At low field strengths e shows no dependence on this variable, as discussed in section 1.3.2. However, on increasing the field strength, the parameter is obtained, where is the difference between the dielectric constant under the high field and under low fields. Malecki et al. (1991) and Dutkiewicz and Dutkiewicz (1993) have reported that this parameter is far more sensitive to changes in polarity and molecular interactions

Dielectric analysis of solutions

63

within solvent systems than the dielectric constant. This approach has not yet been used in the pharmaceutical field and represents a potentially exciting new approach to characterising solutions. 3.3.2.2 Dielectric behaviour of non-aqueous solutions The study of solutions of molecules in non-polar liquids represents one of the most important aspects of dielectric studies to the organic chemist, as by examining the relaxation behaviour of solute molecules it is possible to differentiate the movement of different molecular moieties in that solution. This subject has been reviewed extensively (e.g. Crossley, 1971) and a brief summary will be given here.

Figure 3.5: Plot of relaxation time in p-xylene solution against relative volume for fluorobenzene ( ), chlorobenzene ( ), bromobenzene ( ) and iodobenzene ( ) (reproduced from Crossley (1971) with permission of the Royal Society of Chemistry) Information regarding the shape, rigidity and dipole moment of molecules may be obtained from the relaxation behaviour of solutes in non-aqueous solutes. For example, for rigid molecules of similar shape and dipolar orientation there is a linear relationship between molecular volume and relaxation time in a given solvent, as shown in Figure 3.5. This is in agreement with the relationship proposed by Debye (1.59). Similarly, for an homologous series in which the molecular volume is similar and the direction of the

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64

dipole moment is uniform, the relaxation times are very similar, as found for quinolone, isoquinolone and phthalazine in cyclohexane, all of which reorientate about the same axis (Crossley and Walker, 1968).

However, for a series of molecules whereby the axis of rotation varies, the degree of displacement of surrounding solvent molecules will vary between the solutes and hence the relaxation times will vary accordingly. For example, the relaxation time of 4iodobiphenyl is over six times longer than 2–iodobiphenyl, as in the former case reorientation takes place over the short axis, hence involving a greater degree of solvent disruption.

There is often a considerable discrepancy between the relaxation time calculated from (1.59) and that found experimentally because the macroscopic (measured) viscosity of the solvent may be different to that experienced by the molecule itself due to the presence of a solvent layer sheath around the molecule. In practice, a better agreement between the relaxation time and the viscosity (as predicted by (1.59)) is found for systems in which the solute is large compared to the solvent (Grubb and Smyth, 1961). For less rigid molecules, multiple relaxation times may be seen due to the reorietation of specific moieties in addition to the reorientation of the whole molecule. For example, the response of a substituted phenol is shown in Figure 3.6 (Davies and Meakins, 1957), whereby the low frequency peak represent the molecular reorientation whereas the higher frequency peak corresponds to the reorientation of the substituted groups. A number of modifications have been suggested to take account of the deviations from (1.59) which in themselves yield information concerning the molecules under study, many of which have been reviewed by Illinger (1962). Perrin (1934) and Fischer (1939) took account of non-sphericity of the solute molecules by allowing corrections for ellipsoidal shapes. This analysis predicts the existence of three different relaxation times for such ellipsoidal molecules, which superimpose to produce the broad peaks seen in practice. An alternative approach by Gierer et al. (1953) assumed the existence of a solvent shell around the solute molecules (equivalent to the hydration sheath) which necessitates modification of the viscosity term in (1.59) to account for this layer. A further approach by Le Fevre and Sullivan (1954) included parameters relating to both the shape of the solute molecule and the interactions between solvent and solute. These examples therefore serve to illustrate how dielectric analysis is used to gain

Dielectric analysis of solutions

65

information on a range of molecular characteristics. As far as the relevance of this to the pharmaceutical sciences is concerned, the technique may be used as a means of understanding the molecular mobilities and shapes of drugs. While molecular modelling has allowed the shape of most drug molecules to be assessed relatively easily, the conformation of larger molecules such as proteins is more difficult to predict. As will be discussed in a later section, dielectric analysis may be used as a means of characterising the conformation of such molecules. The principles of analysing these larger molecules are the same as those described here.

Figure 3.6: The dielectric loss of (a) 0.76M 2, 4, 6-tri-tbutylphenol and (b) 0.28M 2, 6-di-t-butyl-4bromophenol in decalin solution (reproduced from Davies and Meakins (1957) through Hasted (1973) with permission of Chapman and Hall Ltd.) 3.4 DIELECTRIC ANALYSIS OF PHARMACEUTICAL LIQUIDS In general, pharmaceutical considerations of the dielectric behaviour of liquids has been almost exclusively confined to the study of the static permittivity (dielectric constant). This has proved to be a useful approach for a number of applications which will be reviewed in this section. A discussion of further uses of static permittivity measurements as well as the potential uses of measuring the frequency dependent behaviour will also be given. 3.4.1 Solubility, cosolvency and dielectric properties The term solubility may be defined as the concentration of solute in a saturated solution

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66

at a given temperature. A knowledge of solubility and solution behaviour is of considerable pharmaceutical importance, particularly in terms of the solubility of solid drugs in liquids. The solubility of a drug partially determines the dissolution rate and ultimately the bioavailablity and biological efficacy of a medicinal substance, as well as being an important practical consideration in the formulation and manufacture of dosage forms. In the formulation of liquid dosage forms such as injectables, the solubility may be sufficiently poor as to necessitate the inclusion of cosolvents, these being non-aqueous liquids which, when added to water, may result in a higher drug solubility than in water alone. In all these cases, therefore, a knowledge of the solubility of drugs is of considerable importance in order to characterise, control and, hopefully, predict the behaviour of a drug substance in a particular systems. It is not possible, at present, to quantitatively predict the solubility of a solute in a solvent. However, there are several approaches which provide a reasonable guide to whether a drug will show appreciable solubility in a given solvent. The most basic and, in many ways, the most useful approach is the ‘like-dissolves-like’ classification, which simply predicts that polar solutes will dissolve in polar liquids and non-polar solutes will dissolve in non-polar liquids. This statement is in many ways intuitively obvious but emphasises the link between solubility and polarity. The approach can be taken further by considering solvents to be polar (e.g. water), semi-polar (e.g. acetone, ethanol) and nonpolar (e.g. liquid hydrocarbons). Polar solvents act by reducing the forces of attraction between polar molecules (either ionic or dipolar), by breaking covalent bonds via acidbase reactions (e.g. HC1 in water) and by solvating solute molecules via hydrogen bonding and other dipolar interactions. Non-polar solvents are unable to reduce the forces of attraction between polar solute molecules, do not break covalent bonds due to their aprotic nature and do not hydrogen bond, hence the high lattice energy of a polar solid will not be overcome in a non-polar solvent. However, non-polar solutes dissolve in nonpolar solvents via weak London forces, hence oils and fats will dissolve in solvents such as carbon tetrachloride. Conversely, a non-polar solute such as a hydrocarbon will not dissolve in water because the interaction between the solute and the individual water molecules is too weak to allow incorporation into the hydrogen-bonded network. Semi-polar solvents, when mixed with water, will reduce the overall polarity of the solvent, thus easing the incorporation of non-polar solutes. Such liquids may therefore act as cosolvents, allowing miscibility of polar and non-polar substances. For example, the presence of propylene glycol increases the mutual solubility of peppermint oil and water, two liquids which are normally immiscible. Cosolvents are also extremely useful model systems for studying solubility behaviour in general, as by varying the proportion of cosolvent it is possible to obtain liquids of different polarities while keeping the nature of the chemical constituents the same. A number of quantitative approaches to the interpretation of solubility phenomena have been described. The simplest of these involves the use of ideal solution theory, i.e. the presumption that Raoult’s law is obeyed in that all intermolecular forces within the system are identical. This gives the ideal solution equation for a solid

Dielectric analysis of solutions

67

(3.7) is the heat of fusion, T0 is the where X2 is the solubility, R is the gas constant, melting point of the solute and T is the temperature of measurement. It should be noted that according to ideal theory, a liquid will be completely miscible with another liquid in all proportions due to the assumption of identical bond strengths in the system once the solid structure has been broken (accounted for by ). While the assumption of identical bond strengths may be reasonable for some non-polar systems, the above equation is clearly insufficient to describe the behaviour of more polar liquids, particularly aqueous systems. This problem was addressed by Hildebrand (1916), who proposed that the solubility of a solute in a solvent will be partly determined by the difference in intermolecular bond strengths between the solvent and solute molecules. This approach is known as regular solution theory. The strength of the intermolecular forces (or, more specifically, the square root of the cohesive energy density) is given by the solubility parameter , via

(3.8) where V2 is the molar volume of solvent, is the volume fraction of solvent and 1,2 are the solubility parameters of the solvent and solute respectively. (3.8) predicts that the solubility of the drug will be at a maximum when the solubility parameters of the drug and the solvent are the same, thus the theory predicts that if 2 is known, the most suitable solvent may be easily selected. This approach is useful for predicting the solubility of drugs in a number of solvents although, significantly, (3.8) is not applicable to aqueous systems due to the high degree of solvent-solute interactions. Martin et al. (1979, 1980) suggested a further modification, the extended Hildebrand approach, which is given by

(3.9) where W is a polynomial related to the solubility parameter of the solvent in the solution in question. This approach has been applied to several polar systems (e.g. Martin et al., 1979, 1980, 1981; Martin and Miralless, 1982) but arguably suffers from two drawbacks. Firstly, the parameter W is empirical and secondly the development of more sophisticated approaches to interpreting solubility data is in some ways self-defeating, as there can be little practical application for a system whereby the estimation of solubility requires a

Dielectric analysis of pharmaceutical systems

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greater number of measurements (to obtain W and , for example) than would be required to measure the solubility directly. However, Rubino and Yalkowsky (1987a,b) have given a thorough discussion of the approaches to solubility prediction and have proposed a linear regression approach which requires only four pieces of information in order to predict the solubility in any solvent: the solubility of the drug in water, the solubility of the drug in two cosolvents and a ‘polarity index’ of the drug such as the dielectric constant or the partition coefficient. While more work is required in order to verify this hypothesis, the possibility of predicting solubilities with such simple measurements has considerable practical significance. 3.4.2 The relationship between dielectric constant and solubility 3.4.2.1 The dielectric constant of cosolvent systems As stated in section 3.4.1, the use of dielectric analysis in the study of pharmaceutical solutions has been largely confined to the examination of dielectric constants rather than relaxation behaviour. While there is therefore a wider range of dielectric information that may be obtained, dielectric constants have proved extremely useful, as will be discussed below. However, in order to examine the relationship between the dielectric constant and drug solubility in cosolvents, it is first necessary to know the dielectric constant of the cosolvent system itself. Within the pharmaceutical literature, a simple mixture approach is used whereby it is assumed that the dielectric constant is linearly related to the proportion of the two pure components, i.e.

(3.10) where the subscripts 1 and 2 refer to the solvent and solute respectively. (3.10) basically predicts that the dielectric constant of the mixture will be the average of the dielectric constants of the individual liquids, weighted to account for changes in proportion of the two (or indeed more) components. This technique is generally ascribed to Moore (1958), although the author himself acknowledges previous work (Barr and Tice, 1957) in which this technique was used. The author also acknowledges that (3.11) will yield only approximate dielectric constants, non-linearity being seen as the polarity (i.e. proportion of aqueous phase) increases. However, the method has been found to give reasonable approximations (e.g. Shihab et al., 1971; Cave et al., 1979). The assumption of linearity between the dielectric constant and composition is essentially the same as for the dielectric decrement approach given in (3.3). However, Sorby et al. (1963) showed considerable variations between calculated and measured values for water-ethanolglycerin and water-ethanol-propylene glycol ternary systems. Furthermore, in a study of the solution properties of tetramethyldicarboxamides, Rebagay and DeLuca (1976) showed non-linearity between dielectric constant and amide concentration, although this system is complicated by the possibility of self-association between the amide molecules. The relationship given in (3.10) therefore gives an indication of the dielectric constant but should be used only as an approximation in many cases. A number of more

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sophisticated approaches are also available, although these have not been extensively utilised within the pharmaceutical sciences. The dielectric constant of a binary mixture may be considered in terms of the excess dielectric constant E, where

(3.11) where is the dielectric constant of the mixture of two components with dielectric constants 1 of 2 and , hence this parameter described the deviation from linearity between the dielectric constant of the mixture and composition. The value of E may be calculated using one of a number of mixture theories such as those of Debye (1912), Onsager (1936) and Kirkwood (1939). These approaches to calculating the dielectric constant of mixtures has been described in more detail by Aminabhavi et al. (1993). A related analysis has been described in the pharmaceutical literature by Amirjahed and Blake (1975), whereby the dielectric constant of a binary solvent was calculated from molar volumes and molar polarisations of the two components, although the fit between the theoretical and measured values for benzene-acetone systems was inferior to that obtained using (3.10). 3.4.2.2 The concept of ‘dielectric requirements’

Figure 3.7: Solubility of phenobarbital in binary systems at 25° C as a function of dielectric constant: 1-propylene glycolethanol, 2-glycerin-ethanol, 3-water-ethanol, 4propylene glycol-water, 5-glycerin-water (reproduced from Lordi et al. (1964) with permission of the

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American Pharmaceutical Association) The possibility of using the dielectric constant of a cosolvent to predict the solubility in that system attracted considerable interest in the late 1950s and early 1960s. The field was developed by Moore (1958), who suggested that the dielectric constant at which the solubility of a drug will be at a maximum will be constant, irrespective of the chemical nature of the cosolvent. This suggestion bears strong similarities to the idea of a solvent having a characteristic solubility parameter which, if the same as that of the drug, will yield maximal solubility. Lordi et al. (1964) suggested the phrase ‘dielectric requirement’ to describe this value of the dielectric constant. A number of studies were conducted to examine the effectiveness of dielectric requirements (DR) as a means of predicting solubility and interpreting solubility behaviour. For example, a value of approximately 15 was reported for salicylic acid (Paruta, 1963), while the solubility of phenobarbital in different cosolvent systems was compiled from published data (Krause and Cross, 1951; Peterson and Ilopornen, 1953) by Lordi et al. (1964) and is presented in Figure 3.8, showing a dielectric requirement of approximately 28. In a later study, Paruta et al. (1964) discussed the solubility of drugs in a series of sugar solutions as a function of dielectric constant. The author found that the solubility of a range of semi-polar drugs increased with sucrose concentration, interpreting this increase in terms of the dielectric constant decreasing and hence approaching the DR for those particular drugs. Figure 3.7 demonstrates that the maximum solubility occurs over a comparatively narrow range of dielectric constants, as predicted by Moore (1958). However, the absolute values of solubility are clearly different at any single dielectric constant, hence the ideal solvent system will not be simply predicted by this parameter. The solubility of phenobarbital in ternary solvent systems have also been reported (Lordi et al., 1964) and again the absolute value of solubility varies considerably in isodielectric solvents. Lordi et al. (1964) concluded that the dielectric requirement gives a useful guide to the optimum solubility in any single cosolvent system, but may prove less effective in predicting the most appropriate chemical (as opposed to stoichiometric) composition of the solvent system. The use of dielectric requirements was further explored by Paruta and Sciarrone (1964, 1965) and by Paruta and Irani (1966 a,b). Using xanthines as model drugs, the dielectric requirements of caffeine, theophylline and theobromine were measured in solutions of dioxane, ethanol, methanol and cellosolve. Figure 3.8 demonstrates that a single drug may show multiple dielectric requirements and that the DR values may vary considerably between different solvent systems. This effect of seeing a different number of peaks in different solvents was also noted for amino-benzoate derivatives. In a study using n-alkyl esters of p-hydroxybenzoic acids, Paruta (1966 a, b) found two dielectric requirements of 14 and 30 in a range of pure solvents and a single value of 10 in dioxane-water mixtures, while a value of 30 was found in ethanol-water mixtures. These studies therefore demonstrate a very significant problem associated with this approach, namely that there is often little consistency between the results obtained using different solvents, even if the dielectric constants are identical. In a later study, Ghosal and Gupta (1979) examined the relationship between the dielectric constant of ethanol-water mixes and the solubility of a

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range of barbiturates in an attempt to correlate the dielectric requirement with the pharmacological activity of the various analogues. Some correlation was found, with the analogues with higher dielectric requirement having longer onsets and duration of action. However, the authors themselves pointed out that such correlations may only be expected to occur for families of drugs whereby the biological activity is dependent largely on the distribution and partitioning behaviour rather than the binding behaviour to a specific receptor. The results are nevertheless interesting and merit further exploration.

Figure 3.8: The solubility of caffeine as a function of the dielectric constant of various binary systems. A: dioxane-water, B: ethyl cellosolve-water, C: ethanolwater, D: methanol-water (reproduced from Paruta and Irani (1966a) with permission of the American Pharmaceutical Association) It may therefore be concluded that while dielectric requirements are of use in giving an approximation of the solubility behaviour of drugs, their use as a means of quantitatively predicting solubilities is limited. The difficulty in relating the solubility of a substance to the dielectric constant of the solvent alone can be predicted on theoretical grounds, especially in aqueous solvents. The dielectric requirement concept can only be of quantitative use in systems for which the dielectric constant is the only, or at least the most important, factor determining the solubility. This is often not the case; for example, a thorough study by Fung and Higuchi (1971) showed that polar non-electrolytes are more soluble in non-polar solvents which they term ‘interactive’ such as chloroform and ether than in inert hydrocarbons, even though the dielectric constants are very similar for

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such solvents. The study illustrates that any approach which presumes that the dielectric constant alone will predict solubility is flawed from the outset. However, it should also be emphasised that no single parameter is likely to be capable of predicting solubility in all cases, hence the dielectric requirement approach may be considered to be useful within the limitations described above. A further development has been the exploration of the quantitative relationship between dielectric constant and solubility using electrostatic theory. This is derived from the Born equation

(3.12) where G°el is the free energy of electrostatic interaction, N is Avogadro’s number, e is the electronic charge, r is the dielectric constant of the solvent and rs is the radius of the molecule. The Born equation was used by Larson and Hunt (1958) to relate solubilities to dielectric constant via

(3.13) where k is the Boltzmann constant and S is the solubility in two media (1 and 2). The above relationship predicts that if a range of cosolvent systems is to be used, then a plot of log solubility against inverse dielectric constant should yield a straight line. Larson and Hunt (1958) themselves pointed out the limitations of this approach, arguing that if extensive hydrogen bonding takes place within the system then (3.13) will not provide a good data fit. However, studies have shown the relationship to be valid for a number of systems, many of which will be mentioned in the following section. In particular, Molzon et al. (1978) studied the solubility of calcium oxalate in a series of ethanol-water systems and in a range of pure straight chain alcohols. The authors suggested that in liquids of low dielectric constant, the salt remained in the undissociated state, while in liquids of higher dielectric constant dissociation occurred. They went on to use (3.13) to predict the ionic radius of the oxalate anion as 2.4 A, which is in good agreement with the value of 2.27 A found from theoretical considerations. It is, however, interesting to note that this approach and the dielectric requirement approach are contradictory, as the empirical DR approach shows a maximum in solubility at a certain dielectric constant, while the Born approach predicts that the solubility will increase or decrease with dielectric constant ad infinitum. This contradiction has not yet been fully explored and may provide some interesting information concerning solvent-solute interactions. 3.4.2.3 The relationship between dielectric constant and solubility parameter The difficulties in predicting the solubility of a drug in a cosolvent system are common to all methods, particularly when studying aqueous systems. However, one approach that has been used comparatively successfully has been the use of solubility parameters, as

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discussed in section 3.4.1. Examination of (3.9) shows that the solubility of a substance in a solvent will be at a maximum when the solubility parameters of the two are identical, hence if the solubility parameter of the solute is known, then theoretically the maximum solubility will be obtained by choosing a solvent with a similar value. In this respect, therefore, the approach is similar to that of the dielectric requirements. However, solubility parameters allow a non-empirical analysis of solubility data, while, in the form described above, the dielectric requirement is essentially an empirical analysis. It is therefore worthwhile to examine any links between the two approaches. The relationship between solubility parameters and dielectric constant was first explored by Maryott and Smith (1951) and Burrell (1955), who showed a linear relationship between the solubility parameter and dielectric constant for a number of common solvents, including water. This is shown in Figure 3.9. The authors reported an approximate relationship

(3.14)

Figure 3.9: A plot of solubility parameters of common solvents and their dielectric constants (reproduced from Paruta et al. (1962) with permission of the American Pharmaceutical Association) This equation is therefore of some significance as it suggests a direct relationship between dielectric constant and solubility parameter. Paruta et al. (1962) pointed out that (3.14) was most applicable to solvents which undergo hydrogen bonding such as water and the short chain alcohols rather than the non-polar solvents such as benzene or

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dioxane. Furthermore, Paruta et al. (1962) suggested that, as solubility parameters are relatively difficult to measure, then measurement of dielectric constants could be used as a simple means of obtaining these values. These studies raise several additional interesting points. Firstly, the observation that there is a better relationship between solubility parameters and dielectric constants for polar or semi-polar solvents than for non-polar solvents is unexpected, as the behaviour of non-polar systems is usually more predictable due to the absence of hydrogen bonding. Indeed, Hildebrand and Scott (1950) stated that the solubility parameter approach was based on the assumption of there being no hydrogen bonding. Secondly, if the dielectric requirement is equivalent to the solubility parameter of the solute, then as well as predicting the solubility parameter from the dielectric behaviour, one should theoretically be able to obtain the dielectric requirement from the solubility parameter of the drug, if known. A further study attempting to link solubility parameters with dielectric behaviour was performed by Sun woo and Eisen (1971) using a series of sulphonamides in alcoholwater, water-glycerin, alcohol-water-propylene glycol and dimethylacetamide (DMAC)water-glycerin. The authors found linear relationships between solubility parameters and dielectric constants (estimated using (3.11)) for all solvent systems except the DMACwater-glycerin system, for which deviation from linearity was observed. These results are summarised in Figure 3.10. The slopes given in Figure 3.10 are in reasonable agreement with the value of 0.22 predicted by (3.14), hence the study indicates that the relationship between solubility parameter and dielectric constant holds for some solvent blends as well as pure solvents. There are, however, also discrepancies in this relationship between solubility parameter and dielectric constant. These issues were addressed in a thorough study by Gorman and Hall (1964). The authors argued that if dielectric constants and solubility parameters were indeed equivalent, then the various relationships between solubility and solubility parameters suggested by Hildebrand should be obeyed by dielectric constants. For example, Hildebrand (1963) reported a linear relationship between the log solubility of iodine in a range of solvents and the square of the difference between the solubility parameters of the solute and solvent. Gorman and Hall (1964) showed that this relationship did not hold for the dielectric constants of the solute and solvents. Hildebrand and Scott (1950) also showed a linear relationship between the log solubility of a range of gases and the solubility parameter of the solvent in which they are dissolved. Again, Gorman and Hall (1964) showed that the relationship between dielectric constant and the log solubility of dissolved gases was poor. However, the authors noted that linearity could be observed between logarithmic solubility and dielectric constant of solvents with related chemical structures such as n-alkyl alcohols or polyethylene glycols. Arnold et al. (1985) found a non-linear relationship between dielectric constant of aqueous polyethylene glycol solutions and concentration, ascribing this effect to the binding of water molecules to the polyethylene glycol backbone. They went on to discuss these effects in terms of the ability of polyethylene glycols to cause membrane fusion, arguing that this effect does not arise due to a direct interaction between the polymer and the membrane but rather due to changes in the conformation of the water surrounding that membrane.

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Figure 3.10: Relationship between solubility parameters and dielectric constants for ( ) glycerin-water ( ) alcoholwater ( ) alcohol-propylene glycol-water and ( ) DMAC-water-glycerin (reproduced from Sunwoo and Eisen, 1971). Slopes given in parenthesis. Munafo et al. (1988) have considered the more sophisticated approach of Keller et al. (1971) for non-regular solutions, i.e. systems which interact by forces other than dispersive forces. The total solubility parameter ( T) is considered to be a combination of several components, i.e.

(3.15) where

d

is the dispersive component,

p

is the polar component and is related to the

dielectric constant, while h is the hydrogen bonding component. This approach is of importance as it recognises that the dielectric constant is related to the solubility parameter of the liquid in question, but at the same time is not the only factor determining the overall value. It is possible to speculate that if the approach of relating dielectric constants to solubility behaviour is to advance, then the analysis outlined in (3.15) is likely to be the most satisfactory means of progressing. 3.4.3 Interactions between species in solution In this section, the considerations outlined previously are extended to consider the use of

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dielectric analysis in the study of interactions between species in solution. These have been broadly classified into solute-solvent interactions, the role of the dielectric constant of recrystallising fluids on the subsequent crystal form and the relationship between dielectric constant and the degree of dissociation. The distinction between work included in this section and that in the previous one is that papers reviewed here describe studies which have been specifically conducted in order to examine solvent interactions, solubility measurements being incidental to this. 3.4.3.1 The use of dielectric analysis in the study of solvent interactions The use of dielectric analysis to examine solvent interactions has been outlined in section 3.3, albeit on a more theoretical basis. In terms of practical applications, dielectric analysis is a standard technique within the chemistry literature for examining complex formation and interactions between solvents and solutes. This approach has been outlined in detail by Kulevsky (1975) and only a brief summary will be given here. There are two principal ways in which the approach may be used. Firstly, the static dielectric constant may be measured and from this the dipole moment of the solute in solution may be calculated. The electron distribution around the solute molecule may then be calculated. For example, iodine (which, in an inert solvent, will have no permanent dipole moment) forms complexes with a number of solvents including benzene, mesitylene and p-dioxan (Brownsell and Price, 1966). This was detected by measuring the static permittivity of the solution and, knowing the dielectric constants of the individual solvents, calculating the contribution made by the complex. Secondly, the relaxation behaviour of the solution may be studied. Using the same example (Brownsell and Price, 1966), the relaxation times of the iodine solutions has been used to verify the existence of complexes between solute and solvent. However, it is also possible to examine the dissociation behaviour of these complexes, as the measured relaxation times are generally smaller than one would expect for a rigid molecule of the same size. The rate of dissociation may be estimated from the magnitude of this discrepancy. It is therefore possible to study complex formation between solvent and solute in terms of the nature of the interaction (from electron distribution calculations) and the dissociation behaviour of the formed complex. The use of dielectric analysis in the study of interactions within the pharmaceutical literature has again concentrated on the use of dielectric constants. One of the earliest pharmaceutical studies was that of Schott (1969) who examined the hydration of dimethylsulphoxide (DMSO), a skin penetration enhancer, in aqueous solutions. The author used a number of techniques, including dielectric constant measurements, refractive indices and molar refraction, fluidity (1/viscosity) and density measurements in order to assess the interaction between DMSO and water. Schott (1969) argued that interactions could be detected by looking at the deviation of the above measurements from ideal behaviour, hence he appreciated that such deviations could be of practical use rather than an inconvenience. The greatest deviation was found for fluidity and dielectric measurements, which showed a maximum at a mole fraction of 0.23 and 0.25 DMSO respectively. The author concluded that fluidity measurements represented the best method of detecting such interactions, as this technique showed the clearest deviation

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from ideal behaviour, but found dielectric constant measurements to be a useful supportive technique, showing greater sensitivity to the interactions than refractive index or density measurements. Further relaxation studies include those of Anderson and Smyth (1963). The behaviour of mixes containing the non-polar electron acceptor 1,3,5-trinitrobenzene and the electron donors triethylamine, tributylamine and triphenylamine were studied. Trinitrobenzene and triethylamine were shown to form a stable complex with a relaxation time which showed little or no contribution from the uncomplexed amine, while tributylamine showed two relaxation regions, attributed to the complex and the free amine. No molecular complex was detected for the triphenylamine system. This study therefore demonstrates the potential of using relaxation time measurements as a means of detecting not only the presence but also the strength of interactions. 3.4.3.2 The relationship between crystal form and dielectric behaviour There is a considerable pharmaceutical need to understand the relationship between the processing method used in the manufacture of a drug and the subsequent crystal form, particularly in terms of the solvents used for recrystallisation. The correlation between solvent dielectric constant and crystal form of hexamethylmelamine (HMM), a drug active against human small-cell lung carcinomas, has been studied by Gonda et al. (1986). The authors have reported an empirical relationship between the aspect ratio of the crystallised drug and the polarity of the solvent used. A further study on the relationship between dielectric properties and crystal form was performed by Wadsten and Lindberg (1989) using the cytotoxic drug estramustine, which was shown to exist in four forms (A to D). Form A was anhydrous and was formed from solvents with a dielectric constant <24. Solvates (forms B and C) were formed from solvents with a higher dielectric constant such as methanol (dielectric constant 32.6), which reverted to form A on storage. In aqueous solvents of high dielectric constant (e.g. acetone:water 2:1) form D was found which was shown to be a monohydrate. These studies therefore indicate that there is a relationship between the dielectric properties of the crystallisation fluid and the subsequent crystal form, although the mechanisms involved are not yet clear. 3.4.3.3 Dissociation and dielectric constant The relationship between pH and ionisation of weak electrolytes has been studied extensively and is described by the well known Henderson-Hasselbach equation. However, dissociation of electrolytes may also be dependent on the dielectric constant of the solvent. Agrawal et al. (1987) studied the thermodynamics of dissociation of a series of sulpha drugs in dioxane-water mixes of varying composition, predicting that the free energy of dissociation could be related to electrostatic forces between ions and other, non-electrostatic processes. The pKa may be calculated by

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(3.16) where G°non is the free energy of non-electrostatic interactions, N is Avogadro’s constant, e is electronic charge, R is the gas constant, T is the absolute temperature, S is the dielectric constant of the solvent mixture, w is the dielectric constant of water and r+/– refers to the ionic radii of the cations and anions. Plots of pKa against 1/ should therefore yield a straight line. However, this is often not found in practice and indicates that more complex solvent interactions are present. For example, the authors showed that sulphamethoxazole and sulphisoxazole both showed deviations from (3.16) in the dioxane-water systems, this being ascribed to concentration-dependent interactions between dioxane and water. Newton et al. (1982) used a related analysis whereby

(3.17) where a is the mean cation/anion diameter, k is the Boltzmann constant and BH is a constant. The above relationship again predicts that plots of pKa will vary inversely with the dielectric constant of the medium (at least at reasonably high water contents), which was indeed found in practice for cyclizine, chlocyclizine and hydroxyzine in watermethanol systems. These two studies therefore demonstrate that dielectric constants may be used to predict dissociation constants in aqueous organic solvent mixes, provided that the interactions between the two solvents are linear with concentration. The relationship between dielectric constant and the degree of ionisation in nonaqueous solvents was examined by Amirjahed and Al-Khamis (1980), who related the dielectric constant of binary non-aqueous solvents to the half-neutralisation potentials (h.n.p., which is a measure of basicity in non-aqueous solvents) of dissolved bases. A series of benzene-acetonitrile solvent systems were selected and the half neutralisation potentials of the bases measured and compared to the pKb values. The objective was to determine a method of choosing the most suitable solvent system for assaying mixtures of bases in non-aqueous media. The authors showed that by controlling the dielectric constant, the differences in h.n.p. of a mixture of bases with similar pKb values could be maximised, hence considerably aiding the assay of such mixtures. The degree of association between drugs and excipients may also be related to the dielectric constant of the solvent. A study by Plaizier-Vercammen (1983) investigated interactions between povidone and a series of salicylic acid derivatives, suggesting an empirical relationship

(3.18)

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where B/F is the ratio of bound to free ligand (i.e. drug) and a1 and b1 are constants. 3.4.4 Reaction rate, drug stability and dielectric analysis The dielectric constant of a medium may affect the rate at which a reaction occurs. The studies that have been conducted in this area may be divided into two broad categories; the rate of reactions in general and secondly the more specific case of the stability of substances, notably the stability of drugs. The importance of the ability to interpret and predict the stability of drugs needs hardly be emphasised, stability testing being one of the first operations performed in the development of a new formulation. The principles of the kinetics of degradation have been extensively reviewed (e.g. Garrett, 1962) and some knowledge of this topic will be assumed here. 3.4.4.1 Theoretical treatment of reaction rate and dielectric constant The relationship between reaction rate and dielectric constant is complex, depending on the mechanism of the reaction and the nature of the active intermediate formed between the two species. A number of treatments of this topic have been described and for a thorough discussion of this subject, the reader is referred to Amis (1949). Virtually every treatment is derived from the Debye-Huckel theory. In the case of a reaction between two ions, the rate constant is given by

(3.19) where k′ is the specific rate constant in a medium of dielectric constant , e is the electronic charge, is the rate constant in a medium of standard dielectric constant, ZA.B are the charges on the two ions, k is the Boltzmann constant, T is the temperature and r is the radius of the complex, which is approximately equal to the sum of radii of the two ions. It should be noted that this equation refers to a system whereby the ionic strength is assumed to be zero. Furthermore, (3.19) is applicable to a system whereby the interaction occurs between two ionised species, hence it follows that if the species are of opposite charge, the rate will vary with 1/ , hence the rate will increase as the dielectric constant is lowered. This is intuitively sound, as it implies that a higher value will increase the screening between reactants, hence the higher the dielectric constant, the greater lower the probability of an interaction. Conversely, for ions of the same charge the rate will decrease with decreasing dielectric constant, as a solvent with a high value may reduce the repulsion between the similarly charged ions. (3.19) may also be written as

(3.20)

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where the reference medium is a gas of dielectric constant of unity. Similarly, a further related expression that is sometimes cited is

(3.21) where is the rate constant in a medium of infinite dielectric constant. (3.20) and (3.21) involve the use of theoretical terms which may be found by extrapolation, while (3.19) has the advantage of having a measurable standard reference rate. In all three cases, however, the equations predict a linear relationship between In k and 1/ . A number of refinements and a discussion of the relationship between dielectric constant and thermodynamic parameters associated with the reaction are given by Amis (1949). In the case of reactions between ions and non-ionic dipolar molecules, the treatment described above becomes considerably more complex. However, a simple expression relating the rate to the dielectric constant gives a reasonable approximation:

(3.22) where rA is the radius of the charges species and rM is the radius of the complex between the two species. Finally, reactions between two non-ionised dipolar molecules may also be analysed using this approach by considering the dipole moments of the species. The rate is approximated by the equation

(3.23) where µ1 and µ2 are the dipole moments of the two components and r is the sum of the radii of the two components. There are certain limitations to the above analysis. Firstly, it is generally only applicable to dilute solutions, as in more concentrated systems ionic strength effects become appreciable. Secondly, the analysis does not account for the orientation of solvent molecules within the system. Water molecules in aqueous cosolvent systems will orientate around the ions in question, thus the dielectric constant in the area surrounding the ion in question will be different from that in the rest of the solution. Plots of In k′ against reciprocal dielectric constant may therefore show considerable variation from linearity for this reason. An example of this is given by Michoel and Kinget (1977), who argued that in the immediate proximity of an ion, the dielectric constant is effectively reduced to a smaller value. Despite these difficulties, the above analysis is of considerable pharmaceutical significance. In the case of a hydrolytic reaction, for example, decreasing the dielectric constant may actually increase the rate of degradation, rather than decrease it, despite there being less water present. Furthermore, the slope of

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the In k′ versus curve may be used to assess the mechanism of the reaction. These applications will be discussed in more detail in the next section. 3.4.4.2 Examples of reaction rate relationships

Figure 3.12: Logarithm of the pseudo-first order rate constant for the oxidative degradation of hydrocortisone in mixed aqueous solvents against reciprocal dielectric constant (0.1M borate buffer containing 0.5 µg/ml of copper (II) chloride, pH 9.79, 37°C). ( ) ethanol, ( ) 2-propanol and ( ) propylene glycol (reproduced from Hansen and Bundgaard, 1981) Several examples of the use of dielectric analysis in the study of reaction rates exist in the pharmaceutical and chemical literature, the majority involving drug degradation studies. Early studies include that of Amis and Holmes (1941), who showed that the rate of acid inversion of sucrose increased when the dielectric constant was lowered by the addition of dioxane. Heimlich and Martin (1960) demonstrated that glucose decomposition in water was dependent on the dielectric constant in dioxane-water systems. Again, lowering the dielectric constant increased the reaction rate. Similarly, Marcus and Taraszka (1959) showed that the rate of chloramphenicol degradation in water-propylene glycol systems increased on lowering the dielectric constant. Prasad et al. (1972) demonstrated that the anti-inflammatory agent intrazole showed increased degradation with decreasing dielectric constant in both acid and alkali media. This was ascribed to hydronium ion attack on the uncharged species in acid media and hydroxyl ion attack on the intrazole anion in basic media.

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Figure 3.11: Variation in the rate of blue tetrazolium reaction with dielectric constant of (x) methylene chloride/methanol, ( ) methylene chloride/ethanol, ( ) 1-propanal/methanol and ( ) ethanol/water (reproduced from Graham et al. (1976) with permission of the American Pharmaceutical Association) Graham et al. (1976) studied the effects of dielectric constant on the ‘blue tetrazolium reaction’, which is used to assay corticosteroids. Blue tetrazolium oxidises the C17 side chain of the corticosteroids in alkaline solution and produces a highly coloured formazan whose concentration may be measured spectrophotometrically. The authors showed that decreasing the dielectric constant decreased the rate of reaction, concluding from this and other data that the rate was controlled by the reaction of oppositely charged ions, in this case the enolate ion formed from the steroid and the tetrazolium cation. Interestingly, however, the authors made no reference to (3.19) and plotted the logarithm of the reaction rate against dielectric constant (as opposed to the inverse of the dielectric constant) and obtained a linear relationship. While the dependence of rate on dielectric constant was followed in all solvent systems studied (both aqueous and non-aqueous), the absolute values differed considerably between different cosolvent-water systems. The rate data is plotted as a function of in Figure 3.11. This behaviour may to some extent be explained by a later study in which the kinetics of degradation of hydrocortisone alone were investigated in alkaline aqueous solutions containing a range of alcohols (Hansen and Bundgaard, 1981). In previous communications, the authors had established that hydrocortisone undergoes two

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degradation mechanisms, again involving the side chain at C17. This moiety may undergo metal-ion catalysed oxidative degradation or a non-oxidative reaction leading to the formation of a variety of products. Previous studies had indicated that both the kinetics of degradation and the relative predominance of the two mechanisms are highly dependent on the composition of the solvent. By measuring the degradation kinetics in a range of ethanol, propanol and propylene glycol-water mixtures containing disodium edetate, the oxidative pathway was blocked due to chelation of the trace metal ions required for this reaction. Under these circumstances, the composition of the solvent (and hence the dielectric constant) made no difference to the rate of degradation by the non-oxidative pathways. In the presence of copper ions, however, a linear relationship was found between the logarithm of the rate constant against inverse dielectric constant, irrespective of the chemical nature of the solutions. Interestingly, the chemical nature of the solvent had little effect on this relationship, as shown in Figure 3.12, possibly due to the chemical similarity of the solvents.

Figure 3.13: Effect of dielectric constant on the reaction rate of cephalosporin C at 35° C at pH 1.15 ( ,) 3.00 ( ) and 7.00 ( ) (reproduced from Lumbreras et al., 1984) It is not clear to what extent the reactions described in the two studies are comparable, although it would be interesting to repeat the studies of Graham et al. (1976) including EDTA in the systems under study. As the degradation mechanism was not known at the time of performing the aforementioned study, no criticism may be attached to the workers

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involved for not investigating this possibility. However, differing levels of trace metal impurities may account for the unusual relationship between reaction rate and dielectric constant and the differences in rate using different solvent systems. Equations (3.19) to (3.21) may also be verified by calculating theoretical values of r for the reacting species and comparing these values with tabulated values. This has been performed by Lumbreras et al. (1984), who studied the effect of dielectric constant on the stability of beta-lactam ring of cephalosporin C in methanol-water mixtures at different dielectric constants and different pH values. Again, a good relationship was found between log k and l/ (Figure 3.13). However, this study is of particular interest as the authors investigated this relationship at a range of pH values, as shown in Figure 3.14. The authors suggested that at the lowest pH values the reaction occurs between the cationic form of the cephalosporin molecule and hydrogen ions. Evidence for this was obtained not only from the slope of Figure 3.14, which indicates a reaction between two ions of like charge, but also from the calculation of the theoretical radius from (3.19). At pH 3.00, the drug is in the zwitterionic form and reacts with the unionised water molecules, while at pH 7.0, the reaction is believed to occur between the anionic form and water.

Figure 3.14: Influence of solvent dielectric constant on the observed rate constants for azathioprine hydrolysis at indicated pH and 80C (reproduced from Singh and Gupta, 1988b) While dielectric constants are therefore of use in examining degradation kinetics, they will not be sufficient to fully interpret degradation behaviour. For example, Singh and Gupta (1988a) looked at the stability of butaperizine dimaleate in aqueous solutions and reported an increase in degradation rate on adding dextrose and sucrose to the medium but a decrease in rate on adding ethanol and glycerine, demonstrating that while the

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dielectric constant was lowered in both sets of cosolvents, the reaction rate kinetics did not follow the same trend. The authors then went on to investigate the degradation of azathioprine in aqueous solutions (Singh and Gupta, 1988b) as part of a study into freezedried azathioprine products for injection. The stability in a range of aqueous solutions was measured in a range of ethanol-water mixtures and the relationship between pH and rate of degradation investigated. The stability and dependence on dielectric constant varied considerable with pH, as shown in Figure 3.14. This behaviour was attributed to azathioprine existing as an undissociated species below pH 3.0, at which a protoncatalysed degradation reaction takes place (Mitra and Narurkar, 1986). However, the lack of dependence on the dielectric constant at pH 10.84 is surprising, as at this pH the reaction is believed to take place between azathioprine anions and hydroxyl ions. This was ascribed to tautometric effects or solvent interactions, with the authors proposing that azathioprine exists in three tautomers at this pH, only one of which reacts extensively with water. The lack of rate dependence on dielectric constant was ascribed to the hydration sheath surrounding the drug molecules being unchanged at the various dielectric constants, thus the dielectric constant of the microenvironment around the drug molecules is itself independent of the bulk dielectric constant. A variety of other systems have been studied and the relationship between rate constant and dielectric constant examined. For example, the oxidation of reserpine by nitric acid in organic solvents was examined by Ahmad et al. (1989), while aspartame stability in water was studied by Sanyude et al. (1991), who showed that the stability decreased with decreasing dielectric constant Yuki et al. (1990) studied the degradation kinetics of sodium azulenesulphonate (SAS) and guaiazulene (GA) in ethanol-water mixtures at 40° C to 80°C. In both cases, log-linear relationships were found between rate constant and inverse dielectric constant, the slope being negative. The degradation pathways of these molecules are complex, hence it was not possible to interpret the dependence of the rate on dielectric constant with certainty. A number of studies have been performed relating dielectric constant to reaction rate. For example, the rate of formation of aspirin from a prodrug was studied by Hussain et al. (1979). The authors showed an excellent correlation between rate and 1/ , with the rate decreasing with decreasing . From this and other evidence the authors concluded that the reaction occurs by the formation of a charged intermediate which then undergoes simple bond cleavage. This represents a further research area which may be of considerable practical application in the pharmaceutical sciences.

3.5 CONCLUSIONS This chapter has outlined some of the uses, both current and potential, of dielectric analysis in the characterisation of pharmaceutical solutions. The contribution made by dielectric studies in characterising the properties of water and ice have been outlined and an introduction given to the principles of using the technique to characterise both aqueous and non-aqueous solutions. From the examples given, it is clear that dielectric analysis may be used to gain a greater understanding of how solutes and solvents interact. In terms of more specifically pharmaceutical examples, it is apparent that dielectric analysis has so

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far been confined to the use and measurement of dielectric constants. While this is of use in the study of drug solubility, stability, dissociation and crystallisation, the studies outlined earlier in the chapter strongly suggest that the frequency-dependent response could add a completely new dimension to these studies. It is indeed surprising that no attempt has yet been made to correlate the relaxation behaviour of a solute (such as a drug) in a solvent and the solubility of that solute. This could avoid many of the problems associated with using dielectric constants, as factors such as hydrogen bonding or the formation of solvent sheaths around the drug would be detected by the relaxation behaviour. There is therefore considerable further scope for using the technique to characterise drug behaviour in solution.

4 The analysis of colloids and suspensions 4.1 INTRODUCTION 4.1.1 The pharmaceutical importance of colloids and suspensions In this chapter, the use of dielectric spectroscopy in the analysis of liquid disperse systems will be outlined. In particular, the analysis of pharmaceutical colloids and suspensions such as liposomes, microcapsules and emulsions will be described. Clearly, this represents a broad range of pharmaceutically important systems and only a brief outline of the uses and dielectric properties of each system may realistically be given. However, it is helpful to give an introduction to the definitions used here and the reasons why these systems are of such pharmaceutical importance. Systems containing a material dispersed in a liquid may be classified into three general categories. Firstly, the material may be molecularly dispersed, thus forming a solution. These systems were covered in the previous chapter. Secondly, the material may be dispersed as fine particles which, while not being considered to be solutions in the strict sense, nevertheless have some properties in common with true solutions. If the particles are less than 0.5 µm in diameter, the systems appear clear, with the particles being maintained in a suspended state due to Brownian motion; these systems are broadly classified as colloids. Systems containing particles greater than approximately 0.5 µm are considered to be suspensions (sometimes referred to as coarse dispersions). These systems are physically unstable and will tend to settle to the bottom of the container due to their larger size. These definitions are extremely broad and many grey areas will exist. In particular, many molecules are known to associate in solution, hence these systems arguably no longer represent true solutions. For example, surface active agents associate to form micellar and other structures; these systems are considered to be colloids rather than solutions. However, there are substances such as ascorbic acid which form dimers in aqueous solution but which are still classified as true solutions rather than colloids. Further difficulties arise when considering macromolecules. Dispersions of many large molecules in water are classified as colloids, even if those molecules are dispersed on a molecular basis. The reason for this is simply that, despite each molecule being separate from its neighbours, the size of that molecule is such that it assumes at least some of the properties of a small solid particle. Colloids have the fundamental property of being two-phase systems, as opposed to solutions which are one-phase. The majority of properties of colloidal systems (including dielectric properties) stem from this single aspect of their physical chemistry. For example, the dispersed particles will have an electrical double layer surrounding the solid

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surface. The properties of this electrical layer are of considerable importance to the physical stability of the system, as by manipulating the charge on the particles it is possible to produce systems with varying tendencies to aggregate (or flocculate). The particles will also tend to scatter light due to the difference in refractive index between the ‘solid’ particle and the surrounding media. This light scattering forms the basis of particle size analysis using laser diffraction techniques. In this chapter, the term ‘colloid’ will be used to represent both colloids and suspensions. It should be appreciated that there are differences between the two types of system, although in dielectric terms the differences are those of magnitude rather than mechanism of response. Dielectric analysis may give information on the size and shape of the particles, the state of water binding in the system, the properties of the colloidsolvent interface and the distribution of ions around the particle surface. Furthermore, information may be obtained concerning the behaviour of molecules within the dispersed particles. While several pharmaceutical examples of the use of this approach will be given, it is hoped that the chapter will also demonstrate the extent to which the technique could be exploited to a much greater extent than is presently the case. 4.1.2 Dielectric analysis of disperse systems 4.1.2.1 The response of heterogeneous systems Before considering the specific example of colloidal dispersions, it is useful to outline the principles associated with the dielectric behaviour of two-phase systems in general. The frequency dependent response of heterogeneous systems is relatively complex. The original analysis was put forward by Maxwell (1873), although Wagner (1914) later modified the approach to account for the behaviour of dispersed spherical systems. The approach is therefore known in general as the Maxwell-Wagner theory. There are several means available to describe this analysis and a modification of the description outlined by Hill and Pickup (1985) will be given here. Other recommended texts which give details of the theory are Daniel (1967) and Hasted (1973).

Figure 4.1: Diagrammatic representation of a heterogeneous system If one imagines two slabs of material in series between two electrodes, their behaviour may be considered to be equivalent to two series RC circuits, as shown in Figure 4.1. If one layer has a large capacitance and a small conductivity (slab 1) and the other to has a

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small capacitance and a large conductivity (slab 2), then the response is given by the plot shown in Figure 4.2. Note that the imaginary component of the complex capacitance (the dielectric loss) is given by G/ω, hence the conductivity is actually constant at low and high frequencies.

Figure 4.2: Typical frequency dependent response of a two-phase system (reproduced from Hill and Pickup (1985) with permission of Chapman and Hall Ltd.) There are several points to note regarding this diagram. Firstly, the behaviour of the sample with a high capacitance (slab 1) will be seen at lower frequencies than the sample with the lower capacitance (slab 2). This may be described by

(4.1) and

(4.2) where R is the resistance (1/G) of the slab. The most important point is that the low and high frequency extremes will represent the behaviour of the two individual components. The region between the two, however, will show a decrease in the capacitance as the frequency is increased and, depending on the characteristics of the two materials, a peak in the loss component. It is therefore possible for a system to exhibit behaviour which is similar to that predicted by Debye for dipolar reorientation, even though no such process has taken place.

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This phenomenon is perhaps best understood by considering the situation on a qualitative basis. When an alternating field is applied to a heterogeneous system, at low frequencies both capacitors (slab 1 and slab 2) will become charged, hence the slab with the highest capacitance (slab 1) will tend to dominate the response, as the amount of charge stored on this slab will be much greater than on slab 2. At high frequencies, however, only the lower value capacitor (slab 2) has time to become charged, as it requires less time for charge to accumulate or discharge, as described in section 1.2.2. Slab 2 will therefore dominate as the frequency is raised. At intervening frequencies, however, the loss will show a peak value, indicating that there is a process occurring which has a relaxation time. As both the high and low frequency capacitors are being charged at these frequencies, there will be a charge imbalance across the interface of the two due to their differing abilities to store charge. Therefore, in order for these charges to balance, a finite time is needed for charges to travel across the interface which will depend on the capacitance of slab 1, as this describes the amount of charge accumulated and the resistance (or conductance) of slab 2, as this determines the ease with which slab 1 may discharge. In a manner similar to that seen for the Debye case, there will be a frequency at which this process occurs most efficiently; at lower frequencies the transfer is much more rapid than the change in field, hence the process is inefficient, while at higher frequencies there is insufficient time for the charge movement to take place between the two slabs. Indeed, as the frequency rises, the higher capacitance slab does not become charged at all. The frequency at which the capacitance and loss are equal is known as the crossover frequency (ωMW) and is given by

(4.3) hence the relaxation time for the process will be R2C1. In general, these effects occur at relatively low frequencies, usually in the kilohertz to millihertz region. 4.1.2.2 The response of colloidal particles In dispersions of pharmaceutical interest, the most important systems are particles with a relatively small conductance in a conducting (aqueous) medium. While the responses of the different systems will be discussed individually, it is helpful to outline the general theories associated with the response. One of the first approaches to the analysis of such systems was given by Schwan et al. (1962). This analysis was based around the observation that the real part of the permittivity had been observed to increase with decreasing frequency in the low frequency region, an example of which is shown in Figure 4.3. The authors discussed the various possible explanations for their results as follows. The Maxwell-Wagner theory described above was not considered to be responsible due to the predicted change in dielectric constant being too small compared to the experimentally observed changes in the real permittivity. They also considered electrophoretic movement of the particles due to surface ionisation of the polystyrene spheres. Again, however, theoretical calculation of the loss frequency and magnitude of

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the dispersion indicated that this factor was unlikely to explain the observed results. They also considered a surface conduction, whereby charges would move tangentially to the surface on application of an electric field, again with little correlation between experimental and predicted data. However, the authors suggested that the data could be interpreted in terms of a surface admittance, whereby the surrounding layer could be considered to have both conductive and capacitive elements. A physical explanation of this was not put forward, although the movement of counterions around the surface of the particles was implicated. This approach was discussed in greater detail by Schwarz (1962), who suggested that a tightly bound layer of adsorbed counterions may be responsible for the observed increase in permittivity at low frequencies and DeLacey and White (1981), who have given a thorough discussion of the relationship between the low frequency response, the zeta potential and the particle size of colloidal systems.

Figure 4.3: Real and imaginary permittivities of polystyrene particles as a function of frequency (reproduced from Schwan et al. (1962) with permission of the American Chemical Society) Lyklema et al. (1983) discussed further modifications to the low frequency analysis described above, particularly in terms of the assumptions that there was no charge exchange between the adsorbed layer and the bulk medium and that only the tightly bound ions needed to be accounted for in the analysis. The authors took account of the diffuse layer of counterions as well as the bound Stern layer, arguing that the low frequency permittivity was a function of both these processes. Furthermore, they suggested that dielectric analysis may be used to characterise the nature of the diffuse double layer surrounding suspended particles. Lyklema et al (1986) later compared theoretical predictions made on the basis of this approach with experimental data

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obtained by Springer et al. (1983). The latter study involved the measurement of preparations containing polystyrene spheres in electrolyte solutions over a frequency range 70 Hz to 3.2 MHz. Lyklema et al. (1986) reported good correlation between theoretical and experimental data, thus lending support to the principal that the diffuse layer of counterions must be accounted for when interpreting low frequency data. In both cases (diffuse layer and bound counterion responses), the frequency dependent behaviour will be characterised by a relaxation time which is a function of the diffusion coefficient of either type of ion. O’Brien (1986) predicted that colloids will in fact exhibit two dispersions; one at low (kilohertz) frequencies which relates to the diffusion of ions in the diffuse layer surrounding the particle (as discussed above) and the higher frequency dispersion (in the megahertz region) which is due to redistribution of ions across the surface of the particle. Midmore et al. (1987) examined the behaviour of polystyrene latex particles in the region 1–10 MHz, a range in which the crossover between the two types of behaviour is expected. Using a model based on the above assumptions, a good correlation was found between experimental data and theory, although the fit was noted to be better at higher volume fractions of spheres (>0.3). However, Rosen and Saville (1991) later studied the response of an amphoteric polystyrene latex and colloidal silica in aqueous media, concluding from the poor correlation between experimental and theoretical data that the dielectric response may not be fully interpreted simply in terms of the particle size and ionic composition of the suspending electrolyte. Recently, the study of the role of ion movement in the Stern layer has been developed further by Rosen et al. (1993), who included such ion migration processes within their model which leads to a greater correlation between predicted and experimental data. While there is therefore good evidence that counterion relaxation is responsible for some dielectric phenomena, the low frequency response is still not fully understood. For example, Lyklema et al. (1986) acknowledge that their model does not fully explain the magnitude of rise in the low frequency permittivity. In many studies, the dispersion at very low frequencies is ascribed to electrode polarisation, which was outlined in section 2.2.3. Electrode polarisation is considered to be due to the accumulation of ions at the electrodes which will lead to an additional low frequency response. This phenomenon has been considered by Hill and Pickup (1985), who argued that the accumulation of ions is only one of a number of types of layer which may accumulate at the electrodes. Furthermore, the authors have suggested that the electrode layer may be considered as a separate layer in series with the bulk response. These layers may therefore be described by the Maxwell-Wagner response described earlier. The composition of the electrode layers may be ionic but may also consist of molecules adsorbed from the bulk, hence these layers may not merely be an artefact but may yield useful information on the sample as a whole. Examples of this will be given in subsequent sections. One aspect of this analysis, however, that has proved particularly useful is that the authors applied the Dissado-Hill theory (Dissado and Hill, 1979) to these low frequency responses. The complete analysis is beyond the scope of this book but the essential features are as follows. The response that is observed experimentally is often similar to that shown in Figure 4.4, whereby the low frequency capacitance is not constant and the high frequency

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capacitance has a log/log slope other than the value of –2 shown in the theoretical case (Figure 4.2). The value of the low frequency capacitance indicates the ‘leakiness’ of the electrode barrier layer. A slope of 0 indicates that the electrode layer is completely effective at blocking charge movement, while deviations from the horizontal (given by the exponent s in Figure 4.4) indicate an increasing lack of physical integrity of the adsorbed layer. This type of measurement may have practical implications, as will be discussed shortly. The approach used by Hill and Pickup (1985) predicts that the sum of the logarithmic high and low frequency slopes should equal –2. This has been shown to be the case on a number of occasions (e.g. Binns et al., 1992; Barker et al., 1994), thus lending support to the approach outlined here.

Figure 4.4: Schematic representation of the low frequency response whereby non-ideal behaviour is observed (reproduced from Hill and Pickup (1985) with permission of Chapman and Hall Ltd.) There are therefore two approaches to the interpretation of the low frequency behaviour, involving either the response of ions surrounding the particles or the presence of electrode layers. These two approaches are not necessarily incompatible; if, for example, the establishment of electrode layers is prevented by the use of four electrode cells (Myers and Saville, 1989a,b) or any of the means outlined in section 2.2.3, then the response of the surrounding ions may be expected to dominate the response. However, this area does represent one in which the lack of definitive agreement has led to different groups pursuing very different lines of research, depending on which of the above two basic assumptions is assumed to be correct. Furthemore, there is evidence for the validity of both approaches. For example, the work of Schwan et al. (1962) predicts an inverse square relationship between the peak loss frequency and particle size, which has been noted by several workers (as described below). However, advocates of the electrode layer Maxwell-Wagner approach would point to the extremely high values of low frequency permittivity shown in Figure 4.3 (approximately 2400) which may not be easily explained by the counterion relaxation theory.

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4.1.2.3 The measurement of particle size and shape A number of studies have described the use of the technique to assess the size and shape of dispersed particles. The former is of particular interest due to the possibility of assessing size in relatively concentrated systems, when light scattering and other techniques are inappropriate. This approach again relates to the work of Schwan et al. (1962), whereby the relaxation peak frequency caused by the counterion redistribution was predicted to be inversely proportional to the square of the particle radius. This approach has been developed by several workers, for example Vogel and Pauly (1988), who suggested the relationship

(4.4) where fc is the loss peak frequency and D is the diffusion coefficient of the counterions (approximately 2×10–5 cm2) around a particle of radius a. A number of further predictive relationships have been suggested, although none have been shown to be fully reliable (Sauer et al., 1990). However, the inverse square proportionality between the relaxation frequency and the radius is supported by numerous studies (e.g. Chew and Sen, 1982; Grosse and Foster, 1987) and therefore represents empirical evidence for the counterion relaxation approach, as mentioned above. There is therefore potential for using the technique as a means of particle size analysis for pharmaceutical systems, particularly for concentrated systems but more work is required to fully develop the theoretical basis of the approach. The possibility of using the method to analyse concentrated systems is, however, certainly worthy of further examination, even if the technique is found to be more suitable as a comparitive method rather than a means of making absolute particle size measurements. This is because pharmaceutical systems such as emulsions may be highly concentrated and therefore unsuited to analysis by many more conventional techniques. However, the use of dilution in order to allow measurement may change the system under study (for example, if flocculation or coalescense processes dominate), hence there is a role for an alternative approach to the measurement of concentrated systems. This will be discussed further in the next chapter. A further aspect of dielectric analysis has been the characterisation of particle shape. One of the first means of analysis on this topic was developed by Sillar (1937). The author assumed the particles to be ellipsoids with axis a in the field direction and equal axes b=c at right angles to the field in a medium of negligible conductivity. Sillar related the relaxation time to a dimensionless parameter via

(4.5) where

is the conductivity of the particles in a non-conducting fluid and

permittivities of the medium and particles respectively.

1/2

are the

is related to the axial ratio of

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the particles, as shown in Figure 4.5. While more sophisticated analyses are available (Daniel, 1967), the study demonstrates the importance of shape on the dielectric response of disperse systems. This discussion has therefore outlined the uses, both present and potential, of dielectric analysis in the study of disperse systems. In subsequent sections, specific examples will be given which describe the investigation of systems with pharmaceutical relevance.

Figure 4.5: The relationship between and axial ratio of conducting particles (reproduced from Sillar, 1937) 4.2 ANALYSIS OF SURFACTANTS AND MICELLES Surface active agents (or surfactants) are characterised by two fundamental properties. Firstly, they contain both hydrophilic and hydrophobic moieties, and secondly they have a tendency to adsorb at interfaces, notably the air-water interface of aqueous surfactant solutions. A further characteristic is a dramatic change in properties of surfactant solutions at a specific concentration, particularly the surface tension, conductivity and osmotic pressure. These phenomena are due to the formation of aggregates (micelles) which may, in the simplest case, be spherical. For either non-ionic surface active agents or high concentrations of ionic surfactants the micelles may be ellipsoidal or rod-like, or may ultimately form a gel system. The importance of surface active agents within the pharmaceutical sciences need hardly be overstated. They are used as wetting agents, as solubilising agents and as cleansing agents amongst other applications. It is therefore of considerable use to have some knowledge of their structure and behaviour. Dielectric analysis has been extensively used in this field, notably in the study of micellar structure, the energetics of micelle formation, micellar shape and the water binding within the micelle structure. A brief discussion of these principles will be given here.

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4.2.1 The dielectric constant of micellar membranes As micelles may be used to incorporate drugs, it is of considerable use to have some understanding of the factors determining the quantity of drug that may be incorporated. Ultimately, this will be a function of the physico-chemical properties of the micelles with respect to those of the drug. One method of studying the former is to use molecular probes in order to characterise the dielectric constant and microviscosity of the micellar membrane. In an excellent review, Grieser and Drummond (1988) discussed the various probe techniques that may be used for this purpose. By knowing the effective dielectric constant within the structure, it is possible to have a greater understanding of how drugs are solubilised. Again, however, this will ultimately rely on an understanding of how the solubility of a drug relates to the dielectric constant of the surrounding media, as discussed in Chapter 3. The most widely used probe techniques for the determination of dielectric constants involves the use of UV/visible spectrophotometry. In the ground state, a number of chromophores are more polar (in the sense of having a greater dipole moment) than in the excited state. Consequently, polar solvents will tend to stabilise the ground (polar) state and thereby increase the energy required for the transition to the excited state (ET). The wavelength of the maximum absorbance (

) of that chromophore is given by

(4.6) where h is Planck’s constant and c is the speed of light. A polar solvent will therefore max of a probe in a range tend to reduce , as ET will be increased. By measuring of solvents of known dielectric constant, it is possible to then assess the dielectric constant within the micelle, simply by measuring in a micellar solution. In a very interesting paper by Nakagaki et al. (1986), the authors measured the dielectric constant of micellar and liposomal membranes, with a view to understanding the solubilising properties of these systems. The authors used fluorescent probes (dialkylthiacarbocyanine dyes) to examine both the microviscosity and the dielectric constant of the membranes. The microviscosity may be measured using a similar approach to that described above in that the relationship between fluorescence and the viscosity of the surrounding medium was measured using standard solvents, after which the probe was incorporated into the micelle. The relationship between the dielectric constant of the surrounding media and the are shown in Figure 4.6. The authors used two probes, 3,3′-diethyl-2,2′-thiacarbocyanine iodide, 3-methyl-3′-octadecyl-2,2′-thiacarbocyanine bromide (C1–18) and 3,3′-dioctadecyl-2,2′-thiacarbocyanine bromide

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Figure 4.6: Relationship between wave number of the absorption maxima for C1–18 and C18–18 with dielectric constant of solvents: C1–18, C18–18 (reproduced from Nakagaki et al, 1986) The authors went on to tabulate the dielectric constant and microviscosity values of a range of systems, an abridged summary of which is given in Table 4.1. In a later study, Matsuzaki et al. (1989) looked at the microenvironments of bile salt micelles and bile salt phosphatidylcholine mixtures in order to gain a greater understanding of the processes involved in absorption of materials from the gut. Bile salts are amphiphilic and form micelles in aqueous media with small aggregation numbers (a phenomenon known as mild cooperativity). These systems may solubilise small amounts of phosphatidylcholine to form mixed micelles at a critical concentration.

Table 4.1—Experimental values of the dielectric constant ( r) for C1−18 and C18– 18 in various micelles and liposomal membranes (Nakagaki et al., 1986) Dye

Surfactant/lipid

Aggregate

r

C1–18

Heptaethylene glycol dodecyl ether Sodium dodecyl sulphate Cetyltrimethylammonium chloride Dodecylsulphobetaine L- -lauroyl lysophosphatidylcholine1 L- -dimyristoylphosphatidylcholine

Micelle " " " " "

18.5 19.3 22.5 22.6 25.0 23.5

Dielectric analysis of pharmaceutical systems

L- -palmitoyl lysolecithin L- -dimyristoyl phosphatidylcholine2 L- -dipalmitoyl lecithin L- -distearoyl lecithin C18–18 Heptaethylene glycol dodecyl ether Sodium dodecyl sulphate Cetyltrimethylammonium chloride Dodecylsulphobetaine L- -lauroyl lysophosphatidylcholine1 L- -dimyristoylphosphatidylcholine L- -palmitoyl lysolecithin L- -dimyristoyl phosphatidylcholine2 L- -dipalmitoyl lecithin 1Lysolecithin 2Lecithin

98

" Liposome " " Micelle " " " " " " Liposome "

22.0 18.9 20.5 21.6 20.0 18.0 20.0 23.0 24.5 23.0 24.5 9.5 12.3

The authors used sodium cholate, sodium deoxycholate, sodium glycholate and sodium taurocholate with egg yolk L- -phosphatidylcholine and two fluorescent probes, pyrene and C18–18. Dielectric constant studies indicated that in bile salt micelles, pyrene experiences a relatively apolar environment (dielectric constants ranged from 5 to 11) compared to typical micelles and vesicles (where the value is approximately 20). As pyrene is known to locate at micellar surfaces (Kalyanasundaram and Thomas, 1977; Ganesh et al., 1982: Turro and Okubo, 1982), the authors interpreted the data in terms of the hydrophobic nature of the micellar surface. The C18–18 probe showed a dielectric constant of approximately 20, probably due to the positively charged probe interacting with the anionic head groups of the bile salts in the surface polar regions. On addition of egg phosphatldylcholine, the dielectric constant increased for pyrene due to the release of the probe from the hydrophobic regions of the micelles into the phospholipid areas of the mixed micelles. The C18–18 systems showed an initial small decrease in dielectric constant on adding egg phosphatidylcholine to the micelles, after which the value remained constant with phospholipid concentration. This may be due to the probe interacting preferentially with the phospholipid. The study therefore shows how different probes may be used to gain complementary information on a system and to gain an idea of what might occur when drugs are added to those micelles. While this approach is extremely effective, it should be remembered that polarity is not the only factor that may influence the value. Molecular interactions such as hydrogen bonding or other solute-solvent interactions may also have an effect. However, these problems aside, there are a number of areas within the pharmaceutical sciences whereby this approach is of use, particularly in the field of colloidal drug delivery and in understanding the role of bile salt micelles in the absorption of drugs through the gastrointestinal tract.

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4.2.2 The conductivity and dielectric constant of micellar solutions

Figure 4.7: Specific conductivity of aerosol OT in benzene as a function of concentration (reproduced from Eicke and Christen (1974) with permission of the Academic Press) In addition to the information regarding the dielectric constant within the micellar systems that may be gained using probe techniques, direct measurement of dielectric properties may yield information on the formation, structure and properties of micellar systems. Conductivity measurements are a standard method of measuring the critical micelle concentration (cmc), with a discontinuity in the conductance against concentration relationship being seen at this concentration due to the surfactant molecules and counterions no longer being free to conduct but instead becoming incorporated into micelles. Consequently, even though the absolute conductance will tend to increase on addition of further surfactant, the concentration dependence of this effect will decrease above the cmc. In addition to identifying the cmc, a variety of further information may be obtained from conductivity measurements. For example, information regarding the mechanism of micelle formation such as the detection of nucleation processes may be obtained. Eicke and Christen (1974) showed that the specific conductivity of aerosol OT in benzene showed a plateau region on addition of the surfactant, as shown in Figure 4.7. The authors interpreted the initial region as being due to the conductivity of the monomers, followed by a plateau region which was attributed to the response of small nuclei which formed prior to micelle formation. In a later study, Singh et al. (1980) studied the micellisation of sodium dodecyl sulphate and cetyltrimethylammonium bromide in organic solvents of different dielectric

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constants by conductivity measurements, concluding that, in general, the cmc tends to decrease with increasing dielectric constant of the medium. However, solvents which exhibited a strong tendency to hydrogen bond did not comply with this relationship, an observation which has parallels in the studies on the relationship between solubility and dielectric constant discussed in the previous chapter. The authors went on to calculate the entropy, enthalpy and free energy of the micellisation process by measuring the cmc over a range of temperatures.

Figure 4.8: The dielectric constant at 1 MHz as a function of the volume fraction of the aqueous micellar core in the L2 phase of the system sodium-octanoate-water decanol (reproduced from Sjoblom et al. (1983) with permission of the Academic Press) While clearly useful in a number of applications, conductivity measurements alone may give only a proportion of the information regarding the electrical properties of a system, with considerably more information being obtained by considering both capacitive and conductive behaviour in these systems. A number of authors have used dielectric constant measurements to characterise surfactant solutions (e.g. Hanai et al., 1959). The study of micellar systems is complicated by the fact that the micelles are not homogeneous spheres, but may be considered to be a sphere surrounded by a shell. An understanding of such systems is of considerable importance, particularly with a view to understanding the behaviour of biological cells; the models used to describe these systems are in fact referred to as cell models. The early theoretical work on such systems was performed by Maxwell (1873), with a later, more detailed model being given by Pauly and Schwan (1959). A number of related approaches have been described and these form the basis for modelling the dielectric behaviour of micelles, liposomes, microcapsules and biological cells. Sjoblom et al. (1983) measured the impedance at 1 MHz for a range of systems

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containing lithium, sodium and potassium octanoate, with octanol and decanol and water as a series of three-phase systems. Three regions were found in the relationship between the dielectric constant and volume fraction of micelles, as shown in Figure 4.8 for sodium octanoate-water-decanol systems. Phase 1 represents the premicellar region, with an initial decrease in dielectric constant due to hydrogen bonding between the water and alcohol molecules. The water is therefore less mobile and does not contribute to the dielectric constant as much as may be expected from water molecules in the free state. The system then forms micelles and the dielectric constant increases, seen in Phase 2, the authors ascribing this to the establishment of the polar micellar core. Phase 3 showed a more rapid increase in dielectric constant which was attributed to the formation of lamellar phases within the system. This study therefore demonstrates that dielectric constant measurements may also yield useful information on the micellisation process. 4.2.3 The frequency dependent response of micelles

Figure 4.9: Dielectric dispersion curves of sodium dodecyl sulphate in water at 25°C above and below the cmc. Surfactant concentrations as follows: ×-12 g/L, -9 g/L, g/L and -2 g/L (reproduced from Beard and McMaster (1974) with permission of the Academic Press) The response of surfactant systems over a range of frequencies has been studied by a number of authors. Beard and McMaster (1974) measured the response of a number of micelle-forming systems in the 0.5 to 350 MHz region, typical results being shown in Figure 4.9. The authors calculated a dielectric increment from the difference in the real permittivity at low frequencies and . The increment showed a discontinuity at a

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concentration corresponding to the cmc, as shown in Figure 4.10. While there was no dispersion in the capacitance at concentrations below the cmc, a dispersion was found above the cmc which was attributed to bound water around the hydrocarbon core of the micelle. Cavell (1977) followed up this study, looking at the response of tetradecyltrimethylammonium bromide over a similar frequency range to that used by Beard and McMaster (1974) but looking at more concentrated solutions of surfactant. The author found more complex concentration-dependent behaviour, attributing this to structure-promoting effects of the micelles in the solvent and to irrotationally bound water molecules associated with the double layers of the micelles.

Figure 4.10: Dielectric increment of sodium dodecyl sulphate in water at 25°C against surfactant concentration (reproduced from Beard and McMaster (1974) with permission of the Academic Press) Abe and Ogino (1981) studied the dielectric response of surfactants over a frequency range of 30 Hz to 6 MHz. Responses for sodium dodecyl sulphate are shown in Figure 4.11. The authors attributed the low frequency response to electrode polarisation, as discussed in section 2.2.3. The authors also estimated the size of the micelles using the theory of Schwarz (1962). In this approach, an dispersion is assumed to exist due to counterion displacement at the colloid-electrolyte interface, as described by Schwan (1957). The frequency of the dispersion ( F) is given by

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(4.7) where u is the mobility of the counterion, k is the Boltzmann constant, T is the absolute temperature and r is the radius of the particle. The authors calculated the radius as being approximately 5 nm, although no particle size data obtained using other techniques was available for comparison. Furthermore, it is not entirely clear from Figure 4.11 what was taken as F.

Figure 4.11: Frequency dependence of the real permittivity of sodium dodecyl sulphate solutions (reproduced from Abe and Ogino (1981) with permission of the Academic Press) In a later study, Abe et al. (1982) addressed the question of solubilisation in micelles by examining the frequency dependent response of materials in sodium dodecyl sulphate using octanoic acid and octane as model solibilisates. The frequency dependence was similar to that shown in Figure 4.11, with a substantial increase in the real permittivity being seen with decreasing frequency, while the conductance showed no discernible frequency dependence. The authors reported that the dielectric response showed little dependence on the concentration of n-octane but an increase in the high frequency dielectric constant with increasing concentration of octanoic acid. Similar dependencies on concentration were also seen for the conductivity. Ogino et al. (1988) later studied the properties of a mixed surfactant system using sodium dodecyl sulphate and alkyl polyoxyethylene ether. The authors again calculated the size of the mixed micelles, proposing that shorter polyoxyethylene chains lead to larger micelles, possibly due to more favourable energetics of micelle formation for shorter chain systems.

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Figure 4.12: The low frequency dielectric response of a) 0.0081 M -cyclodextrin, b) 0.0035M sodium dodecyl sulphate in the presence of 0.0081 M -cyclodextrin, c) 0.021 M sodium dodecyl sulphate in the presence of 0.0081 M cyclodextrin (reproduced from Craig and McDonald (1995) with permission of the American Chemical Society) It is possible that the low frequency increase in permittivity may in fact be interpreted in terms of an electrode layer as described by Hill and Pickup (1985). Craig and MacDonald (1995) investigated the interaction between sodium dodecyl sulphate and cyclodextrin systems, arguing that the establishment of electrode layers results in a low frequency dispersion which may be observed into the kilohertz region, where conductivity measurements are conventionally made. This is shown in Figure 4.12, where the regions at frequencies below the dotted line correspond to electrode layers, while at

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higher frequencies the response is dominated by the bulk dielectric loss G/ and hence conductivity G. Consequently, not only may the low frequency dispersion be interpreted in terms of the establishment of these layers but care must also be taken when using conductivity measurements to assess surfactant behaviour, as at high surfactant concentrations the conductance may be affected by the presence of electrode layers, even in the kilohertz region. A number of other parameters may be studied using dielectric analysis. Desando et al. (1985) studied a series of n-octylammonium carboxylate surfactants in water up to the gigahertz region, relating the dielectric behaviour to relaxation processes of different portions of the surfactant chains. Sjoblom et al. (1984) developed a model relating the dielectric properties of reverse micelles to the micellar shape, showing that at low concentrations of sodium octanoate in decanol-water systems, the data may be interpreted in terms of the micelles assuming a spherical shape. At higher concentrations, however, the data was compatible with the formation of prolate aggregates, with axial ratios calculated from the data. The predicted results were found to be in excellent agreement with X-ray diffraction data. This demonstrates, therefore, that the technique may be used as a means of assessing the shape of micellar systems.

4.3 ANALYSIS OF PHARMACEUTICAL MICROCAPSULES Microcapsules and microspheres have attracted considerable attention as methods of delivering drugs, both for targeting and sustained release purposes. The release kinetics of encapsulated drugs is known to be largely a function of the physical and chemical properties of the microcapsules themselves, hence there is a need to develop techniques with which to study the structure of these systems. A number of studies have been conducted using dielectric analysis as a means of examining the behaviour of microcapsules, most of these originating from Kyoto University in Japan. These studies are described in an excellent review by Sekine et al. (1991), a summary of which is given here. The work is based on a cell model, whereby the microcapsule is considered to be a concentric shelled sphere, as described in section 4.2.2, the difference being that in this analysis, the frequency dependent behaviour is accounted for. In the case of microcapsules, the complex permittivity of the microcapsules themselves is given by

(4.8) where q is the permittivity of the suspension, s the permittivity of the microcapsule shell and i the permittivity of the contents of the microcapsule, the superscript indicating that these quantities are complex. The quantity v is given by

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(4.9) where d is the thickness of the shell and D is the overall diameter of the microcapsule. The complex permittivity ( *) of dilute suspensions of microspheres may be given by

(4.10) where a is the permittivity of the surrounding medium and is the volume fraction of the microcapsules. For concentrated suspensions, (4.10) is modified to

(4.11) Modifications have also been derived to account for the polydispersity of microcapsule size (Kondo, 1979). An analysis was then developed based on three assumptions: that the thickness of the capsule wall is much smaller than the diameter of the microcapsule, that the electrical conductivity of the wall is less than that of the core and suspending medium and finally that the conductivity of the core is higher than that of the suspending medium. While the analysis itself will not be dealt with here, the information that may be obtained includes the volume fraction, the diameter of the microcapsules and the thickness and relative permittivity of the capsule wall. The permittivity of the capsule wall is particularly interesting, as the release of drugs from microcapsules will be dependent on partitioning into the capsule wall, hence a knowledge of the dielectric constant of the wall itself may lead to a greater understanding of release kinetics. The conductivity of the aqueous core may also be calculated from the analysis. Two dispersions are predicted, the frequencies at which they occur being largely a function of the conductivities of the surrounding medium and aqueous core respectively. Studies have been conducted using polystyrene or polymethylmethacrylate spheres (Zhang et al, 1983, 1984; Sekine, 1986, 1987; Sekine and Hanai, 1991). The response of polystyrene spheres containing 2% gelatin solution is shown in Figure 4.13. The two dispersions in permittivity were related to the behaviour of the surrounding medium and the core respectively. Sekine and Hanai (1991) also monitored the release of KC1 from PMMA microspheres, demonstrating the possibility of measuring release characteristics by measuring both the internal and external conductivities of the microcapsule systems. An example of the comparison between the theoretical and observed increase in conductivity of the suspending medium resulting from KC1 release from polymethylmethacrylate (PMMA) microspheres is shown in Figure 4.14, with the terms C-C and box referring to two variations of the model.

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Figure 4.13: Dielectric behaviour of a suspension of polystyrene microcapsules containing a 2% gelatin solution (reproduced from Sekine et al, 1991) Overall, therefore, these studies show that it is possible to study a number of parameters associated with pharmaceutical microcapsules. The versatility of the technique in giving information on the aqueous core, the capsule wall and the behaviour of the surrounding medium is unique to dielectric analysis and should prove to be of considerable interest in the future.

Figure 4.14: The time dependence of the electrical conductivity (κa) of PMMA microspheres containing KCl (reproduced from Sekine and Hanai, 1991) 4.4 ANALYSIS OF GELS Gels are semisolid materials, usually prepared by dispersing a polymeric substance in a

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solvent at high concentrations. Gels are considered to be disperse systems, hence their inclusion here. These systems are of importance pharmaceutically, as drugs may be given in gel vehicles. Furthermore, many dosage forms may form gels within the body, examples being swelling hydrophilic matrices and bioadhesive systems. A number of studies have been conducted on the dielectric properties of gels, with many of the studies of pharmaceutical interest concerning the response in the low frequency (kilohertz and below) frequency region. For example, aqueous gels containing cetostearyl alcohol and cetrimide have been studied (Dissado et al., 1987). These systems are known to form complex, heterogenous structures and are therefore difficult to analyse using the majority of techniques. A diagrammatic representation of the response is shown in Figure 4.15. This response may be interpreted in terms of a number of different processes. Four bulk processes were identified ( C1 to C3) in series with a barrier process (Cs). The parameter C1 was associated with the relaxation of water within the structure, probably as water-filled bulk cavities which have been reported as comprising 24% of the total water content (Loudon et al., 1985). C2 and C3 are associated with ions in the bilayer regions formed by the cetostearyl alcohol, with C3 being associated with transport processes through channels between the cetostearyl bilayers via mobile ions and C2 with regions of the bilayer isolated from those transport paths. Clearly, therefore, it is possible to monitor different regions of complex samples using the technique.

Figure 4.15: Schematic representation of the normalised response of a cetostearyl alcohol/water/cetrimide gel system (reproduced from Rowe et al. (1988) with permission of the Academic Press) The authors (Dissado et al., 1987; Rowe et al., 1988) went on to describe the effects of a number of formulation variations on the dielectric response and to interpret the observed dielectric effects in terms of the structure of the sample. One particularly

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interesting aspect of this study was the investigation of ‘thinning’ effects, whereby storage of the gels resulted in a change from a viscous, opaque system to a low viscosity, pearlescent milky white lotion. The response of the freshly prepared and thinned samples are shown in Figure 4.16. The thinned sample shows less evidence for the C2 process, which is associated with the bilayer structures. The response was therefore interpreted in terms of fusion of the lipid bilayers. These studies therefore demonstrate the possibilities of using the technique to gain structural information on complex gel systems and to use this information as a formulation tool.

Figure 4.16: Dielectric response of cetostearyl alcohol/cetrimide/water systems. freshly prepared, thinned (reproduced from Rowe et al. (1988) with permission of the Academic Press) In a later study, Aliotta et al. (1993) studied the low frequency dielectric response of gels comprising lecithin, isooctane and water (organogels), suggesting that dielectric analysis may be used to characterise the water-assisted interaction between the soybean lecithin molecules which lead to the formation of cylindrical micelles. These structures in turn drive the building of the gel network. Studies have also been conducted on the relationship between the dielectric response of gels and their Theological properties. In a study by Boudakian et al. (1991), the low frequency dielectric responses of surfactant solutions in water, particularly cetyl trimethyl ammonium bromide-salicylic acid (CTMAS) were examined. This compound forms a three-dimensional network above a critical concentration. The authors compared the dielectric data to oscillatory data published in previous studies (Strivens, 1989; Imhof et al., 1990). The dielectric response above approximately 3 Hz was shown to be largely dependent on the dissociation of the surfactant, while at lower frequencies a barrier layer was seen which was a function of the liquid crystalline structure of the gel. It was suggested that the structure detected using Theological techniques could be related to this barrier layer.

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Figure 4.17: Dielectric response of Carbopol 934 (2.5% w/v) in water in the presence and absence of 0.1% w/v chlorhexidine gluconate. , : C,G/ Carbopol 934, , : C,G/ Carbopol 934 with drug (reproduced from Craig et al. (1994a) with permission of Elsevier Science) Binns et al. (1992) examined the dielectric and rheological behaviour of alginate gels. Alginates are widely used in the pharmaceutical and food industries and may form gels at concentrations of approximately 3% w/v. Low frequency dielectric characterisation showed a high frequency conductance in series with a barrier layer. As above, the low frequency response was attributed to adsorbed alginates on the electrodes, while the high frequency conductance was shown to be a function of the movement of ions through the gel matrices, as indicated by the absence of any marked change in the dielectric response when the concentration of alginate was raised above that corresponding to the formation of a viscous gel. However, interestingly, incorporation of a drug (diclofenac sodium) caused an alteration not only in the absolute values of the high frequency conductance, but also in the mechanisms involved, as shown by a change in slope of the loss and capacitance profiles. These results indicated that the drug is interacting with the gel matrix, causing charge movement through the system to become impeded. This study raises some interesting possibilities, as it implies that while viscosity measurements may be used to analyse the structure and interrelationship between the polymer chains, dielectric analysis may give an indication of how small molecules such as drugs move through the system. By using the two techniques together, the behaviour of both the polymer chains and the incorporated drugs may be analysed.

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Craig et al. (1994a) studied the low frequency response of bioadhesive Carbopol 934 gels in conjunction with oscillatory rheometry studies. The authors demonstrated that the addition of a range of agents such as polyethylene glycol resulted in changes in both the Theological and dielectric responses of the gels. In particular, however, the presence of relatively small concentrations of the drug chlorhexidine gluconate caused profound changes in the structure of the gels (Figure 4.17), probably due to an ionic interaction between the basic groups of the drug and the free acid groups of the polymer. The study therefore demonstrates that dielectric analysis may be used to monitor the behaviour of drugs and other additives in gels, particularly when used in conjunction with rheological measurements. Furthermore, the assumption of simple diffusion of drugs through these gels may not necessarily be valid, as the dielectric technique showed that on addition of the drug, the shape of the dielectric spectra altered, indicating a corresponding change in the physical structure of the gel network.

4.5 ANALYSIS OF EMULSIONS AND MICROEMULSIONS Emulsions may be defined as a dispersion of one liquid in another. They may generally be categorised as oil in water or water in oil, depending on the nature of the dispersed and continuous phases. Pharmaceutically, emulsions are of considerable importance, being used orally, parenterally and topically. Most pharmaceutical emulsions are oil in water, consisting of a dispersion of oil droplets surrounded by an interfacial layer within an aqueous continuous phase. The interfacial layer usually contains a surfactant or a mixture of surfactant and a long chain alcohol such as cetosteryl alcohol. The particle size of these emulsions tends to be in the order of a few micrometres, although considerable variation may be seen, depending on the formulation and preparation conditions. For highly viscous emulsions (creams), lamellar phases may exist between the droplets, hence the structure of such systems may be highly complex. In addition to conventional emulsions, there has been growing interest in the use of microemulsions. These are systems which, unlike the thermodynamically unstable macroemulsions described above, form spontaneously (in the thermodynamic, although not necessarily in the kinetic sense), are clear in appearance and are stable on storage. The mechanism of formation of microemulsions is not yet clear, although it is believed that microemulsions will be generated when surfactants lower the interfacial energy between an oil and aqueous phase sufficiently to allow spontaneous curvature of that interface. These systems have attracted considerable interest as potential dosage forms for several reasons. The stability of the formed microemulsions is pharmaceutically advantageous, while the ability of microemulsions to incorporate a variety of poorly soluble drugs has advantages for example in the formulation of aqueous systems for injection, as not only may a greater quantity of drug be incorporated but the particle size of the droplets is sufficiently small to avoid the danger of blocking capillaries once inside the body. Microemulsions show a number of further interesting properties, notable in terms of their phase behaviour. The addition of excess disperse phase to an oil in water or water in oil microemulsion will lead to phase inversion, as found for macroemulsions. However, microemulsion systems will exhibit gel phases during the transition, these gels

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consisting of lamellar phases which exhibit birefringence. There has also been considerable interest in the use of self-emulsifying systems. These are mixes of oils and surfactants which emulsify on contact with water with little or no agitation. While it is not yet clear whether they are effectively identical to microemulsions, these systems have attracted considerable interest as potential oral dosage forms, as poorly soluble drugs may be incorporated into the oil-surfactant mixes. When swallowed, an emulsion forms in the gastrointestinal tract which theoretically allows dissolution of the drug from a large surface area. There may also be an effect due to the enzymatic degradation of the oil facilitating absorption of the drug. However, before discussing these more specialised systems, the dielectric properties of conventional macroemulsions will be outlined. 4.5.1 The dielectric response of conventional emulsions

Figure 4.18: Dielectric response of 70% nujol-carbon tetrachloride in water emulsions at 30°C (reproduced from Hanai et al., 1959) As may be expected, the dielectric responses of o/w and w/o emulsions will show marked differences. Indeed, conductivity measurements are a standard method of assessing which

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form of emulsion is present, as the continuous phase in w/o systems will show a considerably lower conductivity than is found for o/w systems. Studies of the dielectric response of emulsions over a range of frequencies allows a more sophisticated analysis of these systems, with a theoretical treatment having been given by Hanai (1968). The theories relating to dilute suspensions of liquid droplets are similar to those described for micelles and microcapsules, with the interfacial layer acting as the shell surrounding the liquid core. In common with the disperse systems described earlier, there is a large interfacial area between the two phases in emulsions. In practice, the dielectric response of o/w emulsions is relatively difficult to measure due to the high conductivity of the aqueous continuous phase. A study by Hanai et al. (1959) showed a constant conductance and a low frequency dispersion in the real part of the response at low frequencies, as shown in Figure 4.18. The dispersion in the real permittivity at low frequencies may be interpreted either in terms of counterion relaxation or interfacial electrode layers, as discussed in previous sections. In contrast to o/w systems, w/o emulsions may exhibit considerable interfacial polarisation. A number of studies have indicated a loss peak in the low megahertz region which has been ascribed to such effects, as shown in the example given in Figure 4.19. It should be noted, however, that this system does not contain an emulsifier, which would almost certainly complicate the response. Emulsions have been extensively studied since the work described above, the majority of investigations concerning w/o emulsions. Schreirer and Smedley (1986) measured w/o emulsions of KCl in petroleum jelly-heptane and CCl4-heptane mixtures from 5 Hz to 13 MHz, using a cell model to calculate the conductivities of the aqueous phase. Kaneko and Hirota (1985) examined the use of dielectric analysis in the study of the effects of processing and formulation variables on the properties of emulsions; in particular, factors such as particle aggregation on storage were studied. The authors noted that while aggregation took place relatively rapidly after manufacture, the rate of aggregation decreased after storage for one day. This was mirrored by the dielectric constant, which showed a plateau on reaching a maximum value. The authors ascribed the change in dielectric constant to a decrease in the thickness of the interfacial layer between the particles as the particles underwent storage. Using the premise that the value of the dielectric constant reflected the thickness of the interfacial layer, the authors compared the effects of incorporating 23 emulsifying agents within the emulsions. The authors suggested that the technique could be used as a rapid means of selecting the most effective emulsifier, as the agents which resulted in emulsions with the lowest dielectric constant should have the thickest and hence most effective interfacial layer. While this study is largely empirical in nature, the implications for the use of the technique for the manufacture of emulsions is clear.

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Figure 4.19: Frequency dependence of the loss factor of water droplets in wool alcohol at 20°C (from Dryden and Meakins, 1957) Kaneko and Hirota (1985) also examined an anti-inflammatory emulsion with a high disperse phase, showing that different preparation conditions led to different stabilities, again detected by the dielectric method, and that sorbitol was found to stabilise the emulsion systems. In a more recent study, Goggin et al. (1994) used a related approach, changing the conditions used to manufacture a range of o/w and w/o creams and measuring the emulsions using low frequency dielectric analysis and oscillatory rheology. The authors reported parallel changes in the dielectric and Theological profiles of the creams, thus suggesting that the dielectric technique may be used as a means of characterising these complex emulsions which is difficult using most conventional techniques. A further possibility is to monitor the dielectric properties of an emulsion under shear. For example, Hanai (1961) studied nujol/carbon tetrachloride w/o emulsions in a modified rotating cup and bob viscometer, with marked differences being observed between the different shear rates (Figure 4.20) which were interpreted in terms of differences in droplet distribution through the emulsions. The ability to monitor the emulsions under shear has implications for the use of dielectric analysis as an in-process quality control test. 4.5.2 The response of microemulsions Microemulsions have also been extensively studied using the dielectric technique. Clausse et al. (1976) studied benzene in water microemulsions, using Tween 20 and Span 20 as emulsifiers. The authors found good agreement between the theories of Hanai (1968) and the experimental response for concentrated systems, although discrepancies in the predicted and observed permittivities were observed at lower disperse phase volume

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fractions, these being ascribed to the behaviour of the interfacial film. Bostock et al. (1980) examined hydrocarbon-water microemulsions using tetraethylene glycol dodecyl ether as the stabilising agent. The authors also found better agreement between experimental and theoretical results at high concentrations of the disperse phase, although the authors suggested that the discrepancies found at lower concentrations were due to changes in shape of the dispersed droplets.

Figure 4.20: Frequency dependence of ' and " at rest and under various shear conditions for an 80% w/o emulsion (reproduced from Hanai, 1961) Chou and Shah (1981) studied the dielectric response of microemulsions composed of H2O or D2O in oil, using a surfactant/alcohol mixture as the emulsifier. Dielectric studies showed that the charge densities of the emulsions prepared using H2O was three times greater than those using D2O. The authors proposed that the differences in charge density led to a higher alcohol concentration at the interface for the H2O systems, hence this study raises some interesting possibilities regarding the understanding and prediction of the distribution of emulsifiers within disperse systems. Other studies include that of Henze and Schreiber (1985) on aerosol-OT/water/cyclohexane and a theoretical discussion by de Rozieres et al. (1988), in which a model for the permittivity of water-inoil microemulsions has been put forward. In addition, Boned and Peyrelasse (1991) studied percolation phenomena within microemulsions using dielectric analysis.

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Nimtz et al. (1986) studied the phase behaviour of a complex five component microemulsion. Interestingly, these systems are believed to be gel-like bilayer systems below 60°C and microemulsions above this temperature. The authors studied the dielectric response over a range of temperatures and saw marked changes at the transition temperature, as shown in Figure 4.21. These results were interpreted in terms of the water existing in different binding states above and below a transition temperature, indicated by the maxima in the real and imaginary permittivities, hence the technique may be used not only to detect such phenomena but also to allow quantitative characterisation of the processes involved.

Figure 4.21: Real and imaginary permittivities of an emulsion system (10% water, 40% cetylalcohol, 40% stearyl alcohol, 5% sodium cetyl sulphate, 5% sodium stearyl sulphate) against temperature, taken at different frequencies. -1.5 MHz, -6 MHz, ×-72 MHz, -660 MHz (reproduced from Nimtz et al., 1986) Self-emulsifying systems have also been examined using dielectric analysis, particularly in order to gain a greater understanding of the mechanism by which emulsification takes place. Craig et al. (1993a) studied the low frequency response of oil (Labrafil M2125 CS) and surfactant (Tween 80) systems in the presence and absence of a poorly water soluble experimental drug (L-365,260). The spectra showed a decrease in response on addition of the drug, indicating the presence some form of interaction between the drug and the components of the emulsifying vehicle. On forming the emulsions, a bimodal particle size distribution was seen for the systems containing no drug. However, addition of the drug prior to emulsification resulted in a marked shift in the distribution, with a larger proportion being seen in the lower of the two size ranges. In a later study, Craig et al. (1994b) investigated the possibility that emulsification took place via the formation of liquid crystals at the interface between the oil droplets and the continuous phase by examining Imwitor/Tween 80 systems containing 10% v/v water in an attempt to mimic the composition at the interface in formed emulsions. A marked

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change in the low frequency response was seen at oil/surfactant mixtures which corresponded to those which emulsified the most readily, as shown for systems containing 3:7 Imwitor:Tween 80 in Figure 4.22. The spectra corresponding to this composition may be interpreted using the model proposed by Hill and Pickup (1985) and, in terms of the structure of the system, is compatible with the formation of liquid crystals.

Figure 4.22: Dielectric response of Imwitor/Tween 80 selfemulsifying mixtures on addition of 10% v/v water (reproduced from Craig et al. (1994b) with permission of Elsevier Science) 4.6 ANALYSIS OF LIPOSOMES 4.6.1 The structure and uses of liposomes Liposomes are vesicles composed of one or more phospholipid bilayers surrounding an aqueous core. These systems have attracted considerable interest since their discovery, firstly as model membranes and secondly as drug delivery vehicles. The use of liposomes as model membranes stems from the fact that biological membranes are largely composed of phospholipid bilayers. However, such membranes are chemically complex due to the presence of additional molecules such as proteins. By using liposomes, it is possible to gain a greater understanding of how the phospholipid component of biological membranes behaves. Furthermore, the effect of including additives such as cholesterol (also found in cell membranes) may be studied in a controlled manner. The use of liposomes as drug delivery vehicles has been a result of several factors. The hydrophobic nature of the bilayers combined with the presence of the presence of aqueous compartments means that a wide variety of drugs may be entrapped. The release from the

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liposomes may be controlled due to the need for the drug to diffuse through the bilayer, hence allowing sustained release over a period of hours for suitable drugs. Furthermore, as phospholipids are naturally occurring substances, the toxicity associated with introducing foreign molecules into the body is not encountered. A number of applications have been described for liposomes in this context and more information may be found in a number of texts (e.g. Gregoriadis, 1988; Taylor and Craig, 1993). The properties of liposomes (and indeed cell membranes) stem from the unusual chemistry of phospholipids. These molecules comprise two hydrocarbon chains attached to a polar head group. The polar head group is zwitterionic, with no overall charge but considerable charge separation on an atomic scale. The long hydrocarbon chains confer lipophilicity to the molecule, resulting in phospholipids having negligible water solubility. In the presence of water, the phospholipid molecules form bilayers which may then become arranged into spherical liposomes, as indicated in Figure 4.23. The natural phospholipids commonly in use as liposomal materials include egg phosphatidylcholine, phosphatidylserine, phosphatidylglycerol, and sphingomyelin. Synthetic phospholipids include dipalmitoylphosphatidyl choline (DPPC), distearoyl phosphatidylcholine (DSPC) and dimyristoylphosphatidylcholine (DMPC).

Figure 4.23: Diagrammatic representation of a multilamellar liposome The liposomes may be described as multilamellar vesicles (MLVs) if the structure consists several concentric lipid layers (as shown in Figure 4.23). In addition, small unilamellar vesicles (SUVs) may be produced by sonication of MLVs (Huang and Charlton, 1971), by injecting an ethanolic solution of phospholipid into the aqueous phase (Batzri and Korn, 1973), by dialysing a detergent-containing phospholipid solution (Kagawa and Racker, 1971; Milsmann et al., 1978) or by adjusting the pH of the aqueous medium containing the liposomes (Hauser and Gains, 1982). MLVs tend to be in the size

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range 100 nm-10 m, while SUVs are 20–100 nm in diameter. Large unilamellar vesicles (LUVs) may also be produced by the injection method using suitable solvents (Deamer and Bangham, 1976), these vesicles being 70–190 nm in diameter.

Figure 4.24: Real permittivity (a) and conductivity (b) of asolectin liposome suspensions (reproduced from Schwan et al., 1970)

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4.6.2 The dielectric response of liposomal suspensions A number of studies have been conducted on the dielectric properties of liposomes and only a summary is presented here. One of the earliest investigations is that of Schwan et al. (1970). The authors used a cell model, developed earlier (Schwan, 1957), whereby the low frequency dielectric constant ( 0) is given by

(4.12) where r is the permittivity of free space, p is the volume fraction taken up by the liposomes, Cm and Gm are the capacitance and conductivity of the membrane, R is the liposome radius, i and a are the resistivity of the cell interior and the external phase respectively. is the permittivity due to processes other than membrane polarisation and is approximately equal to the permittivity of the suspending medium. Schwan et al. (1970) went on to derive a series of equations to describe the frequency dependence of liposome suspensions. These predict a dispersion in the real part of the permittivity in the region of 10 MHz due to Max well-Wagner effects, as described in section 4.1.2; the interface between the aqueous inner core and the surrounding phospholipid membrane was considered to be responsible for the effect. Schwan et al. (1970) also proposed that a second dispersion in the kilohertz region may be seen due to counterion relaxation around the surface of the liposomes. The authors then examined liposome suspensions of asolectin, a highly complex mixture of phospholipids and lipids (containing approximately 70% phospholipid) in a buffer containing sucrose, EDTA, -thioglycerol and tris(hydroxymethyl)aminomethane. In interpreting the data presented, the complexity of the system used should perhaps be borne in mind, although the authors were limited by the availability of materials at the time of the study. Typical results are shown in Figure 4.24. The data shows two dispersions as described above, denoted and , which are most marked in the real and imaginary data respectively. The interpretation of counterion relaxation ( ) and Maxwell-Wagner polarisation ( ) has formed the basis for a number of additional studies on liposome systems. Redwood et al. (1972) studied vesicles of more uniform size than had been used in previous studies, these liposomes being composed of purified egg yolk phosphatidylcholine in dilute aqueous salt solutions. No dispersion was observed for these systems, although incorporation of stearic acid into the vesicles resulted in a marked dispersion in this region. The authors ascribed this to the negative charge of the vesicles on incorporation of the anionic stearic acid. The study of liposomes has been considerably facilitated by the introduction of pure phospholipids and by the availability of equipment capable of measuring over a wider frequency range. Kaatze et al. (1979, 1984) studied the response of dimyristoylphosphatidylcholine (DMPC) MLV systems over a wide frequency range (3 kHz to 40 GHz). The authors noted that no low frequency (kilohertz) dispersion effects were seen for these samples (shown in Figure 4.25) and suggested that this was due to the

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absence of ionic contaminants within the system. Furthermore, Maxwell-Wagner effects could be excluded due to the absence of significant conductivity in the aqueous phase. In these purified systems, therefore, the dielectric behaviour of the phospholipid molecules themselves could be observed. The authors suggested that the dispersion seen at 80 MHz was due to reorientational polarisation of ionic head groups, while the higher frequency dispersion is due to the relaxation of water.

Figure 4.25: Real permittivity of DMPC liposomes (reproduced from Kaatze et al., 1984) Kaatze and Henze (1980) suggested a theoretical model, outlined in Figure 4.26, whereby the interior of the bilayer is regarded as a homogeneous dielectric with a relative permittivity of approximately 2. The surface is characterised by a surface polarisability, involving the reorientational motions of the cationic tetramethylammonium groups of the polar head group relative to the anionic phosphoryl group. The motion is assumed to be circular, characterised by a radius of the path length . The relaxation time of the head groups will depend upon the degree of correlation between the adjacent molecules. The model suggests the existence of ‘dielectric domains’ on the surface of the liposomes, whereby regions of cooperativity exist. Kaatze et al. (1985) later studied the response of various phospholipids between 500 Hz and 50 GHz, again interpreting the relaxation behaviour in the microwave region as being due to dielectric domains. The study of these domains has interesting implications for understanding features of liposome behaviour such as bilayer flexibility. Furthermore, the effect of including drugs or additives into the bilayer structure could be examined by studying the effects on the size of the cooperative regions. This is particularly important when studying processes such as the incorporation of vaccines into liposomes, as the separation of the vaccine proteins within the liposome structure may have a profound effect on the biological activity (Gregoriadis, 1988). This model therefore has several highly interesting pharmaceutical applications which have yet to be explored. In addition to the studies outlined above, a number of investigations have also been conducted into the low frequency response of these systems. The association between this

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response and the presence of charged material, particularly counterions, is generally accepted. Pottel et al. (1984) demonstrated that very small quantities of added impurities may have a profound effect on the low frequency spectra, as shown in Figure 4.27.

Figure 4.26: Diagrammatic representation of the relaxation behaviour of phospholipid bilayers (reproduced from Kaatze and Henze, 1980)

Figure 4.27: Real permittivity of purified C14-ether-lecithin in the absence (a) and presence (b) of admixed 1 mole % potassium salt of myristic acid at 30°C (reproduced from Pottel et al. (1984) with permission of Elsevier Science)

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The authors suggested that the large dispersion seen on adding the impurity demonstrates the potential for misinterpretation of dielectric data if care is not taken to control the purity of the lipids. The argument can, of course, be used the other way round in that the dielectric response could be used as a means of assessing the purity of these lipids. Uhlendorf (1984) also studied the low frequency dielectric response of liposomes (5 kHz to 100 MHz) with a view to examining the effects of fatty acid contamination on the dielectric response. Again, the author concluded that the low frequency dispersion was due to the presence of ionic impurities. The response of liposomes at even lower frequencies (down to 10−2 Hz) was studied by Barker et al. (1994). The authors found that the low frequency dispersion could be interpreted in terms of the model proposed by Hill and Pickup (1985), whereby a barrier layer forms on the electrode surface. A typical low frequency response is shown in Figure 4.28. It was suggested that the low frequency region corresponds to a layer of liposomes adsorbed onto the electrodes, while the higher frequency response is due to the conductivity of the aqueous medium containing the liposomes. The increase in capacitance (which is equivalent to the real part of the permittivity) was explained in terms of the presence of the barrier layer, rather than as being due simply to electrode polarisation or counterion relaxation. The presence of ionic impurities will result in the barrier layer being seen at higher frequencies, as may be seen from examination of Figure 4.2, where the crossover frequency between the low and high frequency regions is proportional to the conductivity of the bulk phase.

Figure 4.28: Dielectric response of dimyristoyl phosphatidyl choline (DPPC) liposomes (reproduced from Barker et al. (1994) with permission of the Academic Press) This study therefore offers an explanation as to why, in previous studies, the presence of trace impurities led to the observation of the dispersion. The investigation is therefore of interest for two principal reasons. Firstly, it provides an alternative explanation for the low frequency dispersion seen in many studies, and secondly, it predicts that the low frequency response is a function of the composition of the

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liposomes, as this response is a function of the presence of the adsorbed liposome layer. This was found to be the case, as by incorporating different molar ratios of cholesterol into the liposomes, the low frequency capacitance (at 3.162×10−1Hz) was shown to change linearly with mole fraction of cholesterol. This suggests that the technique may be used as an assay method, whereby the presence of the drug in the bilayer may be detected and quantified. Furthermore, if the higher frequency response is a function of the external medium, it may be possible to use this as a means of measuring the rate of release of the drug from the liposomes. This was shown for ascorbic acid, as demonstrated in Figure 4.29.

Figure 4.29: The time dependence of the dielectric loss of an LDPPC liposome suspension containing 2% ascorbic acid at 1 kHz (reproduced from Barker, 1992) The ability of dielectric analysis to monitor the permeability of phospholipid membranes has been studied by Sekine et al. (1983). The authors prepared liposomes containing egg lecithin and cholesterol in a 1:1 ratio and incorporated KCl and glucose in the inner and external phases. The dielectric behaviour was then examined over a frequency range of 1 to 200 MHz. Assuming that the shell may be compared to a semipermeable membrane, the volume of suspended particles is given by

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(4.13) where VR is the ratio of the volume fraction to the volume fraction under isotonic conditions, A is a constant, Vqo is the volume of the suspended particles under isotonic conditions, Ca is the osmolarity of the outer medium and Vd is the ‘osmotically dead’ volume of the particles. These parameters may be calculated from the dielectric data using the model outlined by the authors. If VR is then plotted against the reciprocal external osmolarity, the graph shown in Figure 4.30 is obtained.

Figure 4.30: Relative particle volume against osmolarity of the outer medium containing glucose and KCl (reproduced from Sekine et al., 1983) The authors concluded that when the external phase concentration is hypertonic with respect to the internal phase, the shrinkage of the liposomes follows (4.13), indicating that while water may pass through the membrane, KCl and glucose may not. While this is not surprising in itself, the study demonstrates that dielectric analysis allows an insight into the way in which solutes pass through liposomal, and hence biological, membranes. 4.6.3 The phase transition of phospholipid membranes One feature of liposomes that has been extensively studied has been the phase transition behaviour. At low temperatures, the lipids are arranged in a close-packed form, known as the gel state, as these systems are effectively liquid crystalline. On raising the temperature, the lipid layers become considerably less ordered over a narrow temperature range to form a fluid state. The bilayer becomes thinner and the cross-sectional area per molecule increases due to the greater degree of rotational motion. If cholesterol is

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incorporated within the liposome, the phase transition becomes ‘smaller’, with an abolition of the transition at approximately 50% cholesterol. Papahadjopoulos et al. (1973a) have suggested that the cholesterol renders the membrane more fluid below the phase transition temperature but more ordered above this temperature, thus the difference in structure above and below the temperature is reduced. Whatever the explanation, it is well established that the release rate of most drugs increases from a low level at low temperatures to a high level in the immediate proximity of the phase transition. On further heating the system, the release rate decreases again (Papahadjopoulos et al., 1973b). A knowledge of phase transition phenomena is therefore necessary in order to manipulate the release characteristics of the liposomes, as well as to understand the behaviour of biological membranes. This phenomenon has been studied using dielectric spectroscopy and a summary of the main findings is given below. Shepherd and Buldt (1978) prepared mixes of DPPC and water by using the ratio of phospholipid to water that was reported to result in maximum hydration (25% w/w at 20° C), thus the authors were effectively studying a gel system containing phospholipid bilayers (as confirmed by neutron scattering). The gel systems undergo a pretransition at approximately 35°C, at which temperature the system changes from the lamellar , phase to the phase and on to the lamellar phase above 42°C. The authors studied the change in dielectric response at 50 MHz, the permittivity data being given in Figure 4.31. Clearly, the two transitions may be seen. The authors went on to propose a model whereby the rotation of the zwitterionic head groups could be calculated from the relaxation times found by scanning over a range of frequencies, thus demonstrating that the dielectric technique may be used not only to detect transitions but also to understand the mechanisms by which they occur.

Figure 4.31: Variation in real permitivity at 50 MHz for DPPC/25 wt % H2O; increasing temperature, decreasing temperature (reproduced from Shepherd and Buldt (1978) with permission of Elsevier Science) Kaatze et al. (1979) studied the phase transition of C -lecithin systems between 105 14

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Hz and 1010 Hz, while Stuchly et al. (1988) measured the complex permittivity of DPPC and dipalmitoylphosphatidylglycerol (DPPG) mixed vesicles from 10 kHz to 100 MHz over a temperature range of 25°C to 45°C, which covers the phase transition temperature at 40°C. Both the permittivity and the conductivity showed abrupt changes at the phase transition temperature. The authors suggested that changes in two dielectric phenomena were occurring, the low frequency process being associated with counterion relaxation and the high frequency response to charging of the lipid bilayer. Phase transition phenomena associated with the freezing of water have also been studied using the technique. Grunert et al. (1984) used dielectric studies in the microwave region in combination with differential scanning calorimetry (DSC) to examine the phase transition behaviour of DPPC suspensions over a wide range of temperatures. The results of the dielectric study performed at 9.9 GHz are shown in Figure 4.32.

Figure 4.32: Real and imaginary parts of the dielectric response versus temperature of a DPPC samples containing 36% water. I-subzero, II-pre- and III-main phase transition (reproduced from Grunert et al., 1984) The smaller transitions at higher temperatures corresponded to the pre-transition and main transitions, as shown by DSC. In addition, the lower temperature behaviour is of considerable interest, as the response is associated with the freezing of water. Small angle X-ray diffraction studies indicated that seven molecules of water per lipid molecule remain between the bilayers in a fluid state down to –40° C, while the excess previously bound water is squeezed out of the bilayer and freezes with the surrounding water. Therefore by examining the dielectric response over this temperature range in combination with complementary techniques, the mechanisms by which biological tissues are damaged by freezing may be better understood.

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4.6.4 Miscellaneous dielectric studies In addition to the studies described above, a number of further investigations into the dielectric response of liposomes have been carried out. A detailed description of the effects of external electrical fields on the interfacial properties of phospholipids has been given by Shchipunov and Kolpakov (1991), while Liburdy and Magin (1985) have studied microwave-stimulated drug release from liposomes. A number of studies have also been conducted on phospholipid films, i.e. hydrated bilayers which are not suspended in water as such. Coster and Simons (1970) used membranes formed over a 6.8 mm hole in a polycarbonate septum. There appeared to be several problems associated with the circuit used which resulted in membrane discolouration and eventual rupture, although during the ‘steady state’ whereby no change in film characteristics was observed during the measuring period there appeared to be a low frequency dispersion between 106Hz and 101 Hz. However, these results must be considered alongside the difficulties in sample preparation seen by the authors. In a study by Mueller et al. (1983), the authors implanted microelectrodes into the bilayers of brain lipid vesicles, demonstrating the possibility of using the technique to characterise lipid layers from various tissues within the body on an extremely small scale. Dielectric analysis may also be used to study the interaction of biomolecules with phospholipid membranes. Bellamare and Fragata (1980) studied -tocopherol, a sterol which has been associated with membrane permeability, oxygen scavenging and electron transport, among other functions. The authors used a probe which interacts with tocopherol to study the polarity of the environment in which -tocopherol is located within the lipid bilayer. In a later study, Ruderman and Grigera (1986) investigated the effects of adding a surfactant (Triton X-100) to aqueous liposome suspensions. The authors showed a change in the low frequency response on addition of the surfactant which, given the work of Barker et al. (1994), may possibly be interpreted in terms of changes in the adsorbed layer of liposomes on the electrode surface. Surfactant interactions with biological membranes are of considerable importance, hence the study of Ruderman and Grigera (1986) suggests an interesting area of investigation.

4.7 CONCLUSIONS The studies outlined here have given an overview of the ways in which dielectric analysis may be of use in the study of colloidal systems. There is general agreement that the contents of the colloids, the properties of the shell and the properties of the surrounding media may be analysed in isolation. No other single technique is capable of achieving this. Additional areas such as the relaxation behaviour of charged head groups in liposomes or the use of probe techniques to examine the dielectric properties of colloidal membranes have also been discussed. Clearly, therefore, a great deal of information may be obtained on these systems using the technique. However, in this field perhaps more than in the others covered in this book, there is considerable debate regarding the interpretation of much of the available data. From an applied scientist’s viewpoint, this is undoubtedly a disadvantage when considering practical applications of the technique.

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Nevertheless, there is clearly great potential for the use of dielectric analysis in the study of pharmaceutical colloids, although the work to be performed must contribute to the fundamental interpretation of the data as well as demonstrating practical applications of the method.

5 Dielectric analysis of solids 5.1 INTRODUCTION One of the major applications of dielectric analysis is in the study of solids, particularly insulators and semiconductors. While important fields in their own right, their relevance to the pharmaceutical sciences is often tenuous and hence it would be inappropriate to discuss such systems at length. A discussion of the use of dielectric analysis in the study of solids in general is, however, of considerable pharmaceutical interest, particularly in the field of solid dosage form design. It is arguable that there is a lack of techniques capable of effectively analysing the physical structure of solid materials, which in turn is a drawback in the successful manufacture of dosage forms. For example, it would be useful to know more concerning the distribution of a drug within a tablet, or the crystal forms present within a compacted material. The examples given in this chapter are intended to illustrate that dielectric analysis has a role to play in the characterisation of solid pharmaceuticals in terms of identifying different crystal forms and the transitions between them, characterisation of glass transition phenomena, the measurement of the porosity and particle size analysis of powders, characterisation of solid compacts and studying sorption processes, particularly water uptake by solids. Despite the many applications of dielectric analysis in the study of solids, there are nevertheless difficulties associated with the interpretation of the dielectric data. In Chapter 1 the problems associated with interpreting data when interactions between adjacent molecules are present were outlined. Clearly, this consideration will be particularly pertinent to solid samples, where strong interactions between adjacent molecules are implicit. While the theories concerning this problem may become highly complex, a brief outline is given below in order to facilitate appreciation of the examples given later. Further on in the chapter, examples are given of materials which are either used in pharmaceutical products or else are closely related to pharmaceutical systems in order to demonstrate how the technique is used at present and, more importantly, where the potential for future application lies. The special case of polymeric solid samples is discussed in the next chapter and will not be dealt with here. 5.1.1 General principles of the dielectric behaviour of solids As mentioned above, one of the principal difficulties associated with studying the dielectric behaviour of solid samples is the lack of a generally agreed interpretive model. The Debye model is, in practice, rarely, if ever, applicable to solid systems, the model being based on the assumption of non-interacting dipoles. Cole and Cole (1941) noted that the general features of a peak in the dielectric loss and a dispersion in the real part of

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the susceptibility are commonly observed and suggested that the discrepancy from the Debye model was due to the presence of several, almost overlapping relaxation processes, as discussed in Chapter 1. When one considers the physical basis of this distribution, the obvious explanation would be inhomogeneities in the solid structure such as crystal defects or, on an atomic level, a distribution of orientations of the molecules within that structure. This approach has been criticised by Hill and Jonscher (1983) on the basis that the widths of the relaxation peaks observed in practice are so great that the distribution of relaxation times would have to be extremely wide. The authors have questioned whether the level of inhomogeneity expected in the solids under examination could account for this breadth. An alternative, but in many ways equivalent approach is to consider the dielectric behaviour to be a function of the probability of charges hopping between specific sites in a solid, with the peak broadening being due to a distribution of probabilities (Butcher and Morys, 1973; Butcher and Ries, 1981). A further approach (reviewed by Jonscher, 1983) is the use of correlation functions. This method involves the consideration that in a population of dipoles, there will be a distribution of degrees of reorientation with respect to the external field at any particular moment in time. The difference between this approach and that described above is that it is not so much describing a range of absolute relaxation times but is instead working on the basis of the dipoles being in a range of positions with respect to the external field. In other words, if a ‘snapshot’ could be taken of the dipoles at any particular moment, the range of positions of the dipoles would relate to the relaxation behaviour. Further approaches include local field theory, whereby the dielectric processes associated with the area in the immediate proximity of a relaxing dipole are considered, while interfacial phenomena such as the Maxwell-Wagner effects described in the previous chapter may also be of relevance to some solid systems. A more recent analysis has been described by Jonscher (1983), namely the ‘many-body universal model’ which was mentioned in section 1.3.4. This approach is based on the observation that all dielectric samples exhibit fractional power law behaviour with frequency, i.e. when the data is given as a log/log plot (instead of semi-logarithmically) with frequency, a series of linear relationships are observed, the gradients of which give fractional power law exponents. An example of this is shown in Figure 5.1. In this case, the power law exponents are 0.62 and –0.21. Jonscher (1975b,c) has argued that the universality of power law behaviour is a function of many-body interactions arising from the close proximity of neighbouring molecules in condensed matter. This differs from the theory of relaxation time distributions, as the latter is suggesting a superimposition of separate relaxation processes while the former is describing an interactive process between different relaxing species. The author also suggested that in contrast to the Debye model, whereby the dipoles are considered to be ‘floating’ and to reorientate in a continuous manner from the ground state on application of an electric field, transitions occur abruptly in solid samples and even in some liquids. These sudden transitions cause a chain of responses in neighbouring atoms and molecules, resulting in an interactive response rather than one depending solely on the behaviour of individual molecules.

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Figure 5.1: The dielectric response of polydian carbonate, based on original data from Ishida and Matsuoka, 1965 (reproduced from Jonscher, 1975a) The Dissado-Hill theory outlined in section 1.4.4 may be applied to these systems and the approach has been described in detail by Hill and Jonscher (1983). The basic premise of the approach is that the response of solid samples may be considered to be a result of the formation of dipole clusters, whereby regions of local cooperativity occur. These clusters may be described by incorporating the power law indices m and n into the Debye equations, with m representing the cooperativity between adjacent dipoles in clusters and n representing the correlation between different clusters. As these two parameters have specific physical significance, it is possible to use this approach to gain information concerning the structure of the sample, rather than simply the dielectric properties of the material. The low and high frequency behaviour may be described by

(5.1) and

(5.2) hence at high frequencies, both the real and imaginary components will have parallel slopes of –(1–n) while at low frequencies the loss slope will have a positive slope of m (Dissado and Hill, 1984). This approach has been applied to a number of systems and appears to give a good correlation with observed behaviour (Hill, 1978, 1981).

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Figure 5.2: Representation of anomalous low frequency dispersion behaviour. The dotted curve shows the behaviour expected from a simple parallel RC circuit, with a d.c. conductivity and a frequency independent capacitance (reproduced from Dissado and Hill (1984) with permission of the Royal Society of Chemistry) A further, related, aspect of dielectric theory that has proved difficult to interpret has been the low frequency dispersion in solid samples (LFD), whereby an increase in the capacitance and/or loss has been observed at frequencies below the kilohertz region (Jonscher, 1978). A schematic example of this phenomenon is given in Figure 5.2. This is (seemingly) in contrast to the behaviour shown in Figure 5.1, in which the basic Debye shape is retained, although the high frequency parallel behaviour predicted in (5.1) is still observed. The profile shown in Figure 5.2, however, shows a continuous rise in both the real and imaginary components of the response as the frequency is lowered. Behaviour of this type is found in a wide variety of solid samples, hence it is useful to give a brief outline if the interpretation. It has been observed that if one changes the temperature (Jonscher, 1978), humidity (Jonscher et al., 1979) or measuring field strength (Ramdeen et al., 1984) of a sample showing LFD behaviour, the basic shape of the spectrum remains the same but the position with respect to the axes shifts, both above and below the crossover frequency c. This indicates that the complete spectrum is a reflection of a single process, as if the low (< c) and high (> c) frequency responses were due to separate processes, one would not expect identical dependencies on the variables listed above. Dissado and Hill (1984) have argued that in the majority of studies, the response has only been studied down to the

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frequency represented by the dashed-dotted line shown in Figure 5.2 due to the limited frequency range of the instruments used, hence the limited range of the LFD process that is commonly measured may be confused with the onset of d.c. conduction, whereby the real part of the response will be frequency independent and the imaginary (loss) component will increase with decreasing frequency with a slope of –1 (on a logarithmic plot), as indicated by the dotted lines in Figure 5.2. A further explanation is that the LFD is a result of a thin electrode layer, hence one would be observing a Maxwell-Wagner effect, as described in section 4.1.2. However, one would expect such a response to be of the form given in Figure 4.2, whereby the high frequency capacitance decreases with frequency with a slope of –2 and loss again has a slope of –1. Clearly, this is not the case for the example given in Figure 5.2. Instead, it has been suggested that the LFD is an indication of the bulk properties of the sample. Dissado and Hill (1984) have argued that the LFD occurs in solid samples in which there is a mechanically rigid lattice which contains binding sites for ions or molecules, most of which are occupied. A conceptually simple example which will be discussed in greater detail later is water adsorption onto solids such as sand (Shahidi et al., 1975) or proteins (Eden et al., 1980). The reason that most of the sites are occupied is that for systems in which low occupancy occurs, the charges may move between the sites freely and hence d.c. conductivity will be observed. For high occupancy, the number of pathways is severely reduced, resulting in the formation of clusters, whereby regions of local ordering occur, with a greater degree of unoccupied sites occurring towards the outside of a cluster. These structures will be dynamic and will be constantly breaking down and reforming. The explanation therefore leads back again to the cluster model outlined in the DissadoHill theory, as there will be exchange both within and between separate clusters. One may now consider a solid sample in terms of such occupied sites. In the absence of an electric field, the system will be electrically neutral. When a uniform, unidirectional field is applied, the sample will polarise as charge displacement will occur, depending on the dipole strength. When an alternating field is applied, the fluctuations in charge displacement must be considered in terms of the number of routes available for charge movement and the different time-dependencies of relaxation via these various routes. A mathematical treatment of this process (or, more accurately, the sum of these composite processes) has been given by Dissado and Hill (1984). In qualitative terms, the higher frequency response seen in Figure 5.2 is a reflection of charge displacements within a cluster. The regularity of the cluster is reflected by the exponent n, whereby a small value of n corresponds to a highly irregular cluster while a high value indicates that the cluster has a regular, ordered internal structure. These may in turn be associated with the degree of association with the host lattice, as a highly ordered cluster implies a low degree of binding to the host (as the bound charges may not be simultaneously bound to each other to form a regular cluster and also bound extensively to the host), while the presence of an irregular cluster indicates stronger binding to the host, thus disrupting the uniformity of inter-charge binding. The low frequency slope, designated p, indicates the extent of intercluster exchange, with a large value indicating a near d.c. conduction, while a small value indicates that the clusters are essentially separate and of uniform size, thus a low level of correlation exists between them (Hill et al., 1981). While both n and p give information on the exchange processes between clusters, n refers to the behaviour of

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fixed dipoles while p refers to moving charges. The essential difference between this latter process and d.c. conduction is that charge storage is involved in a LFD process (indicated by the rise in capacitance with decreasing frequency) while there is no energy stored in d.c. conduction process.

Figure 5.3: Dielectric responses of (a) crystal state II and (b) crystal state III cycloheptanol. Plots have been normalised with respect to temperature to produce master response curves (reproduced from Shablakh et al. (1983) with permission of the Royal Society of Chemistry)

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These indices may therefore yield structural information on the sample, although much more work needs to be performed in order to clarify the relationship between the power law indices n and p and the macroscopic properties of the sample. Jonscher (1991) has refined the model to suggest that the LFD may arise as a result of transport through the bulk of a sample, transport across interfaces within a sample, and transport along interfaces (e.g. along adsorbed water paths) and has also suggested that electrochemical as well as electrostatic processes must be considered. 5.1.2 Crystalline solids

Figure 5.4: Time dependent changes in capacitance representing transitions from the supercooled plastic crystal state (I) of cyclo-octanol to crystal state (II) (reproduced from Shablakh et al. (1983) with permission of the Royal Society of Chemistry) As almost all drugs are produced as crystalline solids at some stage of the manufacture of the dosage form, an understanding of the crystal properties of such drugs is of great interest to the pharmaceutical scientist. While a number of techniques are available in order to characterise the crystal structure of drugs and excipients. Indeed, many (if not all) companies have experienced difficulties in finding appropriate methods with which to control the quality of these materials, as both drugs and excipients may show alterations in performance which are not easily predicted using standard techniques such as X-ray diffraction and differential scanning calorimetry. It is therefore pertinent to discuss the ways in which dielectric analysis may be of use in this respect. One investigation which illustrates the potential of using dielectric analysis in the study of crystalline solids is that of Shablakh et al. (1983). The authors studied low molecular weight cyclic alcohols, which are known to crystallise into more than one form on cooling from the melt (Adachi et al., 1972). Amongst other investigations, Shablakh et al.

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(1983) showed that the dielectric spectra obtained were dependent on the polymorphic form present, as shown for cycloheptanol in Figure 5.3. The transitions between the crystal forms in the solid state were shown to be dependent on both temperature and time. For example, in the temperature region 190–215 K, cyclo-octanol undergoes a transition between two crystal forms. By measuring the capacitance at a fixed frequency (1 kHz), the authors showed the time dependence of the transition, as indicated in Figure 5.4. This thorough study involves a detailed analysis of data for a range of samples from cyclopentanol through to cyclo-octanol and includes a discussion of how the dielectric data may be related to the crystal structure of the varies systems studied. The study therefore shows that not only may the dielectric technique distinguish between different crystal forms of the same materials but may also allow structural information, particularly in terms of lattice order, to be obtained. By simple extrapolation from this study, it is clear that the technique may be used to detect phenomena such as polymorphism in drugs, which is by no means always a simple task, and may also be used as a means of measuring the kinetics of transformation between different forms under a range of conditions such as elevated temperatures. This therefore represents an exciting and hitherto unexplored application of the technique within the pharmaceutical sciences. 5.1.3 Glass-forming materials

Figure 5.5: Real and imaginary parts of the complex susceptibility for glycerol at 198 K (reproduced from Shablakh et al. (1982) with permission of the Royal Society of Chemistry) A number of materials will form glasses on cooling from the melt, the most important example being polymeric samples which will be dealt with in the next chapter. In addition, a number of smaller molecules such as alcohols and some drugs will also form glasses, hence it is helpful to have some knowledge of how their behaviour may be studied using dielectric analysis. Indeed, a number of drugs are produced in an

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amorphous form (albeit seldom by choice), hence their characterisation is of use within the pharmaceutical sciences. A number of studies have examined glass-forming systems (e.g. Williams, 1966; Johari, 1976; Shablakh et al., 1982) and it is possible to monitor a range of low temperature transitions associated with changes in molecular mobility. These transitions may be studied by increasing or lowering the temperature and observing changes at a fixed frequency or by scanning the samples in the frequency domain at a single temperature and then changing the temperature. An example of this latter approach has been given by Shablakh et al. (1982), whereby a number of glass-forming systems were studied. This material showed a loss peak in the region of 180 K, depending on the frequency of study, which corresponds to the glass transition behaviour of the sample. An example of this response at 198 K is given in Figure 5.5. The authors examined glasses formed from single materials such as glycerol or isopropylbenzene and noted that while the spectra of these materials showed temperature dependence, the shape parameters (m and n) described in the Dissado-Hill theory (1978) remained unchanged. An example of a typical spectrum is given in Figure 5.5 for glycerol. It should be noted that in the spectra the logarithmic real and imaginary components are parallel at frequencies above that of the loss peak, as predicted by the Dissado-Hill theory discussed in section 5.1.1. Shablakh et al. (1982) also studied compound glasses (44% chlorobenzene in pyridine, 44% toluene in pyridine) and found the shape parameters themselves to be dependent on temperature as the systems were cooled towards the glass transition temperature. This implies that the simple glasses such as glycerol form a relatively rigid structure on cooling from the liquid state which consequently shows little change in the shape parameters with temperature. The compound materials, however, have a structure which changes continuously as the glass transition temperature is approached, as indicated by the changes in n and m as the system is cooled. These changes were interpreted in terms of the degree of interaction and correlation between the relaxation of the constituent molecules in the composite glass-forming structure.

5.2 ANALYSIS OF PARTICULATE SYSTEMS The properties of drug particles such as size, shape and porosity are of fundamental importance in the design of dosage forms. The two most important performance characteristics associated with particle morphology are the flow properties and the dissolution characteristics, both of which are largely associated with the effective surface area of the drug powder. More detailed discussions of the importance of particle characteristics are available in almost any pharmaceutics textbook and hence will not be dealt with further here, other than to emphasise the need for the devlopment of accurate and rapid means of particle characterisation. In terms of dielectric analysis, work has been conducted on the dielectric properties of powders with a view to assessing morphology and porosity, particularly in the study of soils (Arulanandan,1991). The author has outlined a theoretical approach to the measurement of the porosity of particles in liquid media which he suggests may be used

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as a routine method of porosity assessment for geological studies. Clearly, there is no reason why the same reasoning cannot be applied to pharmaceutical systems. As this is a potentially important application of the technique, the approach will be described in some detail. The model used by Arulanandan (1991) is based on the Maxwell theory of conductivity through heterogeneous materials. If one envisages spherical particles in dilute suspension within a liquid medium, the conductivity of the system (κ) is given by

(5.3) where κ1 is the conductivity of the solution, κ2 is the conductivity of the particle and n is the porosity of the particle (not to be confused with the power law index n given in (5.1)). Fricke (1924) extended Maxwell’s theory to account for ellipsoidal particles, while Arulanandan (1991) also accounted for the orientation of the particles within the sample cell. This was achieved by measuring samples in a Teflon cell with parallel electrodes at both the top and the sides of the cell. Consequently, it was possible to measure the sample in both the horizontal and vertical directions. Considering these two to represent the extremes of orientation, the author effectively averaged these two in order to account for differences between the two sets of electrodes. Arulanandan (1991) described the response in terms of a formation factor, which describes the conductivity behaviour of the system of particles at any angle to the field (Fθ), where

(5.4) where is κθ the complex conductivity of the medium when all the particles are orientated at an angle θ to the field. The author derived expressions for the horizontal and vertical formation factors ( and ) at high (infinite) frequencies, in this case 50 MHz. The average formation factor is given by

(5.5)

(5.6) where 1 and 2 are the permittivities of the solution and particle, while Aa and Ab are derived from the axial ratios of the particles. As shown in (5.6), it is therefore not

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difficult to calculate a porosity value for any sample using only the axial ratio and complex dielectric behaviour of the sample, particles and solution, as the use of the average formation factor accounts for the orientation of the particles with respect to the field. Furthermore, if one assumes the particles to be spherical then (5.6) reduces to

(5.7) The author presented data for a number of systems, showing the relationship between sample porosity and the formation factor. An excellent agreement was found between theoretical and experimental data, as shown in Figure 5.6. This work could be of considerable pharmaceutical significance, as the study shows that by making rapid and simple measurements the porosity of a powder sample may be assessed. This is in contrast to the present techniques used pharmaceutically such as gas adsorption which are slow and somewhat tedious to perform.

Figure 5.6: Relationship between Fav and porosity for axial ratios (R) 0.001, 0.1 and 1 (reproduced from Arulanandan (1991) with permission of the American Society of Civil Engineers) Dielectric analysis may also yield information on pore geometry and surface area, as the interfaces between the internal surface of the pore and the surrounding medium will show a frequency dependent response (Ruffet et al., 1991). The authors used a model based on fractal theory (Mandelbrot, 1982). This theory considers irregular objects to be composed of self-similar components which are invariant through changes in scale (fractals). If d is the fractal dimension of an object (i.e. the dimension of repeat) then the whole object can be split into N parts via

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(5.8) where r is a measure of the similarity of the subdivisions. Two models have been proposed relating fractal theory to porosity. The Le Mehaute and Crepy model (Le Mahaute and Crepy, 1983), modified by Ruffet et al. (1991) relates the fractal dimension to the correction on the Cole-Cole plot ( ) via

(5.9) The Cole-Cole parameter is itself related to the specific surface area ( ) via the empirical equation

(5.10) where q and b are constants for a given measuring system. The fractal dimension gives an idea of the irregularity of the pore surface, as a large value of d indicates considerable surface roughness. The use of fractal dimensions as a means of characterising pharmaceutical powders has not been extensively