la physique statistique des membranes biologiques

African Journal Of Mathematical Physics Volume 8(2010)101-114

Casimir force in confined biomembranes K. El Hasnaoui, Y. Madmoune, H. Kaidi, M. Chahid, and M. Benhamou Laboratoire de Physique des Polym`eres et Ph´enom`enes Critiques Facult´e des Sciences Ben M’sik, P.O. 7955, Casablanca, Morocco [email protected]

abstract We reexamine the computation of the Casimir force between two parallel interacting plates delimitating a liquid with an immersed biomembrane. We denote by D their separation, which is assumed to be much smaller than the bulk roughness, in order to ensure the membrane confinement. This repulsive force originates from the thermal undulations of the membrane. To this end, we first introduce a field theory, where the field is noting else but the height-function. The field model depends on two parameters, namely the membrane bending rigidity constant, κ, and some elastic constant, µ ∼ D−4 . We first compute the static Casimir force (per unit area), Π, and find that the latter decays with separation D as : Π ∼ D−3 , with a known amplitude scaling as κ−1 . Therefore, the force has significant values only for those biomembranes of small enough κ. Second, we consider a biomembrane (at temperature T ) that is initially in a flat state away from thermal equilibrium, and we are interested in how the dynamic force, Π (t), grows in time. To do calculations, use is made of a non-dissipative Langevin equation (with noise) that is solved by the time height-field. We first show that the membrane roughness, L⊥ (t), increases with time as : L⊥ (t) ∼ t1/4 (t < τ ), with the final time τ ∼ D4 (required time over which the final equilibrium state is reached). Also, we find that the force increases in time according to : Π (t) ∼ t1/2 (t < τ ). The discussion is extended to the real situation where the biomembrane is subject to hydrodynamic interactions caused by the surrounding liquid. In this case, we show that : L⊥ (t) ∼ t1/3 (t < τh ) and Πh (t) ∼ t2/3 (t < τh ), with the new final time τh ∼ D3 . Consequently, the hydrodynamic interactions lead to substantial changes of the dynamic properties of the confined membrane, because both roughness and induced force grow more rapidly. Finally, the study may be extended, in a straightforward way, to bilayer surfactants confined to the same geometry. Key words: Biomembranes - Confinement - Casimir force - Dynamics.

I. INTRODUCTION

The cell membranes are of great importance to life, because they separate the cell from the surrounding environment and act as a selective barrier for the import and export of materials. More details concerning the structural organization and basic functions of biomembranes can be found in Refs. [1 − 7]. It is well-recognized by the scientific community that the cell membranes essentially present as a phospholipid bilayer combined with a variety of proteins and cholesterol (mosaic fluid model). In particular, the function of the cholesterol molecules is to ensure the bilayer fluidity. A phospholipid is an amphiphile

0

c a GNPHE publication 2010, [email protected] ⃝

101

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

molecule possessing a hydrophilic polar head attached to two hydrophobic (fatty acyl) chains. The phospholipids move freely on the membrane surface. On the other hand, the thickness of a bilayer membrane is of the order of 50 Angstroms. These two properties allow to consider it as a two-dimensional fluid membrane. The fluid membranes, self-assembled from surfactant solutions, may have a variety of shapes and topologies [8], which have been explained in terms of bending energy [9, 10]. In real situations, the biomembranes are not trapped in liquids of infinite extent, but they rather confined to geometrical boundaries, such as white and red globules or liposomes (as drugs transport agents [11 − 14]) in blood vessels. For simplicity, we consider the situation where the biomembrane is confined in a liquid domain that is finite in one spatial direction. We denote by D its size in this direction. For a tube, D being the diameter, and for a liquid domain delimitated by two parallel plates, this size is simply the separation between walls. Naturally, the length D must be compared to the bulk roughness, L0⊥ , which is the typical size of humps caused by the thermal fluctuations of the membrane. The latter depends on the nature of lipid molecules forming the bilayer. The biomembrane is confined only when D is much smaller than the bulk roughness L0⊥ . This condition is similar to that usually encountered in confined polymers context [15]. The membrane undulations give rise to repulsive effective interactions between the confining geometrical boundaries. The induced force we term Casimir force is naturally a function of the size D, and must decays as this scale is increased. In this paper, we are interested in how this force decays with distance. To simplify calculations, we assume that the membrane is confined to two parallel plates that are a finite distance D < L0⊥ apart. The word ”Casimir” is inspired from the traditional Casimir effect. Such an effect, predicted, for the first time, by Hendrick Casimir in 1948 [16], is one of the fundamental discoveries in the last century. According to Casimir, the vacuum quantum fluctuations of a confined electromagnetic field generate an attractive force between two parallel uncharged conducting plates. The Casimir effect has been confirmed in more recent experiments by Lamoreaux [17] and by Mohideen and Roy [18]. Thereafter, Fisher and de Gennes [19], in a short note, remarked that the Casimir effect also appears in the context of critical systems, such as fluids, simple liquid mixtures, polymer blends, liquid 4 He, or liquid-crystals, confined to restricted geometries or in the presence of immersed colloidal particles. For these systems, the critical fluctuations of the order parameter play the role of the vacuum quantum fluctuations, and then, they lead to long-ranged forces between the confining walls or between immersed colloids [20, 21]. To compute the Casimir force between the confining walls, we first elaborate a more general field theory that takes into account the primitive interactions experienced by the confined membrane. As we shall see below, in confinement regime, the field model depends only on two parameters that are the membrane bending rigidity constant and a coupling constant containing all infirmation concerning the interaction potential exerting by the walls. In addition, the last parameter is a known function of the separation D. With the help of the constructed free energy, we first computed the static Casimir force (per unit area), Π. The exact calculations show that the latter decays with separation D according to a power 2 law, that is Π ∼ κ−1 (kB T ) D−3 , with a known amplitude. Here, kB T denotes the thermal energy, and κ the membrane bending rigidity constant. Of course, this force increases with temperature, and has significant values only for those biomembranes of small enough κ. The second problem we examined is the computation of the dynamic Casimir force, Π (t). More precisely, we considered a biomembrane at temperature T that is initially in a flat state away from the thermal equilibrium, and we were interested in how the expected force grows in time, before the final state is reached. Using a scaling argument, we first showed that the membrane roughness, L⊥ (t), grows with time as : L⊥ (t) ∼ t1/4 (t < τ ), with the final time τ ∼ D4 . The latter can be interpreted as the required time over which the final equilibrium state is reached. Second, using a non-dissipative Langevin equation, we found that the force increases in time according to the power law : Π (t) ∼ t1/2 (t < τ ). Third, the discussion is extended to the real situation where the biomembrane is subject to hydrodynamic interactions caused by the flow of the surrounding liquid. In this case, we show that : L⊥ (t) ∼ t1/3 (t < τh ) and Π (t) ∼ t2/3 (t < τh ), with the new final time τh ∼ D3 . Consequently, the hydrodynamic interactions give rise to drastic changes of the dynamic properties of the confined membrane, since both roughness and induced force grow more rapidly. This paper is organized as follows. In Sec. II, we present the field model allowing the determination of the Casimir force from a static and dynamic point of view. The Sec. III and Sec. IV are devoted to the computation of the static and dynamic induced forces, respectively. We draw our conclusions in the last section. Some technical details are presented in Appendix.

102

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114 II. THEORETICAL FORMULATION

Consider a fluctuating fluid membrane that is confined to two interacting parallel walls 1 and 2. We denote by D their finite separation. Naturally, the separation D must be compared to the bulk membrane roughness, L0⊥ , when the system is unconfined (free membrane). The membrane is confined only when the condition L << L0⊥ is fulfilled. For the opposite condition, that is L >> L0⊥ , we expect finite size corrections. We assume that these walls are located at z = −D/2 and z = D/2, respectively. Here, z means the perpendicular distance. For simplicity, we suppose that the two surfaces are physically equivalent. We design by V (z) the interaction potential exerted by one wall on the fluid membrane, in the absence of the other. Usually, V (z) is the sum of a repulsive and an attractive potentials. A typical example is provided by the following potential [22] V (z) = Vh (z) + VvdW (z) ,

(2.0)

Vh (z) = Ah e−z/λh

(2.0)

where

represents the repulsive hydration potential due to the water molecules inserted between hydrophilic lipid heads [22]. The amplitude Ah and the potential-range λh are of the order of : Ah ≃ 0.2 J/m2 and λh ≃ 0.2 − 0.3 nm. In fact, the amplitude Ah is Ah = Ph × λh , with the hydration pressure Ph ≃ 108 −109 Pa. There, VvdW (z) accounts for the van der Waals potential between one wall and biomembrane, which are a distance z apart. Its form is as follows [ ] H 1 2 1 VvdW (z) = − − , (2.0) 2 + 2 12π z 2 (z + δ) (z + 2δ) with the Hamaker constant H ≃ 10−22 − 10−21 J, and δ ≃ 4 nm denotes the membrane thickness. For large distance z, this implies VvdW (z) ∼

W δ2 . z4

(2.0)

Generally, in addition to the distance z, the interaction potential V (z) depends on certain length-scales, (ξ1 , ..., ξn ), which are the interactions ranges. The fluid membrane then experiences the following total potential ( ) ( ) D D D D U (z) = V −z +V +z , − ≤z≤ . (2.0) 2 2 2 2 In the Monge representation, a point on the membrane can be described by the three-dimensional position vector r = (x, y, z = h (x, y)), where h (x, y) ∈ [−D/2, D/2] is the height-field. The latter then fluctuates around the mid-plane located at z = 0. The Statistical Mechanics for the description of such a (tensionless) fluid membrane is based on the standard Canham-Helfrich Hamiltonian [9, 23] ∫ [κ ] 2 H [h] = dxdy (∆h) + W (h) , (2.0) 2 with the membrane bending rigidity constant κ. The latter is comparable to the thermal energy kB T , where T is the absolute temperature and kB is the Boltzmann’s constant. There, W (h) is the interaction potential per unit area, that is W (h) =

U (h) , L2

where the potential U (h) is defined in Eq. (2), and L is the lateral linear size of the biomembrane. Let us discuss the pair-potential W (h). 103

(2.0)

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

Firstly, Eq. (2) suggests that this total potential is an even function of the perpendicular distance h, that is W (−h) = W (h) .

(2.0)

In particular, we have W (−D/2) = W (D/2). Secondly, when they exist, the zeros h0 ’s of the potential function U (h) are such that ( ) ( ) D D V − h0 = −V + h0 . 2 2

(2.0)

This equality indicates that, if h0 is a zero of the potential function, then, −h0 is a zero too. The number of zeros is then an even number. In addition, the zero h0 ’s are different from 0, in all cases. Indeed, the quantity V (D/2) does not vanish, since it represents the potential created by one wall at the middle of the film. We emphasize that, when the potential processes no zero, it is either repulsive or attractive. When this same potential vanishes at some points, then, it is either repulsive of attractive between two consecutive zeros. Thirdly, we first note that, from relation (2), we deduce that the first derivative of the potential function, with respect to distance h, is an odd function, that is W ′ (−h) = −W ′ (h). Applying this relation to the midpoint h = 0 yields : W ′ (0) = 0. Therefore, the potential W exhibits an extremum at h = 0, whatever the form of the function V (h). We find that this extremum is a maximum, if V ′′ (D/2) < 0, and a minimum, if V ′′ (D/2) > 0. The potential U presents an horizontal tangent at h = 0, if only if V ′′ (D/2) = 0. On the other hand, the general condition giving the extrema {hm } is dV dV = . (2.0) dh h= D −hm dh h= D +hm 2

2

Since the first derivative W ′ (h) is an odd function of distance h, it must have an odd number of extremum points. The point h = hm is a maximum, if d2 V d2 V < − , (2.0) dh2 h= D −hm dh2 h= D +hm 2

2

and a minimum, if d2 V d2 V >− . dh2 h= D −hm dh2 h= D +hm 2

(2.0)

2

At point h = hm , we have an horizontal tangent, if d2 V d2 V =− . dh2 h= D −hm dh2 h= D +hm 2

(2.0)

2

The above deductions depends, of course, on the form of the interaction potential V (h). Fourthly, a simple dimensional analysis shows that the total interaction potential can be rewritten on the following scaling form ( ) 1 h ξ1 ξn W (h) = 2Φ , , ..., , (2.0) kB T D D D D where (ξ1 , ..., ξn ) are the ranges of various interactions experienced by the membrane, and Φ (x1 , ..., xn+1 ) is a (n + 1)-factor scaling-function. Finally, we note that the pair-potential W (h) cannot be singular at h = 0. It is rather an analytic function in the h variable. Therefore, at fixed ratios ξi /D, an expansion of the scaling-function Φ, around the value h = 0, yields ( ) γ h2 W (h) = + O h4 . 4 kB T 2D 104

(2.0)

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

We restrict ourselves to the class of potentials that exhibit a minimum at the mid-plane h = 0. This assumption implies that the coefficient γ is positive definite, i.e. γ > 0. Of course, such a coefficient depends on the ratios of the scale-lengths ξi to the separation D. In confinement regime where the distance h is small enough, we can approximate the total interaction potential by its quadratic part. In these conditions, the Canham-Helfrich Hamiltonian becomes ∫ [ ] 1 2 H0 [h] = dxdy κ (∆h) + µh2 , (2.0) 2 with the elastic constant µ=γ

kB T . D4

(2.0)

The prefactor γ will be computed below. The above expression for the elastic constant µ gives an idea on its dependance on the film thickness D. In addition, we state that this coefficient may be regarded as a Lagrange multiplier that fixes the value of the membrane roughness. Thanks to the above Hamiltonian, we calculate the mean-expectation value of the physical quantities, like the height-correlation function (propagator or Green function), defined by G (x − x′ , y − y ′ ) = ⟨h (x, y) h (x′ , y ′ )⟩ − ⟨h (x, y)⟩ ⟨h (x′ , y ′ )⟩ . The latter solves the linear differential equation ( 2 ) κ b∆ + µ b G (x − x′ , y − y ′ ) = δ (x − x′ ) δ (y − y ′ ) ,

(2.0)

(2.0)

where δ (x) denotes the one-dimensional Dirac function, and ∆ = ∂ 2 /∂x2 + ∂ 2 /∂y 2 represents the twodimensional Laplacian operator. We have used the notations : κ b = κ/kB T and µ b = µ/kB T , to mean the reduced membrane elastic constants. From the propagator, we deduce the expression of the membrane roughness ⟨ ⟩ 2 (2.0) L2⊥ = h2 − ⟨h⟩ = G (0, 0) . Such a quantity measures the fluctuations of the height-function (fluctuations amplitude) around the equilibrium plane located at z = 0. We show in Appendix that the membrane roughness is exactly given by L2⊥ =

D2 , 12

(2.0)

provided that one is in the confinement-regime, i.e. D << L0⊥ . Notice that the above equality indicates that the roughness is independent on the geometrical properties of the membrane (through κ). We emphasize that this relation can be recovered using the argument that each point of the membrane has equal probability to be found anywhere between the walls [24]. The elastic constant µ may be calculated using the known relation L2⊥ =

1 kB T . √ 8 µκ

(2.0)

This gives 2

µ=

9 (kB T ) . 4 κD4

(2.0)

This formula clearly shows that this elastic constant decays with separation D as D−4 . The term µh2 /2 then describes a confinement potential that ensures the localization of the membrane around the midplane. Integral over the hole plane R2 of this term represents the loss entropy due to the confinement of the membrane. The value (19) of the elastic constant is compatible with the constraint (17). Therefore, the elaborated model is based on the Hamiltonian (13), with a quadratic confinement potential. We can say that the presence of the walls simply leads to a confinement of the membrane in a region 105

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

of the infinite space of perpendicular size L⊥ . We define now another length-scale that is the in-plane correlation length, L∥ . The latter measures the correlations extent along the parallel directions to the walls. More precisely, the propagator G (x − x′ , y − y ′ ) fails exponentially beyond L∥ , that is for distances d such that d = √ 2 2 (x − x′ ) + (y − y ′ ) > L∥ . From the standard relation L2⊥ =

kB T 2 L , 16κ ∥

(2.0)

we deduce 2 L∥ = √ 3

(

κ kB T

)1/2 D.

(2.0)

In contrary to L⊥ , the length-scale L∥ depends on the geometrical characteristics of the membrane (through κ). The next steps consist in the computation of the Casimir force at and out equilibrium.

III. STATIC CASIMIR FORCE

When viewed under the microscope, the membranes of vesicles present thermally excited shape fluctuations. Generally, objects such as interfaces, membranes or polymers undergo such fluctuations, in order to increase their configurational entropy. For bilayer biomembranes and surfactants, the consequence of these undulations is that, they give rise to an induced force called Casimir force. To compute the desired force, we start from the partition function constructed with the Hamiltonian defined in Eq. (13). This partition function is the following functional integral { } ∫ H0 [h] Z = Dh exp − , (3.0) kB T where integration is performed over all height-field configurations. The associated free energy is such that : F = −kB T ln Z, which is, of course, a function of the separation D. If we denote by Σ = L2 the common area of plates, the Casimir force (per unit area) is minus the first derivative of the free energy (per unit area) with respect to the film-thickness D, that is Π=−

1 ∂F . Σ ∂D

(3.0)

This force per unit area is called disjoining pressure. In fact, Π is the required pressure to maintain the two plates at some distance D apart. In term of the partition function, the disjoining pressure rewrites Π 1 ∂ ln Z 1 ∂µ ∂ ln Z = = . kB T Σ ∂D Σ ∂D ∂µ

(3.0)

Using definition (19) together with Eqs. (23) and (24) yields Π=−

1 ∂µ 2 L . 2 ∂D ⊥

(3.0)

Explicitly, we obtain the desired formula 2

Π=

3 (kB T ) . 8 κD3

(3.0)

From this relation, we extract the expression of the disjoining potential (per unit area) [25] ∫ Vd (D) = −

D

Π (D′ ) dD′ =

∞

106

2

3 (kB T ) . 16 κD2

(3.0)

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

The above expression of the Casimir force (per unit area) calls the following remarks. Firstly, this force decays with distance more slowly in comparison to the Coulombian one that decreases rather as D−2 . Secondly, this same force depends on the nature of lipids forming the bilayer (through κ). In this sense, contrarily to the Casimir effect in Quantum Field Theory [16] and in Critical Phenomena [20], the present force is not universal. Incidentally, if this force is multiplied by κ, then, it will become a universal quantity. Thirdly, at fixed temperature and distance, the force amplitude has significant values only for those bilayer membrane of small bending rigidity constant. Fourthly, as it should be, such a force increases with increasing temperature. Indeed, at high temperature, the membrane undulations are strong enough. Finally, the numerical prefactor 3/8 (Helfrich’s cH -amplitude [9]) is close to the value obtained using Monte Carlo simulation [26]. In Fig. 1, we superpose the variations of the reduced static Casimir force Π/kB T upon separation D, for two lipid systems, namely SOPC and DAPC [27], at temperature T = 18◦ C. The respective membrane bending rigidity constants are : κ = 0.96 × 10−19 J and κ = 0.49 × 10−19 J. These values correspond to the renormalized bending rigidity constants : κ b = 23.9 and κ b = 12.2. The used methods for the measurement of these rigidity constants were entropic tension and micropipet [27]. These curves reflect the discussion made above.

FIG. 1. Reduced static Casimir force, Π/kB T , versus separation D, for two lipid systems that are SOPC (solid line) and DAPC (dashed line), of respective membrane bending rigidity constants : κ = 0.96 × 10−19 J and κ = 0.49 × 10−19 J, at temperature T = 18◦ C. The reduced force and separation are expressed in arbitrary units.

IV. DYNAMIC CASIMIR FORCE

To study the dynamic phenomena, the main physical quantity to consider is the time height-field, h (r, t), where r = (x, y) ∈ R2 denotes the position vector and t the time. The latter represents the time observation of the system before it reaches its final equilibrium state. We recall that the time height function h (r, t) solves a non-dissipative Langevin equation (with noise) [28] δH0 [h] ∂h (r, t) = −Γ + ν (r, t) , ∂t δh (r, t) 107

(4.0)

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

where Γ > 0 is a kinetic coefficient. The latter has the dimension : [Γ] = L40 T0−1 , where L0 is some length and T0 the time unit. Here, ν (r, t) is a Gaussian random force with mean zero and variance ⟨ν (r, t) ν (r′ , t′ )⟩ = 2Γδ2 (r − r′ ) δ (t − t′ ) ,

(4.0)

and H0 is the static Hamiltonian (divided by kB T ), defined in Eq. (13). The bare time correlation function, whose Fourier transform is the dynamic structure factor, is defined by the expectation mean-value over noise ν G (r − r′ , t − t′ ) = ⟨h (r, t) h (r′ , t′ )⟩ν − ⟨h (r, t)⟩ν ⟨h (r′ , t′ )⟩ν ,

t > t′ .

(4.0)

The dynamic equation (28) shows that the time height function h is a functional of noise ν, and we write : h = h [ν]. Instead of solving the Langevin equation for h [ν] and then averaging over the noise distribution P [ν], the correlation and response functions can be directly computed by means of a suitable field-theory, of action [28 − 31] { } ∫ [ ] ∫ δH0 e e A h, e h = dt d2 r e h∂t h + Γe h − hΓh , (4.0) δh so that, for an arbitrary observable, O [ζ], one has ∫ ⟨O⟩ν =

∫

e DhDe hOe−A[h,h] , [dν] O [φ [ν]] P [ν] = ∫ e DhDe he−A[h,h]

(4.0)

where e h (r, t) is an auxiliary field, coupled to an external field h (r, t). The correlation and response functions can be computed replacing the static Hamiltonian H0 appearing in Eq. (13), by a new one : ∫ H0 [h, J] = H0 [h] − d2 rJh. Consequently, for a given observable O, we have ⟨ ⟩ δ ⟨O⟩J =Γ e h (r, t) O . (4.0) δJ (r, t) J=0 [ ] The notation ⟨ . ⟩J means the average taken with respect to the action A h, e h, J associated with the Hamiltonian H0 [h, J]. In view of the structure of equality (33), e h is called response field. Now, if O = h, we get the response of the order parameter field to the external perturbation J ⟨ ⟩ δ ⟨h (r′ , t′ )⟩J ′ ′ R (r − r , t − t ) = =Γ e h (r, t) h (r′ , t′ ) . (4.0) δJ (r, t) J=0 J=0 The causality implies that the response function vanishes for t < t′ . In fact, this function can be related to the time-dependent (connected) correlation function using the fluctuation-dissipation theorem, according to which ⟨ ⟩ Γ e h (r, t) h (r′ , t′ ) = −θ (t − t′ ) ∂t ⟨h (r, t) h (r′ , t′ )⟩c . (4.0) The above important formula shows that the time correlation function C (r − r′ , t − t′ ) = ⟨φ (r, t) φ (r′ , t′ )⟩c may be determined by the knowledge of the response function. In particular, we show that ∫ t ⟨ ⟩ ⟨ 2 ⟩ 2 h (r, t′ ) h (r, t′ ) . (4.0) dt′ e L⊥ (t) = h (r, t) c = −2Γ −∞

The limit t → −∞ gives the natural value L2⊥ (−∞) = 0, since, as assumed, the initial state corresponds to a completely flat interface. Consider now a membrane at temperature T that is initially in a flat state away from thermal equilibrium. At a later time t, the membrane possesses a certain roughness, L⊥ (t). Of course, the latter is timedependent, and we are interested in how it increases in time. 108

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

We point out that the thermal fluctuations give rise to some roughness that is characterized by the appearance of anisotropic humps. Therefore, a segment of linear size L effectuates excursions of size [32] L⊥ = BLζ .

(4.0)

Such a relation defines the roughness exponent ζ. Notice that L is of the order of the in-plane correlation length, L∥ . From relation (20), we deduce the exponent ζ and the amplitude B. Their respective values 1/2 are : ζ = 1 and B ∼ (kB T /κ) . In order to determine the growth of roughness L⊥ in time, the key is to consider the excess free energy (per unit area) due to the confinement, ∆F . Such an excess is related to the fact that the confining membrane suffers a loss of entropy. Formula (27) tells us how ∆F must decay with separation. The result reads [32] 2/ζ

∆F ∼ kB T /L2max ∼ kB T (B/L⊥ )

,

(4.0)

where Lmax represents the wavelength above which all shape fluctuations are not accessible by the confined membrane. The repulsive fluctuation-induced interaction leads to the disjoining pressure Π=−

∂∆F −(1+2/ζ) ∼ L⊥ . ∂L⊥

(4.0)

In addition, a care analysis of the Langevin equation (28) shows that ∂∆F ∂L⊥ −(1+2/ζ) ∼ −Γ = Γ × Π ∼ ΓL⊥ . ∂t ∂L⊥

(4.0)

We emphasize that this scaling form agrees with Monte Carlo predictions [32, 33]. Solving this first-order differential equation yields [34] L⊥ (t) ∼ Γθ⊥ tθ⊥ ,

θ⊥ =

1 ζ = . 2 + 2ζ 4

(4.0)

This implies the following scaling form for the linear size L (t) ∼ Γθ∥ tθ∥ ,

θ∥ =

1 1 = . 2 + 2ζ 4

(4.0)

Let us comment about the obtained result (39). Firstly, as it should be, the roughness increases with time (the exponent θ⊥ is positive definite). In addition, the exponent θ⊥ is universal, independently on the membrane bending rigidity constant κ. Secondly, we note that, in Eq. (39), we have ignored some non-universal amplitude that scales as κ−1/4 . This means that the time roughness is significant only for those biomembranes of small bending rigidity constant. Fourthly, this time roughness can be interpreted as the perpendicular size of holes and valleys at time t. Fifthly, the roughness increases until a fine time, τ . The latter can be interpreted as the time over which the system reaches its final equilibrium state. This characteristic time then scales as τ ∼ Γ−1 L⊥

1/θ⊥

,

(4.0)

where we have ignored some non-universal amplitude that scales as κ. Here, L⊥ ∼ D is the final roughness. Explicitly, we have τ ∼ Γ−1 D4 .

(4.0)

As it should be, the final time increases with increasing film thickness D. On the other hand, we can rewrite the behavior (39) as L⊥ (t) = L⊥ (τ )

( )θ⊥ t . τ

109

(4.0)

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

This equality means that the roughness ratio, as a function of the reduced time, is universal. Now, to compute the dynamic Casimir force, we start from a formula analog to that defined in Eq. (24), that is b b Π (t) 1 ∂ ln Z 1 ∂µ ∂ ln Z = = , kB T Σ ∂D Σ ∂D ∂µ

(4.0)

with the new partition function b= Z

∫

e

DhDe he−A[h,h] .

(4.0)

A simple algebra taking into account the basic relation (35a) gives Π (t) 1 ∂µ 2 =− L (t) , kB T 2 ∂D ⊥

(4.0)

which is very similar to the static relation defined in Eq. (25), but with a time-dependent membrane roughness, L⊥ (t). Combining formulae (43) and (46) leads to the desired expression for the time Casimir force (per unit area) Π (t) = Π (τ )

( )θf t , τ

(4.0)

where Π (τ ) is the final static Casimir force, relation (25). The force exponent, θf , is such that θf = 2θ⊥ =

ζ 1 = . 1+ζ 2

(4.0)

The induced force then grows with time as t1/2 until it reaches its final value Π (τ ). At fixed time and separation D, the force amplitude depends, of course, on κ, and decreases in this parameter according to κ−3/2 . Also, we note that the above equality means that the force ratio as a function of the reduced time is universal. In Fig. 2, we draw the reduced dynamic Casimir force, Π (t) /Π (τ ), upon the renormalized time t/τ .

110

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

FIG. 2. Reduced dynamic Casimir force, Π (t) /Π (τ ), upon the renormalized time t/τ .

Finally, consider again a membrane which is initially flat but is now coupled to overdamped surface waves. This real situation corresponds to a confined membrane subject to hydrodynamic interactions. The roughness now grows as [35] b ⊥ (t) ∼ tθb⊥ , L

θb⊥ =

ζ 1 = . 1 + 2ζ 3

(4.0)

Therefore, the roughness increases with time more rapidly than that relative to biomembranes free from hydrodynamic interactions. In this case, the dynamic Casimir force is such that Πh (t) = Π (τh )

(

t τh

)θbf ,

(4.0)

where Π (τh ) is the final static Casimir force, relation (25). The new force exponent is θbf = 2θb⊥ =

2 2ζ = . 1 + 2ζ 3

(4.0)

There, τh ∼ D3 accounts for the new time-scale over which the confined membrane reaches its final equilibrium state. Therefore, the dynamic Casimir force decays with time as t2/3 , that is more rapidly than that where the hydrodynamic interactions are ignored, which scales rather as t1/2 . As we said before, this drastic change can be attributed to the overdamped surface waves that develop larger and larger humps. We depict, in Fig. 3, the variation of the reduced dynamic force (with hydrodynamic interactions), Πh (t) /Π (τh ), upon the renormalized time t/τh . 111

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

FIG. 3. Reduced dynamic Casimir force (with hydrodynamic interactions), Πh (t) /Π (τ ), upon the renormalized time t/τh .

V. CONCLUSIONS

In this work, we have reexamined the computation of the Casimir force between two parallel walls delimitating a fluctuating fluid membrane that is immersed in some liquid. This force is caused by the thermal fluctuations of the membrane. We have studied the problem from both static and dynamic point of view. We were first interested in the time variation of the roughening, L⊥ (t), starting with a membrane that is inially in a flat state, at a certain temperature. Of course, this length grows with time, and we found that : L⊥ (t) ∼ tθ⊥ (θ⊥ = 1/4), provided that the hydrodynamic interactions are ignored. For real systems, however, these interactions are important, and we have shown that the roughness increases more rapidly b ⊥ (t) ∼ tθb⊥ (θ⊥ = 1/3). The dynamic process is then stopped at a final τ (or τh ) that represents as : L the required time over which the biomembrane reaches its final equilibrium state. The final time behaves as : τ ∼ D4 (or τh ∼ D3 ), with D the film thickness. Now, assume that the system is explored at scales of the order of the wavelength q −1 , where q = (4π/λ) sin (θ/2) is the wave vector modulus, with λ the wavelength of the incident radiation and θ the scattering-angle. In these) conditions, the relaxation rate, τ (q), scales with q as : τ −1 (q) ∼ q 1/θ⊥ = q 4 ( b

or τh−1 (q) ∼ q 1/θ⊥ = q 3 . Physically speaking, the relaxation rate characterizes the local growth of the height fluctuations. Afterwards, the question was addressed to the computation of the Casimir force, Π. At equilibrium, using an appropriate field theory, we found that this force decays with separation D as : Π ∼ D−3 , with a known amplitude scaling as κ−1 , where κ is the membrane bending rigidity constant. Such a force is then very small in comparison with the Coulombian one. In addition, this force disappears when the 112

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

temperature of the medium is sufficiently lowered. The dynamic Casimir force, Π (t), was computed using a non-dissipative Langevin equation (with noise), solved by the time height-field. We have shown that : Π (t) ∼ tθf (θf = 2θ⊥ = 1/2). When the hydrodynamic interactions effects are ( ) important, we found that the dynamic force increases more rapidly as : θbf b b Πh (t) ∼ t θf = 2θ⊥ = 2/3 . Notice that we have ignored some details such as the role of inclusions (proteins, cholesterol, glycolipids, other macromolecules) and chemical mismatch on the force expression. It is well-established that these details simply lead to an additive renormalization of the bending rigidity constant. Indeed, we write κeffective = κ + δκ, where κ is the bending rigidity constant of the membrane free from inclusions, and δκ is the contribution of the incorporated entities. Generally, the shift δκ is a function of the inclusion concentration and compositions of species of different chemical nature (various phospholipids forming the bilayer). Hence, to take into account the presence of inclusions and chemical mismatch, it would be sufficient to replace κ by κeffective , in the above established relations. As last word, we emphasize that the results derived in this paper may be extended to bilayer surfactants, although the two systems are not of the same structure and composition. One of the differences is the magnitude order of the bending rigidity constant.

APPENDIX

To show formula (17), we start from the partition function that we rewrite on the following form ∫ Z=

{

H [h] Dh exp − kB T

}

D/2 ∫

dzΦ (z) .

=

(5.0)

−D/2

Also, it is easy to see that the membrane mean-roughness is given by D/2 ∫

L2⊥ =

dzz 2 Φ (z)

−D/2 D/2 ∫

.

(5.0)

dzΦ (z)

−D/2

The restricted partition function is ∫ Φ (z) =

{ } H [h] Dhδ [z − h (x0 , y0 )] exp − . kB T

(5.0)

Here, H [h] is the original Hamiltonian defined in Eq. (3). Of course, this definition is independent on the chosen point (x0 , y0 ), because of the translation symmetry along the parallel directions to plates. Notice that the above function is not singular, whatever the value of the perpendicular distance. Since we are interested in the confinement-regime, ( ) that is when the separation D is much smaller than the membrane mean-roughness L0⊥ z ∼ h << L0⊥ , we can replace the function Φ par its value at z = 0, denoted Φ0 . In this limit, Eq. (A.2) gives the desired result. This ends the proof of the expected formula.

ACKNOWLEDGMENTS

We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the ”First International Workshop On Soft-Condensed Matter Physics and Biological Systems”, 14-17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank the Professor C. Misbah for fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kinds of hospitalities during his regular visits.

113

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114 REFERENCES

1

M.S. Bretscher and S. Munro, Science 261, 12801281 (1993). J. Dai and M.P. Sheetz, Meth. Cell Biol. 55, 157171 (1998). 3 M. Edidin, Curr. Opin. Struc. Biol. 7, 528532 (1997). 4 C.R. Hackenbrock, Trends Biochem. Sci. 6, 151154 (1981). 5 C. Tanford, The Hydrophobic Effect, 2d ed., Wiley, 1980. 6 D.E. Vance and J. Vance, eds., Biochemistry of Lipids, Lipoproteins, and Membranes, Elsevier, 1996. 7 F. Zhang, G.M. Lee, and K. Jacobson, BioEssays 15, 579588 (1993). 8 S. Safran, Statistical Thermodynamics of Surfaces, Interfaces and Membranes, Addison-Wesley, Reading, MA, 1994. 9 W. Helfrich, Z. Naturforsch. 28c, 693 (1973). 10 For a recent review, see U. Seifert, Advances in Physics 46, 13 (1997). 11 H. Ringsdorf and B. Schmidt, How to Bridge the Gap Between Membrane, Biology and Polymers Science, P.M. Bungay et al., eds, Synthetic Membranes : Science, Engineering and Applications, p. 701, D. Reiidel Pulishing Compagny, 1986. 12 D.D. Lasic, American Scientist 80, 250 (1992). 13 V.P. Torchilin, Effect of Polymers Attached to the Lipid Head Groups on Properties of Liposomes, D.D. Lasic and Y. Barenholz, eds, Handbook of Nonmedical Applications of Liposomes, Volume 1, p. 263, RCC Press, Boca Raton, 1996. 14 R. Joannic, L. Auvray, and D.D. Lasic, Phys. Rev. Lett. 78, 3402 (1997). 15 P.-G. de Gennes, Scaling Concept in Polymer Physics, Cornell University Press, 1979. 16 H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetenschap B 51, 793 (1948). 17 S.K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997). 18 U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 (1998). 19 M.E. Fisher and P.-G. de Gennes, C. R. Acad. Sci. (Paris) S´er. B 287, 207 (1978); P.-G. de Gennes, C. R. Acad. Sci. (Paris) II 292, 701 (1981). 20 M. Krech, The Casimir Effect in Critical Systems, World Scientific, Singapore, 1994. 21 More recent references can be found in : F. Schlesener, A. Hanke, and S. Dietrich, J. Stat. Phys. 110, 981 (2003); M. Benhamou, M. El Yaznasni, H. Ridouane, and E.-K. Hachem, Braz. J. Phys. 36, 1 (2006). 22 R. Lipowsky, Handbook of Biological Physics, R. Lipowsky and E. Sackmann, eds, Volume 1, p. 521, Elsevier, 1995. 23 P.B. Canham, J. Theoret. Biol. 26, 61 (1970). 24 O. Farago, Phys. Rev. E 78, 051919 (2008). 25 We recover the power law Π ∼ D−3 that is known in literature (see, for instance, Ref. [8]), but the corresponding amplitude depends on the used model. 26 G. Gompper and D.M. Kroll, Europhys. Lett. 9, 59 (1989). 27 U. Seifert and R. Lipowsky, in Structure and Dynamics of Membranes, Handbook of Biological Physics, R. Lipowsky and E. Sackmann, eds, Elsevier, North-Holland, 1995. 28 J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. 29 H.K. Jansen, Z. Phys. B 23, 377 (1976). 30 R. Bausch, H.K. Jansen, and H. Wagner, Z. Phys. B 24, 113 (1976). 31 F. Langouche, D. Roekaerts, and E. Tirapegui, Physica A 95, 252 (1979). 32 R. Lipowsky, in Random Fluctuations and Growth, H.E. Stanley and N. Ostrowsky, eds, p. 227-245, Kluwer Academic Publishers, Dordrecht 1988. 33 R. Lipowsky, J. Phys. A 18, L-585 (1985). 34 R. Lipowsky, Physica Scripta T 29, 259 (1989). 35 F. Brochard and J.F. Lennon, J. Phys. (Paris) 36, 1035 (1975). 2

114

African Journal Of Mathematical Physics Volume 8(2010)91-100

Brownian dynamics of nanoparticles in contact with a confined biomembrane Y. Madmoune, K. El Hasnaoui, A. Bendouch, H. Kaidi, M. Chahid, and M. Benhamou Laboratoire de Physique des Polym`eres et Ph´enom`enes Critiques Facult´e des Sciences Ben M’sik, P.O. 7955, Casablanca, Morocco [email protected]

abstract The system we consider is a fluid membrane confined to two parallel reflecting walls that are separated by a finite distance, L, assumed to be small in comparison to the bulk roughness. The attractive membrane is surrounded by small colloidal particles (nanoparticles). The purpose is the study of Brownian dynamics of these particles, under a change of a suitable parameter, such as temperature, T , or colloid-membrane interaction strength, w. The Brownian dynamics is investigated through the knowledge of the time particle density, which solves the Smoluchowski equation. Solving this equation around the midplane, where the essential of phenomenon occurs, we obtain the exact form of the local particle density, as a function of the perpendicular distance and time. In the derived expression, appears some time-scale, τ , which scales as τ ∼ L3 /w. This scale-time can be interpreted as the required time over which the colloidal suspension reaches their final equilibrium state. Also, τ can be regarded as the time-interval over which the particles are trapped in holes and valleys. Key words: Biomembranes - Nanoparticles - Confinement - Brownian dynamics.

I. INTRODUCTION

The biomembranes play a crucial role in life. Indeed, they separate the cell from the surrounding environment, and act as a selective barrier for the import and export of materials. These biological materials are complex systems, but they possess a natural structural organization, where each component has a specific function [1 − 7]. Nowadays, the scientific community recognizes that the cell membranes essentially present as a phospholipid bilayer combined with a variety of proteins and cholesterol. For example, the function of the cholesterol molecules is to ensure the bilayer fluidity. A phospholipid is commonly defined as an amphiphile molecule that is composed of a hydrophilic polar head attached to two hydrophobic (fatty acyl) chains. We note that the phospholipids move freely on the membrane surface. On the other hand, the thickness of the bilayer is of the order of 5 nanometers. This two facts allow to consider this bilayer as a two-dimensional fluid membrane. Experiment shows that the fluid membranes, self-assembled from surfactant solutions, may have a variety of shapes and topologies [8]. These forms have been theoretically explained in terms of bending energy [9, 10]. Usually, the biomembranes are not immersed in liquids of infinite extent, but they are rather confined to geometrical boundaries. Typical examples are provided by white and red globules or liposomes, as drugs

0

c a GNPHE publication 2010, [email protected] ⃝

91

Y. Madmoune et al.

African Journal Of Mathematical Physics Volume 8(2010)91-100

transport agents [11 − 14], in blood vessels. To obtain quantitative results, we consider the situation where the biomembrane is trapped in a liquid delimitated by two parallel reflecting walls, which are a finite distance, L, apart. By finite distance, we 0 mean that the separation L is much smaller than the bulk membrane mean-roughness, ξ⊥ (film geometry). The latter can be regarded as the typical size of humps caused by the thermal fluctuations of the 0 membrane. Of course, the scale ξ⊥ depends on the nature of lipid molecules forming the bilayer. The 0 condition L < ξ⊥ then ensures the confinement of the biomembrane. Such a condition is similar to that usually encountered in confined polymers context [15]. In real situations, the biomembranes are not pure, but they are in the presence of various entities, like proteins, small and macro-ions, or more complex structures [16]. For example, the membrane suspensions used in detergency and cosmetics are usually in contact with numerous additives (macromolecules and colloids), in order to improve their efficiency and to control their viscoelastic properties [17]. A simple way to study this influence consists in regarding these entities as small spherical colloids. This assumption has a physical sense only when one is concerned with those phenomena occurring at scales greater than the characteristic size of neighboring entities (diameter of particles, gyration radius of macromolecules, etc.). The organization of nanoparticles around a fluctuating fluid membrane, embedded in a liquid of infinite extent, is the subject of very recent theoretical works [18 − 20]. The question was addressed to statistical properties of the particles mediated by the membrane undulations. In particular, it was found that these undulations give rise to an aggregation of the beads in the vicinity of the fluid membrane. Such an aggregation is caused by the appearance of mutual attractive forces due to their contact with this membrane. Also, the attention has been paid to the investigation of the phase transition [18] that drives the colloids from a dispersed phase (gas) to a dense one (liquid). In a very recent work [21], one has studied the Brownian dynamics of nanoparticles of very low density, which are in contact with an interacting fluid membrane. The colloids and membrane were assumed to be trapped in a liquid of infinite extent. More precisely, the problem to solve was how these particles are pushed by the external potential towards the interface. The Brownian dynamics was studied through the time evolution of the particle density, when some suitable parameter, such as temperature, pressure, or membrane environment, is changed from an initial value to a final one. We recall that the Brownian motion governs various time-dependent phenomena ranging from suspensions [22 − 27] to polymer solutions [28]. This motion can be investigated using two approaches, namely the Smoluchowski equation solved by the distribution function and Langevin equation. Although the two theoretical formulations are different, but they are physically equivalent. The Smoluchowski equation that is a generalization of the usual diffusion equation, has a clear relevance to thermodynamics of irreversible processes. The Langevin equation, however, has no direct relationship to thermodynamics, but it provides a successful tool for the description of wider classes of stochastic processes. In this paper, the purpose is to extend the study to Brownian dynamics of colloidal particles in contact with an attractive fluid membrane, where the host liquid is delimitated by two impenetrable walls. More precisely, the question is how this dynamics can be affected by confinement. As we shall see below, this confinement induces drastic changes of the statistical properties of beads. To study the Brownian dynamics of a colloidal dispersion around a confined interacting fluid membrane, use is made of a theoretical approach based on the Smoluchowski equation. In fact, this equation describes the evolution of the particle density in time, and involves a known mean-force external potential experienced by the particles [19]. To simplify, the immersed particles are assumed to be point-like and of very low-density. The first assumption remains valid as long as we are concerned with strong membrane undulations, while the second means that the mutual interactions between particles can be ignored. Thus, the only remaining interaction is an external potential originating from the statistical fluctuations of this membrane. In the distance-range of interest, that is around the fluid membrane, we determine the exact form of the local particle density. The latter depends on the perpendicular distance z from the mid-plane located at z = 0, time t, and some characteristic time-scale τ ∼ L3 /w, where L is the separation between the confining walls and w is the colloid-membrane interaction strength. We emphasize that the time-scale τ can be regarded as the required time over which the beads are trapped in the new holes and valleys. This paper is organized as follows. In Sec. II, we present the essential of field theory allowing the calculation of a basic quantity that is the mean-force potential due to the membrane undulations. Brownian dynamics study is the aim of Sec. III. Some concluding remarks are drawn in the last section.

92

Y. Madmoune et al.

African Journal Of Mathematical Physics Volume 8(2010)91-100 II. THEORETICAL FORMULATION

Start with a fluctuating fluid membrane, free from particles, which is confined to two parallel reflecting walls 1 and 2. These are separated by a finite distance L that is assumed to be much smaller than 0 the bulk membrane roughness, ξ⊥ , when the system is unconfined (free membrane). The membrane is 0 confined only when the condition L < ξ⊥ is fulfilled. I n the Monge representation, a point on the membrane can be described by the three-dimensional position vector r = (x, y, z = h (x, y)), where h (x, y) ∈ [−L/2, L/2] is the height-field. The latter then fluctuates around the mid-plane located at z = 0. The Statistical Mechanics for the description of such a (tensionless) fluid membrane is based on the standard Canham-Helfrich Hamiltonian [9, 29] 1 H0 [h] = 2

∫

[ ] 2 dxdy κ (∆h) + µh2 ,

(2.0)

with the elastic constant [30] 2

µ=

9 (kB T ) . 4 κL4

(2.0)

Here, κ is the bending rigidity constant that is directly proportional to the thermal energy kB T . In fact, the term µh2 /2 describes a confinement potential that ensures the localization of the membrane around the mid-plane. Integral over the hole plane R2 of this term represents the loss entropy due to the confinement of the membrane. We recall that formula (2) of the elastic constant is compatible with the roughness expression, i.e. ⟨ ⟩ 1 kB T L2 2 2 ξ⊥ = h2 − ⟨h⟩ = √ = , 8 µκ 12

(2.0)

0 provided that one is in the confinement-regime where L << ξ⊥ . Such a quantity measures the fluctuations of the height-function (fluctuations amplitude) around the equilibrium plane located at z = 0. We recall that the result (3) was recently derived in Ref. [30]. Now, consider an assembly of N colloidal particles moving around a fluctuating fluid membrane. To simplify calculations, the particles are assumed to be point-like. In fact, this assumption makes sense √ only if the particle size is much smaller than the membrane roughness ξ⊥ = L/2 3. Typically, the considered particles have diameter of a few tens of nanometers, while the roughness is of the order of 1 micrometer. In addition, we suppose that there is no direct colloid–colloid interaction. This assumption is valid only when the colloidal dispersion is of very low-density. We recall that, in this paper, we are concerned with the influence of the membrane undulations on the particles movement. Of course, the primitive mutual interactions between nanoparticles should be taken into account when one is interested in their phase transition (colloidal aggregation) near an attractive fluid interface [19]. The total Hamiltonian describing physics of colloids and membrane reads [19]

H [h] = H0 [h] + Hcm [h] ,

(2.0)

where H0 [h] is the bare Hamiltonian defined in Eq. (1). In the above definition, Hcm accounts for the colloid-interface interaction, which is generally a complicated function of particles positions and configurations of the interface. For the sake of simplify, we suppose that the interaction Hcm is of contact type, and depends only on the relative perpendicular distances between particles and surface. Then, the proposed form is N w∑ Hcm [h] =− δ [zi − h (ρi )] . kB T 2 i=1

(2.0)

Here, δ is the one-dimensional Dirac function. The discrete sum is performed over all particles positions, ri = (ρi , zi ), 1 ≤ i ≤ N . In the above definition, w > 0 represents the surface coupling constant that 93

Y. Madmoune et al.

African Journal Of Mathematical Physics Volume 8(2010)91-100

measures the colloid-membrane interaction strength. In fact, w plays the role of an extrapolation length as usually encountered in Surface Critical Phenomena [31 − 33]. We note that the interaction magnitude w may be influenced by a change of temperature or membrane environment. In this model, we suppose that the attractive interface is penetrable and the colloids can accommodate on its both sides. As shown in Ref. [19], the membrane undulations give rise to one, two and more bodies interactions between beads. The exact calculations of these effective interactions were achieved taking advantage of the so-called cumulant method traditionally used in Statistical Field Theory [34, 35]. In this work, we focus our attention on those colloidal suspensions of very low-density (ideal gases), only. In these conditions, the mutual interactions between particles can be neglected. Therefore, the only remaining interaction is the attractive one-body interaction, U (z), which describes the direct potential between colloids and interface. Its expression was found to be [19] } { 6z 2 , (2.0) U (z) = U0 exp − 2 L with the negative amplitude √ U0 = −

3 w kB T . 2π L

(2.0)

The quantity |U0 | is the potential depth. Originally [19], the √ above relations explicitly incorporate the roughness ξ⊥ that we replaced by its expression : ξ⊥ = L/2 3. Let us discuss the above expression of the external potential felt by the nanoparticles. Firstly, in addition to the perpendicular distance z, the interaction potential naturally depends on the separation L between the reflecting walls, and the surface coupling constant w. Secondly, this one-body potential exhibits a minimum at the mid-plane z = 0. In addition, it is symmetric around this point. Thirdly, remark that the potential depth |U0 | depends on three kinds of parameters, which are the absolute temperature T , the surface coupling constant w, and the film thickness L. For instance, if T and w are fixed, the potential depth is inversely proportional to separation L. This means that the external potential experienced by beads has a significant magnitude only for those membranes confined to very narrow geometries. If T and L are now fixed to some values, the potential depth linearly increases with increasing surface coupling constant w. Fourthly, we emphasize that |U0 | must be small in comparison with the √thermal energy kB T . This implies that the coupling constant w is bounded from above, i.e. w < w∗ = 2π/3 × L ≃ 1. 4472 × L. Finally, as it should be, in the absence of the colloid-membrane interaction (w = 0), the one-body potential vanishes. The (reduced) external potential experienced by the nanoparticles, U (z) /kB T , is depicted in Fig. 1, upon the renormalized perpendicular distance z/L, for two values of the surface coupling constant w : w1 = 0.5 × L and w2 = 0.9 × L. As it should be, the curve drawn with parameter w2 is below that associated with w1 < w2 .

94

Y. Madmoune et al.

African Journal Of Mathematical Physics Volume 8(2010)91-100

FIG. 1. Reduced mean-force potential versus the renormalized perpendicular distance z/L, for two values of the surface coupling constant w : w1 = 0.5 × L and w2 = 0.9 × L.

The above expression of the one-body potential is the principal ingredient for the Brownian dynamics study of very low-density particles, which are located near a soft membrane. But, in order to facilitate calculations and get exact results, the above expression for the external √ potential must be simplified. Since the essential of phenomenon occurs in the interval |z| < ξ⊥ = L/2 3, such a potential reduces to [21] U (z) ≃ U0 + W (z) ,

z < L⊥ ,

(2.0)

with the harmonic potential 1 2 kz , 2

W (z) = where the elastic constant k is as follows U0 k = − 2 = 12 ξ⊥

√

3 w kB T > 0 . 2π L3

(2.0)

(2.0)

The above equality shows that the elastic constant k scales with separation L as : k ∼ L−3 . We note that the potential depth |U0 | has as effect to renormalize the density amplitude [21]. III. TIME EVOLUTION OF THE PARTICLE DENSITY

Consider, now, an assembly of colloidal particles moving around a fluctuating fluid membrane. Under a sudden change of a suitable parameter, such as temperature, pressure or membrane environment, the 95

Y. Madmoune et al.

African Journal Of Mathematical Physics Volume 8(2010)91-100

system is out equilibrium. We assume that the system is subject to Brownian dynamics by changing the colloid-membrane interaction strength w, but the temperature and film thickness remain fixed. This change may be caused by affecting the membrane environment. The problem can be studied through the local particle density, n (z, t). The latter represents the number of colloids per unit volume, at distance z and at time t. More precisely, we are interested in how this density evolves in time before the colloidal suspension reaches its final equilibrium state. For simplicity, we will neglect mutual interactions between nanoparticles. This hypothesis makes sense at least for small particle densities. Hence, the only interaction experienced by the beads is an external potential caused by the membrane undulations. Within the harmonic approximation, this potential is defined in Eqs. (9) and (10). To determine the time evolution of the local particle density, use will be made of the Smoluchowski equation [27, 28], which is a linear partial differential equation, of first order and second order, with respect to time and perpendicular distance, respectively. Before writing and solving this equation, we shall need some backgrounds. We first recall the expression of the equilibrium particle density } { U (z) neq (z) = A exp − , (3.0) kB T where A is a normalization constant and U (z) is the one-body potential due to the membrane undulations. If this potential is approximated by its harmonic form, the above definition becomes { } W (z) neq (z) = n0 exp − , (3.0) kB T where n0 is now the value of the particle density at the mid-plane z = 0. When the colloidal dispersion is out of equilibrium, in addition to distance, the density depends on time. This means that the nanoparticles execute Brownian dynamic but in the presence of the harmonic potential W (z). In order to compute this local density, we first recall that the Brownian diffusion is correctly described by the Fick’s law. The latter stipulates that the flux of matter, j (z, t), is directly proportional to the spatial gradient of density, that is j = −D

∂n 1 ∂W − , ∂z ζ ∂z

(3.0)

with D=

kB T ζ

(3.0)

the diffusion constant and ζ the friction coefficient, of which the inverse 1/ζ is the mobility. If we design by a the particle radius and by ηs the solvent viscosity, the friction coefficient ζ can be calculated from hydrodynamics [36] : ζ = 6πηs a. Equality (14) states that the diffusion constant characterizing the thermal motion is related to the quantity ζ, which expresses the response to an external field. Such an equality is a consequence of the well-known dissipation-fluctuation theorem [27, 28]. On the other hand, relation (13) must be combined with the local conservation law of matter ∂n ∂j + =0. ∂t ∂z Combining Eqs. (13) and (15) yields the Smoluchowski equation ( ) ∂n 1 ∂ ∂n ∂W = kB T +n ∂t ζ ∂z ∂z ∂z

(3.0)

(3.0)

solved by the local particle density n (z, t). At equilibrium, that is ∂n/∂t = 0, the above equation reduces to : kB T ∂n/∂z + n∂W/∂z = 0, whose solution is neq (z) = n0 exp exp {−W (z) /kB T }, which is nothing else but the density defined in Eq. (12). Replacing the harmonic potential W (z) by its explicit form (9) gives ∂ 2 n k ∂n k ∂n =D 2 + z + n. ∂t ∂z ζ ∂z ζ 96

(3.0)

Y. Madmoune et al.

African Journal Of Mathematical Physics Volume 8(2010)91-100

This new Smoluchowski equation must be supplemented by two boundary conditions, which are n (z, t = ∞) = nf (z) .

n (z, t = 0) = ni (z) ,

(3.0)

If the temperature T and separation L are fixed, the inial and final equilibrium particle densities, ni (z) and nf (z), are completely determined by the initial and final surface coupling constants wi and wf , respectively. Therefore, the dynamic is caused by a change of the membrane environment. Taking advantage of those mathematical techniques used in Ref. [21], we show that the solution to the Smoluchowski equation (17) is given by { ( t/τ )2 } [ ( )]−1/2 ∫ ∞ ze f − y −2t/τf ( ) [ni (y) − nf (y)] , n (z, t) = nf (z) + 2πDτf 1 − e dy exp − 2Dτf e2t/τf − 1 −∞ (3.0) where the initial and final equilibrium particle densities are given by { } { } ki z 2 z2 i i ni (z) = n0 exp − = n0 exp − , (3.0) 2kB T 2Dτi

nf (z) =

nf0

} } { { kf z 2 z2 f = n0 exp − , exp − 2kB T 2Dτf

(3.0)

with the time-scales τi and τf τi =

ζ 1 = ki 12

√

2π D−1 3 L , 3 wi

τf =

ζ 1 = kf 12

√

2π D−1 3 L , 3 wf

(3.0)

where wi and wf > wi are the initial and final surface coupling constants. This means that the colloidmembrane interaction is suddenly increased from wi to wf . The above relations suggest that the times τi and τf depend on the colloid-membrane interaction and film thickness. In particular, the time-scale τf may be interpreted as the time beyond which the colloidal system reaches its final equilibrium state. Then, a weak colloid-membrane interaction necessitates a great time before the colloidal system tends to its final state. In fact, τf has another physical meaning, and can be regarded as the required time over which the particles are trapped in new holes and valleys. Now, after integration over the y variable, in Eq. (19), we obtain a closer form for the time particle density { } [ ( )]−1/2 1 z2 i −2t/τf ( ) n (z, t) = n0 1 + η e −1 exp − , (3.0) 1 + η e−2t/τf − 1 2Dτi with the reduced time-shift η=

τi − τf wf − wi = >0. τi wf

(3.0)

The quantity η then represents the relative shift of the colloid-membrane interaction strengths wi and wf . The density amplitude, in Eq. (23), was obtained using the matter conservation law ∫

∫

+∞

−∞

ni (z) dz =

+∞ −∞

nf (z) dz ≡ Γ .

(3.0)

Here, Γ represents the adsorbance, which is defined as the total number of colloids (per unit area) located near membrane. Combining Eqs. (20), (21) and (25) yields the relationship √ √ 2πDτf nf0 = 2πDτi ni0 = Γ . (3.0) Let us comment the density expression (23). First, we note that, it is easy to see that the solution (23) satisfies the two boundary conditions (18). 97

Y. Madmoune et al.

African Journal Of Mathematical Physics Volume 8(2010)91-100

The initial and final equilibrium states are defined in Eqs. (20) and (21). Second, when it is reduced by√ ni0 , the time particle density depends on three dimensionless factors, namely the renormalized distance z/ 2Dτi , the time-ratio t/τf and the time-shift η = (τi − τf ) /τi . Therefore, all microscopic details (colloid-membrane interaction) are entirely contained in τi and τf . Finally, we emphasize that the time particle density curve exhibits a maximum at z = 0, and it is symmetric around this point, whatever be the values of t/τf and η. We√depict, in Fig. 2, the reduced time particle density n (z, t) /ni0 versus the renormalized distance z/ 2Dτi , choosing three values of the time-ratio t/τf : 0, 0.5, and ∞. The former corresponds to the initial state and the second to the final one. These curves are drawn with parameter η = 0.5. This value means that the final surface coupling constant wf is two times more important than the initial one wi , that is wf = 2wi .

√ FIG. 2. Reduced local particle density, n (z, t) /ni0 , versus the renormalized distance z/ 2Dτi , with three values of the time-ratio t/τf : 0 (dashed line), 0.5 (solid line), ∞ (line in dots). These curves are drawn choosing the value η = 0.5 (wf = 2wi ).

IV. CONCLUSIONS

This work is dedicated to the Brownian dynamics study of small colloidal particles in contact with an attractive penetrable fluid membrane. The host liquid was assumed to be delimitated by two parallel reflecting walls, which are a finite distance apart. The membrane surrounded by beads is confined only if the film thickness is much smaller than the bulk membrane mean-roughness. Physics was discussed in terms of three relevant parameters, which are the absolute temperature, T , the separation between walls, L, and colloid-membrane interaction strength, w. In our study, we have fixed the temperature and film thickness to some values, and varied the surface coupling constant. This can be experimentally achieved modifying the membrane environment. For the present study, we have started from three hypothesizes : (1) the particles are point-like, (2) they are of very low-density (in order to forget their mutual interactions), and (3) strongly interact with the membrane. To achieve the investigation of the Brownian dynamics, use was made of a theoretical formalism based on the Smoluchowski equation. The latter is solved by the time particle density we were interested in. We have exactly computed this physical quantity, around the mid-plane of the film, where the essential of phenomenon occurs. Within this distance-domain, the mean-force external potential was approximated 98

Y. Madmoune et al.

African Journal Of Mathematical Physics Volume 8(2010)91-100

by an harmonic one. This means that we were in the presence of Brownian particles moving in an harmonic potential, that is, √in addition to the usual diffusion, these experience small oscillations with a frequency ν scaling as ν ∼ w/L3 . Hence, the harmonic approximation used for the mean-force potential is largely justified for those fluid interfaces of small enough colloid-membrane interaction strength. As we have shown, the time particle density depends on a time-scale, τ , scaling as τ ∼ L3 /w. We have interpreted this scale-time as the required time over which the nanoparticles reach their final equilibrium state. Also, τ can be regarded as the time-interval where the particles are trapped in holes and valleys of size slightly smaller than the film thickness L. 0 We note that, when the separation L is much greater than the bulk roughness ξ⊥ , in addition to the above evoked parameters, the physical phenomenon depends on the specific membrane characteristic (via the bending rigidity constant κ). In this case, the Brownian dynamics can be caused by a change of the parameter κ. This situation has less physical interest, since, in this case, finite size effects contribute to the leading behavior only by exponentially small corrections. As last word, we emphasize that the results derived in this paper may be extended to bilayer surfactants, although the two systems are not of the same structure and composition.

ACKNOWLEDGMENTS

We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the ”First International Workshop On Soft-Condensed Matter Physics and Biological Systems”, 14-17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank the Professor C. Misbah for fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kinds of hospitalities during his regular visits.

REFERENCES

1

M.S. Bretscher and S. Munro, Science 261, 12801281 (1993). J. Dai and M.P. Sheetz, Meth. Cell Biol. 55, 157171 (1998). 3 M. Edidin, Curr. Opin. Struc. Biol. 7, 528532 (1997). 4 C.R. Hackenbrock, Trends Biochem. Sci. 6, 151154 (1981). 5 C. Tanford, The Hydrophobic Effect, 2d ed., Wiley, 1980. This book includes, in addition, a good discussion about the interactions of proteins and membranes. 6 D.E. Vance and J. Vance, eds., Biochemistry of Lipids, Lipoproteins, and Membranes, Elsevier, 1996. 7 F. Zhang, G.M. Lee, and K. Jacobson, BioEssays 15, 579588 (1993). 8 S. Safran, Statistical Thermodynamics of Surfaces, Interfaces and Membranes, Addison-Wesley, Reading, MA, 1994. 9 W. Helfrich, Z. Naturforsch. 28c, 693 (1973). 10 U. Seifert, Advances in Physics 46, 13 (1997). 11 H. Ringsdorf and B. Schmidt, How to Bridge the Gap Between Membrane, Biology and Polymers Science, P.M. Bungay et al., eds, Synthetic Membranes : Science, Engineering and Applications, p. 701, D. Reiidel Pulishing Compagny, 1986. 12 D.D. Lasic, American Scientist 80, 250 (1992). 13 V.P. Torchilin, Effect of Polymers Attached to the Lipid Head Groups on Properties of Liposomes, D.D. Lasic and Y. Barenholz, eds, Handbook of Nonmedical Applications of Liposomes, Volume 1, p. 263, RCC Press, Boca Raton, 1996. 14 R. Joannic, L. Auvray, and D.D. Lasic, Phys. Rev. Lett. 78, 3402 (1997). 15 P.-G. de Gennes, Scaling Concept in Polymer Physics, Cornell University Press, 1979. 16 C. Fradin, A. Abu-Arish, R. Granek, and M. Elbaum, Biophys. J. 84, 2005 (2003). 17 D.F. Evans and H. Wennerstr¨ om, The Colloidal Domain, Wiley, New-York, 1999. 18 T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). 2

99

Y. Madmoune et al.

African Journal Of Mathematical Physics Volume 8(2010)91-100

19

H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005) A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1 (2006). 21 A. Bendouch, M. Benhamou, and H. Kaidi, Elec. J. Theor. Phys. 5, 215 (2008). 22 A. Einstein, Ann. Physik. 17, 549 (1905) ; 19, 371 (1906); see, also, Investigation on the Theory of the Brownian Movement, E.P. Dutton and Copy Inc, New York, 1926. 23 N. Wax, Noise and Stochastic Processes, Dover Publishing Co., New York, 1954. 24 M. Lax, Rev. Mod. Phys. 32, 25 (1960) ; 38, 541 (1966). 25 R. Kubo, Rep. Prog. Phys. 29, 255 (1966). 26 C.W. Gardiner, Handbook of Stochastic Methods, 3d ed., Springer, 2004. 27 H. Risken and T. Frank, The Focker-Planck Equation : Methods of Solutions and Applications, 2d ed., Springer, 1989. 28 M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press Oxford, 1986. 29 P.B. Canham, J. Theoret. Biol. 26, 61 (1970). 30 K. El Hasnaoui, Y. Madmoune, H. Kaidi, M. Chahid, and M. Benhamou, Induced forces in confined biomembranes, submitted for publication. 31 K. Binder, in : Phase Transitions and Critical Phenomena, Vol. 8, edited by C. Domb and J.L. Lebowitz, Academic Press, London, 1983. 32 H.W. Diehl, in : Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 10, Academic Press, London, 1986. 33 S. Dietrich, in : Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 12, Academic Press, London, 1988. 34 C. Itzykson and J.M. Drouffe, Statistical Field Theory : 1 and 2, Cambridge University Press, 1989. 35 J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. 36 G.K. Batchelor, An Introduction to Fluid Dynamics, Chap. 4, Cambridge University Press, 1970. 20

100

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright

Author's personal copy Physica A 389 (2010) 3465–3475

Contents lists available at ScienceDirect

Physica A journal homepage: www.elsevier.com/locate/physa

Statistical mechanics of bilayer membranes in troubled aqueous media M. Benhamou ∗ , K. Elhasnaoui, H. Kaidi, M. Chahid Laboratoire de Physique des Polymères et Phénomènes Critiques, Faculté des Sciences Ben M’sik, P.O. 7955, Casablanca, Morocco

article

info

Article history: Received 30 March 2010 Available online 14 April 2010 Keywords: Bilayer membranes Vesicles Impurities Equilibrium Statistical mechanics

abstract We consider a bilayer membrane surrounded by small impurities, assumed to be attractive or repulsive. The purpose is a quantitative study of the effects of these impurities on the statistical properties of the supported membrane. Using the replica trick combined with a variational method, we compute the membrane mean-roughness and the height correlation function for almost-flat membranes, as functions of the primitive elastic constants of the membrane and some parameter that is proportional to the volume fraction of impurities and their interaction strength. As results, the attractive impurities increase the shape fluctuations due to the membrane undulations, while repulsive ones suppress these fluctuations. Second, we compute the equilibrium diameter of (spherical) vesicles surrounded by small random particles starting from the curvature equation. Third, the study is extended to a lamellar phase composed of two parallel fluid membranes, which are separated by a finite distance. This lamellar phase undergoes an unbinding transition. We demonstrate that the attractive impurities increase the unbinding critical temperature, while repulsive ones decrease this temperature. Finally, we say that the presence of small impurities in an aqueous medium may be a mechanism to suppress or to produce an unbinding transition, even the temperature and polarizability of the aqueous medium are fixed, in lamellar phases formed by parallel lipid bilayers. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Usually, the aqueous media supporting biological membranes are assumed to be homogeneous. Actually, any real system inevitably contains impurities. Under well-controlled conditions, the particles can be removed from the surrounding medium. But, if these entities are present, it is also interesting to study their effect on the statistical properties of the biomembranes, such as fluctuations’ spectrum and dynamical behavior. In general, random inhomogeneities tend to disorder the system. It is important to make a distinction between annealed and quenched disorders. The former is used when impurities and host constituents (phospholipids) are in equilibrium [1]. This means that their respective mobilities are comparable. If it is not the case, that is host constituents and impurities are out of equilibrium, the disorder is rather quenched [1]. When the Statistical Mechanics is used, the latter consists to trace over all membrane undulations, before performing the summation over the impurities’ disorder. Although the quenched disorder is harder to analyze, it remains more realistic than its annealed counterpart. Indeed, the thermal and the noise averaging have very different roles. In this paper, the physical system we consider is a fluid membrane (flat or closed) trapped in a troubled aqueous medium. The latter is impregnated by a weak amount of impurities that may be attractive or repulsive regarding the membrane. The aim is to show how these entities can modify the statistical properties of the fluid membrane. These properties will be studied through the fluctuations’ amplitude. To model the system, we suppose that the impurities act as a random external potential with a Gaussian distribution (uncorrelated disorder). In addition, we suppose that the disorder is quenched. To

∗

Corresponding author. E-mail address: [email protected] (M. Benhamou).

0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.03.049

Author's personal copy 3466

M. Benhamou et al. / Physica A 389 (2010) 3465–3475

do calculations, the replica theory [2,3] that is based on a mathematical analytical continuation, usually encountered in Quantum Mechanics [4] and Critical Phenomena will be made use of [5,6]. Our finding are as follows. First, using the above evoked theory, we compute the mean-roughness of an almost-flat membrane, as a function of the primitive parameters of the pure membrane (free from impurities) and a certain parameter depending on the volume fraction of impurities and their interaction strength. The main conclusion is that, attractive impurities increase the shape fluctuations due to the thermal undulations, while repulsive ones tend to suppress these fluctuations and then lead to a strong membrane confinement. Second, we analyze the impurities effects on the equilibrium shape of closed vesicles solving the curvature equation (for spherical vesicles). We show that the vesicle is more stable in the presence of repulsive impurities, in comparison with attractive ones. Thereafter, the study is extended to lamellar phases formed by two parallel fluid membranes. We demonstrate that the presence of random impurities drastically affects the physical properties of the lamellar phase, in particular, the unbinding transition driving the system from a bind state to a state where the two membranes are completely separated. This paper is organized as follows. In Section 2, we describe the fundamentals of the used model. Section 3 deals with the computation of the fluctuations amplitude of a single almost-flat fluid membrane, surrounded by attractive or repulsive impurities. We compute, in Section 4, the equilibrium diameter of a closed vesicle in the presence of impurities. Extension of study to lamellar phases is the aim of Section 5. Finally, some concluding remarks are drawn in the last section. 2. Effective field theory Consider a fluctuating fluid membrane embedded in a three-dimensional liquid surrounded by very small impurities. For the sake of simplicity, we suppose that the impurities are point like. Within the framework of the Monge representation, a point on the membrane can be described by the position-vector (r , z = h (r )), where r = (x, y) ∈ R2 is the transverse vector and h (r ) is the height function. The Statistical Mechanics of fluid membranes free from impurities is based on the Canham–Helfrich Hamiltonian [7]

Z

d2 r

H0 [h] =

hκ 2

(∆h)2 +

µ 2

i

h2 ,

(1)

with κ the membrane bending modulus. The confinement energy (per unit area) µh2 /2 (µ > 0) is responsible for the localization of the membrane in some region of the Euclidean space, where it fluctuates around an equilibrium plane located at h = 0. Therefore, the height h takes either positive and negative values. For simplicity, the membrane is assumed to be tensionless. In fact, this assumption does not change conclusions made below. To model the impurities effects on the statistical properties of the system, we suppose that these tend to reinforce the membrane confinement, if they are repulsive, or to render this membrane more free, if the particles are rather attractive. These tendencies can be explained assuming that the confinement parameter is local in space and making the substitution

µ → µ + V (r ) ,

(2)

in the above Hamiltonian, where the variation V (r ) can be regarded as a random external potential. To simplify, the corresponding probability distribution is supposed to be Gaussian (uncorrelated disorder), that is V (r ) = 0,

V (r ) V (r 0 ) = −vδ2 r − r 0 .




(3)

Here, −v is a positive constant proportional to both concentration of impurities and strength of their interaction potential, and δ2 (r ) denotes the two-dimensional Dirac distribution. Therefore, the new effective Canham–Helfrich Hamiltonian reads

Z H [h] =

d2 r



κ 2

(∆h)2 +

1 2

 (µ + V (r )) h2 ,

(4)

for attractive impurities. For repulsive ones, V (r ) must be replaced by iV (r ), with i2 = −1. Since the disorder distribution is Gaussian, all its odd moments vanish, but the even ones do not. This implies that all physical quantities, calculated with the pure imaginary potential iV , are entire numbers. Since the impurities and membranes are not in equilibrium, the disorder is rather quenched, that is we have to average not the partition function, Z , but its logarithm, ln Z . The latter defines the free energy. For a given (quenched) configuration of impurities, the partition function is

Z Z =

D he−A[h] ,

(5)

with the action A [h] =

H [h] kB T

Z =

2

d r



b κ 2

(∆h) + 2

1 2


2 b b µ + V (r ) h .

(6)

Author's personal copy M. Benhamou et al. / Physica A 389 (2010) 3465–3475

3467

We have used the notations

b κ=

κ kB T

,

µ

b µ=

kB T

V b V = .

,

(7)

kB T

In term of the reduced impurities potential b V , the second disorder law in Eq. (3) becomes: b V (r ) b V (r 0 ) = −b vδ r − r 0 , with




b v = v/ (kB T )2 . A simple dimensional analysis shows that: [b κ ] = L0 , [b µ] = L−4 and [b v ] = L−6 , where L is some length that may be the membrane thickness. The main quantity to consider is the average of the logarithm of the partition function, ln Z , over disorder. To compute such a quantity, we use the replica trick [2]. This method consists to formally write Zn − 1

ln Z = lim

n

n→0

.

(8)

After performing the average over disorder, we get

Z Zn

=

D h1 . . . D hn e−A[h1 ,...,hn ] ,

(9)

with the n-replicated effective action A [h1 , . . . , hn ] =

Z

" d2 r

n b κX

2 α=1

(∆hα )2 +

n b µX

2 α=1

h2α +

n b v X

8

α=1

! h2α

n X

β=1

!# h2β

,

(10)

where Greek indices denote replicas. The last term introduces an additional coupling constant b v < 0, which is directly responsible for the effective interaction between replicas due to the presence of impurities. The above action describes attractive impurities. For repulsive ones, the coupling b v must be replaced by −b v . The additive quartic term in the above action means that the presence of impurities is accompanied by an increase of entropy, when these are attractive, and by an entropy loss, if they are rather repulsive. The following paragraph is devoted to a quantitative determination of the fluctuations spectrum in the presence of impurities. 3. Single almost-flat membrane We start by considering the attractive impurities problem. Of course, the functional integral (9) cannot be exactly computed. One way is the use of a variational method. To this end, we consider a bilayer membrane in the presence of the following bare action Aη [h1 , . . . , hn ] =

Z

" d2 r

n b κX

2 α=1

(∆hα )2 +

n b ηX

2 α=1

# h2α

,

(11)

with the new confinement coupling b η > 0. With this action, the partition function, Zη , is exact. We have Zη = (Z0 )n ,

(12)

with

Z Z0 =

 Z   b κ b η 2 2 2 D h exp − d r . (∆h) + h 2

(13)

2

The latter accounts for the usual partition function of a bilayer membrane free from impurities. Introduce the mean-value of a functional X [h1 , . . . , hn ], calculated with the bare action Aη [h1 , . . . , hn ],

hX i0 =

1

Z

D h1 . . . D hn X [h1 , . . . , hn ] exp −Aη [h1 , . . . , hn ] .



Zη



(14)

In term of this mean-value, the averaged partition function may write





Z n = Zη exp − A − Aη


 0

.

(15)

Using the standard inequality

X e

0

≥ e hX i0 ,

(16)

we get



Z n ≥ Zη exp − A − Aη

 0

.

(17)

Author's personal copy 3468

M. Benhamou et al. / Physica A 389 (2010) 3465–3475

This implies that



1 − Zn

− ln Z = lim

≤ lim

n

n→0

1 − Zη exp − A − Aη

 .

0

n

n→0

(18)

On the other hand, we have



A − Aη

"

Z



d2 r

=

0

n

2 b µ −b ηX

2

α=1

hα

+

0

n X n

b vX

8 α=1 β=1

#  2

h2α hβ

0

.

(19)

It is easy to see that n X

2

α=1

hα

n X n X

= nσ 2 ,

0

α=1 β=1

h2α h2β

 0

= n (n + 2) σ 2


2

,

(19a)

with the squared membrane roughness 1

σ 2 (η) =

1

p . 8 b κb η

(20)

With these considerations,

− ln Z ≤ − ln Z0 + Σ



b µ −b η 2

σ2 +

b v 4

σ2


2



,

(21)

where Σ is the area of the reference plane. Notice that the right-hand side of this inequality is b η-dependent function. Therefore, a good approximate value of −ln Z is



− ln Z



 MF

= min − ln Z0 + Σ η



b µ −b η 2

σ + 2

b v 4

σ

2 2






.

(22)

(The subscript MF is for mean-field theory). The variational parameter b η is such that

   ∂ b µ −b η 2 b v 2
2 − ln Z0 + Σ σ + σ = 0. ∂b η 2 4

(23)

Combining the relationship

σ2 = −

2 ∂ ln Z0 Σ ∂b η

(24)

and Eq. (20) yields the minimum value of b η, which satisfies the following implicit relation

b η=b µ +b vσ 2 .

(25)

The squared membrane roughness, σ 2 , is then given by the parametric equations

 σ 2 = 1 p1 , 8 b κb η  b η=b µ +b vσ 2 .

(26)

Eliminating b η between these equations gives the implicit relation

σ2 =

1

1

q
. 8 b κ b µ +b vσ 2

(27)

p

Introduce the usual squared roughness of a membrane free from impurities, σ02 = 1/8 b κb µ. Then, we have

σ 2 = σ02 p

1 1 +b vσ 2 /b µ

.

(28)

The latter may be rewritten, in terms of dimensionless variables σ 2 /σ02 , b κ , and w = −64b κb v σ02

σ2 1 = q , 2 σ0 1 − wσ 2 /σ02

(attractive impurities) .


3

> 0, as (29)

Author's personal copy M. Benhamou et al. / Physica A 389 (2010) 3465–3475

3469

Table 1 Some values of the coupling w and their corresponding ratios σ 2 /σ02 , for attractive impurities.

w

σ 2 /σ02

0.1 0.15 0.2 0.25 0.3

1.0575 1.0937 1.1378 1.1939 1.2715 √ 3

w∗

Notice that the coupling constant w > 0 is directly proportional to the volume fraction of impurities and their interaction strength. Let us comment about the obtained result. Firstly, this makes sense only when σ 2 /σ02 < 1/w . This condition is fulfilled for very weak disorders (w is small enough). Secondly, since w is positive definite, we have: σ 2 > σ02 . A comparison between this inequality and that just evoked above implies that 0 < w < 1. Thirdly, if we set x = σ 2 /σ02 , the above relation can be rewritten as

w=

x2 − 1 x3

.

(30)

Therefore, w is a direct function of x (Fig. 1). This formula may be experimentally used to estimate the impurities coupling w, knowing the experimental value of the ratio σ 2 /σ02 . If w is fixed to some value, the x-variable then solves the following third-degree algebraic equation

w x3 − x2 + 1 = 0 .

(31)

The existence of√its roots then depends on the value of the renormalized coupling w . We show that there exists a typical value w ∗ = 2/3 3 of the coupling w , such that: (1) For w > w ∗ (strong disorder), we have only one root that is negative ; √ (2) For w = w ∗ (medium disorder), we have one negative root and a positive unique root, which is x = σ 2 /σ02 = 3. Then, the maximal value of the membrane roughness is σ ∗ = 31/4 σ0 . This constraint means that the lipid bilayer cannot support undulations of perpendicular size (hunt size) greater than σ ∗ . This is realized if only if the impurities volume fraction φ is below some value φ ∗ , directly proportional to the threshold w ∗ ; (3) For w < w ∗ (weak disorder), we have one negative root andtwo positive ones. Only one positive root is acceptable (the  smallest one). At this particular root, the free energy −kB T ln Z MF is minimal. We show that the absolute minimum satisfies the inequality

− 4wx6 + 3x4 − 1 > 0.

(32)

In this case, the ratio σ /σ can be obtained by numerically solving Eq. (31). We report in Table 1 some values of the impurities coupling w and the corresponding dimensionless squared membrane roughness σ 2 /σ02 . It is easy to see that, the ratio σ 2 /σ02 increases with increasing coupling w , provided that w is in the interval w < w ∗ . 2

2 0

Now, for repulsive impurities, the fundamental relationship (29) is replaced by

σ2 1 = q , 2 σ0 1 − wσ 2 /σ02

(repulsive impurities) ,

(33)

or equivalently

− wx3 − x2 + 1 = 0,

(34)

with the stability condition 4w x6 + 3x4 − 1 > 0.

(35)

The equality (34) can be transformed into the following direct function (Fig. 2)

w=

x2 − 1 x3

,

(36)

with x < 1 and w < 0. As it should be, the membrane roughness is reduced by the presence of repulsive impurities, that is σ 2 < σ02 , and in addition, σ 2 decreases with increasing impurities coupling −w . As a matter of fact, the effect of these particles is to reinforce

Author's personal copy 3470

M. Benhamou et al. / Physica A 389 (2010) 3465–3475

Fig. 1. Impurities’ coupling w versus the ratio σ 2 /σ02 , for attractive impurities.

Fig. 2. Impurities’ coupling w versus the ratio σ 2 /σ02 , for repulsive impurities.

Table 2 Some values of the coupling w and their corresponding ratios σ 2 /σ02 , for attractive impurities.

w

σ 2 /σ02

−0.1 −0.15 −0.2 −0.25 −0.3 −0.4

0.9554 0.9364 0.9190 0.9032 0.8885 0.8622

the confinement of the considered fluid membrane. Therefore, a bilayer membrane collapses, when it is trapped in a troubled aqueous media with repulsive impurities. In this case, it is easy to see that the algebraic equation has only one positive root. In Table 2, we give some values of the coupling w and the corresponding ratio σ 2 /σ02 .


 = h (r ) h r 0 − hh (r )i h r . The latter measures the fluctuations of the height function h around its mean-value hhi. Here, h.i denotes Another interesting physical quantity is the (connected) height correlation function: G r − r 0




 0

the thermal expectation mean-value, which must not be confused with average over disorder.




Author's personal copy M. Benhamou et al. / Physica A 389 (2010) 3465–3475

3471

To compute the correlation functions, we start from the generating functional



Z Z [J] =

 d r J (r ) h (r ) ,

Z

2

D h exp −A [h] +

(37)

with the action A [h] = H [h] /kB T , where H [h] is the effective Canham–Helfrich Hamiltonian defined in Eq. (4). Here, J is an auxiliary source coupled to the field h. Functional derivatives of Z [J] with respect to source J give all height correlation functions, in particular, the propagator G r −r

0

δ 2 ln Z [J] = δ J (r ) δ J (r 0 )




.

(38)

J =0

This definition indicates that the main object to compute is the averaged connected generating functional ln Z [J]. To compute the functional ln Z [J], as before, we use the replica method based on the limit ln Z [J] = lim

Z n [J] − 1 n

n→0

,

(39)

with

(

Z

D h1 . . . D hn exp −A [h1 , . . . , hn ] +

Z n [J] =

n Z X

) d2 r J (r ) hi (r ) ,

(40)

i=1

where A [h1 , . . . , hn ] is the action, relation (10). Using the standard cumulant method [5,6] based on the approximate formula




hexp {X }i0 = exp hX i0 + (1/2!) X 2 0 − hX i20 + · · · ,

(41)

we find that



Z n [J] ' Zη exp − A − Aη



 Z  Z
0
n 2 2 0 0 + · · · exp d r d r J r G r − r J r , ( ) 0 0

(42)

2

with the usual bare propagator G0 r − r

0




Z =

0 d2 q eiq.(r −r )

κ q4 + b η (2π )2 b

.

(43)

In Eq. (42), we have ignored high-order terms in J that do not contribute to the propagator. After performing the limit n = 0, we find that the averaged connected generating functional reads ln Z [J] ' − ln Z0 + Σ



b µ −b η 2

σ2 +

b v 4

σ2


2

 +

1 2

Z

d2 r

Z

d2 r 0 J (r ) G0 r − r 0 J r 0 .







(44)

By simple functional derivation, we obtain the expected propagator G r −r

0




= G0 r − r

0




Z =

0 d2 q eiq.(r −r )

κ q4 + b η (2π )2 b

.

(45)

Then, the expected propagator identifies with the bare one. Here, the variational parameter b η satisfies the implicit equation (25). We then recover the relationship: σ 2 = G (0), where σ 2 is the squared membrane roughness computed above. The next step consists to extend the study to a closed vesicle surrounded by attractive or repulsive impurities. 4. Single vesicle We start by recalling some basic backgrounds dealt with the equilibrium shape of (spherical) vesicles, which may studied using Differential Geometry techniques. The vesicle is essentially formed by two adjacent monolayers (inner and outer) that are formed by amphiphile lipid molecules. These permanently diffuse with the molecules of the surrounded aqueous medium. Such a diffusion then provokes thermal fluctuations (undulations) of the membrane. This means that the latter experiences fluctuations around an equilibrium plane. Consider a biomembrane of arbitrary topology. A point of this membrane (surface) can be described by two local coordinates (u1 , u2 ). At each point of the surface, there exists two particular curvatures (minimal and maximal), called

Author's personal copy 3472

M. Benhamou et al. / Physica A 389 (2010) 3465–3475

principal curvatures, denoted C1 = 1/R1 and C2 = 1/R2 . The quantities R1 and R2 are the principal curvature radii. With the help of the principal curvatures, one constructs two invariants that are the mean-curvature 1

(C1 + C2 ) , 2 and the Gauss curvature C =

(46)

K = C1 C2 .

(47)

The principal curvatures C1 and C2 are nothing else but the eigenvalues of the curvature tensor [8]. To comprehend the geometrical and physical properties of the biomembranes, one needs a good model. The widely accepted one is the fluid mosaic model proposed by Singer and Nicolson in 1972 [9]. This model consists to regard the cell membrane as a lipid bilayer, where the lipid molecules can move freely in the membrane surface like a fluid, while the proteins and other amphiphile molecules (cholesterol, sugar molecules, . . . ) are simply embedded in the lipid bilayer. We note that the elasticity of cell membranes crucially depends on the bilayers in this model. The elastic properties of bilayer biomembranes were first studied, in 1973, by Helfrich [7]. The author recognized that the lipid bilayer could be regarded as smectic-A liquid crystals at room temperature, and proposed the following curvature free energy (without impurities) F =

κ

Z

(2C )2 dA + κG

2

Z

Z K dA +

γ dA + p

Z

dV .

(48)

where dA denotes the area element, and V is the volume enclosed within the lipid bilayer. In the above definition, κ accounts for the bending rigidity constant, κG for the Gaussian curvature, γ for the surface tension, and p for the pressure difference between the outer and inner sides of the vesicle. The spontaneous curvature is ignored. The first-order variation gives the shape equation of lipid vesicles [10] p − 2γ C + 4κ C C 2 − K + κ∇ 2 (2C ) = 0,




(49)

with the surface Laplace–Bertolami operator

∂ ∇ = √ g ∂ ui



1

2

∂ gg ∂ uj

√

ij



,

(50)




where g ij is the metric tensor on the surface and g = det g ij . For open or tension-line vesicles, local differential equation (49) must be supplemented by additional boundary √ conditions we do not write [11]. The above equation have known three analytic solutions corresponding to sphere [10], 2-torus [12–15] and biconcave disk [16]. For spherical vesicles, the solution to the above curvature equation is exact, and we find that the equilibrium radius is R0 =

2γ p

.

(51)

Now, assume that the vesicle is trapped in a troubled aqueous medium. Usually, to take into account the presence of impurities, a low-order coupling between their volume fraction, φ , and the mean-curvature C is added to the above free energy F , that is

Z F → F ± kB T

φ C dA.

(52)

The positive sign is for attractive impurities and the negative sign for repulsive ones. The new free energy is then F =

κ

Z

(2C ± 2C0 ) dA + κG 2

2

Z

Z K dA +

b γ dA + p

Z dV

(53)

with the notation C0 = k B T

φ , 4κ

b γ = γ − 2κ C02 .

(54)

Then, the impurities generate an extra spontaneous curvature. This means that these give arise to an asymmetry of the vesicle when its membrane is crossed. In addition, these impurities additively renormalize the interfacial tension coefficient. Minimizing this new curvature free energy yields p − 2b γ C + κ (2C ± C0 ) 2C 2 ∓ C0 C − 2K + κ∇ 2 (2C ) = 0.




(55) 2

For a spherical vesicle, the mean curvature is a constant, and we have K = C . In this case, the mean-curvature C is a root of a polynomial of degree 2. This means that we have an exact solution we do not write. In particular, for very small volume fractions (φ  1), we find that the vesicle equilibrium radius writes R R0

=1±

pkB T 8γ

2


φ + O φ2 ,

(56)

Author's personal copy M. Benhamou et al. / Physica A 389 (2010) 3465–3475

3473

or

R − R0 pkB T R ' 8γ 2 φ. 0

(57)

The signs (+) and (−) describe attractive and repulsive impurities, respectively. Here, R0 = 2γ /p denotes the equilibrium radius for a spherical vesicle free from impurities. The above results call the following remarks. Firstly, as it should be, the equilibrium radius of the vesicle increases with increasing volume fraction of impurities, when these are attractive. This means that the vesicle is swollen, but it radius remains close to the unperturbed one R0 , as long as the volume fraction of impurities is very small. For repulsive impurities, however, the vesicle is collapsed. Secondly, formula (57) makes sense only when the volume fraction φ is below some threshold φ ∗ = 8γ 2 /kB Tp. Since the latter must be small√in comparison with unity, the vesicle bears interfacial tensions of coefficients that do not exceed some typical value γ ∗ = kB Tp/8. Finally, at fixed volume fraction φ < φ ∗ , this formula may be used to estimate the experimental value of the ratio p/γ 2 , by a simple measurement of the equilibrium radii R (with impurities) and R0 (without impurities). The following paragraph will be devoted to the study of lamellar phases made of two parallel lipid bilayers. 5. Lamellar phases The natural question is the extension of the study to the case when we have more than one membrane immersed in a aqueous media, which is impregnated by small random impurities. In our analysis, we start from a lamellar phase composed of two parallel (neutral) fluid membranes free from impurities. The effects of these entities on physics will be discussed below. The cohesion between these bilayer membranes is ensured by long-ranged attractive van der Waals forces [17], which are balanced, at short membrane separation, by strong repulsion coming from hydration forces [18] and by steric shape fluctuations ones resulting from the membrane undulations [7]. For two parallel bilayer membranes separated by a finite distance l, the total interaction energy per unit area is V (l) = VH (l) + VW (l) + VS (l) .

(58)

The first part VH (l) = AH e−l/λH

(59)

represents the hydration potential (per unit area) that acts at small separations of the order of 1 nm. The corresponding amplitude AH and potential-range λH are about AH ' 0.2 J/m2 and λH ' 0.3 nm. The second part VW (l) = −

W 12π



1 l2

−

2

(l + δ)2

+

1

(l + 2δ)2

 (60)

is the attractive van der Waals potential (per unit area) that originates from polarizabilities of lipid molecules and water molecules. Here, W accounts for the Hamaker constant that is in the range W ' 10−22 − 10−21 J, and δ for the bilayer thickness. The latter is of the order of δ ' 4 mm. The last part is the steric shape fluctuations potential (per unit area) [7] Vs (l) = cH

(kB T )2 , κ l2

(61)

with kB the Boltzmann’s constant, T the absolute temperature, and κ the common bending rigidity constant of the two membranes. But in the case of two bilayers of different bending rigidity constants κ1 and κ2 , we have κ = κ1 κ2 / (κ1 + κ2 ). There, the coefficient cH is a known numerical coefficient [7]. We note that the lamellar phase remains stable at the minimum of the potential, provided that the potential depth is comparable to the thermal energy kB T . This depends, in particular, on the value of amplitude W of the direct van der Waals energy. The Hamaker constant W may be varied changing the polarizability of the aqueous medium. In a pioneered theoretical paper, Lipowsky and Leibler [19] have shown that there exists a certain threshold Wc beyond which the van der Waals attractive interactions are sufficient to bind the membranes together, while below this characteristic amplitude, the membrane undulations dominate the attractive forces, and then, the membranes separate completely. According to the authors, Wc is in the interval Wc ' (6.3 − 0.61) × 10−21 J when the bending rigidity constant is in the range κ ' (1 − 20) × 10−19 J. We note that the typical value Wc corresponds to some temperature, Tc , called unbinding critical temperature [19,20]. In particular, it was found [19] that, when the critical amplitude is approached from above, the mean-separation between the two membranes, hli0 , diverges according to

hli0 = ξ⊥0 ∼ (Tc − T )−ψ ,

T → Tc− ,

(62)

with ψ a critical exponent whose value is [19]: ψ ' 1.00 ± 0.03. The latter was computed using field-theoretical Renormalization-Group.

Author's personal copy 3474

M. Benhamou et al. / Physica A 389 (2010) 3465–3475

Now, assume that the lamellar phase is trapped in a troubled aqueous medium. As first implication, the impurities drastically modify the unbinding transition phenomenon, in particular, the critical temperature. For simplicity, suppose that the two adjacent fluid membranes are physically identical. Then, we have to consider two distinct physical situations: The impurities attract or push the two membranes. As we have shown above, for attractive impurities, the membrane roughness ξ⊥ is important, and then, the steric shape fluctuation energy dominates. Therefore, we expect that the unbinding transition occurs at a low critical temperature Tc∗ we will determine below. For repulsive impurities, however, we have an opposite tendency, that is the unbinding transition takes place at a high temperature greater than Tc (absence of impurities). This can be explained by the fact that the membrane roughness is less important, and then, the direct van der Waals energy dominates. In any case, the mean-separation hli scales as

hli = ξ⊥ ∼ Tc∗ − T


−ψ

,

T → Tc− .

(63) ∗

The new unbinding critical temperature Tc can be estimated as follows. Using formulae (29) and (33), we show that, at first order in the impurities coupling w ,


ξ⊥ w ' 1 + + O w2 , 0 4 ξ⊥
ξ⊥ w ' 1 − + O w2 , 0 4 ξ⊥

(attractive impurities) ,

(64)

(repulsive impurities) .

(65)

Combing Eqs. (62) to (65), we find that the difference between the two critical temperatures Tc and Tc∗ (without and with impurities) is as follows Tc − Tc∗ ∼ w + O w 2 ,




Tc − Tc∗ ∼ −w + O w

(attractive impurities) ,


2

,

(repulsive impurities) .

(66) (67)

The prefactors in the above behaviors remain unknown. The temperature shift is then proportional to the volume fraction of impurities and their interaction strength through the coupling w . Then, we are in a situation similar to finite size scaling. For charged membranes forming the lamellar phase, it was found [19] that the mean-separation between two adjacent bilayers scales as [19]

hli0 = ξ⊥0 ∼ (χ − χc )−ψ ,

(68)

in the vicinity of χc , with χ the ionic concentration of the aqueous medium and χc its critical value. For instance, for DPPC in CaCl2 solutions, χc is in the interval [21]: χc ' 84 − 10 mM. In the presence of impurities, we state that the mean-separation behaves as

hli = ξ⊥ ∼ χ − χc∗


−ψ

,

(69)

with the same exponent ψ . In this case, we have


χc∗ − χc ∼ w + O w2 , (attractive impurities) ,
χc∗ − χc ∼ −w + O w 2 , (repulsive impurities) .

(70) (71)

We note that the same comments may by done in this case. 6. Conclusions In this paper, we presented a large scope about the effects of impurities on the statistical properties of fluid membranes. In fact, these drastically affect the living systems behavior. As an example, we can quote a very recent experimental study [22], where the authors have undertaken a series of comparative experiments, in order to explore the effect of impurities in the form of proteins and lipids on the crystallization of membrane proteins in vapor diffusion. For the present study, the impurities were assumed to be attractive or repulsive. Using the replica trick combined with a variational method, we have computed the membrane mean-roughness, as a function of the parameters associated with the pure membrane and some parameter that is proportional to the volume fraction of impurities and their interaction strength. The main conclusion is that, attractive impurities increase the shape fluctuations due to the membrane undulations, but repulsive ones tend to suppress these fluctuations. Also, we have computed the equilibrium diameter of (spherical) vesicles surrounded by small random particles solving the curvature equation. Thereafter, we extended discussion to lamellar phase formed by two parallel fluid membranes that are a finite distance apart. This lamellar phase may undergo an unbinding transition. We have shown that, attractive impurities increase the unbinding critical temperature, while repulsive ones tend to decrease this temperature.

Author's personal copy M. Benhamou et al. / Physica A 389 (2010) 3465–3475

3475

We point out that, the incorporation of a small amount of impurities in an aqueous medium may be a mechanism to suppress or to produce an unbinding transition (even the temperature and polarizability of the aqueous medium are fixed) within lamellar phases composed of fluid membranes. Finally, the present study may be extended to bilayer membranes of arbitrary topology. The essential conclusion is that, attractive impurities tend to swell the membrane, while in the presence of repulsive ones, this membrane is collapsed. Also, the used method can be applied to multilayers constituted by several parallel lamellar phases. Acknowledgements We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the ‘‘First International Workshop On Soft-Condensed Matter Physics and Biological Systems’’, 14–17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank Professor C. Misbah for fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kind hospitality during his regular visits. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1996. V.J. Emery, Phys. Rev. B 11 (1975) 239. S.F. Edvards, P.W. Anderson, J. Phys. (France) 5 (1975) 965. L.D. Landau, E.M. Lifshitz, Quantum Mechanics, 3rd edition, Pergamin Press, 1991. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. C. Itzykson, J.-M. Drouffe, Statistical Field Theory: 1 and 2, Cambridge University Press, 1989. W. Helfrich, Z. Natureforsch 28c (1973) 693. K. Wolfgang, Differential Geometry: Curves – Surfaces – Manifolds, American Mathematical Society, 2005. S.J. Singer, G.L. Nicolson, Science 175 (1972) 720. O.-Y. Zhong-Can, W. Helfrich, Phys. Rev. Lett. 59 (1987) 2486. Z.C. Tu, Z.C. Ou-Yang, Phys. Rev. E 68 (2003) 061915. O.-Y. Zhong-Can, Phys. Rev. A 41 (1990) 4517. Z. Lin, R.M. Hill, H.T. Davis, L.E. Scriven, Y. Talmon, Langmuir 10 (1994) 1008. M. Mutz, D. Bensimon, Phys. Rev. A 43 (1991) 4525. A.S. Rudolph, B.R. Ratna, B. Kahn, Nature 352 (1991) 52. H. Naito, M. Okuda, O.Y. Zhong-Can, Phys. Rev. E 48 (1993) 2304. J.N. Israelachvili, Intermolecular and Surface Forces, 2nd edition, Academic Press, London, 1991. R.P. Rand, V.A. Parsegian, Biochim. Biophys. Acta 988 (1989) 351. R. Lipowsky, S. Leibler, Phys. Rev. Lett. 56 (1986) 2541. R. Lipowsky, E. Sackmann (Eds.), An Extensive List of References on the Subject can be Found in: Structure and Dynamics of Membranes: Generic and Specific Interactions, vol. 1B, Elsevier, 1995. [21] L.J. Lis, W.T. Lis, V.A. Parsegian, R.P. Rand, Biochemistry 20 (1981) 1771. [22] A. Christopher, et al., Acta Cryst. D 65 (2009) 1062–1073.

Home

Search

Collections

Journals

About

Contact us

My IOPscience

An extended study of the phase separation between phospholipids and grafted polymers on a bilayer biomembrane

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 Phys. Scr. 83 065801 (http://iopscience.iop.org/1402-4896/83/6/065801) View the table of contents for this issue, or go to the journal homepage for more

Download details: IP Address: 41.137.25.75 The article was downloaded on 20/06/2011 at 23:39

Please note that terms and conditions apply.

IOP PUBLISHING

PHYSICA SCRIPTA

Phys. Scr. 83 (2011) 065801 (6pp)

doi:10.1088/0031-8949/83/06/065801

An extended study of the phase separation between phospholipids and grafted polymers on a bilayer biomembrane M Benhamou, I Joudar, H Kaidi, K Elhasnaoui, H Ridouane and H Qamar Laboratoire de Physique des Polymères et Phénomènes Critiques, Faculté des Sciences, Ben M’sik, PO Box 7955, Casablanca, Morocco E-mail: [email protected]

Received 4 April 2010 Accepted for publication 9 May 2011 Published 26 May 2011 Online at stacks.iop.org/PhysScr/83/065801 Abstract We re-examine here the phase separation between phospholipids and adsorbed polymer chains on a fluid membrane with a change in some suitable parameter (temperature). Our purpose is to quantify the significant effects of the solvent quality and of the polydispersity of adsorbed loops formed by grafted polymer chains on the segregation phenomenon. To this end, we elaborate on a theoretical model that allows us to derive the expression for the mixing free energy. From this, we extract the phase diagram shape in the composition–temperature plane. Our main conclusion is that the polymer chain condensation is very sensitive to the solvent quality and to the polydispersity of loops of adsorbed chains. PACS numbers: 87.16.Dg, 47.57.Ng, 64.60.F

bounded by sphingolipids such as cholera and tetanus toxins. Sphingolipids and cholesterol favor the aggregation of proteins in microdomains called rafts. In fact, these play the role of a platform for the attachment of proteins while the membranes are moved around the cell and also during signal transduction. Proteins (long macromolecules) are another principal component of cell membranes. Transmembrane proteins are amphipathic and are formed by hydrophobic and hydrophilic regions having the same orientation as other lipid molecules. These proteins are also called integral proteins. Their function is to transport substances, such as ions and macromolecules, across the membrane. There exist other types of proteins that may be attached to the cytoplasm surface by fatty acyl chains or to the external cell surface by oligosaccharides. These are termed peripheral membrane proteins. They have many functions; in particular, they protect the membrane surface, regulate cell signaling and participate in many other important cellular events. In addition, some peripheral membrane proteins (those having basic residues) tend to bind electrostatically to negatively charged membranes, such as the inner leaflet of the plasma membrane.

1. Introduction Biological membranes are of great importance to life, because they separate the cell from the outside environment and separate the compartments inside the cell in order to protect important processes and specific events. Nowadays, it is largely recognized that biological membranes are present as a lipid bilayer composed of two adjacent leaflets [1, 2], which are formed by amphiphile molecules possessing hydrophilic polar-heads pointing outward and hydrophobic fatty acyl chains that form the core. The majority of lipid molecules are phospholipids. These have a polar-head group and two non-polar hydrocarbon tails, whose length is of the order of 5 nm. Also, the cell membranes incorporate another type of lipid, cholesterol [1, 2]. The cholesterol molecules have several functions in the membrane. For example, they give rigidity or stability to the cell membrane and prevent crystallization of hydrocarbons. The biomembranes also contain glycolipids (sugars), which are lipid molecules that microaggregate in the membrane, and may be protective and act as insulators. Certain kinds of molecules are 0031-8949/11/065801+06$33.00

Printed in the UK & the USA

1

© 2011 The Royal Swedish Academy of Sciences

Phys. Scr. 83 (2011) 065801

M Benhamou et al

We note that the majority of macromolecules forming the bilayer are simply anchored on the membrane and form a soft branched polymer brush [3, 4]. The study of grafted polymers on soft interfaces was motivated by the fact that they have potential applications in biological materials, such as liposomes [5–8]. These soft materials were discovered by A Bangham. Currently, they are major tools in biology, biochemistry and medicine (as drugs transport agents). Liposomes are artificial vesicles of spherical shape that can be produced from natural nontoxic phospholipids and cholesterol. But the lipid bilayers have a short lifetime, because of the weak stability of the vesicles and their extermination by white blood cells. To have stable vesicles in time, one useful method consists of protecting them with a coat of flexible polymer chains (coat size of the order of 50 nm [1]), which prevents the adhesion of marker proteins [6, 7]. In fact, these polymer chains stabilize the liposomes, due to the excluded volume forces between monomers [8]. Liposomes can also be synthesized from A–B diblock-copolymers immersed in a selective solvent that prefers to be contacted by the polymer A. The hydrophobic B-polymer chains then aggregate and form a thin bilayer, while the hydrophilic polymer chains A float in the solvent. These copolymer-based liposomes have properties that are slightly different from those of the lipid ones [9] (high resistance, high rigidity and weak permeability to water). Depending on the choice of copolymers, these liposomes are resistant to detergents [10]. The grafting of polymers onto lipid membranes was considered in a very recent paper [11]. More precisely, the purpose was to investigate of the phase separation between phospholipids and anchored polymers. As assumptions, the aqueous medium was assumed to be a good solvent, and in addition, the polymer chains were anchored to the interface only by one extremity, a big amphiphile molecule. The latter is chemically different from the phospholipid molecules. In this paper, however, we assume that the polymer chains may be adsorbed on the membrane by many monomers that are directly attached to some polar-heads of the host lipid molecules (mobile anchors), and then organize in polydisperse loops. The surrounding liquid may be a good solvent or a theta solvent. As we shall see below, the loops’ polydispersity and solvent quality drastically affect the phase behavior of the system. Under a change in a suitable parameter, such as temperature, the phospholipids and adsorbed polymer chains phase separate into macroscopic domains alternately rich in the two components. For the determination of the phase diagram shape, we elaborate on a theoretical model that takes into account both the loops’ polydispersity and the solvent quality. This paper is organized as follows. In section 2, we derive the expression for the mixing free energy of the phospholipid–anchor mixture. To investigate the phase diagram shape is the aim of section 3. Finally, some concluding remarks are given in section 4.

bound to the polar-head of a lipid molecule (for many proteins, the lipid anchors are glycosylphosphatidylinositol units) [12, 13]; (ii) by hydrophobic side-groups of the polymers which are integrated into the bilayer [14, 15]; (iii) by membrane spanning hydrophobic domains of the polymer (membrane-bound proteins, for example); and (iv) by a strong adsorption that drives the polymer from a desorbed state to an adsorbed one [16]. For a single polymer chain, the adsorption (on solid substrates) can be theoretically studied using a scaling argument [17], for instance. In addition to the polymerization degree of the polymer chain, N , and the excluded volume parameter, v, the study was based on an additional parameter, δ, which is the energy (per kB T unit) required to adsorb one monomer on the surface. For a strong adsorption, δ −1 defines the adsorbed-layer thickness. An adsorption transition takes place at some typical value δ ∗ of δ scaling as [17]: δ ∗ ∼ RF−1 ∼ a N −ν F (νF = 3/5), where RF is the Flory radius of the polymer chain. For dilute and semi-dilute polymer solutions, the adsorption phenomenon depends, in addition to the parameter δ, on the polymer concentration. In this paper, we consider situation (iv), where each monomer has the same probability of being adsorbed on the membrane surface [18]1 . More precisely, a given monomer is susceptible to becoming linked to a polar-head of a phospholipid molecule (anchor). As a result, the polymer chains form polydisperse loops (with eventually one or two tails floating in the aqueous medium). In fact, this assumption conforms to what was considered in [19]. The case where no loops are present (adsorption only by one extremity) was considered in [11]. Contrary to the adsorption on the solid surface, the anchored polymer chains are mobile on the host membrane and may undergo the aggregation transition that we are interested in. To be more general, the fluid membrane is assumed to be in contact with a good solvent or a theta solvent. The purpose is to write a general expression for the mixing free energy. The latter will allow the determination of the phase diagram related to the aggregation of anchored polymer chains. First, we start with the free energy (per unit area) of the polymer layer (for solid substrates), which is given by [20]  0  Z N F0 1 S (n) 2 β 2 0 ' 2 [b S (n)] + [−b S (n)] ln − dn. kB T b 1 S1 (1) For the configuration study, Guiselin [21] considered that each loop can be viewed as two linear strands (two half-loop). Here, S(n) is the number of strands having more than n monomers per unit area, and b represents the monomer size. The number N denotes the length of the longest strand. Hereafter, we shall use the notation S1 = 8/a, which denotes the total number of grafted chains per unit area, with 8 being the volume fraction of anchors and a their area. If the mixture is assumed to be incompressible, then 1 − 8 is the volume fraction of the phospholipid molecules. Let us come back to the free energy expression (1); note that the first term of the right-hand side represents the

2. Mixing free energy

1

The adsorption of an adequate polymer on a fluid membrane made of two phospholipids of different chemical nature may be a mechanism of phase separation between these two unlike components as was pointed out.

Polymers can adhere to biomembranes in several ways: (i) by lipid anchors; that is, the polymer is covalently 2

Phys. Scr. 83 (2011) 065801

M Benhamou et al

contribution of the (two- or three-body) repulsive interactions between monomers belonging to the polymer layer. The second term is simply the entropy contribution describing all possible rearrangements of the grafted chains in the polymer layer. There, the exponent β depends on the solvent quality [20]. When the grafting is accomplished in a dilute solution with a good solvent or a theta solvent, the values of β are β = 11/6 (blob model) or β = 2, respectively. For polydisperse polymer layers, the distribution S(n) is known in the literature [20]. This can be obtained by minimizing the above free energy with respect to this distribution. Without presenting details, we simply sketch the general result that [20] 8 S(n) ∼ 1/(β−1) . (2) an For usual solvents, we have S(n) ∼

8 an 6/5

(good solvents),

8 (theta solvents). an We shall rewrite the distribution S(n) as S(n) ∼

S(n) =

8 f (n), a

Explicitly, we have ζ (N ) = 1 (polymer brushes), ζ (N ) ∼ N β/(1−β)

(3)

(4)

(5)

F = 8 ln 8 + (1 − 8) ln(1 − 8) + χ 8(1 − 8) kB T + u8β + η(N ) 8.

(6)

 β N 1/(1−β) ln N . β −1

(7)

where the positive coefficients C and D are such that (in dimension 2) [23] Z ∞ π D = −π rU (r ) dr, C = σ 2 (covolume). (15) 2 σ

In the asymptotic limit N → ∞, this coefficient goes to η(N ) → a −1 [β + ln(β − 1)],

(8)

provided that β > 1 (note that β = 11/6, for good solvents, and β = 2, for theta ones). This asymptotic limit is always positive definite and inversely proportional to the polar-head area a. In fact, the positive sign of the coefficient η(N ) agrees with the entropy loss due to the polymer chains grafting. As we shall see below, this linear contribution does not change the phase diagram in the composition-critical parameter plane. Here, the coupling constant u is as follows: b2 a

β−1

1 N

Z

 u= with ζ (N ) =

N

N ζ (N ),

[ f (n)]β dn.

(13)

We note that, for polymer chains anchored by one extremity, a big amphiphile molecule (the polymer brush case), the first contribution of entropy, 8 ln 8, should be divided by some factor q, which represents the ratio of the anchor area to the area of polar-heads of the host phospholipids. In the present case, we have q = 1. In equality (11), χ accounts for the Flory interaction parameter   1 D χ = χ0 − 2 C − (14) , A kB T

where the linear term in 8 describes the contribution of entropy to the free energy, where the coefficient η(N ) is as follows (see the appendix): 
−1 [β + ln(β − 1)] 1 − N 1/(1−β) η(N ) = a +

(12)

Note that ζ (N ) < 1, since, in all cases, β > 1. Therefore, the polydispersity of loops decreases the repulsive interaction energy. Now, we have all the ingredients for the determination of the expression for the mixing free energy. To this end, we imagine that the interface is present as a two-dimensional (2D) Flory–Huggins lattice [17, 22], where each site is occupied by an anchor or by a phospholipid molecule. Hence, we can regard the volume fraction of anchors 8 as the probability that a given site is occupied by an anchor. Therefore, 1 − 8 is the occupation probability of phospholipids. Before the determination of the desired mixing free energy (per site), we note that the latter is the sum of three contributions, which are the mixing entropy (per site), the volume free energy (per site) and the interaction energy (per site) coming from the membrane undulations. Actually, the induced attractive forces due to the membrane undulations are responsible for the condensation of anchors. These forces balance the repulsive ones between monomers along the connected polymer chains. We then write

with f (n) ∼ 1/n 1/(β−1) , for polydisperse polymer layers, and f (n) = 1 (for all n), for polymer brushes. Therefore, F0 can be approximated by F0 = a −1 u8β + η(N )8, kB T

(polydisperse systems).

(11)

Here, U (r ) is the pair potential induced by the membrane undulations ([11]; [24] and references therein)  r < σ, ∞,  σ 4 U (r ) = (16) −AH , r > σ, r where σ is the hard disc diameter, which is proportional to the square root of the anchor area a. There, the potential amplitude AH plays the role of the Hamaker constant. It was found that the latter decays with the bending rigidity constant according to ([11]; [24] and reference therein)

(9)

(10)

AH ∼ κ −2 .

1

3

(17)

Phys. Scr. 83 (2011) 065801

M Benhamou et al

But this amplitude is also sensitive to temperature. We remark that the above mixing free energy is not symmetric under the change 8 → 1 − 8. Some straightforward algebra gives the following expression for the attraction parameter D: D = AH

π κ −2 ∼ . 2σ 2 σ2

interactions widen the compatibility domain, and then, the separation transition appears at low temperature. Now, to see the influence of the solvent quality, we rewrite the interaction parameter u as u = u 0 ζ (N ) ∼ u 0 N β/(1−β) < u 0 , where u 0 is the interaction parameter relative to a monodisperse system. Thus, the polydispersity of loops has a tendency to reduce the compatibility domain in comparison with the monodisperse case. The critical volume fraction, 8c , can be obtained by minimizing the critical parameter χ (8) with respect to the 8-variable. We then obtain

(18)

Therefore, those membranes with a small bending modulus induce significant attraction between anchors. In formula (12), χ0 > 0 is the Flory interaction parameter describing the chemical segregation between amphiphile molecules that are phospholipids and anchors. This segregation parameter is usually written as χ0 = α0 +

1 1 = 0. − 2 + β(β − 1) (β − 2) u8β−3 c 2 8 (1 − 8c ) c For good solvents (β = 11/6), we have

γ0 , T

(19) 55 1 1 − 2− u8−7/6 = 0. c 2 8 216 (1 − 8c ) c

where the coefficients α0 and γ0 depend on the chemical nature of the various species. Also, the total interaction parameter χ can be written as χ =α+

γ , T

C , A2

γ = γ0 +

(20)

D . kB A2

(24)

Therefore, the critical volume fraction is the abscissa of the intersection point of the curve of the equation (2x − 1)/ x 5/6 (1 − x)2 and the horizontal straight line of the equation y = (55/216)u. Note that this critical volume fraction is unique, and in addition, it must be greater than the value 1/2 (for mathematical compatibility). The coordinates of the critical point are (8c , χc ), where 8c solves the above equation and χc = χ (8c ). The latter can be determined by combining equations (19) and (21). For theta solvents (β = 2), the coordinates of the critical point are exact,

with the new coefficients α = α0 −

(23)

(21)

These coefficients then depend on the chemical nature of the unlike components and on the membrane’s characteristics through its bending modulus κ. If we admit that the coupling constant u has a slight dependence on temperature, we will draw the phase diagram in the plane of variables (8, χ). Indeed, all of the temperature dependence is contained in the Flory interaction parameter χ.

8c = 12 ,

χc = 2 + u,

(25)

where the interaction parameter u scales as u=

3. Phase diagram

b2 −1 N . a

(26)

The above relation clearly shows that the polymer chains’ condensation rapidly takes place only when the characteristic mass N is high enough. The same tendency is also seen in the case of good solvents. It is straightforward to show that the critical fraction and the critical parameter are shifted to lower values in the case of polydisperse systems, whatever be the quality of the surrounding solvent. In figure 1, we present the spinodal curve for monodisperse (with no loops) and polydisperse (with loops) systems, with a fixed parameter N . We have chosen the good solvents situation. For theta solvents, the same tendency is seen. We present in figure 2 the spinodal curve for a polydisperse system (with loops), at various values of the parameter N . As expected the critical parameter is shifted to higher values when we augmented the typical polymerization degree. Finally, we compare, in figure 3, the spinodal curves for a polydisperse system (with loops) for the case of theta solvents and those for the case of good solvents, at fixed parameters b and N . All the curves in the figure reflect our discussions above.

With the help of the above mixing free energy, we can determine the shape of the phase diagram in the (8, χ )-plane that is associated with the aggregation process that drives the anchors from a dispersed phase (gas) to a dense one (liquid). We focus only on the spinodal curve along which the thermal compressibility diverges. The spinodal curve equation can be obtained by equating to zero the second derivative of the mixing free energy with respect to the anchor volume fraction 8; that is, ∂ 2 F/∂82 = 0. Then, we obtain the following expression for the critical Flory interaction parameter:   1 1 χ(8) = + β(β − 1) u8β−2 . (22) 2 8(1 − 8) Above this critical interaction parameter appear two phases: one is homogeneous and the other is separated. Of course, the linear term in 8 appearing in equality (11) does not contribute to the critical parameter expression. We remark that, in usual solvents, this critical interaction parameter is increased due to the presence of (twoor three-body) repulsive interactions between monomers belonging to the polymer layer. This means that these 4

Phys. Scr. 83 (2011) 065801

χ

M Benhamou et al

χ

80

20

16

60

12 40

8 20

4

0 0,0

0,2

0,4

0,6

0,8

0 0,0

1,0

Φ

12

8

4

0,4

0,6

0,8

0,8

1,0

polymer chains. The latter were taken into account through the well-known form of the chains’ length distribution. In doing so, we first computed the expression for the mixing free energy by adopting the Flory–Huggins lattice image usually encountered in polymer physics [17, 22]. Such an expression shows that there is competition between four contributions: entropy, chemical mismatch between unlike species, interaction energy induced by the membrane undulations and the interaction energy between monomers belonging to the grafted layer. Such a competition governs the phases succession. We emphasize that the present work and a previous work [11] differ from another previous work that was concerned with the same problem, but in which the substrate was assumed to be a rigid surface [3]. Therefore, the membrane undulations were neglected. As we have seen, these undulations increase the segregation parameter χ by an additive term, χm , scaling as κ −2 . This means that the phase separation is accentuated due to the presence of thermal fluctuations. Compared with the previous work [11], which was concerned with soft brushes with monodisperse end-grafted polymers, the present work is more general, since it takes into account both the polydispersity of loops forming the adsorbed polymer chains and the solvent quality. Thus, the present study was achieved in a unified way. As we have seen, these details drastically affect the phase diagram architecture. Now, let us discuss further the influence of solvent quality on the critical phase behavior. We recall that the solvent quality appears in the free energy (11) through the repulsion parameter u, defined in equation (7). We find, in the N → ∞ limit, that u g ∼ N 1/5 u θ , where the subscripts g and θ stand for good and theta solvents, respectively. This implies that the good solvent plays the role of a stabilizer regarding the phase separation. Finally, this work must be considered as a natural extension of a study reported previously [11], which was concerned with monodisperse end-grafted polymer chains

16

0,2

0,6

Figure 3. Superposition of spinodal curves for a polydisperse system for the case of theta solvents (solid line) on those for the case of good solvents (dashed line). For these curves, we chose N = 100 and b2 = 0.5a.

20

0 0,0

0,4

Φ

Figure 1. Spinodal curves (in a good solvent) for monodisperse (dashed line) and polydisperse (solid line) systems, when N = 100, with the parameter b2 = 0.5a.

χ

0,2

1,0

Φ Figure 2. Spinodal curves for a polydisperse system, when N = 50 (solid line), 100 (dashed line) and 150 (dotted line), with the parameter b2 = 0.5a . We assumed that the surrounding liquid is a good solvent. For theta solvents, the tendency is the same.

4. Discussion and conclusions This paper is devoted to the thermodynamic study of the aggregation of the polymer chains adsorbed on a soft surface. Such an aggregation is caused by competition between the chemical segregation between phospholipids and grafted polymer chains, their volumic interactions and the membrane undulations. More precisely, we addressed the question of how these polymer chains can be driven from a dispersed phase (gas) to a dense one (liquid), under a change in a suitable parameter, e.g. the absolute temperature. To be more general, we achieved the study in a unified way; that is, we have considered more realistic physical situations: the solvent quality (good or theta solvents) and the polydispersity of the adsorbed loops formed by the grafted 5

Phys. Scr. 83 (2011) 065801

M Benhamou et al

trapped in a good solvent. Therefore, the present work presents a wide perspective on the phenomenon of segregation between the host phospholipids and grafted polymer chains on bilayer membranes.

References [1] Lipowsky R and Sasckmann S (ed) 1995 Structure and Dynamics of Membranes: From Cells to Vesicles vol 1A (New York: Elsevier) [2] Benhamou M 2008 Lipids Insights 1 2 [3] Nicolas A and Fourcade B 2003 Eur. Phys. J. E 10 355 [4] Aubouy M et al 2000 Phys. Rev. Lett. 84 4858 Aubouy M et al 1996 Macromolecules 29 7261 [5] Ringsdorf H and Schmidt B 1986 How to bridge the gap between membrane, biology and polymers science Synthetic Membranes: Science, Engineering and Applications ed P M Bungay et al (Dordrecht: D Reidel) p 701 [6] Lasic D D 1992 Am. Sci. 80 250 [7] Torchilin V P 1996 Effect of polymers attached to the lipid head groups on properties of liposomes Handbook of Nonmedical Applications of Liposomes vol 1, ed D D Lasic and Y Barenholz (Boca Raton, FL: RCC Press) p 263 [8] Joannic R, Auvray L and Lasic D D 1997 Phys. Rev. Lett. 78 3402 [9] Discher B M et al 1999 Science 284 1143 [10] Elferink M G L et al 1992 Biochim. Biophys. Acta 1106 23 [11] Benhamou M, Joudar J and Kaidi H 2007 Eur. Phys. J. E 24 343 [12] Blume G and Cev G 1990 Biochim. Biophys. Acta 1029 91 [13] Lasic D D, Martin F J, Gabizon A, Huang S K and Papahadjopoulos D 1991 Biochim. Biophys. Acta 1070 187 [14] Decher G et al 1989 Angew. Makromol. Chem. 166 71 [15] Simon J, Kuhner M, Ringsdorf H and Sackmann E 1995 Chem. Phys. Lipids 76 241 [16] Garel1 T, Kardar M and Orland H 1995 Europhys. Lett. 29 303 Chatellier X and Andelman D 1995 Europhys. Lett. 32 567 Lipowsky R 1995 Europhys. Lett. 30 197 Xie A F and Granick S 2002 Nature Mater. 1 129 [17] de Gennes P-G 1979 Scaling Concept in Polymer Physics (Ithaca, NY: Cornell University Press) [18] Binder W H, Barragan V and Menger F M 2003 Angew. Chem. Int. Edn Engl. 42 5802 [19] Manghi M and Aubouy M 2001 Adv. Coll. Interf. Sci. 94 21 [20] Manghi M and Aubouy M 2003 Phys. Rev. E 68 041802 [21] Guiselin O 1992 Euro. Phys. Lett. 17 225 Guiselin O 1992 PhD Thesis Paris VI University [22] Flory P J 1953 Principles of Polymer Chemistry (Ithaca, NY: Cornell University Press) [23] Balian R 2006 From Microphysics to Macrophysics—Methods and Applications of Statistical Physics (Berlin: Springer) [24] Marchenko V I and Misbah C 2002 Eur. Phys. J. E 8 477

Acknowledgments We are indebted to Professors T Bickel, J-F Joanny and C Marques for helpful discussions during the First International Workshop on Soft-Condensed Matter Physics and Biological Systems (14–17 November 2006, Marrakech, Morocco). MB thanks Professor C Misbah for fruitful correspondence and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kind hospitality during his visit. We are grateful to the referee for critical remarks and useful suggestions that helped us to improve the scientific content of this paper.

Appendix The aim is to determine the coefficient η(N ) appearing in formula (6). We start with the entropy contribution to the free energy  0  Z N  2 0  Fentropic 1 S (n) ' 2 −b S (n) ln − dn, (A.1) kB T b 1 S1 where the loop-size distribution is defined in equation (2). Then, the above expression can be rewritten as Fentropic = η(N ) 8, kB T

(A.2)

with N

   1 n β/(1−β) dn. n β/(1−β) ln β −1 1 (A.3) Straightforward algebra gives 
η(N ) = a −1 [β + ln(β − 1)] 1 − N 1/(1−β) η(N ) = a −1

1 β −1

+

Z

 β N 1/(1−β) ln N . β −1

(A.4)

This concludes the determination of the coefficient η(N ).

6

African Journal Of Mathematical Physics Volume 10(2011)55-64

Statistics of a single D-manifold restricted to two parallel biomembranes or a tubular vesicle M. Benhamou∗ , K. Elhasnaoui, H. Kaidi, M. Chahid Laboratoire de Physique des Polym`eres et Ph´enom`enes Critiques Facult´e des Sciences Ben M’sik, P.O. Box 7955, Casablanca, Morocco ∗ [email protected]

abstract The purpose is an extensive conformational study of a single polymer immersed in an aqueous medium (good solvent) delimitated by bilayer membranes. To be more general, we assume that the polymer is of arbitrary topology we term D-polymeric fractal or Dmanifold, where D is the spectral dimension (for instance, D = 1, for linear polymers, and 4/3, for branched ones). The main quantity to consider is the parallel extension of the confined polymer. To make explicit calculations, we suppose that the polymer is restricted to a tubular vesicle or two parallel biomembranes. We first show that, for the first geometry, the polymer is confined only when the tubular vesicle is in equilibrium state. For the second geometry, the confinement is possible if only if the two parallel membranes are in their binding state, that is below the unbinding or adhesion temperature. In any case, the parallel gyration radius of the confined polymer is computed using an extended Flory-de Gennes theory. As result, this radius strongly depends on the polymer topology (through the spectral dimension D) and on the membranes sizes, which are the equilibrium diameter (function of bending modulus, pressure difference between inner and outer sides of the membrane, and interfacial tension coefficient), for the first geometry, and the mean-separation (function of temperature and interaction strength between the adjacent membranes), for the second one. Finally, we give the expression of the confinement free energy, as a function of the polymer size, and discuss the effects of external pressure or lateral tension on the radius expression for two confining parallel membranes. Key words: D-Polymeric fractals, Biomembranes, Vesicles, Confinement.

I. INTRODUCTION

The polymer confinement finds many applications in various domains, such as biological functions, filtration, gel permeation chromatography, heterogeneous catalysis, and oil recuperation. The physics of polymer confinement is a rich and exciting problem. Recently, much attention has been paid to the structure and dynamics of polymer chains confined to two surfaces, or inside cylindrical pores [1 − 7]. Thereafter, the study has been extended to more complex polymers, called D-polymeric fractals or D-manifolds [8] that are restricted to the same geometries. Here, D is the spectral dimension that measures the degree of the connectivity of monomers inside the polymer [9]. For instance, this intrinsic dimension is 1 for linear polymers and 4/3 for branched ones. The D-manifolds may be polymerized (or

0

c a GNPHE publication 2011, [email protected] ⃝

55

M. Benhamou et al.

African Journal Of Mathematical Physics Volume 10(2011)55-64

crumpled) vesicles [10]. The polymer confinement between two surfaces and in cylinders with sinusoidal undulations was also investigated [11, 12]. The confining geometries may be soft-bodies, as bilayer biomembranes and surfactants, and spherical and tubular vesicles. A particular question has been addressed to polymer confinement in surfactant bilayers of a lyotropic lamellar phase [13], where the authors reported on small-angle X-ray scattering and freefracture electron microscopy studies of a nonionic surfactant/water/polyelectrolyte system in the lamellar phase region. The fundamental remark is that, the polymer molecules cause both local deformation and softening of the bilayer. In the same context, it was experimentally demonstrated [14] that the polymer confinement may induce a nematic transition of microemulsion droplets. More precisely, the authors showed that upon confinement, spherical droplets deform to prolate ellipsoid droplets. The origin of such a structural transition may be attributed to a loss of the conformational entropy of polymer chains due to the confinement. In other experiments [15 − 17], a new phase has been observed adding an neutral hydrosoluble polymer (PVP) in the lyotropic lamellar phase (CPCI/hexanol/water). Also, one has studied the effect of a neutral water-soluble polymer on the lamellar phase of a zwitterionic surfactant system [18]. The polymer confinement is also relevant for living systems and governs many biological processes. As example, we can quote membrane nanotubes that play a major role in intercellular traffic, in particular for lipid and proteins exchange between various compartments in eukaryotic cells [19 − 22]. The traffic of macromolecules and vesicles in nanotubes is ensured by molecular motors [23]. Also, these are responsible for the extraction of nanotubes [24]. The formation of tubular membrane tethers or spicules [25 − 33] results from the action of localized forces that are perpendicular to the membrane. These forces may originate from the polymerization of fibers [34], as actin [35], tubulin [36], or sickle hemoglobin [37]. Inspired by biological processes, as macromolecules and vesicles transport, we aim at a conformational study of confined polymers in aqueous media delimitated by biomembranes. More precisely, the purpose is to see how these biomembranes can modify the conformational properties of the constrained polymer. To this end, we choose two geometries : a tubular vesicle (Geometry I), and two parallel biomembranes (Geometry II). To be more general, we consider polymers or arbitrary connectivity called D-polymeric fractals or D-manifolds [38]. Linear and branched polymers, and polymerized vesicles constitute typical examples. We note that this conformational study in necessary for the description of dynamic properties of polymers restricted to these geometries. As we shall below, the polymer confinement is entirely controlled by the confining biomembranes state. Indeed, for Geometry I, the polymer may be confined only when the confining tubular vesicle is at equilibrium. For Geometry II, the confinement is possible if only if the two parallel membranes are in the binding state. This paper is organized as follows. In Sec. II, we briefly recall the conformational study of unconfined D-polymeric fractals in good solvent. Sec. III deals with the conformational study of polymers confined inside a tubular vesicle. In Sec. IV, we extend the study to D-polymeric fractals confined to two parallel biomembranes. Some concluding remarks are drawn in the last section.

II. UNCONFINED POLYMERIC FRACTAL

Consider a single polymeric fractal of arbitrary topology (linear polymers, branched polymers, polymer networks, ...). We assume that the considered polymer is trapped in a good solvent. We denote by RF ∼ aM 1/dF

(1)

its gyration (or Flory) radius, where dF is the Hausdorff fractal dimension, M is the molecular-weight (total mass) of the considered polymer, and a denotes the monomer size. The mass M is related to the linear dimension N by : M = N D , where D is the spectral dimension [9]. The latter is defined as the Hausdorff dimension corresponding to the maximal extension of the fractal. Naturally, the Hausdorff dimension depends on the Euclidean dimensionality d, the spectral dimension D and the solvent quality. When the polymer is ideal (without excluded volume forces), its Hausdorff dimension, d0F , is a known simpler function of D [9] d0F =

2D . 2−D 56

(2)

M. Benhamou et al.

African Journal Of Mathematical Physics Volume 10(2011)55-64

For linear polymers (D = 1), d0F = 2 [2], for ideal branched ones (D = 4/3), d0F = 4 [9], and for crumpled membranes (D = 2), d0F = ∞. Because of the positivity of the Hausdorff dimension, the above expression makes sense only for D < 2. Indeed, this condition is fulfilled for any complex polymer with spectral dimension in the interval 1 ≤ D < 2 [9]. A polymeric fractal in good solvent is swollen, because of the presence of the excluded volume forces. The polymer size increases with increasing total mass M according to the power law (1). The first implication of the polymer swelling is that, the actual Hausdorff dimension dF is quite different from the Gaussian one, defined in Eq. (2). However, there exists a special value of the Euclidean dimensionality called upper critical dimension duc [39, 40], beyond which the polymeric fractal becomes ideal. This upper dimension is naturally a D-dependent function, which can be determined [39] using a criterion of Ginzburg type, usually encountered in critical phenomena [41, 42]. According to Ref. [39], duc is given by duc =

4D . 2−D

(3)

For instance, the upper critical dimension is 4 for linear polymers [2], and 8 for branched ones [39]. We emphasize that, in general, the Hausdorff fractal dimension dF cannot be exactly computed. Many techniques have been used to determine its approximate value, in particular, the Flory-de Gennes (FD) theory [2]. Using a generalized FD approach, it was found that the fractal dimension is given by [39] dF = D

d+2 , D+2

(4)

below the critical dimension, and it equals the Gaussian fractal dimension d0F described above. For dimension 3, we have dF (3) =

5D . D+2

(5)

For instance, for linear polymers, dF (3) = 5/3, and dF (3) = 2, for branched ones (animals). In the following paragraph, we shall focus our attention on the conformational study of a single polymeric fractal confined to a long tubular vesicle.

III. CONFINED POLYMERIC FRACTAL IN GEOMETRY I A. Useful backgrounds

Before studying the conformation of a single polymeric fractal, we recall some basic backgrounds concerning the equilibrium shape of tubular vesicles. This can be done using Differential Geometry machineries. The tubular vesicle is essentially formed by two adjacent leaflets (inner and outer) that are composed of amphiphile lipid molecules. These permanently diffuse with the molecules of the surrounded aqueous medium. Such a diffusion then provokes thermal fluctuations (undulations) of the membrane. This means that the latter experiences fluctuations around an equilibrium plane we are interested in. Consider a biomembrane of arbitrary topology. A point of this membrane can be described by two local coordinates (u1 , u2 ). From surfaces theory point of view, at each point, there exists two particular curvatures (minimal and maximal), called principal curvatures, denoted C1 = 1/R1 and C2 = 1/R2 . The quantities R1 and R2 are the principal curvature radii. With the help of the principal curvatures, one constructs two invariants that are the mean-curvature C=

1 (C1 + C2 ) , 2

(6)

and the Gauss curvature K = C1 C2 .

57

(7)

M. Benhamou et al.

African Journal Of Mathematical Physics Volume 10(2011)55-64

We recall that C1 and C2 are nothing else but the eigenvalues of the curvature tensor [43]. To comprehend the geometrical and physical properties of the biomembranes, one needs a good model. The widely accepted one is the fluid mosaic model proposed by Singer and Nicholson in 1972 [44]. This model consists to regard the cell membrane as a lipid bilayer, where the lipid molecules can move freely in the membrane surface like a fluid, while the proteins and other amphiphile molecules (cholesterol, sugar molecules, ...) are simply embedded in the lipid bilayer. We note that the elasticity of cell membranes crucially depends on the bilayers in this model. The elastic properties of bilayer biomembranes were first studied, in 1973, by Helfrich [45]. The author recognized that the lipid bilayer could be regarded as smectic-A liquid crystals at room temperature, and proposed the following curvature free energy ∫ ∫ ∫ ∫ κ 2 (2C + 2C0 ) dA + κG KdA + γdA + p dV . (8) F = 2 where dA denotes the area element, and V is the volume enclosed within the lipid bilayer. In the above definition, κ accounts for the bending rigidity constant, C0 for the spontaneous curvature, κG for the Gaussian curvature, γ for the surface tension, and p for the pressure difference between the outer and inner sides of the vesicles. The first order variation gives the shape equation of lipid vesicles [46] ( ) p − 2γC + κ (2C + C0 ) 2C 2 − C0 C − 2K + κ∇2 (2C) = 0 , (9) with the surface Laplace-Bertlami operator ) ∂ gg , (10) ∂uj ( ) where g ij is the metric tensor on the surface and g = det g ij . For open or tension-line vesicles, local differential equation (9) must be supplemented by additional boundary conditions we√do not write [47]. The above equation have known three analytic solutions corresponding to sphere [46], 2-torus [48 − 51] and biconcave disk [52]. For cylindrical (or tubular) vesicles, one of the principal curvature is zero, and we have 1 ∂ ∇ =√ g ∂ui 2

C=−

( √

ij

1 , R

K=0,

(11)

where R is the radius of the cylinder. If we ignore the boundary conditions (assumption valid for very long tubes), the uniform solution to equation (9) is ( H=2

4κ p

)1/3 ,

(12)

where H is the equilibrium diameter. We have neglected the surface tension and spontaneous curvature contributions, in order to have a simplified expression for the equilibrium diameter. The above relation makes sense as long as the pressure difference is smaller than a critical value pc that scales as [53] : pc ∼ κ/R3 . The latter is in the range 1 to 2 Pa. The meaning of the critical pressure is that, beyond pc , the vesicle is unstable. This implies that the equilibrium diameter must be greater than 1/3 the critical one Hc = 2 (4κ/pc ) . In what follows, we shall use the idea that consists to regard the tubular vesicle as a rigid cylinder of effective diameter H that depends on the characteristics of the bilayer through the parameters κ and p.

B. Parallel extension to the cylinder-axis

Consider now a polymeric fractal of arbitrary topology confined inside a tubular vesicle of equilibrium diameter H. The host solvent is assumed to be a good solvent. The polymer is confined if only if its three-dimensional Flory radius RF 3 is much larger than the diameter H, that is H << RF 3 . At fixed parameters κ and p, this condition implies that the polymer mass M must be greater than some typical 5D/3(D+2) value M ∗ that scales as M ∗ ∼ (κ/p) . Now, if the parameter M and p are fixed, the polymer is 58

M. Benhamou et al.

African Journal Of Mathematical Physics Volume 10(2011)55-64

confined is only if κ << κ∗ ∼ pM 3(D+2)/5D . This behavior clearly indicates that the confinement is more favorable when the tubular vesicle is of small bending rigidity constant. We note that the confinement condition depends on the nature of the polymeric fractal through its spectral dimension D. Also, the solvent quality is another factor that influences this confinement. The confinement of polymers of arbitrary topology inside a rigid tube is largely investigated in Ref. [8]. In this paper, to achieve the conformational study of a confined polymer, we use a generalized FD theory based on the following free energy [8] R∥2 M2 F = 2 +v , kB T R0 R∥ H 2

(13)

where R∥ denotes the polymer parallel extension to the tube-axis, and v is the excluded volume parameter. 0 There, R0 ∼ aM 1/dF is the ideal radius and R∥ H 2 represents the volume occupied by the fractal. Minimizing the above free energies with respect to R∥ yields the desired result R∥ ∼ aM

(D+2)/3D

)−2/9 ( −3 κ . a p

(14)

We have used relationship (12). Let us comment about the obtained result. Firstly, notice that, in any case, the parallel radius naturally depends on polymer and tubular vesicle characteristics, through M and parameters (κ, p), respectively. Secondly, at fixed polymer mass M , the parallel extension is important for those tubular vesicles of small bending modulus. Finally, the above behavior is valid as long as the parallel extension R∥ remains below the maximal extension of the polymer, that is we must have R∥ < aM 1/D (maximal extension). This gives a minimal tube diameter Hmin ∼ aM (D−1)/2D .

(15)

Therefore, the confinement of the polymeric fractal implies that the tube diameter is in the interval Hmin < H < RF 3 . For instance, for linear polymers, Hmin ∼ a, and Hmin ∼ aM 1/8 , for branched ones. The minimum pore size for linear polymers is then independent on the molecular-weight. This means that the linear polymers find their way even through a very small pore. It is not the case for branched polymers, for which the minimum pore size increases with the total mass. As pointed out in Ref. [8], it should be possible to construct a porous medium that may separate a mixture of branched and linear molecules on a basis of their connectivity. Indeed, one can choose an appropriate minimum pore size through which linear polymers can pass, whereas branched polymers cannot. Now, combining relations (12) and (15) yields a minimal value for the bending rigidity constant (at fixed pressure difference) κmin ∼ a3 pM 3(D−1)/2D .

(16)

Thus, a tubular membrane confines a polymeric fractal only if its bending modulus is in the interval κmin < κ < κ∗ , where the typical value κ∗ is that given above. The following paragraph is devoted to the confinement of a polymer of arbitrary topology to two parallel fluctuating fluid membranes.

IV. CONFINED POLYMERIC FRACTAL IN GEOMETRY II A. Basic backgrounds

Consider a lamellar phase formed by two parallel bilayer membranes. The cohesion between these lipid bilayers is provided by long-ranged van der Waals forces [54]. But these attractive interactions are balanced, at short membrane separation, by strong repulsion coming from hydration forces for uncharged bilayers [55]. In addition, for bilayers carrying electric charges, the attractive interaction is reduced by 59

M. Benhamou et al.

African Journal Of Mathematical Physics Volume 10(2011)55-64

the presence of electrostatic forces [54, 56, 57]. For two parallel bilayer membranes that are a finite distance l apart, the total interaction energy (per unit area) is the following sum V (l) = VH (l) + VE (l) + VW (l) .

(17)

The first part represents the hydration energy. The hydration forces that act at small separation of the order of 1 nm, have been discovered for multilayers under external stress [55]. The adopted form for the hydration energy is an empirical exponential decay VH (l) = AH e−l/λH .

(18)

The typical values of amplitude AH and potential-range λH are AH ≃ 0.2 J/m2 and λH ≃ 0.3 nm. The electrostatic energy between two charged membranes also decays exponentially, that is VE (l) = AE e−l/λE ,

(19)

provided that the separation l is greater than the Debye-H¨ uckel length. The potential amplitude and its range depend on ionic concentration in the aqueous solution, χ, and the surface charge density at membranes, σ0 . More precisely, λE and AE scale as : λE ∼ χ−1/2 and AE ∼ σ02 χ−1/2 . The last part is the attractive van der Waals energy that results from polarizabilities of lipid molecules and water molecules. This interaction energy has the standard form [ ] W 1 2 1 VW (l) = − − . (20) 2 + 2 12π l2 (l + δ) (l + 2δ) The Hamaker constant is in the range W ≃ 10−22 −10−21 J. In the above expression, the bilayer thickness δ is of the order of δ ≃ 4 mm. In principle, one must add a steric interaction energy that originates from membranes undulations. According to Helfrich [45], this energy (per unit area) is 2

Vs (l) = cH

(kB T ) , κl2

(21)

with kB the Boltzmann’s constant, T the absolute temperature, and κ the common bending rigidity constant of the two membranes. But, for two bilayers of different bending rigidity constants κ1 and κ2 , we have κ = κ1 κ2 / (κ1 + κ2 ). The precise value of coefficient cH remains an open debate. Indeed, Helfrich gave for this coefficient the value cH ≃ 0.23 [45], but computer simulations predict smaller prefactors, namely, cH ≃ 0.16 [58], cH ≃ 0.1 [59], cH ≃ 0.07 [60] and cH ≃ 0.08 [61]. Therefore, the steric interaction energy is significant only for those membranes of small bending modulus. Of course, this energy vanishes for rigid interfaces (κ → ∞). We note that the lamellar phase remains stable at the minimum of the potential, provided that the potential-depth is comparable to the thermal energy kB T . This depends, in particular, on the value of W -amplitude. From a theoretical point of view, Lipowsky and Leibler [62] had predicted a phase transition that drives the system from a state where the membranes are bound to a state where they completely separated. Such a phase transition is first-order if the steric repulsive energy is taken into account. But, if this energy is ignored (for relatively rigid membranes), the transition is rather second-order. We restrict ourselves to second-order phase transitions, only. The authors have shown that there exists a certain threshold Wc beyond which the van der attractive interactions are sufficient to bind the membranes together, while below this characteristic amplitude, the membrane undulations dominate the attractive forces, and then, the membranes separate completely. In fact, the critical value Wc depends on the parameter of the problem, which are temperature T , and parameters AH , λH , δ and κ. For room temperatures and AH ≃ 0.2 J/m2 , λH ≃ 0.3 nm, and δ ≃ 4 nm, one has Wc ≃ (6.3 − 0.61) × 10−21 J, when the bending rigidity constant is in the range κ ≃ (1 − 20) × 10−19 J. For instance, for egg lecithin, one has [63] κ ≃ (1 − 2) × 10−19 J, and the corresponding threshold Wc is in the interval Wc ≃ (6.3 − 4.1) × 10−21 J. We note that the typical value Wc corresponds to some temperature, Tc , called unbinding critical temperature [62, 64].

60

M. Benhamou et al.

African Journal Of Mathematical Physics Volume 10(2011)55-64

Let us first consider uncharged membranes, and notice that the Hamaker constant may be varied changing the polarizability of the aqueous medium. It was found [62] that, when the critical amplitude is approached from above, the mean-separation between the two membranes diverges according to ( ) −ψ −ψ H ∼ (Tc − T ) ∼ (W − Wc ) , W → Wc+ or T → Tc− . (22) Here, ψ is a critical exponent whose value is [62] ψ ≃ 1.00 ± 0.03 .

(23)

Such an exponent was computed using field-theoretical Renormalization-Group. From an experimental point of view, critical fluctuations in membranes were considered in some experiment [65], and in particular, the mean-separation was measured. We have now all ingredients to study the conformation of a single polymeric fractal of arbitrary topology, which is confined to two parallel fluid membranes forming a lamellar phase.

B. Polymer parallel extension

First, we note that the polymer is confined only when its three-dimensional gyration radius RF 3 ∼ −ψ aM (D+2)/5D is much greater than the mean-separation H ∼ (Tc − T ) , that is H << RF 3 . This condition implies that the polymer confinement is possible only when temperature T is below some value T ∗ = Tc − aM −(D+2)/5Dψ . This temperature is then smaller than the critical unbinding one Tc , and the shift Tc − T ∗ essentially depends on the polymer mass M and its spectral dimension D. The first implication of the polymer confinement is that, its behavior becomes two-dimensional. This means that the polymer can be regarded as a two-dimensional polymeric fractal formed by blobs (pancakes) of size H. To determine the parallel extension of the polymer, R∥ , as before, we start from a FD free energy R∥2 F M2 = 2 +v 2 , kB T R0 R∥ H

(24) 0

where v denotes the excluded volume parameter. There, R0 ∼ aM 1/dF is the ideal radius and R∥2 H represents the volume occupied by the fractal. Minimization of the above free energy with respect to R∥ gives ψ/4

R∥ ∼ aM (D+2)/4D (W − Wc )

ψ/4

∼ aM (D+2)/4D (Tc − T )

,

(T << T ∗ ) .

(25)

We have used Eq. (22). The above result calls the following remarks. First, the expression of the polymer parallel extension combines two critical phenomena, one is related to the long-mass limit of the polymeric fractal and the other to the vicinity of the unbinding transition of the membranes. Second, in this formula, naturally appears the fractal dimension (D + 2) /4D of a two-dimensional polymeric fractal. Third, the parallel radius becomes more and more smaller as the unbinding transition is reached. In other word, this radius is important only when the two adjacent membranes are strongly bound. Finally, the parallel radius expression may be used to determine the unbinding critical exponent ψ, in X-ray experiment, for instance.

V. CONCLUDING REMARKS

In this paper, we have two objectives, namely the conformational study of a polymeric fractal inside a tubular vesicle or between two parallel membranes forming an equilibrium lamellar phase. For the former, the length scale is the equilibrium diameter that depends on the characteristics of the membrane, which are the bending modulus and the pressure difference between inner and outer aqueous media. For 61

M. Benhamou et al.

African Journal Of Mathematical Physics Volume 10(2011)55-64

the second geometry, the length scale is the mean-separation between the adjacent membranes. The main quantity to consider was the parallel extension of the confined polymeric fractal. Such a quantity was computed within the framework of an extended Flory-de Gennes theory. Notice that the same result may be recovered using scaling argument or blob model [8]. Another physical quantity to consider is the confinement free energy, denoted ∆F . It is the reduced free energy of the polymer, measured from the state with H = ∞. Here, H is the tube diameter or the mean-separation. The confinement free energy must have the scaling form : ∆F = kB T f (RF 3 /H), where the unknown scaling function f (x) has the following features : f (x) ∼ 0, for x << 1 and f (x) ∼ xm , for x >> 1. The exponent m can be obtained using the fact that ∆F must be extensive as a function −5D/(D+2) of the total mass M . This gives m = dF (3) = 5D/ (D + 2). Therefore, ∆F = kB T M (H/a) , 1/3 −ψ with H ∼ (κ/p) , for tubular vesicles, and H ∼ (W − Wc ) , for lamellar phases. Notice that ∆F may be measured by comparison of the two concentrations in the pore (tube or slab) and in the bulk solution. We have Cpore /Cbulk = α exp (−∆F/kB T ), where α is a certain coefficient depending on the ratio RF 3 /H. We emphasize that for charged membranes forming the lamellar phase, it was demonstrated [62] that the −ψ mean-separation between two adjacent bilayers scales as : H ∼ (χ − χc ) , as χ → χ+ c , where χ is the ionic concentration of the aqueous medium and χc is its critical value. Of course, the latter depends on the nature of the lipid system. For instance, for DPPC in CaCl2 solutions, experimental measurements [66] showed that χc is in the range χc ≃ 84 − 10 mM. In this case, the parallel radius of the polymer ψ/4 scales as : R∥ ∼ aM (D+2)/4D (χ − χc ) . We said above that, for T > Tc , the shape fluctuations drive the membranes forming lamellar phase apart even in the presence of the direct attractive forces. In this case, the system recovers its bound state by a simple application of an external pressure or a lateral tension. In the presence of an external pressure P , it was found [64] that the mean-separation H scales as : H ∼ ξ⊥ ∼ P −1/3 (ξ⊥ being the membrane mean-roughness). Such a behavior agrees with MC data [67]. In this case, the parallel extension of the polymer obeys the scaling law : R∥ ∼ aM (D+2)/4D P 1/12 . As it should be, this extension increases with increasing external pressure. The role of a lateral tension is to suppress the bending undulations and the fluctuation-induced repulsion. In fact, the latter becomes short-ranged, and the long-ranged van der Waals attraction then dominates [64]. For this case, it was found [64] that the mean-separation behaves as : H ∼ ξ⊥ ∼ Σ−1/2 , where Σ represents the lateral tension. Then, the resulting parallel extension is : R∥ ∼ aM (D+2)/4D Σ1/8 . As expected, the parallel radius of the polymer increases as the lateral tension is augmented. For tubular vesicles, we have assumed that they are tensionless. If it is not the case, and if the pressure difference between inner and outer aqueous media can be neglected, the equilibrium diameter is : H = 1/2 2 (2κ/γ) , which is solution to Eq. (9). Here, γ is the interfacial tension coefficient. In this case, the ( )1/3 parallel extension of the polymer is : R∥ ∼ aM (D+2)/3D a2 γ/κ . Naturally, this extension increases with increasing interfacial tension coefficient. The minimal diameter corresponds to a typical value γmin ∼ a−2 κM (1−D)/D of γ (κ is fixed). Therefore, the polymer is confined if only if γ < γmin . Finally, further questions in relation with the subject are under consideration.

ACKNOWLEDGMENTS

We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the ”First International Workshop On Soft-Condensed Matter Physics and Biological Systems”, 14-17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank the Professor C. Misbah for fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kinds of hospitalities during his regular visits.

62

M. Benhamou et al.

African Journal Of Mathematical Physics Volume 10(2011)55-64 REFERENCES

1

E.F. Cassasa, J. Polym. Sci. B 5, 773 (1967). P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, 1979. 3 M. Daoud and P.-G. de Gennes, J. Phys. (France) 38, 85 (1977). 4 F. Brochard and P.-G. de Gennes, J. Phys. (France) 40, L-399 (1979). 5 E. Raphael and P. Pincus, J. Phys. II (France) 2, 1341 (1992). 6 J.T. Brooks and M.E. Cates, J. Chem. Phys. 99, 5467 (1993). 7 J. Lal, S.K. Sinha, and L. Auvray, J. Phys. II (France) 7, 1597 (1997). 8 T.A. Vilgis, P. Haronska, and M. Benhamou, J. Phys. II (France) 4, 2139 (1994). 9 S. Orbach and R. Orbach, J. Phys. (France) 43, 1341 (1992). 10 B. Duplantier, Phys. Rev. Lett. 62, 23372341 (1989), and references therein. 11 T. Dotera and Y.Y. Suzuki, Phys. Rev. E 62, 53185323 (2000). 12 Y.Y. Suzuki, T. Dotera, and M. Hirabayashi, Simulation of Gaussian and Excluded-Volume Chains in Curved Slits, 3rd International Symposium on Slow Dynamics in Complex Systems, AIP Conference Proceedings, Volume 708, pp. 257-258, 2004. 13 E.Z. Radlinska et al., Phys. Rev. Lett. 74, 42374240 (1994). 14 K. Nakaya et al., Europhys. Lett. 71, 494 (2005). 15 G. Bouglet, Thesis, Montpellier II University, France, 1997. 16 C. Ligoure, G. Bouglet, and G. Porte, Phys. Rev. Lett. 71, 3600 (1993). 17 C. Ligoure, G. Bouglet, G. Porte, and O. Diat, J. Phys. II (France) 7, 473 (1997). 18 A. Maldonado et al., Journal of Colloid and Interface Science 296, 365 (2006). 19 P.A. Halban and J.C. Irminger, Biochem. J. 299, 1 (1994). 20 M.G. Farquhar and H.P. Hauri, Protein Sorting and Vesicular Traffic in the Golgi Apparatus, in The Golgi Apparatus, E.G. Berger and J. Roth, Eds., pp. 63-129, 1997. 21 B. Alberts, Transport des Prot´eines dans la Cellule : Essentiel de la Biologie Cellulaire, Garland Eds., 1999. 22 C. Leduc et al., PNAS 101, 17095 (2004). 23 J.C. Howard, Mechanics of Motors Proteins and the Cytoskeleton, Sinauer, Sunderland, MA, 2001. 24 A. Roux et al., Proc. Nath. Acad. Sci. USA 99, 5394 (2002). 25 I. Derenyi, F. Julicher, and J. Prost, Phys. Rev. Lett. 88, 238101 (2002). 26 B. Bozic, V. Heinrich, S. Svetina, and B. Zeks, Eur. Phys. J. E. 6, 91 (2001). 27 D.J. Bukman, J.H. Yao, and M. Wortis, Phys. Rev. E. 54, 5463 (1996). 28 V. Heinrich, B. Bozic, S. Svetina, and B. Zeks, Biophys. J. 76, 2056 (1999). 29 S. Zhang, J. Phys. Soc. Japan. 68, 3603 (1999). 30 A. Smith, E. Sackmann, and U. Seifert, Phys. Rev. Lett. 92, 208101 (2004). 31 T.R. Powers, G. Huber, and R.E. Goldstein, Phys. Rev. E. 65, 041901 (2002). 32 D. Raucher and M.P. Sheetz, Biophys. J. 77, 1992 (1999). 33 D.K. Fygenson et al., Phys. Rev. Lett. 79, 4497 (1997). 34 More details can be found in : D.R. Daniels and M.S. Turner, Phys. Rev. Lett. 95, 238101 (2005). 35 D. Boal, Mechanics of the Cell, Cambridge University Press, 2001. 36 B. Alberts et al., Molecular Biology of the Cell, Garland, New York, 2002. 37 R.W. Briehl et. al., J. Mol. Biol. 245, 710 (1995). 38 See Ref. [8], and references therein. 39 D. Lhuillier, J. Phys. (France) 49, 705 (1988). 40 T.A. Vilgis, J. Phys. (France) 49, 1481 (1988). 41 J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. 42 C. Itzykson and J.-M. Drouffe, Statistical Field Theory : 1 and 2, Cambridge University Press, 1989. 43 K. Wolfgang, Differential Geometry : Curves - Surfaces - Manifolds, American Mathematical Society, 2005. 44 S.J. Singer and G.L. Nicolson, Science 175, 720 (1972). 45 W. Helfrich, Z. Natureforsch 28c, 693 (1973). 46 O.-Y. Zhong-Can and W. Helfrich, Phys. Rev. Lett. 59, 2486 (1987). 47 Z.C. Tu and Z.C. Ou-Yang, Phys. Rev. E 68, 061915 (2003). 48 O.-Y. Zhong-Can, Phys. Rev. A 41, 4517 (1990). 49 Z. Lin, R.M. Hill and H.T. Davis, L.E. Scriven, and Y. Talmon, Langmuir 10, 1008 (1994). 50 M. Mutz and D. Bensimon, Phys. Rev. A 43, 4525 (1991). 2

63

M. Benhamou et al.

African Journal Of Mathematical Physics Volume 10(2011)55-64

51

A.S. Rudolph, B.R. Ratna, and B. Kahn, Nature 352, 52 (1991). H. Naito, M. Okuda, and O.Y. Zhong-Can, Phys. Rev. E 48, 2304 (1993). 53 Z.C. Tu, L.Q. Ge, J.B. Li, and Z.C. Ou-Yang, Phys. Rev. E 72, 021806 (2005). 54 J.N. Israelachvili, Intermolecular and Surface Forces, 2nd edition, Academic Press, London, 1991. 55 R.P. Rand and V.A. Parsegian, Biochim. Biophys. Acta 988, 351 (1989). 56 E.J.W. Verwey and J.Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. 57 B.V. Derjaguin, N.V. Churaev and V.M. Muller, Surface Forces, Consultants Bureau, New York, 1987. 58 R. Lipowsky and B. Zielinska, Phys. Rev. Lett. 62, 1572 (1989). 59 N. Gouliaev and K.J.F. Nagle, Phys. Rev. Lett. 81, 2610 (1998). 60 W. Janke and H. Kleinert, Phys. Rev. Lett. 58, 144 (1987); Phys. Lett. A 117, 353 (1986). 61 G. Gompper and D.M. Kroll, Europhys. Lett. 9, 59 (1989). 62 R. Lipowsky and S. Leibler, Phys. Rev. Lett. 56, 2541 (1986). 63 W. Harbich and W. Helfrich, Chem. Phys. Lipids 36, 39 (1984). 64 An extensive list of references can be found in : Structure and Dynamics of Membranes : Generic and Specific Interactions, R. Lipowsky and E. Sackmann, Eds, Volume 1B, Elsevier, 1995. 65 R. Zhang et al., Phys. Rev. Lett. 74, 2832 (1995). 66 L.J. Lis, W.T. Lis, V.A. Parsegian, and R.P. Rand, Biochemistry 20, 1771 (1981). 67 R. Lipowsky and B. Zielinska, Phys. Rev. Lett. 62, 1572 (1989). 52

64

Fluctuation spectra of supported membranes via long-flexible polymers

T. El Hafi 1, M. Benhamou 1,2, K. Elhasnaoui 1, H. Kaidi 1,3 1

2

LPPPC, Sciences Faculty Ben M’sik, P.O. Box 7955, Casablanca, Morocco ENSAM, Moulay Ismail University, P.O. Box 25290, Al Mansour, Meknes, Morocco 3 CRMEF, P.O. Box 255, Meknes, Morocco ([email protected])

Abstract. We consider a fluid membrane that is supported to a solid substrate via longflexible polymers (as proteins or lipopolymers). In this paper, we aim at a quantitative investigations of the associated fluctuation spectra. The study is achieved through the knowledge of the height correlation function between points on the membrane. First, we compute exactly such a function for all pairs of points of the membrane, versus the parameters of the problem, which are the elastic constants of the membrane (bending modulus and interfacial tension) and the molecular-weight of the connected polymer chains. Second, from this correlation function, we extract the exact expression of the fluctuations amplitude and discuss its dependence on molecular-weight. Third, we write exact scaling laws for both height correlation function and fluctuations amplitude. Finally, we emphafsize that the presence of the connected polymers may be a new mechanism to bind membranes in the vicinity of substrates for specific interests. Keywords : Supported membranes; Tethers; Polymers; Correlation function; Fluctuation spectra.

1. Introduction Lipid-bilayer membranes supported on solid substrates are widely used as cellsurface models connecting biological and artificial materials. Usually, the supported biomembranes can be placed either directly on solids or on ultrathin polymer supports that play the role of the extracellular matrix [1]. The main goal is to control, organize and study the properties and function of membranes and membrane-associated proteins. Supported lipid-membranes have other practical interests [2]. Indeed, they permit the biofunctionalization of inorganic solids and polymeric materials, and provide a natural environment for the immobilization of proteins (such as 1

hormone receptors and antibodies). They allow the preparation of ultrathin, high-electric-resistance layers on conductors and the incorporation or receptors into these insulating layers for the design of biosensors. In addition, they are used in Electrochemistry for the detection of protein activity by measuring the resistance and capacitance changes [3-6]. Currently, supported membranes can be assemled as follows : (i) the inner monolayer of the lipid-bilayer fixed to the substrate by covalent chemical bonds or by ion bridges, (ii) freely supported lipid-protein bilayers separated from the substrate by ultrathin water layers (about ten nanometers), and (iii) bilayer membranes linked to the substrate by grafted or adsobed polymers, as lipopolymers [7]. The fixation of these lipopolymers on flat surfaces was studied using Surface Plasmon Spectroscopy. The film thickness of the adsorbed lipopolymer is about 12 to 20Å. Statistical Mechanics of supported membranes has been the subject of many theoretical investigations [8]. The majority of these works considered tethers are springs. In this paper, however, the tethers are long-flexible polymer chains. We shall assume that the two extremities of each polymer chain on the inner monolayer of the supported bilayer membrane and wall remain parallel each to other. The purpose is the study of the fluctuation spectra of supported membrane pairs. This is done through the computation of the height-height correlation function. We have computed this function exactly, from which we extracted an important quantity that is the fluctuations amplitude, even when the polymer chains are polydisperse. Such an amplitude is naturally space-dependent function. The remaining of presentation proceeds as follows. Section 2 deals with the description of the used extended Hamiltonian. In Section 3, we present the exact computation of the correlation function of the supported membrane. Section 4 is dedicated to the fluctuation spectra and discussion. Some concluding remarks are drawn in the last section. Finally, some technical details are relegated in Appendices A and B.

2

2. Extended Canham-Helfrich Hamiltonian Start with a fluctuating fluid membrane that is embedded in a three-dimensional homogeneous aqueous medium, delimitated by a plan rigid substrate (semiinfinite geometry). We assume that the membrane is linked to the solid surface by polymer chains. We denote by the position-vector of the second extremity of chain on the supported interface. For simplicity, we suppose that the two extremities of a given chain remain along the same axis that is perpendicular to the fluid membrane and substrate. To be more general, the positions of anchoring extremities are assumed to be distributed at random. Within the framework of the Monge representation, a point on the membrane can be described by , = ℎ , where = , is the two-dimensional transverse vector and ℎ is the height-function (perpendicular distance to the surface). The Statistical Mechanics of the supported fluid membrane is based on an extended Hamiltonian [8] ℎ + ∑

ℎ =

ℎ

,

(1)

with the bare Canham-Helfrich Hamiltonian [9,10] !

ℎ =

" ∆ℎ

+ $ ∇ℎ

,

(2)

where " is the membrane bending modulus, $ is the interfacial tension coefficient. The second part in the r.h.s of Hamiltonian expression (1) accounts for the elastic contribution due to the presence of the connected hydrocarbon denotes the elastic constant of chains (springs), which is not local. There, polymer chain . The generalized Hamiltonian (1) describes the so-called model D in literature [8]. It will be convenient to rewrite the total Hamiltonian (1) as follows ℎ =

!

" ∆ℎ

+ $ ∇ℎ

+ &ℎ

,

(3)

with the local function '

=∑

( 3

−

.

(4)

Here, ( denotes the two-dimensional Dirac-function. Then, the term &ℎ /2 plays the role of a local confinement potential that maintains the position of the fluid membrane around some typical distance, ℎ+ . 3. Exact height-correlation function With the help of the above total Hamiltonian, we compute the expectation meanvalue of various functionals of the height-field ℎ, in particular the two-point correlation function , , ′ = 〈ℎ

ℎ ′ 〉 − 〈ℎ

〉〈ℎ ′ 〉 .

(5)

Such a function measures the height-fluctuations. As we shall see below, this correlation function allows the determination of the local membrane-roughness. Notice that the presence of tethers breaks the translation symmetry in the parallel directions to the supported membrane. Now, the aim is the computation of the expected correlation function. To this end, we start from the following second-order differential equation "̂ ∆1 − $2Δ1 + '2

, , ′ =(

− ′ ,

(6)

+ 4 /4 . We with the two-dimensional Laplacian operator Δ1 = 4 /4 have introduced the rescaling parameters : "̂ = "/ 5 6, $2 = $/ 5 6, 7 = / 5 6, where 6 is the absolute temperature and 5 is the Boltzmann’ constant. There, '2 = '/ 5 6. Differential equation (6) is directly obtained from Hamiltonian (1) by simple functional derivatives. To compute the height-correlation function, the first step consists to transform differential equation (6) into the following integral equation solved by the propagator ,, , , ′ =,

− ′ − !

′′,

− ′′ '2 ′′ , ′′ − ′ ,

(7)

or equivalently, , , ′ =,

− ′ −∑

, 4

−

,

− ′ ,

(7a)

where , − ′ is the (bare) propagator of a non-supported fluid membrane =0 , ,

− ′ = 〈ℎ

ℎ ′ 〉 − 〈ℎ

〉 〈ℎ ′ 〉 =

;. A ; BA ;C 8 9 :; = >?

E : F GH E :9 < 9D

.

(8)

The notation 〈… 〉 means the thermal average, which is performed with the defined in Eq. (2). The general expression for Canham-Helfrich Hamiltonian this propagator we do not recall is known in literature [11-13]. The particular value of the propagator , 0 = JK defines the mean-roughness of the membrane, JK . The latter and the in-plane correlation length (beyond which the two-point correlation function exponentially fails), J∥ , are related by : JK = J∥ /16"̂ . To solve integral equation (7a), we set = is arbitrary. It is straightforward to show that ∑

Q

N

,

N

=(

N,

− ′ =,

where the subscript 1 ≤ P ≤ − ′ ,

N

(9)

with the matrix elements Q

N

+7 ,

−

N

.

(10)

− ′ ,

(11)

.

(12)

The inversion of Eq. (7a) gives − ′ = ∑N

,

RS

N

,

N

with the matrix coefficients R

N

= QN = (

N

+ 7N ,

−

N

Matrix R is then the transpose of matrix Q. Combining now relationships (7a) and (11) yields the expression for the height-correlation function , , ′ =,

− ′ −∑

,N

, 5

−

T N,

N

− ′ ,

(13)

with coefficients T

N

=

7 + 7N RS

N

.

(14)

These define a squared matrix, T. In Appendix A, we show that this matrix is symmetric and positive definite. The main result (13) calls the following remarks. Firstly, as it must be, the height-correlation function is not invariant under translations in the parallel directions to the fluid membrane and substrate, but the bare correlation function , does. The translation symmetry breaking can be directly seen on integral equation (7). Secondly, since matrix T is symmetric, the propagator is invariant permutting the points and ′. Thirdly, all dependence of this propagator in elastic constants 's is entirely contained in matrix elements T N 's. Fourthly, the positivity property of the second term in the r.h.s of expression (13) indicates that the presence of tethers considerably reduces the thermal fluctuations. Sixthly, the expected propagator is completely determined by the knowledge of matrix T and bare propagator , . Finally, as it should be, at infinity, that is → ∞ and ′ → ∞, , , ′ goes to zero. 4. Fluctuation spectra and discussion The local membrane mean-roughness, JK , is simply the value of propagator , , ′ for = ′. From expression (13), we obtain the following exact result JK

= JK

−∑

,N

,

−

T N,

−

N

.

(15)

Matrix elements T N ’s depend on all distances between tethers W − N W’s and their elastic constants ’s. Before discussing the above finding, we rewrite it on the following scaling form

6

JK

= JK X1 − ∑

Y

,N

Z NY T

−

−

N

,

(16)

with the dimensionless quantities

Y

−

Z T

N

= JK T

=,

−

N

,

(17)

/ JK

.

(18)

Z N ’s The latter is small than unity and Y 0 = 1. Notice that coefficients T depend only on dimensionless elastic constants 7 JK ’s and renormalized bare

propagators Y − N ’s. Now, let us discuss the obtained result. Firstly, to comprehend the general expression (16), we examine some cases : (i) One tether: Z is a scalar quantity, and we Let be the vector-position. In this case, matrix T have Z = 7 JK /[1 + 7 JK \ . T

(19)

Combing this equality with formula (13) yields JK

= JK ]1 −

a 9 7 _` ^

a 9 7 _` G^

Y

−

.

(20)

(ii) Two tethers: We denote by and the position-vectors of the two tethers. In this case, we Z are as follows find that the matrix elements of T Z T

=

a _`

b

9

[ 7 + 7 7 JK \ , Z T

Z =T

=−

Z T

a _`

b

F

with 7

=

7 7 Y

a _`

b

9

−

[ 7 + 7 7 JK \ , (21a) ,

(21b)

c = 1 + 7 JK

+ 7 JK

+ 7 7 JK

d

− 7 7 JK d Y

−

. (21c)

Then, the associated local roughness is JK

Z Y Z Y 1−T − −T − = JK e Z Y −2T − Y −

f

/

.

(22)

(iii) The tethers have the same length and their extremities coincide at one : point ≡ This case corresponds to a -watermelon tether, and we find that the associated local roughness is given by JK

= JK h1 − iY

−

,

(23)

with the constant i=∑

,N

Z T

N

.

(24)

Z. A Such a constant is nothing else but the sum of all coefficients of matrix T comparison between relations (20) and (23) suggests that a -watermelon tether Z , such that is equivalent to one tether of effective elastic constant j Z JK /[1 + j Z JK \ . i=j

(25)

i = 7 JK /[1 + 7 JK \ .

(26)

Z= 7. j

(27)

Z to that of one tether, 7 . We Such a equality relates the new elastic constant j show in appendix B that the quantity i is exactly given by

By comparison, we have

Secondly, in all cases, the positivity of the second term in the r.h.s of equality (16) implies that the local roughness, JK , remains above the bulk one, JK , at 8

any point of the membrane. Therefore, the presence of tethers has tendency to suppress the shape fluctuations of the supported fluid membrane. Thirdly, for any number of tethers, when the considered point is far from the positions of the tethers extremities ’s, that is | − | ≫ J∥ , the local roughness JK fails exponentially to the bulk one, JK . Finally, let us assume that the tethers are long-flexible polymer chains (in dilute solution) of common polymerization degree, m. In this case, the associated elastic constants are identical and scale as noS ~qS m S , where q represents the monomer size. For infinitely long-polymer chains, that is m → ∞, Z goes to JK /no r, where r is the matrix unity. In this limit, to first matrix T order in m S , the general result (16) becomes JK

= JK s1 −

JK /no

∑

Y

−

+ t mS u .

(28)

The above equality can be seen as a Taylor series in JK /no -variable. 5. Concluding remarks

We recall that the purpose of this paper is the investigation of correlations between points belonging to a supported membrane that is linked to a solid surface via long-flexible polymer chains. The presence of the connected macromolecules can be regarded as a new mechanism to bind the membrane for specific interests. This binding phenomenon depends naturally on the value of the molecular-weight of the connected polymer chains. High-molecular-weights have tendency to increase the shape-fluctuations amplitude. But, low-molecular-weights suppress these fluctuations. According to formula expressing the local fluctuations amplitude, the quality constitutes a pertinent parameter of discussion. In fact, the presence of a good solvent reduce Our predictions reveal that the fluctuations amplitude is local, and it is more important in regions of the membrane with low-grafting density. This means that such amplitude depends strongly on the distribution of the anchoring points. In our description, we assumed that the anchoring points on the inner monolayer of the supported membrane are immobile. In fact, these points free move since they are fixed on polar-heads of some phospholipids. Such a problem will be discussed elsewhere. 9

For simplicity, we have ignored primitive interactions between the supported membrane and the wall, as van der Waals, hydration and shape-fluctuations interactions. Under certain physical circumstances, theses interactions can be neglected (rigid tethers…). Finally, questions such as dynamics, extension of study to more than one membrane and other developments are under consideration.

10

Appendix A The aim is to demonstrate that matrix T, defined in Eq. (14), is symmetric and v and ,̅ , of positive definite. To this end, we introduce two matrices, j respective elements 7 ( N and , − N , with indices 1 ≤ x ≤ and 1 ≤ P ≤ , and ( N = 1, if x = P, and ( N = 0, otherwise. It is easy to see that the matrix B expresses as follows v, R = r + ,̅ j

(A.1)

where r is the matrix unity. On the other hand, the matrix T can be rewritten as v r + ,̅ j v v RS = j T=j

S

.

(A.2)

v S and ,̅ are Such a form clearly suggests that Ty = T, since both matrices j symmetric. Then, matrix T is symmetric. To show that T is positive definite, we consider its inverse matrix TS = v S . Explicitly, we have Rj v S + ,̅ . TS = j

(A.3)

The associated matrix elements are TS

N

= 7S (

N

+,

−

.

N

(A.4)

To prove that matrix T is positive definite, it will be sufficient to show that z≡∑

TS

,N

N N

>0,

(A.5)

for any set of integer variables ,…, , and z = 0, if only if ,…, = 0, … ,0 . Start from the Fourier integral representation of the bare propagator ,

−

N

=

8 9 :; =

; .|A ; } BA ; ~• >?

E : F GH E :9 < 9 D

We show that 11

.

(A.6)

z=∑

• ; .A ; } •………………………………………………………… >? ; .A ;} „? ∑• 8 9 :; |∑}‚ƒ €} = }‚ƒ €~ =

+

E : F GH E :9 D

< 9

.

(A.7)

Here, ̅ denotes the complex conjugate of , and ̅ > 0 is its squared module. Since the elastic constants "̂ , $2 of the fluid membrane are positive, z > 0. The latter vanishes, if only if = 0 x = 1, … , . Matrix TS is then positive definite. Therefore, T is a positive definite matrix. This ends the proof of the desired mathematical properties of matrix T. Appendix B To show formula (25), use is made of relation (A.4) of Appendix A. We start with identity ∑

T

,N

N

TS

N†

=(

†

.

(B.1)

According to Eq. (A.4), we have ∑

,N

T N [ 7 S (N† + JK \ = (

A double summation over indices x and ‡ gives ∑

,N

Z T

N

†

.

= i = 7 JK /[1 + 7 JK \ .

We have Eq. (17). This ends the proof of formula (25).

12

(B.2)

(B.3)

References [1] More details can be found in : M. Tanaka and E. Sackmann, Nature 437, 656 (2005). [2] E. Sackmann, Science 271, 43 (1996). [3] S. Terrettaz, M. Mayer and H. Vogel, Langmuir 19, 55675569 (2003). [4] P.C. Gufler, D. Pum, U.B. Sleytr and B. Schuster, Biochimica et Biophysica Acta 1661, 154 (2004). [5] V. Atanasov, P.P. Atanasova, I.K. Vockenroth, N. Knorr and I. Köper, Bioconjugate Chemistry 17, 631 (2006). [6] L.J.C. Jeuken, R.J. Bushby and S.D. Evans, Electrochemistry Communications 9, 610 (2007). [7] P. Théato, R. Zentel and S. Schwarz, Macromolecular Bioscience 2, 387 (2002). [8] R.-J. Merath and U. Seifert, Eur. Phys. J. E 23, 103 (2007), and references therein. [9] P.B. Canham, J. Theoret. Biol. 26, 61 (1970). [10] W. Helfrich, Z. Natureforsch 28c, 693 (1973). [11] T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). [12] H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005). [13] A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1 (2006).

13

Statistical Theory and Method Abstracts – Zentralblatt Database c 2012 FIZ Karlsruhe

ZMATH 1197.76166 Madmoune, Y.; El Hasnaoui, K.; Bendouch, A.; Kaidi, H.; Chahid, M.; Benhamou, M. Brownian dynamics of nanoparticles in contact with a confined biomembrane. Afr. J. Math. Phys. 8, No. 1, 91-100, Article No. 1010, electronic only (2010). Summary: The system we consider is a fluid membrane confined to two parallel reflecting walls that are separated by a finite distance, L, assumed to be small in comparison to the bulk roughness. The attractive membrane is surrounded by small colloidal particles (nanoparticles). The purpose is the study of Brownian dynamics of these particles, under a change of a suitable parameter, such as temperature, T , or colloidmembrane interaction strength, w. The Brownian dynamics is investigated through the knowledge of the time particle density, which solves the Smoluchowski equation. Solving this equation around the mid-plane, where the essential of phenomenon occurs, we obtain the exact form of the local particle density, as a function of the perpendicular distance and time. In the derived expression, appears some time-scale, τ , which scales as τ ∼ L3 /w. This scale-time can be interpreted as the required time over which the colloidal suspension reaches their final equilibrium state. Also, τ can be regarded as the time-interval over which the particles are trapped in holes and valleys. Classification: 76Z05 82D99 92C30 60K35 Keywords: biomembranes; nanoparticles; confinement; Brownian dynamics http://www.fsr.ac.ma/GNPHE/ajmpVolume8N1-2010/ajmp1010.pdf

–1–

November 13, 2012

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH

Vol. 5. No. 3. May, 2013

T. El Hafi, M. Benhamou, K. Elhasnaoui, H. Kaidi. Fluctuation spectra of supported membranes via long-flexible polymers. International Journal of Academic Research Part A; 2013; 5(3), 5-10. DOI: 10.7813/2075-4124.2013/5-3/A.1

FLUCTUATION SPECTRA OF SUPPORTED MEMBRANES VIA LONG-FLEXIBLE POLYMERS 1

1,2

1

T. El Hafi , M. Benhamou , K. Elhasnaoui , H. Kaidi

1,3

1

2

LPPPC, Sciences Faculty Ben M’sik, P.O. Box 7955, Casablanca, ENSAM, Moulay Ismail University, P.O. Box 25290, Al Mansour, Meknes, 3 CRMEF, P.O. Box 255, Meknes (MOROCCO) [email protected] DOI: 10.7813/2075-4124.2013/5-3/A.1

ABSTRACT We consider a fluid membrane that is supported to a solid substrate via long-flexible polymers (as proteins or lipopolymers). In this paper, we aim at a quantitative investigations of the associated fluctuation spectra. The study is achieved through the knowledge of the height correlation function between points on the membrane. First, we compute exactly such a function for all pairs of points of the membrane, versus the parameters of the problem, which are the elastic constants of the membrane (bending modulus and interfacial tension) and the molecularweight of the connected polymer chains. Second, from this correlation function, we extract the exact expression of the fluctuations amplitude and discuss its dependence on molecular-weight. Third, we write exact scaling laws for both height correlation function and fluctuations amplitude. Finally, we emphafsize that the presence of the connected polymers may be a new mechanism to bind membranes in the vicinity of substrates for specific interests. Key words: Supported membranes; Tethers; Polymers; Correlation function; Fluctuation spectra 1. INTRODUCTION Lipid-bilayer membranes supported on solid substrates are widely used as cell-surface models connecting biological and artificial materials. Usually, the supported biomembranes can be placed either directly on solids or on ultrathin polymer supports that play the role of the extracellular matrix [1]. The main goal is to control, organize and study the properties and function of membranes and membrane-associated proteins. Supported lipid-membranes have other practical interests [2]. Indeed, they permit the biofunctionalization of inorganic solids and polymeric materials, and provide a natural environment for the immobilization of proteins (such as hormone receptors and antibodies). They allow the preparation of ultrathin, high-electric-resistance layers on conductors and the incorporation or receptors into these insulating layers for the design of biosensors. In addition, they are used in Electrochemistry for the detection of protein activity by measuring the resistance and capacitance changes [3-6]. Currently, supported membranes can be assemled as follows : (i) the inner monolayer of the lipid-bilayer fixed to the substrate by covalent chemical bonds or by ion bridges, (ii) freely supported lipid-protein bilayers separated from the substrate by ultrathin water layers (about ten nanometers), and (iii) bilayer membranes linked to the substrate by grafted or adsobed polymers, as lipopolymers [7]. The fixation of these lipopolymers on flat surfaces was studied using Surface Plasmon Spectroscopy. The film thickness of the adsorbed lipopolymer is about 12 to 20Å. Statistical Mechanics of supported membranes has been the subject of many theoretical investigations [8]. The majority of these works considered tethers are springs. In this paper, however, the tethers are long-flexible polymer chains. We shall assume that the two extremities of each polymer chain on the inner monolayer of the supported bilayer membrane and wall remain parallel each to other. The purpose is the study of the fluctuation spectra of supported membrane pairs. This is done through the computation of the height-height correlation function. We have computed this function exactly, from which we extracted an important quantity that is the fluctuations amplitude, even when the polymer chains are polydisperse. Such an amplitude is naturally space-dependent function. The remaining of presentation proceeds as follows. Section 2 deals with the description of the used extended Hamiltonian. In Section 3, we present the exact computation of the correlation function of the supported membrane. Section 4 is dedicated to the fluctuation spectra and discussion. Some concluding remarks are drawn in the last section. Finally, some technical details are relegated in Appendices A and B.

Baku, Azerbaijan| 5

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH

Vol. 5. No. 3. May, 2013

2. EXTENDED CANHAM-HELFRICH HAMILTONIAN Start with a fluctuating fluid membrane that is embedded in a three-dimensional homogeneous aqueous medium, delimitated by a plan rigid substrate (semi-infinite geometry). We assume that the membrane is linked to the solid surface by polymer chains. We denote by ⃗ the position-vector of the second extremity of chain on the supported interface. For simplicity, we suppose that the two extremities of a given chain remain along the same axis that is perpendicular to the fluid membrane and substrate. To be more general, the positions of anchoring extremities are assumed to be distributed at random. Within the framework of the Monge representation, a point on the membrane can be described by ⃗, = ℎ( ⃗) , where ⃗ = ( , ) is the two-dimensional transverse vector and ℎ( ⃗) is the height-function (perpendicular distance to the surface). The Statistical Mechanics of the supported fluid membrane is based on an extended Hamiltonian [8] [ℎ] =

[ℎ] + ∑

ℎ (⃗ ),

(1)

with the bare Canham-Helfrich Hamiltonian [9,10] [ℎ] =

⃗ [ (∆ℎ) + (∇ℎ) ],

∫

(2)

where is the membrane bending modulus, is the interfacial tension coefficient. The second part in the r.h.s of Hamiltonian expression (1) accounts for the elastic contribution due to the presence of the connected hydrocarbon chains (springs), which is not local. There, denotes the elastic constant of polymer chain . The generalized Hamiltonian (1) describes the so-called model D in literature [8]. It will be convenient to rewrite the total Hamiltonian (1) as follows [ℎ] =

∫

⃗ [ (∆ℎ) + (∇ℎ) + ℎ ],

(3)

with the local function ( ⃗) = ∑

( ⃗ − ⃗ ).

(4)

Here, ( ⃗) denotes the two-dimensional Dirac-function. Then, the term ℎ /2 plays the role of a local confinement potential that maintains the position of the fluid membrane around some typical distance, ℎ . 3. EXACT HEIGHT-CORRELATION FUNCTION With the help of the above total Hamiltonian, we compute the expectation mean-value of various functionals of the height-field ℎ, in particular the two-point correlation function ( ⃗, ⃗′) = 〈ℎ( ⃗)ℎ( ⃗′)〉 − 〈ℎ( ⃗)〉〈ℎ( ⃗′)〉.

(5)

Such a function measures the height-fluctuations. As we shall see below, this correlation function allows the determination of the local membrane-roughness. Notice that the presence of tethers breaks the translation symmetry in the parallel directions to the supported membrane. Now, the aim is the computation of the expected correlation function. To this end, we start from the following second-order differential equation ̂ ∆ − Δ + ( ⃗)

( ⃗, ⃗′) =

( ⃗ − ⃗′),

(6)

)+( / ). We have introduced the rescaling with the two-dimensional Laplacian operator Δ = ( / parameters : ̂ = / , = / , = / , where is the absolute temperature and is the Boltzmann’ constant. There, = / . Differential equation (6) is directly obtained from Hamiltonian (1) by simple functional derivatives. To compute the height-correlation function, the first step consists to transform differential equation (6) into the following integral equation solved by the propagator , ( ⃗, ⃗′) =

( ⃗ − ⃗′) − ∫

( ⃗, ⃗′) =

( ⃗ − ⃗′) − ∑

⃗′′

( ⃗ − ⃗′′) ( ⃗′′) ( ⃗′′ − ⃗′),

(7)

( ⃗ − ⃗ ) ( ⃗ − ⃗′),

(7a)

or equivalently,

where

( ⃗ − ⃗′) is the (bare) propagator of a non-supported fluid membrane (

6 | PART A. NATURAL AND APPLIED SCIENCES

= 0),

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH

Vol. 5. No. 3. May, 2013

( ⃗ − ⃗′) = 〈ℎ( ⃗)ℎ( ⃗′)〉 − 〈ℎ( ⃗)〉 〈ℎ( ⃗′)〉 = ∫ (

⃗.( ⃗ ⃗′)

⃗ )

.

(8)

The notation 〈… 〉 means the thermal average, which is performed with the Canham-Helfrich Hamiltonian defined in Eq. (2). The general expression for this propagator we do not recall is known in literature [11-13]. The particular value of the propagator (0) = ( ) defines the mean-roughness of the membrane, . The latter and the in-plane correlation length (beyond which the two-point correlation function exponentially fails), ∥ , are related by : (

) = ∥ /16 ̂ . To solve integral equation (7a), we set ⃗ = ⃗ , where the subscript 1 ≤ to show that ∑

( ⃗ − ⃗′) =

≤

is arbitrary. It is straightforward

⃗ − ⃗′ ,

(9)

with the matrix elements =

⃗ −⃗ .

+

(10)

The inversion of Eq. (7a) gives [

( ⃗ − ⃗′) = ∑

]

⃗ − ⃗′ ,

(11)

with the matrix coefficients =

=

⃗ −⃗ .

+

(12)

Matrix is then the transpose of matrix . Combining now relationships (7a) and (11) yields the expression for the height-correlation function ( ⃗, ⃗′) =

( ⃗ − ⃗′) − ∑

,

(⃗ − ⃗ )

⃗ − ⃗′ ,

(13)

with coefficients =

+

[

]

.

(14)

These define a squared matrix, . In Appendix A, we show that this matrix is symmetric and positive definite. The main result (13) calls the following remarks. Firstly, as it must be, the height-correlation function is not invariant under translations in the parallel directions to the fluid membrane and substrate, but the bare correlation function does. The translation symmetry breaking can be directly seen on integral equation (7). Secondly, since matrix is symmetric, the propagator is invariant permutting the points ⃗ and ⃗′. Thirdly, all dependence of this propagator in elastic constants 's is entirely contained in matrix elements 's. Fourthly, the positivity property of the second term in the r.h.s of expression (13) indicates that the presence of tethers considerably reduces the thermal fluctuations. Sixthly, the expected propagator is completely determined by the knowledge of matrix and bare propagator . Finally, as it should be, at infinity, that is ⃗ → ∞ and ⃗′ → ∞, ( ⃗, ⃗′) goes to zero. 4. FLUCTUATION SPECTRA AND DISCUSSION The local membrane mean-roughness, ( ⃗), is simply the value of propagator expression (13), we obtain the following exact result ( ⃗) = (

) −∑

,

(⃗ − ⃗ )

⃗− ⃗ .

( ⃗, ⃗′) for ⃗ = ⃗′. From

(15)

Matrix elements ’s depend on all distances between tethers ⃗ − ⃗ ’s and their elastic constants Before discussing the above finding, we rewrite it on the following scaling form ( ⃗) =

1−∑

,

(⃗ − ⃗ )

⃗− ⃗ ,

’s.

(16)

with the dimensionless quantities

Baku, Azerbaijan| 7

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH =(

)

Vol. 5. No. 3. May, 2013

,

(⃗ − ⃗ ) =

(17)

( ⃗ − ⃗ )/( ) .

(18)

The latter is small than unity and (0) = 1. Notice that coefficients ’s depend only on dimensionless ⃗ − ⃗ ’s. elastic constants ( ) ’s and renormalized bare propagators Now, let us discuss the obtained result. Firstly, to comprehend the general expression (16), we examine some cases : (i) One tether: Let ⃗ be the vector-position. In this case, matrix is a scalar quantity, and we have = ( ) / 1+ (

) .

(19)

Combing this equality with formula (13) yields ( ⃗) =

of

(

1−

) (

(⃗ − ⃗ ) .

)

(20)

(ii) Two tethers: We denote by ⃗ and ⃗ the position-vectors of the two tethers. In this case, we find that the matrix elements are as follows =

+

( ) , =

=

( ) ,

+

(21a)

( ⃗ − ⃗ ),

=−

(21b)

with =1+

( ) +

( ) +

( ) −

( )

( ⃗ − ⃗ ).

(21c)

Then, the associated local roughness is ( ⃗) =

(⃗ − ⃗ ) − (⃗ − ⃗ )

1− −2

(⃗ − ⃗ ) (⃗ − ⃗ )

/

.

(22)

(iii) The tethers have the same length and their extremities coincide at one point ⃗ ( ⃗ ≡ ⃗ ) : This case corresponds to a -watermelon tether, and we find that the associated local roughness is given by ( ⃗) =

(⃗ − ⃗ ),

1−

(23)

with the constant =∑

.

,

(24)

Such a constant is nothing else but the sum of all coefficients of matrix . A comparison between relations (20) and (23) suggests that a -watermelon tether is equivalent to one tether of effective elastic constant , such that =

( ) / 1+ ( ) .

Such a equality relates the new elastic constant quantity is exactly given by =

(

(25)

to that of one tether, . We show in appendix B that the

) / 1+

(

) .

(26)

By comparison, we have =

.

(27)

Secondly, in all cases, the positivity of the second term in the r.h.s of equality (16) implies that the local roughness, ( ⃗), remains above the bulk one, , at any point ⃗ of the membrane. Therefore, the presence of tethers has tendency to suppress the shape fluctuations of the supported fluid membrane.

8 | PART A. NATURAL AND APPLIED SCIENCES

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH

Vol. 5. No. 3. May, 2013

Thirdly, for any number of tethers, when the considered point ⃗ is far from the positions of the tethers extremities ⃗ ’s, that is | ⃗ − ⃗ | ≫ ∥ , the local roughness ( ⃗) fails exponentially to the bulk one, . Finally, let us assume that the tethers are long-flexible polymer chains (in dilute solution) of common polymerization degree, . In this case, the associated elastic constants are identical and scale as ~ , where represents the monomer size. For infinitely long-polymer chains, that is → ∞, matrix goes to ( / ) , where is the matrix unity. In this limit, to first order in , the general result (16) becomes ( ⃗) =

1− (

/

) ∑

(⃗ − ⃗ ) + (

The above equality can be seen as a Taylor series in

/

).

(28)

-variable.

5. CONCLUDING REMARKS We recall that the purpose of this paper is the investigation of correlations between points belonging to a supported membrane that is linked to a solid surface via long-flexible polymer chains. The presence of the connected macromolecules can be regarded as a new mechanism to bind the membrane for specific interests. This binding phenomenon depends naturally on the value of the molecular-weight of the connected polymer chains. High-molecular-weights have tendency to increase the shape-fluctuations amplitude. But, low-molecular-weights suppress these fluctuations. According to formula expressing the local fluctuations amplitude, the quality constitutes a pertinent parameter of discussion. In fact, the presence of a good solvent reduce Our predictions reveal that the fluctuations amplitude is local, and it is more important in regions of the membrane with low-grafting density. This means that such amplitude depends strongly on the distribution of the anchoring points. In our description, we assumed that the anchoring points on the inner monolayer of the supported membrane are immobile. In fact, these points free move since they are fixed on polar-heads of some phospholipids. Such a problem will be discussed elsewhere. For simplicity, we have ignored primitive interactions between the supported membrane and the wall, as van der Waals, hydration and shape-fluctuations interactions. Under certain physical circumstances, theses interactions can be neglected (rigid tethers…). Finally, questions such as dynamics, extension of study to more than one membrane and other developments are under consideration. REFERENCES 1. More details can be found in : M. Tanaka and E. Sackmann, Nature 437, 656 (2005). 2. E. Sackmann, Science 271, 43 (1996). http://dx.doi.org/10.1126/science.271.5245.43. 3. S. Terrettaz, M. Mayer and H. Vogel, Langmuir 19, 55675569 (2003). http://dx.doi.org/10.1021/ la034197v. 4. P.C. Gufler, D. Pum, U.B. Sleytr and B. Schuster, Biochimica et Biophysica Acta 1661, 154 (2004). http://dx.doi.org/10.1016/j.bbamem.2003.12.009. 5. V. Atanasov, P.P. Atanasova, I.K. Vockenroth, N. Knorr and I. Köper, Bioconjugate Chemistry 17, 631 (2006). http://dx.doi.org/10.1021/bc050328n. 6. L.J.C. Jeuken, R.J. Bushby and S.D. Evans, Electrochemistry Communications 9, 610 (2007). http://dx.doi.org/10.1016/j.elecom.2006.10.045. 7. P. Théato, R. Zentel and S. Schwarz, Macromolecular Bioscience 2, 387 (2002). http://dx.doi.org/ 10.1002/1616-5195(200211)2:8<387::AID-MABI387>3.0.CO;2-5. 8. R.-J. Merath and U. Seifert, Eur. Phys. J. E 23, 103 (2007), and references therein. http://dx.doi.org/ 10.1140/epje/i2006-10084-2. 9. P.B. Canham, J. Theoret. Biol. 26, 61 (1970). http://dx.doi.org/10.1016/S0022-5193(70)80032-7. 10. W. Helfrich, Z. Natureforsch 28c, 693 (1973). 11. T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). http://dx.doi.org/10.1103/ PhysRevE.70.051404. 12. H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005). http://dx.doi.org/10.1209/ epl/i2004-10305-4. 13. A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1 (2006).

Baku, Azerbaijan| 9

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH

Vol. 5. No. 3. May, 2013

APPENDIX A The aim is to demonstrate that matrix , defined in Eq. (14), is symmetric and positive definite. To this end, ⃗ − ⃗ , with indices 1 ≤ ≤ and we introduce two matrices, and ̅ , of respective elements and 1 ≤ ≤ , and = 1, if = , and = 0, otherwise. It is easy to see that the matrix B expresses as follows = + ̅

,

(A.1)

where is the matrix unity. On the other hand, the matrix = ( + ̅

=

can be rewritten as

) .

(A.2) and ̅ are symmetric. Then, matrix

Such a form clearly suggests that = , since both matrices is symmetric. To show that is positive definite, we consider its inverse matrix

=

. Explicitly, we have

+ ̅ .

=

(A.3)

The associated matrix elements are [ To prove that matrix

]

=

⃗ −⃗ .

+

(A.4)

is positive definite, it will be sufficient to show that ≡∑

[

,

for any set of integer variables ( , … , integral representation of the bare propagator ⃗ −⃗

]

> 0,

), and

= 0, if only if ( , … ,

⃗. ⃗

⃗ )

= ∫(

(A.5) ) = (0, … ,0). Start from the Fourier

⃗

.

(A.6)

We show that =∑

⃗ )

+ ∫(

⃗. ⃗

∑

⃗. ⃗

∑

.

(A.7)

Here, ̅ denotes the complex conjugate of , and ̅ > 0 is its squared module. Since the elastic constants ( ̂ , ) of the fluid membrane are positive, > 0. The latter vanishes, if only if = 0 ( = 1, … , ). Matrix is then positive definite. Therefore, is a positive definite matrix. This ends the proof of the desired mathematical properties of matrix .

APPENDIX B To show formula (25), use is made of relation (A.4) of Appendix A. We start with identity ∑

,

∑

,

[

]

=

.

+(

)

(B.1)

According to Eq. (A.4), we have

A double summation over indices ∑

,

and =

=

.

(B.2)

gives =

(

) / 1+

We have Eq. (17). This ends the proof of formula (25).

10 | PART A. NATURAL AND APPLIED SCIENCES

( ) .

(B.3)

J Membrane Biol (2013) 246:383–389 DOI 10.1007/s00232-013-9544-9

The Quantum Casimir Effect May Be a Universal Force Organizing the Bilayer Structure of the Cell Membrane Piotr H. Pawlowski • Piotr Zielenkiewicz

Received: 1 November 2012 / Accepted: 29 March 2013 / Published online: 24 April 2013 Ó The Author(s) 2013. This article is published with open access at Springerlink.com

Abstract A mathematic–physical model of the interaction between cell membrane bilayer leaflets is proposed based on the Casimir effect in dielectrics. This model explains why the layers of a lipid membrane gently slide one past another rather than penetrate each other. The presented model reveals the dependence of variations in the free energy of the system on the membrane thickness. This function is characterized by the two close minima corresponding to the different levels of interdigitation of the lipids from neighbor layers. The energy barrier of the compressing transition between the predicted minima is estimated to be 5.7 kT/lipid, and the return energy is estimated to be 3.1 kT/lipid. The proposed model enables estimation of the value of the membrane elastic thickness modulus of compressibility, which is 1.7 9 109 N/m2, and the value of the interlayer friction coefficient, which is 1.9 9 108 Ns/m3. Keywords effect

Cell membrane  Lipid bilayer  Casimir

Introduction Casimir-Polder (Casimir 1948; Casimir and Polder 1948) forces are universal physical forces arising from a quantized field. They act even between two uncharged metallic P. H. Pawlowski (&)  P. Zielenkiewicz Institute of Biochemistry and Biophysics, Polish Academy of Sciences, PAS, Pawinskiego 5a, 02-106 Warszawa, Poland e-mail: [email protected] P. Zielenkiewicz Plant Molecular Biology Laboratory, Warsaw University, Warsaw, Poland

plates in a vacuum, placed a few micrometers apart, without any external electromagnetic field. The idea that Casimir forces may play an important role in different biomembrane systems is quite new. Based on the fundamental work of Lifshitz (1956) dealing with retarded van der Waals forces between macroscopic bodies, first it was applied to the formation of cellular ‘‘rouleaux’’ (Bradonjic´ et al. 2009) and confined biomembranes (El Hasnaoui et al. 2010). In a natural manner, it supplements the theory of nonretarded van der Waals interactions in a lipid–water system (Parsegian and Ninham 1970), sometimes offering an interesting counterproposal. Thus, forces related to the zero-point energy of quantum fluctuations may play an important role in biology, and the analysis of these forces offers a new view into biological phenomena at the cellular level. Herein, we propose a simple second-quantized explanation for the fact that cell membrane bilayer leaflets slip (Otter and Shkulipa 2007) past one another, rather than penetrate each other. This membrane feature is of great importance as it determines the anisotropy of the membrane’s rheological properties. The relative freedom of movement of molecules along membrane leaflets and the relative restriction of displacement in the transverse direction account for the great lateral fluidity and the small perpendicular compressibility of a membrane (Evans and Hochmuth 1978). These dual mechanical properties, both fluid-like and solid-like, have an impact on the structure and function of the proteins embedded in the lipid bilayer matrix (Andersen and Koeppe 2007). This impact finally determines the status of the cell membrane as an active barrier, a natural organizer and an important participant in all processes of life. The proposed model of the interaction of the cell membrane bilayer leaflets considers the membrane interior to be a three-layer dielectric sandwich. The free energy of

123

384

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

the system is related to the electromagnetic field excitations in the ground state. The free energy depends on the varying thickness of the central layer, where lipid chains penetrate the opposite layers and where the density of lipid chains, and the dielectric permittivity, varies in space. The theoretical values of the two energy minima, the membrane elastic thickness modulus of compressibility and the interlayer friction coefficient, were set according to this model.

The Model The cell membrane interior was considered to be a threelayer dielectric sandwich, which consists of parallel slabs as in Fig. 1. The two lateral peripheral regions of this system are fully occupied by all the lipids of the local leaflet. The remaining central region contains the hydrocarbon tails that penetrate from the neighboring layers. The dielectric constant, ec, of this central layer differs from the dielectric constant, ep, of the other parts of the membrane. We assumed that the length of the lipids in each leaflet may vary within a narrow range, L ± dL/2, where L represents the average lipid length and dL represents the width of the length distribution. The distribution of the lipid lengths was assumed to be uniform. Thus, the perpendicular cross section of a membrane bilayer matrix resembles ‘‘the two overlapping combs with some broken teeth.’’

ep

L

ec

ep

δL

Fig. 1 Three-layer dielectric sandwich model of a cell membrane. Only the longest and shortest lipids are shown. L average lipid length, dL width of the lipid length range, ec dielectric constant of the central region, ep dielectric constant of the peripheral region

123

The thickness of the central region, dc, falls to a minimum value equal to dL in the configuration in which the total membrane thickness, d, equals 2L (Fig. 2a). This configuration we called ‘‘configuration MTCR’’ (minimal thickness of the central region). To clarify further considerations, configuration MTCR was treated as the reference configuration. In this configuration, dc may increase both with an increase (Fig. 2b) and with a decrease (Fig. 2c) in the total membrane thickness. Thus, the possible variation in the thickness of the central region in configuration MTCR is unidirectional. For convenience, we introduced the variable x to describe the difference between the actual total membrane thickness and the thickness of the membrane in configuration MTCR; that is, x = d-2L. Then to formalize the description of the central region, one may write dc = dL ? |x|, where |x| denotes the absolute value. The dielectric constant, ec, was assumed to vary in space within the central region due to variations in the lipid density. Variations along the direction perpendicular to the membrane plane were postulated. At first approximation, ec was simply characterized by the spatial average heci. In general, when the total membrane thickness differs from the thickness of the reference configuration by x, the average heci changes as described by the following formula:  8   jxj=dL > x\0 < 1 þ ep  1 1 þ 1þjxj=dL hec i ¼ ep ð1Þ x¼0 > : 1 þ  e  1 1 x[0 p 1þx=dL For details, see Appendix. The plot of Eq. 1 indicates (Fig. 3) that the average heci increases with the compression of the membrane thickness (x \ 0) and decreases with membrane thickness extension (x [ 0). For the membrane in the configuration MTCR (x = 0), heci strictly equals ep. Quantum electrodynamic considerations reveal that two dielectric plates separated by a dielectric medium may be attracted as a result of the decrease in the zero-point energy of the quantum electromagnetic field excitations (Srivastava et al. 1985). In the case of the cell membrane, by approximating the dielectric constant of the central region with the average value heci, the free energy, F, of the field per unit area may be described by the following equation (Bradonjic´ et al. 2009):   p2 ep  hec i 2 hc pffiffiffiffiffiffiffiffi F¼ 3 720 ep þ hec i hec iðdLÞ ð1 þ jxj=dLÞ3

ð2Þ

where h is the reduced Planck constant and c is the speed of light in a vacuum; the dependence of heci on x is described by Eq. 1.

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

385

d = 2L

d > 2L

d < 2L

MTCR

dc = δ L

a

b

dc > δ L

Fig. 2 Unidirectional variations in the thickness of the central region of the membrane. Only the longest and shortest lipids are shown. a Configuration MTCR. Total membrane thickness d = 2L. Central region thickness dc = dL. b Bilayer leaflets farther apart than in configuration MTCR. Total membrane thickness d [ 2L. Central

dc > δ L

c

region thickness dc [ dL. c Bilayer leaflets closer together than in configuration MTCR. Total membrane thickness d \ 2L. Central region thickness dc [ dL. L average lipid length, dL width of the lipid length range, MTCR minimal thickness of the central region

According to the model, in both cases, the thickness of the ˚) central region of the membrane (dc = 9.0 and dc = 7.9 A is larger than the assumed thickness in the configuration ˚ ). MTCR (dc = 5 A

εc 2εp - 1

εp

Discussion 1

-10

0

10

x/ δ L

Fig. 3 The space average of the dielectric constant in the central region, heci, as a function of the ratio x/dL. x the difference between the total membrane thickness and the membrane thickness in configuration MTCR, dL width of the lipid length range, ep dielectric constant in peripheral regions of the membrane

We assumed that the central region of the membrane may be treated as a kind of leaflet interface. One interesting question is how the interface is stabilized to avoid leaflet penetration and sticking. The proposed model may offer a simple explanation based on the Casimir effect in dielectrics. At reasonable values of the applied parameters (see ‘‘Results’’ section), the model predicts two free energy minima. For the membrane organized in the lower energy ˚ ), its leaflets are separated by 4 A ˚ minimum (x = 4 A

Results

0,03 0,01 2

F [J/m ]

Equations 1 and 2 show that, for ideally flat leaflets (dL = 0), Casimir forces couple membrane layers at zero distance (x = 0), with the free energy tending to negative ˚ , ep = 2), infinity. More reasonable calculations (dL = 5 A taking into account the heterogeneity of the lipid lengths (Blaurock 1982) and specifying the dielectric constant of the peripheral regions of the membrane (Huang and Levitt 1977) were also performed. These calculations indicate (Fig. 4) that there are two finite minima of free energy, both for the membrane in configurations close to the MTCR. The first (lower) minimum (F = -4.68 9 10-2 J/ ˚ ), and m2) is for leaflets being slightly separated (x = 4.0 A -2 2 the second one (F = -2.54 9 10 J/m ) is for leaflets ˚ ). being moderately pushed together (x = -2.9 A

0,05

-0,01 -0,03 -0,05 -0,07 -50

-30

-10

10

30

50

70

90

x [A] Fig. 4 The free energy, F, of the field per unit area as a function of a change, x, in the membrane thickness relative to the thickness of the MTCR membrane

123

386

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

greater than in configuration MTCR. At the assumed het˚ ), this result means erogeneity of the lipid lengths (dL = 5 A that 20 % of lipids (1-x/dL) penetrate into the neighboring ˚ (dL-x). leaflet (Fig. 5a) but by no more than 1 A According to the results presented in Fig. 4, transition ˚ ) requires into the second (higher) minimum (x = –2.9 A approximately 5.7 kT/lipid (calculated for T = 300 K and ˚ 2). In this minimum energy conarea per lipid a = 50 A figuration, the membrane leaflets are in closer contact. All ˚ lipids penetrate the opposite layer but no deeper than 7.9 A (Fig. 5b). The return to the other minimum requires 3.1 kT/ lipid. The revealed characteristic energies are several times higher than those estimated assuming the average tension c (0.003 mN/m) of living cells (Blanchard and Rauch 2012). The result ca  kT indicates that the Casimir effect may be an important contributor to the membrane dynamics, along with the hydrophobic effect. To study the mechanical properties of the presented model membrane, the elastic thickness modulus of

a

100%

n1

-4.5

n2

-0.5 0.5

4.5

ξ [A]

3.95

ξ [A]

b

100%

n1

-3.95

n2

-1.05

1.05

Fig. 5 The area densities n1 and n2 of lipids belonging to a given leaflet as a function of a distance, n, from the membrane midplane. The range occupied by the longest and shortest lipids is shown above. ˚ . b For the a For the first minimum (lower) of free energy, x = 4 A ˚ second minimum (higher) of free energy, x = -2.9 A

123

compressibility, k, was numerically estimated using a differential second derivative of the free energy at the first minimum (k = d F00 ). Assuming a bilayer thickness of d = 5 nm (Kuchel and Ralston 1998), k = 1.7 9 109 N/ m2; i.e., this estimated value of k is of the same order as that measured using volume dilatometry of lipid bilayers (Srinivasan et al. 1974). This result indicates that the central region of the cell membrane resembles an ‘‘incompressible’’ core that does not allow lipid interleaflet penetration. The results of the presented model also enable estimation of the value of the interlayer friction coefficient, f (friction force per unit area and unit velocity). Approximating the end of the lipid penetrating the neighbor leaflet with a hemisphere of a radius r that experiences the action of the Stokes force from the liquid passing with the velocity 2v, one may obtain f = 6pgr/a, where g is the membrane shear viscosity. Taking the typical value of viscosity g = 0.1 Ns/m2 estimated from the diffusion coefficient of membrane-spanning proteins in phospholipid bilayers (Waugh 1982) and assuming that the membrane is ˚ , f = 1.9 9 108 Ns/ in the first minimum with r = 0.5 A 3 m , which is within the range of reported experimental values (Shkulipa et al. 2005; Otter and Shkulipa 2007). For the second minimum, one may expect an eightfold higher value because of a deeper penetration. According to the model, a membrane bilayer in a basic state (lower minimum) should possess relatively small interleaflet friction. The probability that, due to thermal fluctuations, some regions of the membrane reach second minimum is relatively small. At first glance, one may worry about some physical and mathematical problems with the proposed approach. It is obvious that some points require additional discussion, especially phenomena at a physical level neglected in our model. First of all, is it justified to assume that the lipids of opposite leaflets interpenetrate each other at all? Steric interactions seem to be the dominant suppressor of interdigitation. However, a spontaneous or induced interdigitated phase of bilayers consisting of double-tail lipids was confirmed in computer simulation and differential scanning calorimetry experiments (Kranenburg 2004; Kranenburg et al. 2004; Mavromoustakos et al. 2011). Moreover, a simple estimation below shows that the possible steric effect is not energetically dominant, as one may expect. Let us assume that during interdigitation the part p of lipids is compressed and their length decreases by an assumed certain value, k. Let us also assume that each of two lipid acyl chains contains the number b of C–C bonds characterized by a certain bond stiffness, g. Then, elastic deformation energy per lipid molecule may be calculated as pgk2/b. For typical conditions, p = 1/10 (meaning that 50 % of penetrating lipids in the lower minimum are

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

387

˚´ and b = 15; this compressed), g = 100 N/m, k = 0.5 A energy equals 0.8 kBT. These are only 7 % of the predicted value of the energy barrier in a lower minimum and, as such, may be neglected at first approximation. The next question is, how much does the derived shape of the free energy, F (Fig. 4), depend on the specific way that the interpenetration occurs? It was assumed that the distribution of lipid lengths was uniform and, in this way, interdigitation varied linearly with a distance from the membrane midplane (Figs. 5, 6). What will change if we assume the more spectacular variation? In extreme cases, when a single-point distribution (dL = 0) is assumed, one infinite minimum of energy at midplane will be obtained and the energy will increase with distance, like –1/|x|3. Thus, narrowing the distribution lowers minima and approaches them together. It should be stressed that the parameters heci, ep and dL in Eq. 2 may also effectively describe a more diverse system. Applied formula for the free energy per unit area, Eq. 2, assuming ep ? ? and heci ? ?, gives the famous Casimir result for the energy of attraction between ideal mirrors. This energy is a result of the change in the zero-point energy of an empty quantum vacuum. From the other hand, it is well known that real cell membranes are under the permanent influence of electrostatic interactions. Assuming a natural transmembrane electric potential V = 100 MV, a membrane thickness dm = 10-8, an effective membrane dielectric constant em = 2 and a vacuum permittivity e0 = 10-11 [F/m], it is easy to estimate the area density of the energy of an electric field, 0.5e0em (V2/dm). It is equal to 10-5 J/m2 . This result is three orders less than the energy of the considered Casimir effect.

np

n1

n2

ξ -L-x /2

−δ L/2-x /2

−δ L/2+x/2

0

δ L/2-x /2

δ L/2 +x/2

L +x/2

Fig. 6 The area densities n1 and n2 of lipids belonging to a given leaflet as a function of the distance, n, from the membrane midplane. The range occupied by the longest and shortest lipids is shown above. np total number of lipids belonging to a given leaflet per unit surface area, L average lipid length, dL width of the lipid length range, x = d - 2L, where d is the total membrane thickness

It is necessary to underline that the discussed formula is correct only within certain assumed constraints. One of them, zero temperature approximation (Landau and Lifshitz 1960) may be justified for considered conditions (kBT  hc/dc). The second assumption, i.e., the same dielectric constant at all frequencies, is formally valid for ideal dielectrics or in the case of large separations. However, for values of constant permittivity not so far from unity (vacuum value), possible error for small distances is disregarded. Moreover, the bulk term in energy is disregarded, which may be dominant for large separations. As the ratio of energies of attraction of water slabs and lipid slabs (both estimated using Eq. 2) is as small as 1:10, for the sake of simplicity, the influence of water outside the membrane was not considered. Anisotropy in the dielectric constant is also beyond the scope of this article, and the model for the dielectric constant in the region of interdigitation is very simple, based on a linear superposition of dielectric constants in terms of the effective densities of tails. We realize that real membranes, especially cell membranes, are obviously complex, heterogeneous, nonideal dielectrics with complicated frequency responses and real conductivity. They are certainly not perfectly plain and smooth plates. Despite the above simplifications, the general predictions of our model, i.e., the magnitude of the free energy and the existence of two energy minima, seem to be still reasonable and wait for more detailed further investigation and confirmation. A lateral Casimir effect for corrugated planes or nonretarded local pairwise van der Waals forces might provide a better description of the physics there and provide a more accurate description. For example, the last one can replace an inverse-cube law, describing variation in energy with distance, by an inverse-square law. Using this method, a simple estimation of energy of so-called hydrophobic bonding, at a Hamaker function ´˚ equal to 7 9 10-21 J and a distance of 50 A (Parsegian and Ninham 1970), gives the value of the density of free energy close to 10-5 J/m2. This quantity is three orders less than the energy of lipid–lipid interactions and two orders less than the energy of water–water interactions estimated in our model. We think that future numerical brute-force simulations might make an important contribution toward a better understanding of the mentioned discrepancy. In light of these findings, it is evident that the Casimir effect may play an important role in many biological phenomena and may be a universal force that organizes the tensegrity structure of biological systems. Some scientists might expect spectacular ‘‘levitation’’ forces between the leaflets, but it appears that there is instead a ‘‘quantum trap’’ preventing membrane leaflet interdigitation and collapse as well as maintaining a significant gap between leaflets, which leaves molecules a plane with freedom of

123

388

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

movement. Even without taking into account interlayer lipid collisions, hydrophobic interactions and the stabilizing role of proteins, Casimir forces may prevent lipid chain mishmash or molecular escape. This hypothesis appears to be fruitful and is worth experimental verification. Acknowledgements We thank Prof. Bogdan Lesyng and Dr. Marek Kalinowski from the University of Warsaw for inspiration and valuable discussion. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

8 > <0   n nx=2 n2 ¼ 2p 1 þ dL=2 > : np

ð6Þ Independently of the sign of x, the central region consists of three layers: 8 < dL=2  j xj=2  n\  dL=2 þ j xj=2 dL=2 þ j xj=2  n  dL=2  j xj=2 ð7Þ : dL=2  j xj=2\n  dL=2 þ j xj=2 Calculating the sum in Eq. 4, with the help of Eqs. 5, 6 in respective layers, depending on the sign of x, one can obtain:   8 n > np þ 2p 1 þ nx=2 > dL=2 > <   x n ¼ np 1  dL >   > > : n þ np 1  nþx=2 p dL=2 2

Appendix The Average Value of the Dielectric Constant in the Central Region of a Lipid Bilayer

L  x=2  n\  dL=2 þ x=2 dL=2 þ x=2  n  dL=2 þ x=2 dL=2 þ x=2\n  L þ x=2

dL=2  jxj=2  n\  dL=2 þ j xj=2 dL=2 þ jxj=2  n  dL=2  jxj=2

x\0

dL=2  jxj=2\n  dL=2 þ j xj=2

ð8Þ Let variable n represent the distance from the membrane midplane; then, the central region may be defined as the membrane layer within the range dL=2  j xj= 2  n  dL=2 þ j xj=2. Here, dL represents the width of the distribution of lipid lengths, and x = d - 2L, where d is the total membrane thickness and L is the average lipid length. In general, x may be positive (thickness expansion), negative (thickness compression) or zero (configuration MTCR) and falls within the range x [ dL - 2L. Assuming that the dielectric constant ec in the central region varies locally with n, ec depends on the number of lipids, n, that are passing through the unit area of plane n = constant. This dependence may be described as n ec ¼ 1 þ ðep  1Þ ð3Þ np where ep is the dielectric constant of the peripheral region of the membrane fully occupied only by the lipids belonging to a given leaflet and np is the total number of lipids belonging to a given leaflet per unit surface area. The area density n of lipids at distance n can be described by the equation n ¼ n1 þ n2

ð4Þ

where n1 and n2 are area densities of lipids belonging to a given leaflet. For a uniform distribution of lipid lengths, the constituent densities n1 and n2 (Fig. 6) can be described as follows: 8 > < np  L  x=2  n\  dL=2  x=2 np nþx=2 n1 ¼ 2 1  dL=2 dL=2  x=2  n  dL=2  x=2 > : 0 dL=2  x=2\n  L þ x=2 ð5Þ

123

n ¼ np

dL=2  n  dL=2

 8  np nþx=2 > 1  > dL=2 2 > <   x n ¼ np 1  dL >   > > : np 1 þ nx=2 2 dL=2

x¼0

ð9Þ

dL=2  x=2  n\  dL=2 þ x=2 dL=2 þ x=2  n  dL=2  x=2

x[0

dL=2  x=2\n  dL=2 þ x=2

ð10Þ The space average of n in the range dL=2  j xj=2  n  dL=2 þ j xj=2 is calculated as dL=2þ R jxj=2

hni ¼

ndn

dL=2j xj=2

dL þ j xj

and according to Eqs. 8, 9 and 10 equals  8  j xj=dL > x\0 < np 1 þ 1þjxj=dL hni ¼ np x¼0 > :n 1 x[0 p 1þx=dL

ð11Þ

ð12Þ

Then, the space average of the dielectric constant ec according to Eqs. 3 and 12 can be described as  8   jxj=dL > x\0 < 1 þ ep  1 1 þ 1þjxj=dL hec i ¼ ep ð13Þ x¼0 > : 1 þ  e  1 1 x[0 p 1þx=dL

References Andersen OS, Koeppe RE (2007) Bilayer thickness and membrane protein function: an energetic perspective. Annu Rev Biophys Biomol Struct 36:107–130

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect Blanchard A, Rauch C (2012) Membrane lipid asymmetry and permeability to drugs: a matter of size. In: Deavaux PF, Herrmann A (eds) Transmembrane dynamics of lipids. Wiley, Hoboken, pp 251–274 Blaurock AE (1982) Evidence of bilayer structure and of membrane interactions from X-ray diffraction analysis. Biochem Biophys Acta 650:167–207 Bradonjic´ K, Swain JD, Widom A, Srivastava YN (2009) The Casimir effect in biology: the role of molecular quantum electrodynamics in linear aggregations of red blood cells. J Phys Conf Ser 161:12035 Casimir HBG (1948) On the attraction between two perfectly conducting plates. Proc R Netherlands Acad Arts Sci 51:793–795 Casimir HBG, Polder D (1948) The influence of retardation on the London-van der Waals forces. Phys Rev 73:360–372 El Hasnaoui K, Madmoune Y, Kaidi H, Chahid M, Benhamou M (2010) Casimir force in confined biomembranes. Afr J Math Phys 8:101–114 Evans EA, Hochmuth RM (1978) Mechanical properties of membranes. Curr Topics Membr Transport 10:1–65 Huang W, Levitt DG (1977) Theoretical calculation of the dielectric constant of a bilayer membrane. Biophys J 17:111–128 Kranenburg M (2004) Phase transitions of lipid bilayers. A mesoscopic approach. Thesis, University of Amsterdam

389 Kranenburg M, Vlaar M, Smit B (2004) Simulating induced interdigitation in membranes. Biophys J 87:1596–1605 Kuchel PW, Ralston GB (1998) Schaum’s outline of theory and problems of biochemistry. McGraw-Hill, New York Landau LD, Lifshitz EM (1960) Electrodynamics of continuous media. Pergamon Press, Oxford Lifshitz EM (1956) The theory of molecular attractive forces between solids. Sov. Phys 2:73–83 Mavromoustakos T, Chatzigeorgiu P, Koukoulista C, Durdagi S (2011) Partial interdigitation of lipid bilayers. Int J Quantum Chem 111:1172–1183 Otter WK, Shkulipa SA (2007) Intermonolayer friction and surface shear viscosity of lipid. Biophys J 93:423–433 Parsegian VA, Ninham BW (1970) Temperature-dependent van der Waals forces. Biophys J 10:664–674 Shkulipa SA, Otter WK, Briels WJ (2005) Surface viscosity, diffusion, and intermonolayer friction: simulating sheared amphiphilic bilayers. Biophys J 89:823–829 Srinivasan KR, Kay RL, Nagle JF (1974) The pressure dependence of the lipid bilayer phase transition. Biochemistry 13:3494–3496 Srivastava Y, Widom A, Friedman MH (1985) Microchips as precision quantum electrodynamic probes. Phys Rev Lett 55:2246–2248 Waugh RE (1982) Surface viscosity measurements from large bilayer vesicle tether formation experiments. Biophys J 38:29–37

123

Colloidal aggregation in critical crosslinked polymer blends M. Benhamou∗ , F. Elhajjaji, K. Elhasnaoui, and A. Derouiche Polymer Physics and Critical Phenomena Laboratory Sciences Faculty Ben M’sik, P.O. Box 7955, Casablanca, Morocco

We consider a low-density assembly of small colloidal particles that are immersed in a critical crosslinked polymer blend made of two polymers of different chemical nature. We assume that, near the spinodal temperature at which the mixture exhibits a microphase separation, the particles preferentially adsorb one of the two polymers. The consequence is that, the beads aggregate in the non-preferred phase. The aim is an extensive study of the thermodynamical properties of the colloidal aggregation, under a change of a suitable parameter such as temperature. This phase transition drives the colloids from a dispersed phase (gas) to a dense one (liquid). To this end, we elaborate a new field theory that allows the determination of the effective free energy of the mobile colloids, as a functional of their density. From this effective free energy, we draw the phase diagram in particle volume fractiontemperature plane, and investigate all critical properties of the colloidal flocculation, from a static and dynamic point of view. Finally, the discussion is extended to impregnated crosslinked polymer blends in solution.

I. INTRODUCTION

The aggregation phenomenon within the traditional colloidal solutions are due to the van der Waals attractive forces, which generally originate from the fact that the particles possess a dipolar moment [1]. The same phenomenon may produces within the so-called critical systems with immersed colloidal particles, such as a fluid near the liquid-gas critical

∗ Author

for correspondence; E-mail : [email protected]

1

point, a mixture of simple liquids (or polymers) near the consolute point, liquid 4 He near the λ-transition or liquid-crystals. For these systems, the critical fluctuations of the order parameter generate long-range attractive forces between colloids, termed critical Casimir forces [2]. The word Casimir is attributed to the well-known Casimir effect [3], according to which two perfectly conducting parallel metals attract each other, due to the vacuum quantum fluctuations. The latter play the role of the critical fluctuations for physical systems exhibiting a critical point. Physics of the colloidal aggregation in critical two-component liquid mixtures is a very exciting and rich problem, and has received a great deal of attention from a theoretical and experimental point of view. Theoretically, the critical Casimir effect has been studied by several methods, such as conformal invariance in dimension 2 [4 − 8], field-theoretical Renormalization-Group (RG) in dimension 3 [2, 9 − 15], and Monte Carlo simulation [16, 17]. Recently, the study was successfully extended to polymer blends [18]. Experimentally, the critical Casimir force was measured in a series of experiments [19 − 27]. Among these, we can quote certain experiments [19 − 23] concerning the behavior of silica beads of small diameter, immersed in a binary liquid mixture made of lutidine and water. In fact, the silica beads have tendency to adsorb rather lutidine, when one is in the vicinity of the consolute point of the mixture free from particles. This is the critical adsorption [28 − 41]. The experimental observation shows that, near criticality, the colloids undergo a reversible aggregation in the water-rich side. It was then natural to consider more complex systems such as crosslinked polymer blends (CPBs) confined to two parallel walls [42] or with immersed colloids [43]. CPBs are made of two chemically incompatible polymers that are densely crosslinked in their one-phase region, by γ-irradiation [44], for instance. When the temperature is lowered, below some characteristic temperature depending on the reticulation dose, the CPB phase separates into mesoscopic phases alternatively rich in unlike polymers. This is the microphase separation (MPS). We note that the first theory of MPS in CPBs was introduced by de Gennes [45], followed by several extended works [46]. The incorporation of a small amount of particles in a CPB may have some relevance for industry. Indeed, these particles can reinforce the mechanical properties of the considered

2

CPB. But, in most practical realizations, the immersed particles are instable and flocculate under some special physical conditions. It is then useful to study their stability conditions. In fact, the presence of colloids influences the primitive segregation interactions between unlike connected polymer chains and their chemical potentials, and the internal rigidity of the polymer network. In this paper, we consider a low-density assembly of colloidal particles, immersed in a critical CPB made of two polymers A and B, of different chemical nature. It is assumed that, near the spinodal temperature at which the CPB undergoes a MPS, the colloids preferentially adsorb one polymer, saying A. Thus, we are in the presence of a critical adsorption. The consequence is that, the particles aggregate in the non-preferred B-rich phase. For instance, the system may be a hydrogenated polyolefin-deuterated polyolefin mixture incorporating silicon particles, considered in some recent experiment by Wendlandt and coworkers [47]. The induced force responsible for the colloidal aggregation was computed in Ref. [43], at the spinodal point. The main result is that, the induced force decays as r−3 exp (−r/ξ ∗ ), where r is the interparticle distance and ξ ∗ ∼ an1/2 is the microdomains size (mesh size) [45]. Here, a denotes the monomer size and n is the mean number of monomers per strand (section of chains between consecutive crosslinks). The purpose is a study of the thermodynamical properties of the colloidal aggregation, from a static and dynamic point of view. This colloidal aggregation is a phase transition driving the colloids from a dispersed phase (gas) to a dense one (liquid). To this end, we first elaborate a field theory governing the flocculation phenomenon. Combining this field theory with the usual cumulant method [48, 49], we derive the expression of the effective free energy of the moving colloids. The latter is naturally a functional of the volume fraction of particles. We assume that the colloidal system is disordered. The problem is then the specification of the disorder nature. To be more realist, we choose the most difficult one, that is a quenched disorder [50]. This means that one must average the logarithm of the partition function over the disorder. The opposite case is an annealed disorder, where the partition function itself is averaged over the external disorder variables. This disorder is relatively simple, and assumes that the colloids and polymers are at equilibrium. In fact, such an assumption is not reliable, since, in principle, the diffusion time of colloids is different

3

from that of polymers. To do explicit calculations, we suppose that the disorder is rather quenched and follows a Gaussian distribution. After averaging over disorder, we find that the effective free energy of colloids is similar to that relatively to the traditional Flory-Huggins theory usually encountered in Polymer Physics [51, 52]. The obtained effective free energy of the dispersed colloids allows the determination of the phase diagram shape in the particle volume fraction-temperature plane and all critical properties of the aggregation transition. Finally, the discussion is extended to troubled CPBs in solution. The remaining of presentation proceeds as follows. We derive, in Sec. II, the expression of the effective free energy of the immersed colloids. The phase diagram is investigated in Sec. II. We discuss, in Sec. III, the static and dynamic scattering properties. Some concluding remarks are drawn in the last section.

II. EFFECTIVE FREE ENERGY

Consider M spherical colloidal particles immersed in a CPB. For simplicity, these particles are assumed to be small and of the same radius. Thus, we are concerned with a monodisperse system. The fundamental assumption is that, near the bulk spinodal temperature, the particles preferentially attract the polymer A. This means that one is in the presence of a critical adsorption, where the colloids are surrounded by the preferred polymer A. The consequence is that, the particles located in the non-preferred B-rich microdomains aggregate. In fact, this aggregation is due to a long-ranged attractive force experienced by the colloids, which was computed in Ref. [43]. To investigate the aggregation phenomenon, we introduce an order parameter or composition fluctuation, ϕ, which is simply the difference of compositions, ΦA and ΦB , of the two polymers, that is ϕ = ΦA − ΦB . The order parameter ϕ (r) is then a scalar field depending on the position vector, r, of the representative point of the mixture. To investigate the physical properties of the troubled CPB, we must precise the form of the Hamiltonian, which is a generalization of that introduced by de Gennes for studying MPS in pure CPBs [45]. In the absence of colloids, the de Gennes’ (bare) Hamiltonian writes [45]

4

Hb [ϕ] = a−d kB T




d

d r



t 2 a2 C ϕ + (∇ϕ)2 + P 2 2 2 2



.

(1)

The latter results from a competition between the usual phase separation and the elastic properties of the polymer network. In the above definition, the integration is performed over the d-dimensional infinite euclidean space Rd . In the above definition, T stands for the absolute temperature, kB for the Boltzmann’s constant, and a for the monomer size. The notation t=

1 (χc − χ) < 0 2

(1a)

denotes the distance from the critical temperature, where χ is the standard Flory interaction parameter, which is inversely proportional to the absolute temperature T , and χc = 2/N is its critical value if the system is uncrosslinked. Here, N denotes the common polymerization degree of the polymer chains before they are crosslinked. Therefore, we are concerned with a monodisperse system. The first two terms in relation (1) constitute the expansion of the traditional Flory-Huggins free energy [51, 52] around the critical composition. The gradient term accounts for the interfacial energy between A-rich and B-rich phases, and the third one describes the gel elasticity. The elastic contribution in this relation was first introduced by de Gennes by analogy with the polarization of a dielectric medium. The internal rigidity constant, C, was related to the average number n of monomers per strand by [45] C ∼ a−2 n−2 .

(1b)

The quantity 1/n was interpreted as the reticulation dose [44]. In equality (1), the vector P denotes the average displacement between the centres of masses of A and B strands; it is the analog of the polarization in the dielectric problem. The monomers A and B play the role of the positive and negative charges of the dielectric medium, respectively. The charge fluctuations correspond to the fluctuations of composition. The quantities P and ϕ are not independent each other, but related by the Maxwell’s relation: divP = −ϕ (r). Such a relation enables us to transform the gel elasticity contribution as follows

C C d 2 d rP → − dd rϕ∆−1 r ϕ , 2 V 2 V where ∆−1 r is simply the inverse Laplacian operator, defined by 5

(1c)

∆−1 r ϕ (r)

=−




dd r′ K (r, r′ ) ϕ (r′ ) ,

(1d)

with the Green function K (r, r′ ), such that −∆r K (r, r′ ) = δd (r − r′ ) .

(1e)

It was found [45] that the system phase separates at the spinodal temperature, that is for 1/2  ts = −2 Ca2 ∼D ,

(1f)

where D = n−1 is the reticulation dose. This relation then defines a critical line in the (C, t)plane along which the CPB undergoes a MPS. For t > ts , the CPB is in the disordered phase, and for t < ts , it is rather in the ordered one. We recall that the microdomains size or mesh −1/4

size is as follows [45]: ξ ∗ = a (Ca2 )

∼ an1/2 .

For a CPB with immersed point-like colloids, the proposed Hamiltonian is M

M

  H [ϕ] Hb [ϕ] ϕ2 (ri ) + C0 P 2 (ri ) . = + t0 kB T kB T i=1 i=1

(2)

Here, M denotes the total number of particles dispersed in the medium. The two last terms in the right-hand side describe the contribution of colloids to the Hamiltonian, where the positive quantities t0 and C0 measure the perturbation of the temperature parameter t and rigidity constant C due to the presence of the moving particles. For shapely particles, t0 and C0 express the interactions between the mixture and their surface. A simple dimensional analysis shows that: [t0 ] = L0 and [C0 ] = L−2 , where L is some length. It will be convenient to rewrite the above deformed Hamiltonian on the following form
H [ϕ] Hb [ϕ] = + dd rρ (r) O (r) , (3) kB T kB T with the local particle density ρ (r) =

N 

δd (r − ri )

(3a)

i=1

and local field operator O (r) = t0 ϕ2 (r) + C0 P 2 (r) . 6

(3b)

Now, to determine the free energy of a CPB, F, where the particles positions, (r1 , ..., rM ), are fixed in space, we shall need the expression of the partition function, Z. The latter is defined by the following functional integral
 d 1 Z (r1 , ..., rM ) = Dϕe−Ab [ϕ]+ d rρ(r)O(r) , M!

(4)

with the bare action Ab [ϕ] = Hb [ϕ] /kB T . The above functional integral is performed over all possible configurations of ϕ-field. The quantity M ! accounts for the usual symmetry factor. It will be useful to introduce the bulk expectation mean-value
−1

Xb = Zb × DϕX [ϕ] e−Ab [ϕ] ,

(5)

with the partition function of the free CPB
Zb = Dϕe−Ab [ϕ] .

(5a)

With these considerations, the partition rewrites as  
 Zb d exp Z (r1 , ..., rM ) = d rρ (r) O (r) . M! b

(6)

We note that the bulk mean-value in the above expression can be computed using the cumulant method usually encountered in Statistical Field Theory [48, 49], based on the approximative formula

X 2 2 e b = eX b +(1/2!)[ X b −X b ]+... .

(7)

Then, we have Zb Z (r1 , ..., rM ) = exp M!




1 d rρ (r) O (r)b + 2 d




d

d r




 d r ρ (r) ρ (r ) O (r) O (r )b,c + ... , d ′

′

′

(8) where

O (r) O (r′ )b,c = O (r) O (r′ )b − O (r)b O (r′ )b

(9)

accounts for the connected two-point correlation function constructed with the field operator O (r). In relation (8), only one and two-body interactions are taken into account. High order terms describing the three-body interactions and more are then ignored. 7

To compute the desired free energy, we start from the standard formula F = −kB T ln Z and find F [ρ] Fb Fint [ρ] = + − S/kB , kB T kB T kB T

(10)

where Fb = −kB T ln Zb represents the free energy of the host mixture, and Fint is the contribution of the effective interactions that can be written as


Fint [ρ] 1 d d = − d rρ (r) O (r)b,c − d r dd r′ ρ (r) ρ (r′ ) O (r) O (r′ )b,c + ... . kB T 2

(11)

Then, this formula is a combination of a one-body potential, O (r)b , and a two-body one,

O (r) O (r′ )b,c . We have  



O (r)b,c = t0 ϕ2 (r) b + C0 P 2 (r) b ,

(12)

  





O (r) O (r′ )b,c = t20 ϕ2 (r) ϕ2 (r′ ) b,c + 2t0 C0 ϕ2 (r) P 2 (r′ ) b,c + C02 P 2 (r) P 2 (r′ ) b,c . (13) In equality (10), S accounts for the entropy of particles, and it is essentially the logarithm of the symmetry factor M !, to which the Stirling’s formula will be applied. Now, the aim is the computation of the effective free energy of the trapped colloids. To do calculations, we first assume that their positions in space are disordered at random, with a quenched disorder. The latter is more realistic than the annealed one, since the monomers and colloids are not in equilibrium. We denote by ρ0 = M/Ω the number density of particles, where Ω is the total volume occupied by the troubled CPB. According to the known central limit theorem, the random density fluctuations around the mean-value ρ0 can be assumed to be governed by a Gaussian (or bimodal) distribution, that is   1 2 P [ρ (r)] = P0 exp − 2 [ρ (r) − ρ0 ] . 2ρ0

(14)

Here, the difference ρ (r) − ρ0 is the density fluctuation and P0 is a normalization constant. Then, the first and second moments of this distribution are as follows ρ (r) = ρ0 ,

(15)

ρ (r) ρ (r′ ) = ρ20 + ρ0 δd (r − r′ ) .

(16)

8

Therefore, we are concerned with a non-correlated disorder. For a quenched disorder, we have to average not the partition function Z but its logarithm ln Z = −F [ρ] /kB T , where F [ρ] is the free energy of expression (11). Then, we have Fb F [ρ] Fint [ρ] = + − S/kB , kB T kB T kB T

(17)

with

1 d d rρ (r) O (r)b,c − d r dd rρ (r) ρ (r′ ) O (r) O (r′ )b,c 2





ρ0 Ω ρ20 2 d d d = −ρ0 d r O (r)b,c − d r O (r) b,c − d r dd r′ O (r) O (r′ )b,c . (18) 2 2

Fint [ρ] ≃− kB T




d

To determine the architecture of the phase diagram of the colloidal system, we need the effective free energy (per site), ∆F , that can be obtained from vF [ρ]/Ω by ignoring the constant Fb and terms proportional to density ρ0 . Here, v denotes the volume of colloids. In order to precise the form of ∆F , we first introduce the dimensionless particle density (volume fraction of colloids) Ψ = ρ0 × v. After symmetrization (Ψ → 1 − Ψ), the entropy (per site) writes [51, 52] vS = −Ψ ln Ψ − (1 − Ψ) ln (1 − Ψ) . kB Ω

(19)

With these considerations, the effective free energy (per site) is given by ∆F = Ψ ln Ψ + (1 − Ψ) ln (1 − Ψ) + uΨ (1 − Ψ) , kB T with the colloid interaction parameter
1 u= dd r O (0) O (r)b,c > 0 2v

(20)

(21)

that essentially presents as the spatial integral of the two-point correlation function of the field operator O. The derived effective free energy calls the following remarks. Firstly, its expression clearly shows its analogy with that relatively to the standard FloryHuggins theory (FH) [51, 52] describing the free polymer blends; u then plays the role of the Flory interaction parameter. Therefore, the present theory of free energy (20) is a lattice model, where a given site is occupied by one particle, with a probability Ψ. Thus, 1 − Ψ is 9

the probability to have an empty site. In fact, if Ψ is the composition of the dense phase (liquid phase), that of the dispersed one is 1 − Ψ. Secondly, formula (20) indicates that the critical fluctuations of composition give arise to an effective interaction energy between colloids that is responsible for their aggregation. Thirdly, we note that the dimensionless parameter u is temperature-dependent and always positive definite. This means that, above some critical value, u∗ , defined later, one assists to a coexistence between dilute (gas) and dense (liquid) phases. Finally, we emphasize that the interaction parameter u is a linear combination of three bulk correlation functions (integrated over space coordinates), that is u=

 1 2 t0 X00 + 2t0 C0 X01 + C02 X11 , 2v

with the coefficients


 X00 = dd r ϕ2 (0) ϕ2 (r) b,c , X11 =




X01 = X10 =




 dd r ϕ2 (0) P 2 (r) b,c ,

 dd r P 2 (0) P 2 (r) b,c .

Replacing P 2 by −ϕ∆−1 r ϕ and using the Wick theorem [48, 49] yields
X00 = 2 dd r [ ϕ (0) ϕ (r)b ]2 , X01 = X10 = 2

X11 = 2









dd r ϕ (0) ϕ (r)b ϕ (0) ∆−1 r ϕ (r) b ,

 −1 dd r ϕ (0) ϕ (r)b ∆−1 r ϕ (0) ∆r ϕ (r) b .

(22)

(22a)

(22b)

(22c)

(22d)

(22e)

Explicitly, we have X00 (T, C) = 2sd a

2d




∞

q d−1 dq , (a2 q 2 + t + C/q 2 )2

0

2d

X01 (T, C) = X10 (T, C) = 2sd a




0

2d

X11 (T, C) = 2sd a




∞ 0

∞

q d−3 dq , (a2 q 2 + t + C/q 2 )2

q d−5 dq . (a2 q 2 + t + C/q 2 )2

10

(22f)

(22g)

(22h)

Here, sd = 2π d/2 Γ (d/2) denotes the area of the unit sphere embedded in the d-dimensional euclidean space Rd , where Γ (z) is the Euler gamma function [53]. In mean-field theory, the critical behavior emerges in the d → 4 limit, and we find, near the spinodal point (t → t+ s ), the following expressions for the coefficients Xij ’s X00 = 2π 3

a7 ξ ∗−3

X01 = X10 = π 3

X11 = π 3

a7 ξ ∗−1 (t − ts )

a7 ξ ∗ (t − ts )

3/2

t → t+ s ,

,

(t − ts )3/2

3/2

,

,

t → t+ s ,

t → t+ s .

Therefore, the colloid interaction parameter scales as

ts − t −3/2 u = u0 , t → t+ s , ts

(23a)

(23b)

(23c)

(24)

with the positive amplitude u0 = We have used the notation

 a4 π3  2 2   2t + 2t C + C . 0 0 0 0 25/2 v 0 = C0 ξ ∗ 2 . C

(24a)

(24b)

The central result (24) call two remarks.

Firstly, as it should be, the colloid interaction parameter becomes more and more important as the spinodal temperature (at which the CPB undergoes a MPS) is reached (from above), and diverges at this same temperature. Secondly, all dependance on the perturbation parameters (t0 , C0 ), due to the presence of fillers, is entirely contained in the positive amplitude u0 defined in Eq. (24a). This dependence is quadratic in these parameters. In addition, this amplitude increases with increasing variables t0 and C0 . In Fig. 1, we draw the reduced colloid interaction parameter, u/u0 , upon the reduced (positive) temperature shift ∆t = (ts − t) /ts . Remark that this curve is universal, independently on the physical and geometrical properties of colloids and their interactions strength with the host CPB. 11

The constructed effective free energy allows the determination of the phase diagram shape in the composition-temperature plane. This is precisely the aim of the following paragraph.

III. PHASE DIAGRAM

Now, the equilibrium volume fraction of particles solves the equation ∂∆F/∂Ψ = 0. Explicitly, we have

Ψ 1 ln . u= 2Ψ − 1 1−Ψ

(25)

This equation defines the coexistence curve in the (Ψ, u)-plane, along which the dispersed and dense phases coexist. On the other hand, the spinodal curve, along which the isotherm compressibility diverges, can be obtained solving ∂ 2 ∆F/∂Ψ2 = 0. Then, the spinodal curve equation is u=

1 . 2Ψ (1 − Ψ)

(26)

Finally, the location of the critical point, K, can be determined solving the two simultaneous equations ∂ 2 ∆F/∂Ψ2 = 0 and ∂ 3 ∆F/∂Ψ3 = 0. We find that its coordinates (Ψ∗ , u∗ ) are such as Ψ∗ =

1 , 2

u∗ = 2 .

(27)

The critical value u∗ = 2 corresponds to a well-defined critical temperature, T ∗ , which satisfies the implicit equation  1 2 t0 X00 (T ∗ , C) + 2t0 C0 X01 (T ∗ , C) + C02 X11 (T ∗ , C) = 2 . 2v

(28)

Combining the dominant parts of the coefficients X00 , X01 and X11 we computed above with equality (28) yields the critical temperature, T ∗ , at which the colloids flocculate   u 2/3  0 ∗ t = ts 1 − . 2

(29)

We have ts < t∗ < 0, provided that u0 < 2. Notice that t∗ = (χc − χ∗ ) /2, with χc = 2/N and χ∗ ∼ T ∗ −1 . Here, the amplitude u0 is defined in equality (24). The above relationship 12

then defines a critical surface in the (t0 , C0 , t∗ )-space on which the colloidal system presents a phase transition. Therefore, at T = T ∗ and Ψ∗ = 0.5, the colloidal system exhibits a second order phase transition from a dispersed phase to a dense one, at the critical temperature T ∗ , which is very close to the spinodal temperature Ts , since the perturbation parameters t0 and C0 are assumed to be small enough. Come back to Eq. (25) defining the coexistence curve and try to rewrite it in the vicinity of the critical point. Close to this point, we obtain u = u∗ +

8 (Ψ − Ψ∗ )2 , 3

Ψ → Ψ∗ ,

(30)

or equivalently, 1 Ψ=Ψ ± 2 ∗



3 (u − u∗ )1/2 , 2

u → u∗ .

(30a)

Notice that u − u∗ ∼ T ∗ − T . The phase diagram of the colloidal system is drawn in Fig. 2, where the solid and dashed lines represent the binodal and spinodal, respectively. These lines meet at the critical point K = (Ψ∗ , u∗ ). The region between the two curves is a metastable domain. We note that the phase diagram architecture we investigated above has been determined using the FH approach that is a mean-field theory (MFT), which underestimates the fluctuations of the composition that are strong near criticality. To go beyond this theory, and in order to get a correct critical phase behavior, use will be made, in the next section, of the scaling approach that is a direct consequence of the RG theory [48, 49].

IV. SCATTERING PROPERTIES A. Static properties

Now, consider another main physical quantity that is the structure factor, S (q), which informs us on the critical properties of the colloidal system. This can be measured in lightscattering and neutron-scattering experiments. The structure factor is nothing else but the Fourier transform of the two-point correlation function G (r) = Ψ (0) Ψ (r)− Ψ (0) Ψ (r), which measures the fluctuations of the particle composition Ψ. We write 13

S (q) =




dd q (2π)d

eiq.r G (r) ,

(31)

where q denotes the wave-vector, whose amplitude is given by |q| = (4π/λ) sin (θ/2), where λ is the wavelength of the incident radiation and θ is the scattering-angle. In particular, the zero-scattering-angle limit (q → 0) of the structure factor defines the thermal compressibility of the colloidal system, κT , that is S (0) = kB T κT .

(32)

This compressibility diverges as the critical temperature T ∗ is approached, and we have κT ∼ |T − T ∗ |−γt ,

(32a)

where γt is a critical exponent whose value depends only the space dimensionality d and not on the chemical details of the system. Other critical exponents can be defined. For example, the order parameter ψ = Ψ − Ψ∗ behaves, around T ∗ , as  0 , T ≥ T∗ , ψ∼  (T ∗ − T )βt , T ≤ T ∗ .

(33)

with a second critical exponent βt , in the limit ∆µ = µ − µc → 0, where µ is the chemical potential and µc is its critical value [48, 49]. In term of the colloid interaction parameter u and composition Ψ, we have u − u∗ ∼ (Ψ − Ψ∗ )1/βt

(33a)

that is the equation of the coexistence curve close to the critical point. At T = T ∗ , the order parameter behaves as ψ ∼ |∆µ|1/δt , with a critical exponent δt . On the other hand, the size of domains rich in colloids or the thermal correlation length measuring the correlations extent, ξt , diverges at criticality according to ξt ∼ σ |T − T ∗ |−νt ,

(34)

with the critical exponent νt and an atomic-scale σ that may be the particle diameter. Also, the specific heat diverges at the critical point with some critical exponent denoted 14

αt . Finally, we define the last critical exponent, ηt , which characterizes the small-distance behavior of the correlation function, that is G (r) ∼ r2−d−ηt ,

r << ξt .

(35)

Therefore, the critical phase behavior is completely determined by the knowledge of the thermal critical exponents (αt , βt , γt , δt , νt , ηt ). To compute these exponents, we first remark that the order parameter ψ plays the role of the magnetization of a magnetic material exhibiting a paramagnetic-ferromagnetic transition, and ∆µ, the role of an external magnetic field. Also, the free energy defined in Eq. (20) resembles the Bragg-Williams one usually encountered in magnetic materials [48, 49]. The conclusion is that, the present second order phase transition belongs to the same universality class as the paramagnetic-ferromagnetic one. Thus, the three-dimensional values of the various critical exponents are those of Ising model we do not recall [48, 49]. Naturally, these good values must be compared to those obtained within the framework of MFT used in the last section, which are α0t = 0, βt0 = 1/2, γt0 = 1, δt0 = 3, νt0 = 1/2, and ηt0 = 0. The difference comes from the fact that MFT underestimates the strong fluctuations of composition of colloids near the critical point. We recall that the real critical exponents have been computed using the RG-techniques. In particular, the realistic coexistence curve equation is that defined in Eq. (33a), with a non-trivial critical exponent βt , different from the mean-field one, βt0 = 1/2. Finally, the structure factor obeys the scaling law S (q) = q ηt −2 f (qξt ) ,

(36)

with the scaling-function f (x).

B. Dynamic properties

Assume that the temperature of the system is suddenly lowered from an initial value to a final one very close to the critical temperature T ∗ . Then, the system is out of equilibrium and the colloid volume fraction relaxes from an initial equilibrium value Ψi to a final one Ψf . We are interested in the time order parameter ψ (r, t) = Ψ (r, t) − Ψ∗ , where r ∈ Rd

15

denotes the position-vector and t the time. The latter represents the time observation of the system before it reaches its final equilibrium state. We recall that the time composition fluctuation ψ (r, t) solves a generalized nondissipative Langevin equation [49] we do not write. Applying the stochastic RG-techniques yields the scaling law for the dynamic structure factor. In particular, the latter depends on the following time-scale R (t) ∼ Γ1/z t1/z ,

(37)

z = 4 − ηt = 3.469 ± 0.001 .

(37a)

with the critical dynamic exponent

We have used the best value for the critical exponent ηt , that is ηt = 0.031 ± 0.001 [48, 49]. There, Γ accounts for the kinetic coefficient. The length R (t) can be interpreted as the size of domains rich in colloids at time t. For MFT, z0 = 4. The final equilibrium state is reached at a characteristic time, τ , which scales as τ ∼ Γξtz ,

(38)

where ξt is the thermal correlation length or the size of domains at the final temperature (the subscript t is for thermal that must not be confused with time). The above behavior can be obtained equating R (t) to ξt . Finally, we write the scaling law for the dynamic structure factor S (q, t) = q ηt −2 g (qξt , qR (t)) ,

(39)

where g (x, y) is a two-factor scaling-function. Of course, the time t must not be confused with the temperature parameter appearing in Eqs (1), (1a), (23a-c) and (24).

V. DISCUSSION AND CONCLUSIONS

In this work, we were interested in the reversible aggregation of colloids immersed in a critical CPB. We assumed that the colloids prefer to be surrounded by one of the two polymers, close to the spinodal point of the free mixture. The aggregation phenomenon 16

results from the existence of long-ranged attractive forces. The question we asked was the investigation of the phase diagram and critical properties of the immersed colloids. To do calculations, use was made, first, of MFT, and second, the scaling theory, in order to get a correct leading phase behavior. In addition, it was assumed that the positions of particles are disordered, and the disorder is quenched and follows a Gaussian distribution. We first determined the expression of the effective free energy of colloids, and showed that the latter is of Flory-Huggins type. From this effective free energy, we drawn the complete shape of the phase diagram. Second, the scaling theory revealed that the critical exponents have non classical values in comparison to those relatively to MFT. Third, the discussion was extended to dynamic properties of the colloidal flocculation. The essential result is that, the colloidal aggregation, which is a phase transition driving the system from dilute to dense phases, occurs when one is close to the spinodal point of the crosslinked polymer mixture free from fillers. For the description of the colloidal aggregation, we assumed that the colloids are spheric and of the same diameter. This means that we were concerned only with monodisperse systems. Actually, the particles are never geometrically identical. They may have different shapes and sizes. Nevertheless, the present approach remains valid for very small particles, whatever is their shape. An interesting problem that we have not undertaken is the effect of an added (good) solvent on the colloidal flocculation. One interesting way is to use the blob model [54, 55]. Quantitatively, all results dealt with the aggregation phenomenon are similar to that relatively to impregnated CPBs in the molten state (no solvent is present) replacing the monomer size a by the screening length, ξ ∼ aΦ−3/4 [52, 56], the polymerization degree N by the number of blobs per connected chain, Z (Φ) ∼ NΦ5/4 [52, 56], the segregation parameter χ by the effective one, χ ∼ χΦ0.26 [55], and the mesh size, ξ ∗ ∼ an1/2 , by its homologous ∗

ξ ∼ an1/2 Φ−1/8 . For theta-solvents, ξ ∼ aΦ−1 [52, 56], Z (Φ) ∼ NΦ2 [52, 56], χ ∼ χΦ0.87 ∗

[57], and ξ ∼ ξ ∗ ∼ an1/2 . We would have the same discussions. But the presence of the solvent drastically affects the critical phase behavior. In particular, the critical tempera∗

ture, T , at which the aggregation transition occurs, depends, in addition to the parameters (N, C, t0 , C0 ) relatively to the molten state, on the monomer volume fraction Φ. For good

17

solvents, for instance, the swelling of strands due to the excluded volume forces alter the aggregation phenomenon. It is well-known that a good solvent plays a stabilizer role. For the description of the colloidal aggregation, we have not taken into account the primitive interactions between the immersed colloids. This assumption makes sense as long as the colloid density is very small. Generally, we show that these primitive interactions contribute to the colloid interaction parameter u by an extra term, up , which presents as ∞ [58]: up = −b0 + a0 /kB T > 0, where b0 is the covolume and a0 = −2π r0 r2 U (r), with the

primitive pair-potential U (r). Here, r0 is the unique zero of U (r), that is U (r0 ) = 0. In this case, the total colloid interaction parameter is then u + up . The presence of the primitive interactions naturally induces a change of the phase behavior. In particular, the location of the critical flocculation temperature is shifted to its higher values. In other word, the critical fluctuations of the host CPB have tendency to accentuate the colloidal aggregation. All conclusions concerning the scattering properties remain the same. Finally, this work must be regarded as a natural extension of a published work dealt

with the colloidal aggregation in uncrosslinked polymer blends with immersed colloids [59].

ACKNOWLEDGMENTS

We are much indebted to Professors Daoud for stimulating correspondences. One of us (M.B) would like to thank the Laboratoire L´eon Brillouin (Saclay) for their kinds of hospitality during his regular visits.

[1] J.N. Israelachvili, Intermolecular and Surface Forces, 2nd edition, Academic Press, London, 1991. [2] M. Krech, The Casimir Effect in Critical Systems, World Scientific, Singapore, 1994. [3] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetenschap B 51, 793 (1948). [4] H.W.J. Bl¨ote, J.L. Cardy, and M.P. Nightingale, Phys. Rev. Lett. 56, 742 (1986). [5] I. Affleck, Phys. Rev. Lett. 56, 746 (1986).

18

[6] J.L. Cardy, Nucl. Phys. B 275, 200 (1986). [7] T.W. Burkhardt and T. Xue, Phys. Rev. Lett. 66, 895 (1991); Nucl. Phys. B 345, 653 (1991). [8] T.W. Burkhardt and E. Eisenriegler, Nucl. Phys. B 424 [FS], 487 (1994). [9] M. Krech and S. Dietrich, Phys. Rev. A 46, 1886 (1992). [10] M. Krech, Phys. Rev. E 56, 1642 (1997). [11] K. Symanzik, Nucl. Phys. B 190 [FS], 1 (1981). [12] M. Krech and S. Dietrich, Phys. Rev. Lett. 66, 345 (1991); 67, 1055 (1991); Phys. Rev. A 46, 1922 (1992). [13] E. Eisenriegler and M. Stapper, Phys. Rev. B 50, 10009 (1994). [14] T.W. Burkhardt and E. Eisenriegler, Phys. Rev. Lett. 74, 3189 (1995). [15] E. Eisenriegler and U. Ritschel, Phys. Rev. B 51, 13717 (1995). [16] K.K. Mon, Phys. Rev. Lett. 54, 2671 (1985) ; M. Krech and D.P. Landau, Phys. Rev. E 53, 4414 (1996). [17] R.R. Netz, Phys. Rev. Lett. 76, 3646 (1996). [18] M. Benhamou, Casimir Effect in Polymer Physics, Recent Res. Devel. Physics 8, 93 (2009). [19] D. Beysens and D. Est`eve, Phys. Rev. Lett. 54, 2123 (1985). [20] V. Gurfein, D. Beysens, and F. Perrot, Phys. Rev. A 40, 2543 (1989). [21] M.L. Broide, Y. Garrabos, and D. Beysens, Phys. Rev. E 47, 3768 (1993). [22] T. Narayanan, A. Kumar, E.S.R. Gopal, D. Beysens, P. Guenoun, and G. Zalczer, Phys. Rev. E 48, 1989 (1993). [23] D. Beysens, J.-M. Petit, T. Narayanan, A. Kumar, and M.L. Broide, J. Phys. Chem. 98, 382 (1994). [24] P.D. Gallagher, M.L. Kurnaz, and J.V. Maher, Phys. Rev. A 46, 7750 (1992).

19

[25] M.L. Kurnaz and J.V. Maher, Phys. Rev. E 51, 5916 (1995). [26] H. Gr¨ ull and D. Woermann, J. Phys. Chem. 101, 814 (1997). [27] A list of recent experimental works can be found in: G.A. Williams, Physica C 404, 415 (2004). [28] M.E. Fisher and P.-G. de Gennes, C. R. Acad. Sci. (Paris) S´erie B 287, 207 (1978) ; also, P.-G. de Gennes, C. R. Acad. Sci. (Paris) II 292, 701 (1981). [29] H.W. Diehl, in: Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 10, Academic Press, London, 1986. [30] S. Dietrich, in: Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 12, Academic Press, London, 1988. [31] D. Beysens and S. Leibler, J. Phys. Lett. (Paris) 43, L-133 (1982). [32] C. Franck and S.E. Schnatterly, Phys. Rev. Lett. 48, 763 (1982). [33] M. Schlossman, X.-L. Wu, and C. Franck, Phys. Rev. B 31, 1478 (1985). [34] J.A. Dixon, M. Schlossman, X.-L. Wu, and C. Franck, Phys. Rev. B 31, 1509 (1985). [35] An extensive list of recent experiments using some new techniques can be found in : A. Hanke, M. Krech, F. Schlessener, and S. Dietrich, Phys. Rev. E 60, 5163 (1999). [36] S. Bl¨ umel and G.H. Findenegg, Phys. Rev. Lett. 54, 447 (1985). [37] E. Br´ezin and S. Leibler, Phys. Rev. B 27, 594 (1983); S. Leibler, PhD Thesis, Paris University XI, 1984. [38] J. Rudnik and D. Jasnow, Phys. Rev. Lett. 48, 1595 (1982). [39] S. Leibler and L. Peliti, J. Phys. C 15, L-403 (1982). [40] L. Peliti and S. Leibler, J. Phys. C 16, L-2635 (1983). [41] Critical adsorption on curved objects, such as single spherical and rod-like colloidal particles, has been investigated in: A. Hanke and S. Dietrich, Phys. Rev. E 59, 5081 (1999); A. Hanke,

20

Phys. Rev. Lett. 48, 2180 (2000). [42] M. Benhamou, A. Agouzouk, H. Kaidi, M. Boughou, S. El Fassi, and A. Derouiche, Electronic Journal of Theoretical Physics 5, 141, (2008). [43] S. El Fassi, M. Benhamou, M. Boughou, H. Kaidi, M. El Yaznasni, and H. Ridouane, Acta Physica Polinica 118, 333 (2010). [44] R.M. Briber and B.J. Bauer, Macromolecules 21, 3296 (1988). [45] P.-G. de Gennes, J. Phys. Lett. (Paris) 40, 69 (1979). [46] An exhaustive list of works dealt with MPS in CPBs can be found in: M. Benhamou, A. Derouiche, A. Bettachy, and F. Elhajjaji, Eur. Phys. J. E 34, 3 (2011). [47] M. Wendlandt, T. Kerle, M. Heuberger, and J. Klein, J. Polymer Science: Part B 38, 831 (2000). [48] C. Itzykson and J.-M. Drouffe, Statistical Field Theory: 1 and 2, Cambridge University Press, 1989. [49] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. [50] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge, University Press, 1996. [51] P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, 1953. [52] P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, New York, 1979. [53] I.S. Gradstein and I.M. Ryzhik, Tables of Integrals, Series and Products, Academic, New York, 1980. [54] J.-F. Joanny, L. Leibler, and R. Ball, J. Chem. Phys. 81, 4640 (1984). [55] D. Broseta, L. Leibler, and J.-F. Joanny, Macromolecules 20, 1935 (1987). [56] J. des Cloizeaux and G. Jannink, Polymers in Solution: Their Modelling and Structure, Oxford

21

University Press, 1999. [57] L. Sch¨afer, U. Lehr, and C. Kappeler, J. Phys. I (Paris) 1, 211 (1991); A.I. Mudrov and K.B. Varnashev, J. Phys. A: Math. Gen. 34, L-347 (2001). [58] R. Balian, From Microphysics to Macrophysics - Methods and Applications of Statistical Physics, Springer Verlag, 2006. [59] M. Benhamou, H. Ridouane, A. Derouiche, and M. Rahmoune, J. Chem. Phys. 122, 244913 (2005).

22

FIGURE CAPTIONS

Fig. 1 : Variation of the reduced colloid interaction parameter, u/u0 , upon the temperature shift ∆t = (ts − t) /ts .

Fig. 2 : Phase diagram shape, where the solid line is the binodal and the dashed one is the spinodal. The two curves meet at the ”liquid-gas” critical point, K, of coordinates (Ψc = 0.5, u∗ = 2).

23

u/u0 20

15

10

5

0 0,0

0,2

0,4

0,6

0,8

1,0

∆t Figure 1

u 7 6 5 .

4 3

K ..

2 0,00

0,25

0,50

0,75

Figure 2

1,00

Ψ