Cheese Rheology and Texture

Cheese Rheology and Texture Sundaram Gunasekaran M. Mehmet Ak

CRC PR E S S Boca Raton London New York Washington, D.C. © 2003 by CRC Press LLC

Library of Congress Cataloging-in-Publication Data Gunasekaran, Sundaram, 1957Cheese rheology and texture / Sundaram Gunasekaran, M. Mehmet Ak. p. cm. Includes bibliographical references (p. ). ISBN 1-58716-021-8 (alk. paper) 1. Cheese—Texture. I. Ak, M. Mehmet. II. Title. TX382 .G86 2002 637′.3—dc21

2002034861

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Visit the CRC Press Web site at www.crcpress.com © 2003 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 1-58716-021-8 Library of Congress Card Number 2002034861 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

Dedication To: My parents, Raga Palanisamy Sundaram and Kamala Sundaram, for inspiring me to always strive for excellence. My wife, Sujatha, and children, Suvai and Suman, for their love, support, and patience. — SG

My father, Haci Ak, and mother, Zeynep Ak, for giving me the opportunities they never had. My wife, Nese, who continuously supported my efforts and patiently endured the time I spent working on this book. My daughter, Asli, and my son, Efe, who cheered me up in times the situation looked hopeless. — MMA

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Foreword Two complex scientific areas, cheese and rheology, create an exponential increase in complexity when combined. This text makes a significant contribution to an understanding of this complexity. It underscores limitations and considerations in evaluating and conducting research on cheese rheology, points out some important gaps in our understanding of cheese rheology, and thoroughly reviews methods, theories, and applications of rheology in general and specifically for cheese. Rheologists will gain a better understanding of the physicochemical properties of cheese, and cheese researchers will be exposed to the wide range of rheological methods and the theoretical bases of those methods. Both groups should realize the need for collaborative research after exposure to the individual complexities of cheese and rheology. The diversity of observations, and seemingly contradictory observations, on the physical and chemical properties of cheese that appear in this text should not be surprising since many of the observations were made before instruments were improved and were specifically adapted to deal with unique properties of cheeses. Also, confusion resulted from: cheese scientists who used techniques inadequate to definitively measure physical properties of cheese; rheologists who chose test samples of cheese that did not possess comparable chemical properties except for the property to be measured; and inadequately defining the chemical properties of cheese. The authors have discussed unique characteristics of cheese that rheologists should be cognizant of in designing experiments. Comments on merits and deficiencies of wide range of rheological test methods as applied to cheese should assist cheese scientists in appropriately using the procedures. The chronology of cheese rheology research outlined in this book is encouraging as evidenced by the increase in collaborative research groups or research groups with better understanding of both research areas. The physicochemical properties of cheese have always been an important component in assessing cheese quality and value. The assessment was usually done by sensory evaluation, which was quite adequate because cheese was generally consumed in its “original” state. Development of heat-processed cheese products in the early 1900s prompted some research on the physical properties of cheese, primarily by modifying chemical properties, however, only limited research was done on rheological properties. The last several decades have greatly changed the forms and uses of cheese in the market place. Cheese has to be sliced or shredded by highspeed cutting devices; the melt and flow properties of cheeses have to be more carefully controlled; flavor intensities and flavor profiles have to be modified without adversely affecting physical properties of cheese; and cheese products must possess adequate stability, often under wide ranges of environments. This myriad of desired properties greatly increases the need for procedures to independently control specific properties and the need for adequate methods to measure the properties specifically © 2003 by CRC Press LLC

being controlled. The authors of this book have facilitated attainment of these goals by their thorough review of the present status of cheese rheology research and by providing guidance for further research efforts. Norman F. Olson Department of Food Science and Center for Dairy Research University of Wisconsin-Madison

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Preface Rheology of cheese has been studied since the early 1950s. In fact, “Cheese Rheology” is the name of a chapter in the 1958 FAO Report*. Since then, many advances have taken place both in cheese technology and rheology. As cheese became an important part of the diet in many parts of world, the cheese industry responded by manufacturing new types of cheeses with varying textures to suit varied needs and to promote cheese use both as a table cheese and as an ingredient food. This flurry of new cheeses and applications and cheesemaking technologies has also brought about an acute need to characterize the rheological and textural attributes of cheeses to ensure their high quality. Thus, for food rheologists and food scientists, cheese is among the most popular subjects of study. In this book, we have attempted to summarize the vast literature available on cheese rheology and texture. Needless to say, the sheer volume of information available and the complexity of both cheese and rheology made this a particularly difficult task. Our goal was to bring together many of the dispersed published information on cheese rheology and texture in one book to serve as a comprehensive reference source. A unique aspect of this book is that it contains detailed descriptions of several methods to study rheology of foods in general and cheese in particular. This is to provide the interested readers the necessary basic information on many techniques reported in the literature which often do not have adequate explanation. Chapter 1 provides an overview of cheesemaking technology. Fundamental rheological test methods are described in much detail in Chapter 2. This chapter will facilitate the readers to gain a deeper understanding of the various rheological test methods. The uniaxial testing, one of the most widely used classes of rheological and texture testing methods, is the focus of Chapter 3. The fracture mechanics are an extension of the uniaxial test methods. These are discussed in Chapter 4. In Chapter 5, linear viscoelastic methods are described. This is now among the most popular rheological test performed on cheeses, and is also known as dynamic testing. Both the theory and applications are discussed in a manner benefiting those who are already familiar and those who are new to the subject. Chapter 6 focuses on nonlinear viscoelasticity of cheeses. This subject has not received much attention due to the lack of available instrumentation and the complexity of data analysis. This chapter will be more useful to those familiar with rheological analysis than to the casual reader. The discussion on cheese texture in Chapter 7 is limited to mechanical texture of cheese, as it is more in line with rheological measurements. Cheese meltability and stretchability, two of the most important properties of cheese used in prepared foods, are the topics of Chapters 8 and 9. The emphasis in these chapters is on measurement methods. The effects of various factors on cheese functional properties are addressed in Chapter 10. * Kosikowski, F.V. and G. Mocquot. 1958. Advances in Cheese Technology, FAO Studies No. 38. Food and Agriculture Organization of the United Nations. Rome, Italy. © 2003 by CRC Press LLC

Acknowledgments We would like to acknowledge many individuals who have contributed directly or indirectly toward making this book a reality. First and foremost, we would like to express gratitude to Professor Norman F. Olson, who was instrumental in helping us to initiate our first project on cheese rheology in 1989, when S.G. was a new assistant professor and M.M.A. was a graduate research assistant. Since then, with his expert knowledge and friendly personality, Professor Olson has been a source of great support. Thanks are also due to Dr. Mark Johnson, Dr. Rusty Bishop, John Jaeggi, and other past and current staff at the Wisconsin Center for Dairy Research. These people are invaluable resources for cheese research. This book draws from much of the research performed in S.G.’s laboratory. As such, the efforts of many graduate students and post-doctoral research associates are deeply appreciated. They include: Chyung Ay, James Colby, Kexiang Ding, Chang Hwang, Sun Young Kim, Sanghoon Ko, Gul Konuklar, Meng-I Kuo, Laura Marschoun, Kasiviswanathan Muthukumarappan, Hongxu Ni, Ramesh Subramanian, Salman Tariq, Deepa Venkatesan, Ya-Chun Wang, and Chenxu Yu. Thanks are also due to S.G.’s colleagues, Professors A. Jeffrey Giacomin and Daniel Klingenberg at the Rheology Research Center, University of Wisconsin-Madison, and Professor Karsten B. Qvist of KVL, Denmark. Thanks to Hallie Kirschner for typing parts of the manuscript. The financial support of Wisconsin Milk Marketing Board and Dairy Management Inc. for many of S.G.’s projects is also deeply appreciated. M.M.A. wishes to thank each member of his family for their full support and patience during the writing of this book. He expresses appreciation to the following: Suat Yasa and Murat Yasa of Aromsa Limited Company, for their interest in the book; friends Elsie and Warren Sveum, Sarah and Alvaro Quinones, Mar GarcimartinAkgul, and Arzu and Yann LeBellour for their constant encouragement; and former students Filiz Lokumcu and Metin Yavuz for their valuable assistance in gathering some of the publications. Sundaram Gunasekaran M. Mehmet Ak

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Table of Contents Chapter 1

Cheesemaking — An Overview

Cheese Types Cheesemaking Milk Pretreatment Coagulation Syneresis Shaping and Salting Ripening Process Cheese References Chapter 2

Fundamental Rheological Methods

Definition of Rheology Basic Concepts Strain Stress Strain Rate Fundamental Methods Uniaxial Compression Uniaxial Tension Bending Test Specimen with a Rectangular Cross-Section Specimen with a Circular Cross-Section Torsion Test Vane Method Stress-Relaxation Test Analysis of Relaxation Behavior Creep Test Analysis of Creep Behavior Shear Rheometry Sliding-Plates Geometry Concentric-Cylinders Geometry Cone-and-Plate Geometry Parallel-Plate Geometry Capillary Rheometry Extensional Rheometry Lubricated Squeezing Flow Equations for Different Fluids in Lubricating Squeezing Flow References © 2003 by CRC Press LLC

Chapter 3

Uniaxial Testing of Cheese

Uniaxial Compression Measurements Structure and Composition Effects Stress-Relaxation Measurements Torsion Measurements Tension Measurements Creep Measurements Bending Measurements Vane Measurements Shear Measurements Lubricated Squeezing Flow Measurements References Chapter 4

Fracture Properties of Cheese

Fracture Mechanics Brittle Fracture Griffith Criterion Determination of KI Fracture Tests on Cheese Notch Tests Cutting, Slicing, and Shredding Cutting with Wire and Blade Eye/Slit Formation and Growth References Chapter 5

Linear Viscoelasticity of Cheese

Mathematical Relations in Linear Viscoelasticity Types of SAOS Measurements Strain (or Stress) Sweep Frequency Sweep Temperature Sweep Time Sweep Time–Temperature Superposition Application of SAOS in Cheese Rheology Linear Viscoelastic Region of Cheeses Cheddar Cheese Gouda Cheese Mozzarella Cheese Mozzarella: Time–Temperature Superposition Example Feta Cheese Imitation Cheese Quarg Cheese Processed Cheese Cox–Merz Rule References © 2003 by CRC Press LLC

Chapter 6

Nonlinear Viscoelasticity of Cheese

Pipkin Diagram Sliding Plate Rheometer Large Amplitude Oscillatory Shear Flow Spectral Analysis Discrete Fourier Transform Determining Material Properties Amplitude Spectrum Stress–Shear Rate Loops Effect of Wall Slip Constitutive Model for Cheese Relaxation Modulus Obtained from SAOS Relaxation Modulus Conforming to LAOS References Chapter 7

Cheese Texture

Texture Development in Cheese Cheese Manufacturing Factors that Affect Texture Textural Changes during Storage Measurement of Texture Texture Profile Analysis TPA Testing of Cheese Uniaxial Tests for Cheese Texture Measurement Compression Test Wedge Fracture Test Torsion Test and Vane Rheometry Texture Map Dynamic Tests Empirical Tests Crumbliness Cone Penetrometer Stringiness References Chapter 8

Measuring Cheese Melt and Flow Properties

Meltability Empirical Tests Objective Tests Steady Shear Viscometry Capillary Rheometry Squeeze-Flow Rheometry UW Meltmeter Viscoelasticity Index for Cheese Meltability Dynamic Shear Rheometry Helical Viscometry © 2003 by CRC Press LLC

Cheese Melt Profile Measurement UW Melt Profiler Determination of Melt Profile Parameters Graphical Method Modeling Melt Profile Constant Temperature Test Transient Temperature Test Conduction Heating References Chapter 9

Measuring Cheese Stretchability

Empirical Methods Instrumented Methods Vertical Elongation Horizontal Extension Compression Tests Helical Viscometry Fiber-Spinning Technique The Weissenberg Effect References Chapter 10 Factors Affecting Functional Properties of Cheese Properties of Milk Cheesemaking Procedures Addition of Starter Culture and Coagulants Curd Handling Cooking, Stretching, and Cooling Cheese Composition Moisture Content Fat Content Salt Content pH Post-Manufacturing Processes Aging/Ripening Freezing and Frozen Storage Heat Processing Other Factors References

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1

Cheesemaking — An Overview

Cheese is one of the first and most popular manufactured food products. What perhaps started out as an accidental curdling of milk has been further refined into cheesemaking. Over several thousand years, cheesemaking has advanced from an art to near science. Cheese varieties have proliferated to suit varied conditions and requirements, especially during the last decade or so. It is estimated that more than 2000 varieties exist (Olson, 1995), and the list may still be growing. Cheese is now an important part of foods consumed in many countries (Table 1.1). In a recent survey, after spices, cheese was named the top ingredient that makes cooks feel more creative (Doeff, 1994). Several cheeses satisfy varied requirements in order to be used as suitable ingredients in various dishes from baby foods to baked products (Table 1.2). Battistotti et al. (1984) described the history of cheese and cheesemaking in much detail. This chapter provides a broad overview of cheesemaking. For further details, readers are referred to many recent books on the subject (Scott et al., 1998; Spreer, 1998; Law, 1999; Walstra et al., 1999; Fox et al., 2000).

CHEESE TYPES Today’s wide array of cheeses may be classified according to the country of origin, manufacturing process, or some end-use property. Classifying cheeses based on manufacturing and maturation processes by Olson (1979) produces a succinct list. A classification based on firmness and the maturation agent used produces a longer list but may be more relevant if textural and rheological properties are important (Figure 1.1). A classification based on the distinctive manufacturing process involved is also useful to understand the effect of the process on the cheese texture (Table 1.3). Other classifications of cheeses, e.g., according to milk source, overall appearance (color, size, shape), chemical analysis, etc., are also possible. Davis (1965) recognized the difficulty in classifying cheeses and attempted to group them based on the nature and extent of chemical breakdown during ripening or according to flavor. Such a classification is still not available. Fox (1993) proposed that the products of proteolysis could be most useful for classification. One of the main difficulties when using classification schemes is that differences exist in the moisture range allowed within various categories published in the literature (Banks, 1998). Davis (1965) assigned some empirical texture/rheological parameter values to the terms from very hard to soft (Table 1.4). The United States Code of Federal Regulations (CFR, 1998) stipulates certain standards of identity for cheeses classified according to their consistency, as listed in Table 1.5. The typical composition of milk and several cheese varieties is given in Table 1.6. In the United States, the cheese market is dominated (almost equally) by Cheddar and Mozzarella cheeses. They comprise about twothirds of the total cheese production over the past several years (Figure 1.2).

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TABLE 1.1 Consumption of Cheese in Selected Countries Consumption per Capita (kg) Country North America Canada Mexico United States South America Argentina Brazil Venezuela Western Europe Denmark France Germany Ireland Italy Netherlands Spain Sweden Switzerland United Kingdom Central Europe Poland Balkans Romania Eastern Europe Russia Ukraine North Africa Egypt Southern Asia Japan South Korea Oceania Australia New Zealand a

1995

2000a

10.86 1.47 12.26

10.76 1.61 13.75

10.35 2.82 3.48

11.06 2.72 2.76

16.84 21.51 11.90 5.54 18.66 14.68 5.46 16.14 14.28 8.75

16.22 22.49 12.50 6.70 20.38 14.97 6.25 16.12 14.31 9.88

2.90

3.87

4.05

4.26

2.03 1.26

1.41 0.79

5.27

5.83

1.46 0.27

1.77 0.70

8.25 8.17

11.10 8.48

Preliminary

Source: After International Dairy Federation (www.dairyinfo.gc.ca/).

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TABLE 1.2 Typical Requirements of Cheese as a Food Ingredient Requirement Crumbles when rubbed Sliceability

Shreddability

Flows freely when shaken

Examples of Food Applications Mixed salads Soup Filled cheese rolls (finger foods) Sandwiches (filled, open, toasted) Cheese slices in burgers Cheese slices on crackers Consumer packs of sliced cheese Pizza pie (frozen/fresh baked) Pasta dishes (lasagna, macaroni and cheese) Cheese sprinklings (on lasagna) Snack coating (e.g., popcorn) Dry soup/sauce mixes

Flowability when blended with other raw materials Ability to “cream” or to form a paste when sheared Nutritional value

Fresh cheese desserts

Meltability upon grilling or baking

All cooked dishes (including sauces, fondues, pizza pie)

Flowability upon grilling or baking

Most cooked dishes (e.g., pizza pie, cheese slices on burgers) Chicken cordon-bleu Deep-fried breaded cheese sticks Deep-fried burgers with cheese inserts Fried cheese dishes

Flow resistance upon deep-frying

Stretchability when baked or grilled Chewiness when baked or grilled Limited oiling-off when baked or grilled Limited browning when baked or grilled

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Cheesecake Tiramisu Homemade desserts Baby foods

Examples of Cheese or Cheese-based Ingredient Feta, Cheshire, Stilton Stilton Swiss-type, Gouda, Edam Swiss-type, Cheddar, Mozzarella Cheddar Cheddar Swiss-type, Cheddar, Mozzarella Mozzarella, Provolone, Cheddar, analog pizza cheese, Monterey Cheddar, Romano, Provolone Grated Parmesan and Romano Cheese powders Cheese powders, enzyme-modified cheese Quark, Fomage frias, Cream cheese

Pizza pie

Cream cheese, Ricotta Mascarpone Cream cheese Dried cheeses, esp. rennet-curd varieties (high in calcium) Mozzarella, Cheddar, Raclette, Swiss, Romano, analog pizza cheese, PCPsa Mozzarella, Cheddar, Swiss, Romano, analog pizza cheese PCPs, Cream cheese PCPs, analog pizza cheese, custom-made Mozzarella or string cheese PCPs, analog pizza cheese Paneer, acid-coagulated Queso Blanco Mozzarella, Kashkaval, young Cheddar, analog pizza cheese Halloumi, Mozzarella, Provolone, Kashkaval, young Cheddar Mozzarella, Kashkaval

Macaroni and cheese Lasagna Pizza pie

Cheddar, Romano Cheddar, Romano, Parmesan Mozzarella, analog pizza cheese

Pizza pie Pizza pie

TABLE 1.2 (continued) Typical Requirements of Cheese as a Food Ingredient Requirement

Examples of Food Applications

Viscosity

Soups Sauces Cheesecake

Flavor

Most cheese dishes, soups Baked products Snack coatings Dressings Baby food Ready-made meals

a

Examples of Cheese or Cheese-based Ingredient Cheese powders, PCPs Cheese powders, Cheddar, Blue cheese, PCPs Cream cheese Cheddar, Romano, Swiss-type, Parmesan Cheese powders, enzyme-modified cheese Cheese powders Cheese powders Dried cheese Cheese powders

Pasteurized process cheese products.

Source: After Fox et al., 2000. With permission.

CHEESEMAKING Though there are numerous cheese varieties, the manufacturing processes of most of them share several common steps. Variations at one or more steps during manufacture produce cheeses of different textures and flavors. The essential steps in cheesemaking and some variations for a few types of cheeses are schematically illustrated in Figure 1.3. These steps are as follows.

MILK PRETREATMENT Milk used for cheesemaking is normally standardized and heat treated. In some cases, milk is homogenized. An acid-producing starter culture is then added. The standardization of milk has become necessary to ensure that milk obtained from several producers or dairies is of a “standard” composition and condition throughout the year. This is critical in cheesemaking because the legal standards of various cheeses specify certain fat-to-protein ratios. The fat-to-protein ratio is determined mainly by the fat-to-casein ratio in the milk (Fox et al., 2000) which can be modified by removing fat or by adding cream or skim milk or skim milk powder, etc. It is also common to add color (annatto or carotene) and calcium (in the form of CaCl2) to the milk and to adjust milk pH to a desired level, known as preacidification. Adding calcium speeds up coagulation or reduces the amount of rennet needed and produces a firmer gel. Heat treatment of milk is primarily intended to destroy the harmful microbial population and enzymes in raw milk to assure product safety and quality. Pasteurization is the most commonly used heat treatment (72°C with 15 s holding time). It © 2003 by CRC Press LLC

Acid Coagulated Soft Cottage Cream Quark Queso Blanco Baker’s Neufchatel Ricotta (Acid and heat coagulated from whey)

Surface Ripened

Semi-soft Brick Bel Paese Havarti Limburger Munster Oka Port du Salut St. Paulin Trappist Taleggio Tilsiter Soft Liderkranz

Concentrated (from Whey)

TYPES OF NATURAL CHEESES

Gjetost Myost Primost Enzyme Coagulated

Internal Bacteria Ripened Very Hard Asiago Grana Parmesan Parmigiano Romano Sabrinz Sardo Hard Caciocavallo Cheddar Cheshire Colby Graviera Ras Cheese with eyes Edam Emmental (Swiss) Gouda Gruyere Maasdam Samsoe

Mould Ripened

Internal Mould Semi-soft Blue Danablu Gorgonzola Roquefort Hard Stilton

Surface Mould Soft Brie Camembert Coulommiers Carre de l’Est

Semi-soft

Soft Salt-cured/ Pickled Domiati Feta

Caerphilly Mahon Monterey Jack Pasta filata Mozzarella Provolone Caciocavallo

FIGURE 1.1 Natural cheeses classified according to the maturation agent used and firmness. (After Vedamuthu and Washam, 1983; Fox et al., 2000.)

not only destroys most of the bacteria present, including lactic-acid bacteria, but also inactivates many enzymes. A gentle heat treatment, known as thermization (60 to 65°C with 15 to 30 s holding time) may also be used advantageously before or after pasteurization (Spreer, 1998). However, many cheeses are still produced from raw milk, especially in Europe (Fox et al., 2000). If the cheeses are made from unpasteurized milk, they must be cured for at least 60 days at not less than 1.7°C (35°F), and the label should indicate the manufacturing date or state “held for more than 60 days.” (NDC, 2000). In traditional cheesemaking, the acid produced by microorganisms present in raw milk lowers the milk pH to a level sufficient for subsequent coagulation. However, if the milk undergoes a heat treatment, selected cultures of lactic-acid bacteria

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TABLE 1.3 Classification of Cheeses by the Distinctive Manufacturing Process Involved Distinctive Process Involved Curd particles matted together Curd particles kept separate Bacteria-ripened throughout interior Prolonged curing period Pasta filata Mold-ripened throughout interior Surface-ripened mainly by bacteria and yeasts Surface-ripened mainly by mold Curd coagulated mainly by acid

Characteristics

Example Cheeses

Close texture, firm body Slightly open texture Gas holes or eyes with eye formation throughout cheese Granular texture; brittle body Plastic curd; stringy texture Visible veins of mold (blue-green or white); piquant, spicy flavor Surface growth; soft, smooth, waxy body; mild to robust flavor Edible crust; soft, creamy interior; pungent flavor Delicate soft curd

Cheddar Colby, Monterey Jack Swiss (large eyes), Edam or Gouda (small eyes) Parmesan, Romano Mozzarella, Provolone Blue, Gorgonzola, Roquefort Brick, Limburger Brie, Camembert Cottage, Cream, Neufchatel

Source: After NDC, 2000. With permission.

TABLE 1.4 Empirical Texture/Rheological Parameter Values Used in Cheese Classification Logarithmic Scale Values Cheese Type Very Hard Hard Semihard Soft

Moisture (%)

Viscosity Factor

Elasticity Factor

Springiness Factor

< 25 25–36 36–40 > 40

>9 8–9 7.4–8 < 7.4

> 6.3 5.8–6.3 < 5.8 < 5.8

> 2.3 2–2.3 1.8–2 < 1.8

Source: After Davis, 1965.

TABLE 1.5 United States Federal Standards for the Maximum Moisture and Minimum Milk Fat for Classes of Cheese Designated by Consistency Consistency

Maximum moisture content (%)

Minimum milk fat in solids (%)

Hard grating Hard Semisoft Semisoft part skim Soft

34 39 50 (>39) 50 Not specified

32 50 50 45 (<50) 50

Source: After CFR, 1998.

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TABLE 1.6 Typical Composition (% by Weight) of Milk and Some Cheese Varieties Moisture

Protein

Total Fat

Total Carbohydrate

Fat in Dry Matter

Ash

Calcium

Phosphorus

Salt

pHa

Cow Goat Sheep Buffalo

87.3 87.7 80.7 82.8

3.4 2.9 4.5 4.8

3.7 4.5 7.4 7.5

Milk 4.8 4.1 4.8 4.8

29.1 36.6 38.3 41.7

0.7 0.8 1.0 0.8

0.12 — — —

— — — —

0.90 0.95 1.10 0.85

6.7 — — — —

Dry curd cottage Creamed cottage Quark Cream Neufchatel

79.8 79.0 72.0 53.7 62.2

17.3 12.5 18.0 7.5 10.0

0.42 4.5 8.0 34.9 23.4

Acid Coagulated 1.8 2.7 3.0 2.7 2.9

2.1 21.4 28.5 75.4 62.0

0.7 1.4 — 1.2 1.5

0.03 0.06 0.03 0.08 0.07

0.10 0.13 0.35 0.10 0.13

nil 1.00 — 0.73 0.75

5.0 5.0 4.5 4.6 4.6

Chhana Queso Blanca, acid Ricotta from 3%-fat milk Ricotone from whey and milk

53.0 55.0 72.2 82.5

17.0 19.7 11.2 11.3

25.0 20.4 12.7 0.5

Heat-Acid Coagulated 2.0 3.0 3.0 1.5

53.2 44.8 45.7 2.9

— — — —

— — — —

— — — —

— 3.00 <0.5 <0.5

— 5.4 5.9 5.8

Queso Blanco-rennet Queso de Frier Italian fresh cheese

52.0 52.4 49.0

23.0 23.0 28.0

Unripened-Rennet Coagulated 20.0 — 42.0 19.5 — 41.0 16.0 — 31.4

— — — —

— — —

— — —

2.50 3.00 nil

5.8 5.8 6.5

Type and Cheese

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TABLE 1.6 (continued) Typical Composition (% by Weight) of Milk and Some Cheese Varieties Type and Cheese

Total Fat

Total Carbohydrate

Fat in Dry Matter

Ash

Calcium

Phosphorus

Salt

pHa

50.3 47.5 50.0 50.0

3.7 5.2 5.1 5.0

0.39 0.49 0.53 —

0.35 0.34 0.39 —

2.10 — 3.50 —

6.9 4.4 6.5 — —

51.7 46.9 47.6 44.6 46.9 51.6

3.4 3.9 4.2 3.3 2.8 3.7

0.68 0.70 0.73 — — 0.72

0.46 0.55 0.54 — — 0.47

0.65 0.82 0.96 1.20 2.20 1.80

5.3 5.8 5.7 5.6 5.9 6.2

24.9 28.1 25.6 19.4

Hard Cheese Low Temperature 33.1 1.3 52.4 26.9 — 45.2 26.6 2.1 45.1 21.6 2.2 47.1

3.9 3.6 4.7 2.6

0.72 — 0.76 0.52

0.51 — 0.50 0.37

1.80 1.50 2.20 1.00

5.5 5.8 5.4 5.3

35.7 31.8 28.4 24.8

Hard Cheese High Temperature 25.8 3.2 36.5 26.9 3.6 39.0 27.4 3.4 43.7 28.3 — —

6.0 6.7 3.5 4.7

1.18 1.06 0.96 —

0.69 0.76 0.60 —

3.00 3.00 1.20 —

5.4 5.4 5.6 5.2

Moisture

Protein

Camembert Feta Blue Gorgonzola

51.8 55.2 42.0 36.0

19.8 14.2 21.0 26.0

Soft Ripened High Acid 24.3 0.5 21.3 — 29.0 2.3 32.0 —

Colby Gouda Edam Fontina Havarti-Danish Munster

40.0 41.5 41.4 42.8 43.5 41.8

25.0 25.0 25.0 24.2 24.7 23.4

31.0 27.4 27.8 25.5 26.5 30.0

Cheddar Manchego, Spain Provolone Mozzarella

36.7 37.9 40.9 54.1

Parmesan Romano Swiss Kaflatyri, Greece

29.2 30.9 37.2 34.2

a

pH at time of retailing.

Source: After Hill, 1995; Fox et al., 2000.

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Semihard Washed 2.0 2.2 1.4 — — 1.1

FIGURE 1.2 United States total (excludes cottage cheese) and Cheddar and Mozzarella cheese production trends. (After Annual Summary of Dairy Market Statistics of years 1997 through 2001. Agricultural Marketing Service, USDA. Mozzarella data from University of Wisconsin Dairy Marketing Web site: www.aae.wisc.edu/future.)

must be added. The type of bacteria added depends on the cheese type and cheesemaking protocol used. These bacteria break down the milk sugar, lactose. Lactic acid produced during this process lowers the pH. An alternative to adding starter culture is to acidify the milk directly by adding lactic acid or hydrochloric acid or gluconic acid-δ-lactone, an acidogen. Though this direct acidification allows better control, starter culture remains active in the cheese during ripening, months after cheese manufacture, and contributes to cheese flavor. Therefore, direct acidification is used primarily when manufacturing cheese varieties for which texture is more important than flavor, e.g., cottage cheese, quark, Mozzarella, etc. (Fox et al., 2000). Walstra and Jenness (1984) reported an increase in cheese yield when using pasteurized milk. This is due to casein–whey protein interaction and greater moisture retention. One disadvantage of pasteurization, however, is that aged cheeses develop their flavors more slowly and to a lesser extent than cheeses made with raw milk (Kristoffersen, 1985). This has led many cheesemakers to use milk heated to 60 to 68.5°C for 15 s or less instead of pasteurized milk (Johnson, 1998).

COAGULATION Since pretreating milk is a fairly recent practice relative to the history of cheesemaking, many consider coagulation as the first and most important step in cheesemaking. Coagulation is the step during which milk undergoes a profound physical and rheological change, that is gelation. Milk gel is formed by aggregation of milk protein, the caseins. This can be accomplished by: 1. The action of a proteolytic enzyme 2. Lowering the pH below the isoelectric point of protein (~ 4.6) 3. Heating to about 90ºC at a pH of about 5.2 (i.e., higher than the isoelectric point)

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Milk

Pretreatment (Standardization, Homogenization, Heat treatment, Starter addition)

Coagulation (Rennet /Coagulant addition)

Syneresis (Cutting, Stirring, Scalding/cooking, Whey removal)

Soft Cheese (e.g., Camembert)

Semi Hard Cheese (e.g., Gouda)

Hard Cheese (e.g., Cheddar)

Pasta Filata Cheese (e.g., Mozzarella)

Moulding

Hot water Washing

Cheddaring

Heating & Stretching

Brining

Pressing & Moulding

Milling

Moulding

Storage

Brine Salting

Dry Salting

Brine Salting

Turning

Waxing & Wrapping

Ripening

Packing

FIGURE 1.3 Major steps in cheesemaking (actual steps and/or conditions for a particular cheese may vary). (After Scott et al., 1998; Fox et al., 2000.)

Among these, enzymatic coagulation is the most popular. Acid coagulation via food-grade acidulants is used to manufacture quark, cottage, and cream cheeses. Heat coagulation is used for Ricotta and Queso Blanco cheeses (Johnson and Law, 1999; Fox et al., 2000). Enzymatic coagulation is accomplished by enzymes from animal (e.g., calf rennet, porcine pepsin), plant (e.g., Cynara Cardunculus from Cardom, Circium and

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Carlina Spp. from thistle), or microbial (e.g., Endothia parasitica, Rhizomucor miehei) origin. Enzymatic coagulation consists of two phases. During the first or primary phase, the hydrophilic hairy structure, stabilized by steric hindrance, of κ-casein is cleaved off at Phe105-Met106 bond. The secondary or clotting phase is initiated as 85 to 90% of the κ-casein is cleaved and results in the aggregation of the altered protein micelles. The κ-casein loses its ability to stabilize the remainder of the caseinate complex. The result is soluble glycomacropeptides (residues 106–169), and hydrophobic, para-κ-casein (residues 1–105). As the protein micelles continue to aggregate, a loose network forms, entrapping fat globules, water, and water-soluble materials. The para-κ-casein left on the micelle is still connected to α- and β-casein, but it is highly hydrophobic and basic, leading to destabilization of the micelle. Gel formation by association of the modified micelles in the secondary phase is highly dependent on the milk’s temperature and calcium content. The coagulation rate is also highly dependent on the concentration and activity of the enzyme solution. Increases in both of these factors shorten coagulation time and increase firmness. Although it is not clear how the micelles aggregate, there are two hypotheses. One is that hydrophobic bonding occurs between the para-κ-casein. The other is that calcium and calcium phosphate bonding occurs in α- and β-caseins. Other factors that affect aggregation are casein concentration and milk pH. The aggregation rate is proportional to the square of casein concentration (Lomholt and Qvist, 1999). As discussed previously, the effect of renneting action strongly depends on milk pH. Each milk-clotting enzyme has an optimum pH at which it is most active. Extremes in acid or base also denature the enzymes but not as irreversibly as high temperatures. Lowering the pH leads to a decrease in coagulation time mainly due to increased enzyme activity, but rate of aggregation is also affected (Lomholt and Qvist, 1999). The aggregation of casein micelles forms strands of casein particles of about three particles wide and 10 particles long, alternated by some thicker nodes of particles (Walstra et al., 1999). After this, the aggregates grow more compact (Bauer et al., 1995). The time when aggregates become visible is known as the flocculation time or rennet coagulation time (RCT). When the flocs grow to occupy the entire volume, the gel is said to have been formed. The gel network is very irregular, with many pores several micrometers in width (Walstra et al., 1999). Aggregation of casein micelles into chains, then into strands and clusters, and eventually into an amorphous mass has been observed by microscopic evaluation in both acid- and enzyme-coagulated systems (Kimber et al., 1974; Glaser et al., 1980). From a rheological standpoint, casein aggregation and gel formation represent an increase in viscosity and gel modulus, respectively. The viscosity increase in renneted milk, however, is observed after an initial lag time (~ 60% RCT) during which the viscosity may actually decrease slightly due to a decrease in voluminosity of the casein micelles following the release of macropeptides (Fox et al., 2000). After this, the viscosity increases exponentially up to the onset of gelation (i.e., 100% RCT). The viscosity increase and the concomitant change in physical properties have been used to identify the RCT (Kopelman and Cogan, 1976; Ay and Gunasekaran, 1994; Fox and McSweeney, 1998; Konuklar and Gunasekaran, 2002). The modulus of the gel increases markedly at gelation time. In fact, gelation time is defined as the time at which the gel modulus increases rapidly. The initial

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102 80 101 70

G′ G′′ δ

gel point

10 –1

10 –2

60

50

δ(°)

G′, G′′ (Pa)

100

40

10 –3

30

10 –4

20 500

1000

1500

2000

2500

3000

3500

Time (s)

FIGURE 1.4 Changes in storage (G′) and loss (G″) moduli and phase angle (δ) of the rennetted milk during coagulation. The gel point is identified at G′–G″ crossover which occurs at δ = 45°. (After Uludogan, 1999.)

increase in modulus is due to the increase in number of contacts between micelles. Subsequently, the strengthening of intermicellar bonds translates into increased gel modulus (Walstra et al., 1999). It has been premised that the increase in gel firmness is due to the increase in the number of bonds with time (Lomholt and Qvist, 1999). This premise was based on the observation that, though the modulus continues to increase, the phase angle stays relatively constant, i.e., the nature of the bonds does not change (Dejmek, 1987; Lopez et al., 1998). Figure 1.4 shows a typical plot depicting changes in viscoelastic moduli and the phase angle of the coagulating milk gel system. As more micelles aggregate, they may fuse together and strengthen the bonds (Lomholt and Qvist, 1999). The modulus continues to increase for several hours after gelation time, signifying gel firming. The microstructure of the gel has been observed to become coarser with larger pores and thicker strands (Lomholt and Qvist, 1999). Carlson et al. (1987) presented a detailed analysis of all aspects of milk coagulation kinetics in a four-part series of papers.

SYNERESIS Due to its porous nature, the coagulum has the propensity to contract and expel entrapped liquid. This is known as syneresis, an important step in concentrating the milk. To a great extent, the success of the remaining cheesemaking steps depends on satisfactorily draining the whey. Also, most of the lactose, a substrate for postproduction microbial activity, is lost in the whey, which helps to prevent some

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adverse effects (Scott et al., 1998). In an undisturbed gel, however, syneresis occurs very slowly. Therefore, during cheesemaking, syneresis is accelerated by cutting the coagulum into small cubes, which increases the surface area and reduces the distance for the diffusion process to facilitate whey removal. Syneresis can also be enhanced by decreasing the pH or increasing the temperature of the coagulum (Walstra et al., 1999). Cutting the coagulum to facilitate faster whey removal must be timed precisely. If the coagulum is cut too soon, some milk solids leave the curd along with whey. Whey normally carries water-soluble components including lactose, whey proteins, salts, peptides, and other nonprotein nitrogenous substances (Scott et al., 1998). If it is cut too late, more water gets trapped in the matrix, resulting in high-moisture cheese. Therefore, cheesemakers have been striving for many years to identify the correct curd-cutting time. Since the coagulum firmness continues to increase uneventfully over several hours, it is hard to determine an optimal curd-cutting time. Many instrumented and so-called objective curd-cutting-time predictions have been made (Hori, 1985; Payne et al., 1993; Gunasekaran and Ay, 1996; O’Callahan et al., 1999). Some commercial units are available based on some of these techniques (Fox and McSweeney, 1998; O’Callahan et al., 1999). However, there is still no universal procedure to identify optimal curd-cutting time. Most large factories apply a set time schedule, depending on the cheese type, to cut the curd after adding the rennet. In many smaller cheesemaking facilities, cutting time is still determined by the subjective judgment of the cheesemaker. Recently, Konuklar and Gunasekaran (2002) reported a novel rheological technique for identifying the curd-cutting time. They observed that the viscosity versus time curves during coagulation under continuous steady shear exhibit several abrupt peaks. The first peak over 40 kPa.s coincides with cutting time determined by an experienced cheesemaker during Cheddar, Swiss, and Gouda cheesemaking (Figure 1.5.) Syneresis is the process that a cheesemaker can use to closely control the moisture content of the cheese and hence the microbial and enzymatic activity in the cheese, which affects ripening, stability, and quality of the cheese (Fox et al., 2000). Therefore, it is specific to a particular cheese type or cheese family. Walstra et al. (1999) listed the following factors as affecting syneresis: firmness of gel at cutting; surface area of the curd; any applied pressure; acidity; temperature; composition of the milk; and other variables. Pearse and Mackinlay (1989) discussed the mechanism and biochemical aspects of syneresis. Stirring exerts pressure, causing curd particles to collide, and facilitates their compression for a short time. Stirring also keeps the curd from settling in the vat. For Cheddar- and Swiss-type cheeses, the cut coagulum is not stirred immediately after cutting. The curd–whey mixture is cooked (at about 40ºC for Cheddar-type and 50ºC for Swiss-type) and vigorously agitated during cooking. For soft cheeses, the curd is ladled and hooped which allows whey to drain without stirring. Cooking the curd, also known as scalding, enhances syneresis by facilitating contraction of the protein matrix. Heating further enhances acid production by the starter organisms. Lowering pH, combined with increased temperature, not only helps to expel more whey but also affects the dissolution of calcium phosphate, and thus has major implications for characteristics of the cheese (Johnson and Law, 1999). The scalding

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FIGURE 1.5 Viscosity (η) of coagulating milk system vs. time after rennetting measured under a continuous steady shear stress of 0.2 Pa. The first viscosity peak over 40 kPa.s coincided with the cutting point (CTP) determined manually during (a) Cheddar; (b) Swiss; and (c) Gouda cheesemaking. (After Konuklar and Gunasekaran, 2002. With permission.)

step can be used to distinguish four major groups of cheese — excluding soft cheeses, some of which may be scalded (Scott et al., 1998): 1. Textured cheeses such as Cheddar or Cheshire 2. Pasta filata types or kneaded cheeses 3. Cheeses untextured in the vat (e.g., Edam and Gouda) and those which acquire texture later (e.g., Tilsiter and Emmental) 4. Blue-veined cheeses

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To manufacture some cheeses (e.g., Edam, Gouda, or Havarti), the curd is washed by adding water to the curd–whey mixture. This accomplishes two things: 1. It adjusts the pH of the cheese independently of its moisture content by removing lactose and other solubles from the curd. 2. It enhances whey removal by adding hot water to raise the curd temperature, as is the case during direct heating. It should be noted that using hot water to stretch pasta filata cheeses (e.g., Mozzarella, Provolone, etc.) is not considered as washing (Scott et al., 1998).

SHAPING

AND

SALTING

When the curd is at the desired moisture content and pH, it is separated from the whey. The curd particles are subsequently shaped into some form and salted (primarily by NaCl), not necessarily in that order. These steps, though common for most cheeses, are performed very differently, depending on the cheese type. As Johnson and Law (1999) stated, the manner in which cheese curd and whey are separated can affect texture as well as color and flavor. When manufacturing hard cheeses such as Cheddar, the curd–whey slurry is pumped into a vat with a perforated bottom for whey removal. The curd is “cheddared” for about 90 min. Cheddaring is the process in which curd particles are allowed to fuse or “mat” together. The mats are then cut into slabs and stacked on top of each other. Physical properties and pH of the curd at this stage affect curd fusion and appearance of the finished cheese (Olson, 1995). When the desired pH has been reached, the slabs are milled into small pieces. At this stage, the curd may be sprayed with warm water and stirred for further whey removal. Salt is sprinkled on at a level of about 2 to 3% which expels additional whey. The salted curd is then hooped in molds and pressed overnight. Manufacturing steps for Mozzarella and other pasta filata cheeses differ markedly after the milling stage described above. The milled curd is “kneaded,” i.e., heated and stretched in warm water (about 60 to 70°C) using an open-channel, single-screw or twin-screw extruder/auger. This transforms the curd into a cohesive, viscoelastic mass. Due to the conveying action of the auger, the curd mass gets stretched into a continuous stream of molten material. This stretching step is unique to Mozzarella manufacturing. It imparts the characteristic oriented microstructure and related textural attributes of these cheeses (Oberg et al., 1993; Ak and Gunasekaran, 1997). The molten cheese is then placed into molds and cooled. When the cheese is cool enough to keep its shape, the mold is removed and the cheese is salted by dropping it in a nearly saturated brine solution (about 25% salt) at 1 to 4°C. The cold brine temperature cools the cheese further. In fact, much of the total cooling of Mozzarella occurs during brining (Nilson, 1968). Brine salting is a slow process, taking several days for uniform salt distribution within a cheese block. It should be noted that, concomitant with salt intake, the cheese loses moisture. The salt and moisture gradients in a cheese during salting are opposite of each other (Turhan and Gunasekaran, 1998; Walstra et al., 1999;

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Fox et al., 2000). Though brine salting is the traditional method, salting of Mozzarella can also be done by adding salt directly to the curd just before stretching, during stretching, or between stretching and molding. This direct salting reduces the subsequent brining time. Another major variation is surface salting. Salt is rubbed directly on the cheese surface (e.g., Romano and Gorgonzola). This is repeated for several days so the salt diffuses throughout. In many other cheeses, surface salting and brining are used in combination (e.g., Gruyere and Emmental). Regardless of the method used, salting is a vital step in cheesemaking because unsalted cheese is virtually tasteless (Olson, 1995). Salt also plays a major role in the texture, flavor, and microbial quality of cheese (Kindstedt et al., 1992; Paulson et al., 1998; Fox et al., 2000). Salt inhibits the growth of certain bacteria, which are harmful to the cheese and cause spoilage, especially on the surface. It further assists in dissolving the casein and in rind formation, as well as in slowing down enzyme activity. Salt concentration in cheese varies greatly from less than 1% in Emmental to 7 to 8% in Domiati (Fox and McSweeney, 1998). The salt content may also vary considerably within a cheese block due to the slow diffusion of salt. Thus, there is more water and less salt at the center of a cheese block compared to at the surface (Prentice, 1993). This unevenness in the salt (and water) distribution also leads to variation in the rheological properties of the cheese within the block (Visser, 1991). As already noted, hard and semihard cheeses are shaped by applying external pressure. Pressing expels whey and facilitates faster curd fusion into an integral mass of a desired shape with a rind. Though simple enough, pressing is perhaps the least understood step in cheesemaking (Scott et al., 1998). The time, pressure, and efficiency of pressing are related to the condition of the curd at pressing time and the decrease in pH during pressing (Johnson and Law, 1999). Sometimes pressing is done in conjunction with vacuum to force out any entrapped air. The complex nature of the interrelationships among many of the cheesemaking parameters makes controlling cheese properties very hard. Tables 1.7a to 1.7d present how various cheesemaking and technological factors affect cheese quality. This set of four tables was prepared in 1961 for the Danish Samso cheese (Birkkjaer et al., 1961), but the information it contained is generally valid for other hard/semihard cheeses (e.g., Gouda).

RIPENING Ripening is the natural process of microbial and biochemical reactions that occurs in a cheese after its manufacture and during storage. Ripening gives different cheeses their unique flavors, textures, and appearances. Except for some soft cheeses (e.g., cottage cheese, cream cheese, quark, etc.), almost all cheeses are held under controlled conditions to develop distinct attributes. Ripening essentially results from the action of microorganisms present within the curd mass and on its surface. Ripening is also influenced by residual enzymes present in the cheese curd. Cheeses are ripened over a range of time from several days (e.g., Mozzarella) to more than a year (e.g., Cheddar).

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Fox et al. (2000) list the following ripening agents in cheese: 1. 2. 3. 4.

Coagulant — chymosin or other suitable proteinase Milk — some indigenous enzymes contained in milk, e.g., plasmin Starter culture — host of enzymes released upon cell death and lysis Secondary microflora — microflora that perform some specific secondary function (e.g., propionic acid, bacteria, and yeasts and molds) 5. Exogenous enzymes — proteinases, peptidases, and lipases added by cheesemakers to accelerate ripening

TABLE 1.7A Effect of Cheesemaking Parameters on Cheese Quality (Prepared for Danish Samso Cheese, a Gouda-Type Semihard Cheese) Modifications Needed to Produce a Harder Cheese and Their Primary and Secondary Effects Factor No. Efficiencya 1

2

+

3c

+

4

+

5c

(+)

6c

+

Modifications

Primary Effect

Secondary Effectb

Use fresh milk or Slightly improves whey Cheese becomes slightly pasteurize at expulsion less acid. Ca content approx. 70ºC increases slightly (158ºF) Reduce or omit Improves whey expulsion Cheese becomes more acidic. Ca content addition of water to increases the milk Add CaCl2 to the Improves whey expulsion Ca content increases. milk Adding more than 40 g/100 kg milk (0.68 oz/110 lbs milk) may give an off flavor Increase amount of Slightly improves whey Cheese becomes more acidic. culture/starter or expulsion from the curd Ca content decreases. prolong preToo much culture/starter or ripening period of too long a preripening the milk makes cheese sour, short, and flaky Lower renneting At the same cooking A renneting temperature temperature temperature, whey drain that is too low results in in the vat increases weak curd and thus a slightly due to a greater bigger loss in the whey. rise in temperature. If no Ca content decreases cooking occurs, whey is reduced, and cheese becomes softer Cut curd into Improves whey expulsion Very fine cutting may result smaller cubes in a bigger loss in whey. Many of the “grains” may retain some of the whey during molding/pressing, so cheese may be softer

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TABLE 1.7A (continued) Effect of Cheesemaking Parameters on Cheese Quality (Prepared for Danish Samso Cheese, a Gouda-Type Semihard Cheese) Modifications Needed to Produce a Harder Cheese and Their Primary and Secondary Effects Factor No. Efficiencya 8

+

9

+++

10

+++

12

++

13c

+++++d

14

++

15c

+

Modifications

Primary Effect

Secondary Effectb

Reduce or omit addition of water during cooking

Improves whey expulsion Cheese becomes sour and of curd, esp. by reducing may eventually become relatively large additions too firm in the curd and of water thus will often break. Ca content increases Increase cooking Increases whey expulsion Cheese becomes less acidic and tougher At high temperature in the vat temperatures, esp. above 40ºC (105ºF); cheese may develop an off flavor Avoid a temperature Increases whey expulsion Cheese becomes less acidic. drop during final in the vat and during Ca content increases stirring pressing Reduce or omit salt Cheese “grains” swell less Cheese becomes less acidic. addition to whey and thus retain less whey Ca content increases. Brine salting may be prolonged during final stirring to get adequate salt content Leave cheese at Cheese liberates a Ca content decreases cooking relatively large quantity considerably temperature in of whey before rind is water or whey after closed pressing in the vat Increase Increases amount of whey Cheese becomes slightly temperature in draining during pressing less acidic. If it is not pressing room cooled longer, the risk of cracked rind and gas from coliforms may increase Prolong pressing Increases amount of whey You might see adhesion, time (possibly until draining during pressing especially when using the next morning) cotton cloths and a relatively high pressing temperature. Counteract this by using nylon cloths and cooling during last part of pressing

a

+ Represents relative efficiency, the higher the better. (+) means that the effect depends on other conditions. Shaded effects reduce acidity; bold-faced effects increase acidity. c Factor numbers 3, 5, 6, 13, and 15 show modifications that influence either acidity or firmness but not both. All other factors influence both. d An extraordinary change in technique, 3 hours at 38ºC. b

Source: After Birkkjaer et al., 1961. With permission.

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TABLE 1.7B Effect of Cheesemaking Parameters on Cheese Quality (Prepared for Danish Samso Cheese, a Gouda-Type Semihard Cheese) Modifications Needed to Produce a Softer Cheese and Their Primary and Secondary Effects Factor No. Efficiencya 1

Modifications

Primary Effect

Pasteurize at 65ºC Slightly reduces whey (150ºF) or above expulsion (above 80ºC 75ºC (166ºF), esp. [175ºF] somewhat) above 80ºC (175ºF)

2

+

3c

+

4

+

5c

(+)

6c

+

8

+

Add water to the milk Reduce or omit addition of CaCl2 to the milk Reduce amount of culture/starter or shorten or omit preripening period of the milk

Reduces whey expulsion Reduces whey expulsion

Slightly reduces whey expulsion

Secondary Effectb Cheese becomes (above 80ºC [175ºF] somewhat) more acidic. Ca content decreases. High pasteurization temperatures often lead to weak eye formation Cheese becomes less acidic. Ca content increases Ca content decreases

Cheese becomes less acidic. Ca content decreases. Not enough culture/starter or a pre-ripening that is too short may produce a tough cheese with an off flavor A renneting temperature that is Raise renneting At the same cooking too high will cause cutting temperature temperature, whey problems since the coagulum expulsion in the vat is slightly reduced due to a will be too firm during cutting. Ca content increases smaller rise in slightly temperature. If no cooking occurs, whey expulsion increases and cheese is firmer Cut curd into bigger Reduces whey expulsion Big curd cubes can be easily cubes stirred into smaller pieces, causing greater whey drain than intended and greater loss in the whey. Many “grains” may retain some whey when cheese is molded so cheese becomes softer than intended Add more water Reduces whey expulsion, Cheese becomes less acidic. Ca content decreases. If you during cooking esp. when relatively add more than 20% of the large amounts of water quantity of milk, cheese often are added develops an off flavor

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TABLE 1.7B (continued) Effect of Cheesemaking Parameters on Cheese Quality (Prepared for Danish Samso Cheese, a Gouda-Type Semihard Cheese) Modifications Needed to Produce a Softer Cheese and Their Primary and Secondary Effects Factor No. Efficiencya

Primary Effect

Secondary Effectb

Reduces whey expulsion

Cheese becomes more acidic. Cheese which is already acidic may crack easily Cooling without water leads to acidic cheese. Cooling with water leads to small or no change in acidity, depending on amount of water. Ca content decreases Cheese becomes more acidic. Ca content decreases. Shorten brine salting so cheese is not too salty. Heavy salting in the vat may restrain fermentation, producing cheese with high pH Ca content increases

Modifications

9

+++

Lower cooking temperature

10

+++

Cool curd cubes for Reduces whey expulsion about 15 min before in the vat and during final stirring ends pressing

12

++

Add more salt to whey during final stirring

Curd cubes swell more and retain more whey

13c

+++++d

Rind closes earlier, slowing whey expulsion

14

++

Reduce pressing temperature or drain the whey faster Lower temperature in pressing room

15c

+

Use low pressure to start or shorten pressing time

Less whey is pressed out of the cheese

Reduces whey expulsion during pressing

a

Moderate cooling yields more acidic cheese. Cooling too soon retards fermenting and gives high pH cheese. Cooling after or during last part of pressing, if long enough, slows rind cracking and coliform production Low pressure and short pressing time may cause bad rind closing, fermenting in rind (cracked rind) and open texture

+ Represents relative efficiency, the higher the better. (+) means that the effect depends on other conditions. Shaded effects reduce acidity; bold-faced effects increase acidity. Shaded and bold-faced effects may result in more or less acidity. c Factor numbers 3, 5, 6, 13, and 15 show modifications that influence either acidity or firmness but not both. All other factors influence both. d An extraordinary change in technique, three hours at 38ºC. b

Source: After Birkkjaer et al., 1961. With permission.

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TABLE 1.7C Effect of Cheesemaking Parameters on Cheese Quality (Prepared for Danish Samso Cheese, a Gouda-Type Semihard Cheese) Modifications Needed to Produce a Less Acidic Cheese and Their Primary and Secondary Effects Factor No. Efficiencya

Modifications

Primary Effect

Use raw milk or Slightly improves whey pasteurize at approx. expulsion 70ºC (158ºF) Add water to the milk Reduces whey expulsion

Secondary Effectb

1

+

2

++

4

++

Reduce amount of culture/starter, or shorten preripening period of the milk

7c

++

8

+++

9

++++++

10

++

11c

+

Start cooking earlier Development of lacticby shortening acid bacteria is prestirring or restrained and whey intermediate stirring, is expelled and prolong final stirring correspondingly Add more water Greater diffusion of sugar Adding more than 20% during cooking and Ca from curd cubes water may produce a to the whey cheese with an off flavor. Increasing the water added produces cheese with a higher water content and lower Ca content Raise cooking Development of lacticCheese becomes firmer temperature acid bacteria in the vat is and tougher. At high slowed. Whey is temperatures, especially expelled earlier above 40ºC (105ºF), cheese develops an off flavor Avoid a drop in Development of lacticCheese becomes firmer. temperature during acid bacteria is slowed Ca content increases final stirring Prolong the time for Acidity is slightly Cheese consistency final stirring so total changed because becomes more supple stirring time is longer development of lactic(flexible), so cheese is acid bacteria is slowed. easier to cut But because curd cubes are kept in the whey longer, more Ca is discharged

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Less acidification

Cheese becomes slightly firmer. Ca content increases slightly Cheese becomes softer. Ca content decreases Cheese becomes a little softer. Ca content increases. Not enough culture/starter and a weak preripening produces a tough cheese with an off flavor Ca content increases

TABLE 1.7C (continued) Effect of Cheesemaking Parameters on Cheese Quality (Prepared for Danish Samso Cheese, a Gouda-Type Semihard Cheese) Modifications Needed to Produce a Less Acidic Cheese and Their Primary and Secondary Effects Factor No. Efficiencya 12

14

Modifications Reduce or omit addition of salt to whey during final stirring

+

Cool cheese early, or carry through pressing at high temperature 40ºC (105ºF)

Primary Effect

Secondary Effectb

Curd cubes swell less and Cheese becomes firmer. thus retain less whey Ca content increases. Brine salting should be prolonged to get an adequate salt content Development of lacticCooling results in a softer acid bacteria is slowed cheese. Cooling too early may result in high pH cheese. High pressing temperature makes a firmer cheese, and the danger of a cracked rind and fermentation is increased if it is not cooled after pressing

a

+ Represents relative efficiency, the higher the better. Shaded effects result in softer cheese; bold-faced effects result in firmer cheese. Shaded and bold-faced effects may result in softer or firmer cheese. c Factor numbers 7 and 11 show modifications that influence either acidity or firmness but not both. All other factors influence both. b

Source: After Birkkjaer et al., 1961. With permission.

Various methods of influencing cheese ripening are summarized in Table 1.8. The primary factors in this process are (Scott et al., 1998): 1. Storage temperature and humidity, humidity being less important for cheeses hermetically packed (e.g., with a wax coating). 2. Chemical composition of the curd — fat content, level of amino acids, fatty acids, and other by-products of enzymatic action. 3. Residual microflora of the curd — primarily from the starter culture. The cheesemaker can do little to influence it except in the case of blue-veined or surface-ripened cheeses. Temperature and humidity are factors that cheesemakers can control during ripening. In general, higher temperatures increase the microbial growth rate and other biochemical reactions occurring in the curd. Thus, cheeses matured at different temperatures can have different flavor profiles. Accordingly, proper control of storage temperature is essential. Variety-specific storage temperature control protocols have

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TABLE 1.7D Effect of Cheesemaking Parameters on Cheese Quality (Prepared for Danish Samso Cheese, a Gouda-Type Semihard Cheese) Modifications Needed to Produce a More Acidic Cheese and Their Primary and Secondary Effects Factor No. Efficiencya 1

+

2

++

4

++

7c

++

8

+++

9

++++++

10

++

11c

+

Modifications

Primary Effect

Secondary Effectb

Pasteurize at 65ºC Slightly reduces whey Cheese becomes slightly (150ºF) or above 75ºC expulsion (above 80ºC softer (above 80ºC [175ºF] (166ºF), esp. above [175ºF] somewhat) somewhat). Ca content 80ºC (175ºF) decreases slightly Reduce or omit addition Improves whey expulsion Cheese becomes a little of water to the milk from the curd firmer. Ca content increases Increase amount of Increases acid production Cheese becomes a little culture/starter or firmer. Ca content prolong preripening of decreases. Too much the milk culture/starter and too strong preripening causes cheese to be sour, short, and flaky Cook longer by Lactic-acid bacteria have Ca content decreases. A prolonging prestirring better growth conditions long intermediate stirring or intermediate stirring, in the vat. Whey time increases the loss in and shorten final stirring expulsion occurs later the whey because curd is correspondingly easily stirred into pieces Reduce or omit addition Less diffusion of sugar Cheese retains more Ca and of water during and Ca from curd cubes less water. It may become cooking to the whey too acidic and stiff and break. Cheese becomes softer. Cheese which is already acidic may crack Lower cooking Lactic-acid bacteria have Cheese becomes softer. temperature better growth conditions Ca content decreases in the vat. Whey expulsion occurs later Cool curd cubes for Lactic-acid bacteria have A short stirring time about 15 min before better growth conditions produces a tough cheese. end of stirring. A small Cut surface often gets amount of water is horny and greasy after optional storage Shorten the time of final Acidity is changed Cheese becomes softer. stirring so total stirring slightly because Ca content decreases. time is shorter development of lacticBrine salting should be acid bacteria is shortened so cheese is not improved. But because too salty curd cubes are kept in the whey for a shorter time, less Ca is discharged

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TABLE 1.7D (continued) Effect of Cheesemaking Parameters on Cheese Quality (Prepared for Danish Samso Cheese, a Gouda-Type Semihard Cheese) Modifications Needed to Produce a More Acidic Cheese and Their Primary and Secondary Effects Factor No. Efficiencya 12

14

Modifications Increase amount of salt added to the whey during final stirring

+

Primary Effect

Secondary Effectb

Curd cubes swell more and thus retain more whey

Too heavy salting in the vat restrains acidification, which may produce a high-pH cheese Take care that cheese Lactic-acid bacteria have Since rind fermentation can temperature during better growth conditions be prevented rather pressing stays near the during pressing. Pressing effectively by cooling, optimum temperature at high temperatures, or cheese should be cooled of the bacteria cooling too early to low after pressing or during temperatures in the curd, the last part of pressing, restrains acidification if this is long enough

a

+ Represents relative efficiency, the higher the better. Shaded effects result in softer cheese; bold-faced effects result in firmer cheese. c Factor numbers 7 and 11 show modifications that influence either acidity or firmness but not both. All other factors influence both. b

Source: After Birkkjaer et al., 1961. With permission.

been developed to optimize cheese quality. For example, Swiss-type Emmental is held at a low temperature initially (10 to 15°C) to facilitate the growth of lactic-acid bacteria. Later, the temperature is increased (20 to 24°C) so that propionic bacteria can grow. These are essential for the characteristic Emmental flavor and “eyes.” For blue-veined cheeses (e.g., Gorgonzola, Roquefort, Stilton), warm-temperature storage is followed by low-temperature storage (Scott et al., 1998). Prevailing relative humidity during storage (80 to 85%) helps to control the moisture content of cheeses not covered with moisture barriers such as a wax coating. The moisture equilibrium in the cheese changes due to reactions occurring that require or release water. Increase in moisture content during storage affects the solute concentration and microbial growth rate. In general, higher moisture content promotes more vigorous growth of microorganisms than does lower moisture content. In addition to temperature and moisture, other factors such as curd pH, inhibitory substances (e.g., antibodies and salts), and oxidation–reduction potential affect the microbial population in the cheese (Scott et al., 1998; Fox et al., 2000). The enzymes relevant for maturation in most hard cheeses are active in the pH range of 4.9 to 5.5, and in soft cheeses from pH 5.3 to 6.0 (Scott et al., 1998). Protein, fat, and lactose are hydrolyzed (i.e., proteolysis, lypolysis, and glycolysis, respectively) to varying extents during cheese ripening. Among these, proteolysis of casein is the most important. Proteolysis of α- and β-casein occurs due to any

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TABLE 1.8 Methods of Influencing Cheese Ripening and Their Advantages and Disadvantages Method of Influence

Advantages

Increased storage temperature Increased inoculation level in starter culture

Easy to perform No aspects determined by law Natural enzyme balance No aspects determined by law

Proteases/peptidases Lipases, animal, and microbial

Addition of Enzymes Relative low cost Specific effect

Lactobacillus/pediococci Brevibacterium lines Mold Propionic-acid bacteria

Cold/warm treated Lysozyme treated Nonacidic producing

Special Cultures Naturally balanced No aspects determined by law

Modified Starter Cultures Natural enzyme balance Easy to conform to

Disadvantages No specific effect Risk of destroying bacteria Influences pH and consistency

Few usable enzymes Risk of over-ripening Aspects determined by law Use of whey

Opposite effect on pH and consistency Different taste profile

Technologically complex

Source: After Kristensen, 1999. With permission.

residual rennet from what was added for coagulation, natural proteases, and proteases and polypeptidases from starter or adventitious bacteria (Scott et al., 1998). This is essential for cheese flavor development. Fat contains lipophilic flavor compounds, which develop or are released by microbial or enzymatic action through oxidation, decarboxylation, and eventually reduction of decarboxyl compounds. Glycolysis is also initiated by adding a starter culture and reaches its peak in the milk during the preripening stage. Here lactic acid, acetic acid, and CO2 are produced. In some cheeses, citrate is also metabolized into citric acid. Proteolysis is also mainly responsible for changes in the body and texture of cheeses. The breakdown of proteins first involves the conversion of casein fractions into large peptides. These peptides are later broken down to lower molecular weight products. The primary proteolysis in ripening has been defined as the changes in caseins, which can be detected by polyacrylamide gel electrophoresis. The products of secondary proteolysis include the peptides and amino acids that are soluble in the aqueous phase of the cheese. In mature Cheddar, approximately one-third of the protein has been broken down to forms that are soluble at pH 4.6 (Banks, 1998).

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PROCESS CHEESE Process cheese is manufactured from one or more of the natural cheeses described thus far. The basic premise is to stabilize the proteins that are normally affected already during one or more of the cheesemaking steps (Shimp, 1985). This is accomplished by heating and mixing cheeses with some emulsifying salts. The careful selection of cheeses, emulsifying salts, and processing factors allows making process cheeses of varied textures suitable for many end uses. The primary reasons for manufacturing process cheese are (Spreer, 1998): 1. 2. 3. 4.

Long shelf life due to heat treatment and hot filling Wide variety due to a multitude of ingredients and composition Efficient utilization due to spreadable consistency and small portions Upgrading of defective rennet cheese products if the defects limit the shelf life but the products are still edible

The basic steps in the manufacture of process cheese are selecting and blending raw materials, heat processing, and forming and packaging. The raw materials include the natural cheeses, emulsifying salts, and other ingredients. Using the appropriate cheeses in the blend is very critical to obtain the desired texture and flavor. The emulsifying salts, primarily phosphates and citrates, are selected for their ability to disperse and increase hydration of the cheese proteins, which creates smoothness and fat emulsification (Olson, 1995). Other ingredients vary, from dairy and nondairy products such as skim milk powder, whey protein concentrate, spices and vegetables, and muscle food ingredients, etc. In general, good quality raw materials ensure good quality process cheese. Process cheeses can be grouped into three major categories based on composition and consistency: process cheese block, process cheese food, and process cheese spread. The selection of type of heat processing and raw materials for each are done accordingly. A fourth group, process cheese analog based on vegetable fat-casein blend, is also manufactured. Specific manufacturing steps, ingredient selection, etc., are detailed in Caric and Kalab (1993). The manufacturing conditions for sliceable and spreadable process cheese are summarized in Table 1.9. The heat-processing step converts the raw material into a homogeneous product. Heating is performed under atmospheric pressure or vacuum at 85 to 95°C or under pressure at 105 to 120°C. Temperatures under 90°C are desirable to avoid a browning reaction when the raw materials are high in lactose. During heating, the mix is continuously stirred at 60 to 140 rpm. The process duration varies from 4 to 8 min for processed cheese blocks to 8 to 15 min for processed cheese spread (Caric and Kalab, 1993). After heat processing, the melt is conveyed to filling machines where it is molded into different shapes or put into containers. It can also be spread on conveyor belts and sliced. The cheese is then cooled. Cooling is performed fairly slowly (10–12 h) for process cheese blocks and very quickly (15–30 min) for process cheese spread to facilitate softening of the product.

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TABLE 1.9 Manufacturing Conditions for Sliceable and Spreadable Process Cheese Process Cheese Type Condition Average age of the raw material Relative casein content structure Melting salt

Water, how added Temperature Time for melting, stirring pH Process cheese Whole milk powder or whey powder Homogenization Packaging (filling) Cooling Treatment

Sliceable

Spreadable

Fresh to half-mature; mostly fresh 75–90%, mostly long

Combination of fresh, half-mature, and over-ripe 60–75%, short to long

Structure: not creamy Emulsifier: high molecular Polyphosphate, etc.

Structure: creamy Emulsifier: lower or medium molecular Polyphosphate, etc. 20–45%, in portions 85–98ºC/150ºC (185–208ºF/302ºF) 8–15 min, fast 5.7–5.9 5–20% 5–10% Desirable

10–25%, all at once 80–85ºC (176–185ºF) 4–8 min, slow 5.4–5.6 0–2% 0 None 5–15 min Slow (10–20 h) at room temperature Very careful

Fast (15–30 min) in freezing conditions (cool air) Intensive (powerful)

Source: After Kristensen, 1999. With permission.

REFERENCES Ak, M.M. and S. Gunasekaran. 1997. Anisotropy in tensile properties of Mozzarella cheese. Journal of Food Science 62(5):1031–1033. Ay, C. and S. Gunasekaran. 1994. An ultrasonic attenuation measurement for estimating milk coagulation time. Transactions of the ASAE 37(3):857–862. Banks, J.M. 1998. Cheese. In The Technology of Dairy Products, Ed. R. Early, 81–122. London, U.K.: Blackie Academic and Professional. Battistotti, B. et al. 1984. Cheese: A Guide to the World of Cheese and Cheesemaking. New York: Facts on File Publications. Bauer, R. et al. 1995. The structure of casein aggregates during renneting studied by indirect Fourier transformation and inverse Laplace transformation of static and dynamic light-scattering data, respectively. Journal of Chemical Physics 103:2725–2737. Birkkjaer, H.E. et al. 1961. The influence of the cheesemaking technique upon the quality of cheese. Report No. 128, Danish Government Research Institute for Dairy Industry, Hillerod, Denmark. (Translated from Danish and reprinted with permission in the Dairy Pipeline, 1998, Center for Dairy Research, University of Wisconsin-Madison, Madison, WI, 53706, U.S.A.)

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Caric, M. and M. Kalab. 1993. Processed cheese products. In Cheese: Chemistry, Physics and Microbiology, Vol. 2, Major Cheese Groups, 2nd ed., Ed. P.F. Fox, 467–505. New York: Chapman and Hall. Carlson, A., C.G. Hill, Jr., and N.F. Olson. 1987. Kinetics of milk coagulation: I-IV. Biotechnology and Bioengineering 29(4):582–589, 590–600, 601–611, 612–624. CFR, 1998. Cheeses and Related Cheese Products. Code of Federal Regulations, Title 21, Part 33, pp 294–346. United States Department of Health and Human Services, Food and Drug Administration, Washington. Davis, J.G. 1965. Cheese, Vol. 1, Basic Technology. London: Churchill Livingstone. Dejmek, P. 1987. Dynamic rheology of rennet curd. Journal of Dairy Science 70:1325–1330. Doeff, G. 1994. Cheese in the U.S.A. Dairy Foods Sept., p. D. Fox, P.F. 1993. Cheese: an overview. In Cheese: Chemistry, Physics and Microbiology, Vol. 1, General Aspects, Ed. P.F. Fox, 1–32. London: Chapman and Hall. Fox, P.F. et al. 2000. Fundamentals of Cheese Science. Gaithersburg, MD: Aspen Publishers, Inc. Fox, P.F. and P.L.H. McSweeney. 1998. Dairy Chemistry and Biochemistry. London: Blackie Academic and Professional. Glaser, J., P.A. Carroad, and W.L. Dunkley. 1980. Electron microscopic studies of casein micelles and curd microstructure in cottage cheese. Journal of Dairy Science 63:37–48. Gunasekaran, S. and C. Ay. 1996. Milk coagulation cut-time determination using ultrasonics. Journal of Food Process Engineering 19(3):331–342. Hill, A.R. 1995. Chemical species in cheeses and their origin in milk components. In Chemistry of Structure-Function Relationships in Cheese, Eds. E.L. Malin and M.H. Tunick, 43–58. New York: Plenum Press. Hori, T. 1985. Objective measurements of the process of curd formation during rennet treatment of milks by hot wire method. Journal of Food Science 50:911–917. Johnson, M.E. 1998. Part II — Cheese chemistry. In Fundamentals of Dairy Chemistry, Ed. N.P. Wong, 634–654. New York: Van Nostrand Reinhard Co. Johnson, M. and B.A. Law. 1999. The origins, development and basic operations of cheesemaking technology. In Technology of Cheesemaking, Ed. B.A. Law, 1–32. Sheffield, England: Sheffield Academic Press Ltd. Kimber, A.M. et al. 1974. Electron microscope studies of the development of structure in Cheddar cheese. Journal of Dairy Research 41:389–396. Kindstedt, P.S., L.J. Kiely, and J.A. Gilmore. 1992. Variation in composition and functional properties within brine-salted Mozzarella cheese. Journal of Dairy Science 75:2913–2921. Konuklar, G. and S. Gunasekaran. 2002. Rennet-induced milk coagulation by continuous steady shear stress. Journal of Colloid and Interface Science (in press). Kopelman, I.J. and U. Cogan. 1976. Determination of clotting power of milk clotting enzymes. Journal of Dairy Science 59(2):196–199. Kristensen, J.M.B. 1999. Cheese Technology — A Northern European Approach. Aarhus, Denmark: International Dairy Books. Kristoffersen, T. 1985. Development of flavor in cheese. Milchwissenschaft 40:197–199. Law, B.A. (ed.) 1999. Technology of Cheesemaking. Boca Raton, FL: Sheffield Academic Press. Lomholt, S.B. and K.B. Qvist. 1999. The formation of cheese curd. In Technology of Cheesemaking, Ed. B.A. Law. Sheffield, England: Sheffield Academic Press Ltd. Lopez, M.B., S.B. Lomholt, and Q.B. Qvist. 1998. Rheological properties and cutting time of rennet gels: effect of pH and enzyme concentration. International Dairy Journal 8:289–293.

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NDC. 2000. Newer Knowledge of Dairy Foods/Cheese. National Dairy Council (www.nationaldairycouncil.org). Managed by Dairy Management Inc., Rosemont, IL. Nilson, K.M. 1968. Some practical problems and their solutions in the manufacture of Mozzarella cheese. In Proc. 5th Annual Marschall Italian Cheese Seminar, 1–7. Madison, WI. Oberg, C.J., W.R. McManus, and D.J. McMahon. 1993. Microstructure of Mozzarella cheese during manufacture. Food Structure 12:251–258. O’Callaghan, D.J., C.P. O’Donnell, and F.A. Payne. 1999. A comparison of on-line techniques for determination of curd setting time using cheese milks under different rates of coagulation. Journal of Food Engineering 41(1):43–54. Olson, N.F. 1979. Cheese. In Microbial Technology II, Eds. H.J. Peppler and D. Perlman, 39–77. New York: Academic Press. Olson, N.F. 1995. Cheese. In Biotechnology, Vol. 9, Eds. H.-J. Rehm and G. Reed, 355–384. Weinheim, Germany: Verlag Chemie. Paulson, B.M., D.J. McMahon, and C.J. Oberg. 1998. Influence of sodium chloride on appearance, functionality and protein arrangements in non-fat Mozzarella cheese. Journal of Dairy Science 81:2053–2064. Payne, F.A. et al. 1993. Fiber optic sensor for predicting the cutting time of coagulating milk for cheese production. Transactions of the ASAE 36(3):841–847. Pearse, M.J. and A.G. Mackinlay. 1989. Biochemical aspects of syneresis: A review. Journal of Dairy Science 72:1401–1407. Prentice, J.H. 1993. Cheese rheology. In Cheese: Chemistry, Physics & Microbiology, Vol. 1, General Aspects, Ed. P.F. Fox, 303–340. Elsevier Applied Science, London. Scott, R., R.K. Robinson, and R.A. Wilbey. 1998. Cheesemaking Practice. Gaithersburg, MD: Aspen Publishers, Inc. Shimp, L.A. 1985. Process cheese principles. Food Technology 39(5):63–69. Spreer, E. 1998. Milk and Dairy Product Technology. New York: Marcel Dekker, Inc. Turhan, M. and S. Gunasekaran. 1998. Analysis of moisture diffusion in white cheese during salting. Milchwissenschaft 54(8):446–450. Uludogan, G. 1999. Evaluation of Milk Coagulation Using Ultrasonic and Rheological Methods, Ph.D. thesis, University of Wisconsin-Madison. Vedamuthu, E.R. and C. Washam. 1983. Cheese. In Biotechnology — A Comprehensive Treatise, Vol. 5, Eds. H.-J. Rehm and G. Reed, 231–313. Weinheim, Germany: Verlag Chemie. Visser, J. 1991. Factors affecting the rheological and fracture properties of hard and semihard cheese. Bulletin of the International Dairy Federation No. 268, 49–61, IDF, Brussels, Belgium. Walstra, P. and R. Jenness. 1984. Dairy Chemistry and Physics. New York: John Wiley and Sons. Walstra, P. et al. 1999. Dairy Technology — Principles of Milk Properties and Processes. New York: Marcel Dekker, Inc.

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2

Fundamental Rheological Methods

Fundamental rheological methods are performed under well-defined and controlled conditions. Though some assumptions about the materials and test methods may be made, calculations of material properties are based on well-defined rheological terms (e.g., strain, stress). Moreover, material properties determined by fundamental methods are independent of the apparatus used for measurements, which allows comparison of data from different research groups (Mitchell, 1984; Shoemaker et al., 1987; Tunick and Nolan, 1992; Tunick, 2000). These fundamental methods help researchers study cheese properties and effects of many manufacturing factors, and eventually develop cheeses with desired and consistent textural and rheological properties. Few reviews have been published on the fundamental rheological methods employed in cheese research (van Vliet, 1991a; Konstance and Holsinger, 1992; Luyten et al., 1992).

DEFINITION OF RHEOLOGY The term rheology was coined by Professor E.C. Bingham to represent a new branch of mechanics concerned with the study of deformation and flow of matter (Reiner, 1964). This definition was accepted at the inaugural meeting of the Society of Rheology (then, the American Society of Rheology) in 1929. Although significant rheology research has been performed prior to this date, the progress in the field of rheology seems to have greatly accelerated after its inception in 1929 as a separate discipline (Doraiswamy, 2002). Rheology is now a well-recognized field with many applications in different industries. Professionals from various disciplines (e.g., physicists, chemists, biologists, engineers, mathematicians) are interested in the theoretical and practical aspects of rheology. As stated in the definition, rheology aims at measuring those properties of materials that control their deformation and flow behavior when subjected to external forces. Thus, rheology is mainly concerned with the relationship between strain, stress, and time. When subjected to external forces, solids (or truly elastic materials) will deform, whereas liquids (or truly viscous materials) will flow. However, contemporary rheology is more interested in the behavior of real materials with properties intermediate between those of ideal solids and ideal liquids (Doraiswamy, 2002). These industrially important materials are called viscoelastic materials, which include almost all real materials.

BASIC CONCEPTS Rheology deals with the relationship between three variables: strain, stress, and time. Strain and stress are related to deformation and force, respectively. Strain accounts

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for the size effect on material deformation due to difference in length (or height) of specimens, whereas stress accounts for the size effect on applied force due to difference in cross-sectional area of specimens. Using strain and stress, rheologists are able to obtain true material properties independent of the sample size and geometry, and compare test results for samples of different sizes and geometries. Many rheological terms are defined and described by van Vliet (1991b).

STRAIN When a material is subjected to an external force, individual points of the body will move relative to one another causing a change in the size and shape of the material. The deformation is the measure of such a change in size and shape. Deformation, however, is not uniquely related to force as illustrated in the following example. Two specimens (A and B), made of the same material and having identical shape and size, are subjected to the same axial force, F, applied perpendicular to the material surfaces (Figure 2.1). This will result in the same amount of extension Lo A

Before deformation

F

F

L A

After deformation

Deformation = ∆L

∆L/2

∆L/2 Lo

F

Before deformation

F

B

L After deformation

B

Deformation = ∆L

∆L/2

∆L/2 2 Lo

F

Before deformation

A

Before deformation

B

F

2L After deformation ∆L

A

After deformation

B

∆L Total deformation = 2∆L

FIGURE 2.1 Effect of test specimen length on force–extension relationship. (After Hall, 1968).

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(or deformation), denoted by ∆L. Let us now imagine that we join the two specimens end to end and again apply the same axial force F. Since each specimen will experience the same axial force as they did when stretched separately, they each will extend to the same amount as before (i.e., ∆L) giving a total extension of 2∆L. However, when we divide the total extension 2∆L by the total original length 2Lo, the resulting quantity has the same numerical value as when the specimens were stretched separately (i.e., ∆L/Lo). This way, we arrive at a quantity that is independent of original specimen length (or height) and is referred to as strain. Thus, strain is a quantitative measure of the intensity of deformation. When the deformation is divided by the initial length of a specimen, as illustrated above, the resulting strain is known as the engineering strain (or nominal strain, Cauchy strain). The average engineering axial strain is then given by: ε=±

L − Lo ∆L =± Lo Lo

(2.1)

This expression defines the tensile engineering strain (+) when L (current length or height) is greater than Lo (original length or height), or compressive engineering strain (-) when L is smaller than Lo. Both of these strains are known as normal strains or axial strains. There is also another kind of strain, the shear strain, which exists when the force is applied parallel to the material surfaces. This is depicted in Figure 2.2. A shear strain is defined as the change in angle between two lines originally at right angles in the undeformed state, that is, γ ≡ θ (Fletcher, 1985). The angle θ may be difficult to measure. Hence, the average shear strain (γ) is obtained by dividing the deformation, δ, by sample height, h: γ=

∆δ δ = = tan θ ∆h h

(2.2) δ

F

θ

h

∆δ

∆h

FIGURE 2.2 Simple shear deformation. When the deformation is uniform, shear strain is independent of size of element taken. Thus, shear strain γ = (∆δ/∆h) = (δ/h).

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It is assumed that the strain is uniform and each small element of the material is subjected to the same local deformation, and that is also equal to the overall strain. Hence, we can write (∆δ/∆h) = (δ/h). In writing Equation (2.2), it is also assumed that the deformation, δ, is small. From the definition of strain it is evident that both normal strain and shear strain are dimensionless quantities. Sometimes, normal strain is expressed in units of, for instance, mm/mm or as a percentage, and shear strain in radian. The utility of engineering strain is limited to very small deformations (typically < 1% in tension), and the meaning becomes distorted when L becomes appreciably larger than Lo during the test (and also in fluid flows). The following two cases demonstrate why engineering strain is not appropriate for large deformations. First, we consider a tensile test where the specimen is stretched from Lo to 2Lo. According to Equation (2.1), the engineering strain is equal to 1 (i.e., ε = 1). However, let us now consider that the test is done in two steps, such that the specimen is first stretched from Lo to 1.4Lo and then from 1.4Lo to 2Lo. In the first step, the engineering strain is calculated to be ε1 = 0.4, whereas in the second step it is ε2 = 0.43. When we add these two strains we obtain ε = ε1 + ε2 = 0.83, which is less than the expected value of 1. Second, when engineering strain is equal to 1 in a tensile experiment, it means L = 2Lo, that is 100% extension. On the other hand, an engineering strain of 1 in compression gives an awkward result as: ε = −1 =

L − Lo Lo

(2.3)

This yields L = 0, which is physically impossible. There are a number of ways to measure strain when the deformations are large. For up to about 25% deformation all measures of strain yield similar stress–strain relationships in an ideal compression (Peleg, 1984). However, when deformations involved are large, the stress–strain relationships differ depending on how strain (and stress) is measured. In food rheology, the so-called Hencky strain (or true strain, natural strain) is most commonly used when large deformations are involved. Hencky strain is a better measure of strain than engineering strain because deformations are referenced to the current specimen length (or height) rather than to the initial specimen length. Using Hencky strain can eliminate the problems associated with engineering strain such as the ones encountered in the two illustrative examples above. Let us consider that a uniaxial extension experiment is performed in many small steps from the initial specimen length Lo to the final length L. In each step, one can define an incremental engineering strain (∆ε) as follows: ∆ε =

∆Li Li

(2.4)

where, ∆Li is the differential increase in the length during the ith step and Li is the length of specimen at the beginning of that step. The total strain is obtained by summing all the incremental strains from Lo to L, thus: © 2003 by CRC Press LLC

εH =

∑

L

L ∆Li dL = = ln   Li L  Lo 

∫

(2.5)

Lo

This is known as Hencky strain, εH. If we reconsider the two examples given earlier, we see that Hencky strain is additive. That is:  2L  ε H ,total = ln  o  = ln[2] and,  Lo  ε H = ε H1 + ε H 2

(2.6)

1.4 Lo   2 Lo  = ln   + ln   = ln[1.4] + ln[2] − ln[1.4] = ln[2]  Lo  1.4 Lo 

Furthermore, when εH = –1, this means that L = 0.37Lo, which is a meaningful result. Both relations, Equations (2.1) and (2.5), represent average quantities and valid only when strain is uniform all along L as illustrated in Figure 2.3. Whenever there is necking in the gage length in a tensile specimen, or bulging in a compressive specimen (Figure 2.4), these strain calculations become inaccurate or invalid. It can be shown that, engineering strain (ε) and Hencky strain (εH) are related as: ε H = ln(1 + ε ) = ε −

ε2 ε3 + ... 2 3

(2.7)

Equation (2.7) is valid for both in tension and compression as long as the magnitude of ε is used with appropriate sign (i.e., plus (+) for tension and minus (–) for compression). We should point out that Hencky strain is larger in compression and smaller in tension than the corresponding engineering strain. The difference between Hencky strain and engineering strain steadily increases with deformation (Figure 2.5). Hencky strain is approximately equal to engineering strain up to about Before deformation A

F

B

C

F

D

L

After deformation A + ∆A

B + ∆B

C + ∆C

D + ∆D

L + ∆L

FIGURE 2.3 Description of uniform deformation along a tensile test piece. For uniform strain: ∆L/L = ∆A/A = ∆B/B = ∆C/C = ∆D/D.

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TENSION Gage length F

F

Necking F COMPRESSION

Bulging

FIGURE 2.4 Necking in tension and bulging in frictional compression. 0.25 Engineering

0.2

Hencky

Strain

0.15

0.1

0.05

0 0

5

10

15

20

Deformation (%)

FIGURE 2.5 Comparison of engineering strain (ε) and Hencky strain (εH) of a specimen in tension (Note: in tension εH < ε; in compression εH > ε).

5% deformation, but the difference increases so that, for instance, ε = –0.25 (25% compressive strain) corresponds to nearly εH = –0.29 (about 16% difference). The use of Hencky strain is also convenient when considering the constant-volume assumption, which is frequently used in the calculation of true stress (see below). For this assumption Hencky strain yields: ε H , x + ε H , y + ε H ,z = 0

(2.8)

where, the subscripts x, y, and z for εH refer to the three orthogonal directions of strain in a volume element.

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F

F

∆L1

L0

∆L2

L0

D

Thicker specimen

d

Thinner specimen ∆L1 < ∆L2

FIGURE 2.6 Effect of cross-sectional area of test piece on force–deformation relationship.

STRESS Stress is defined as force per unit area over which the force is applied. Thus, the unit of stress in SI system is Pa ( = N/m2). The concept of stress is developed to eliminate the artificial effect of sample size on the material properties. Let us consider two cylindrical specimens of same material but different diameters subjected to an axial force, F (Figure 2.6). We will measure different deformations although specimens are made of the same material. The reason is that although the force is the same on both specimens the intensity of the force, or force per unit area, is higher for the thinner specimen. Therefore, stress, instead of force, is a parameter that includes the effect of specimen dimensions and can be used to evaluate the mechanical response of a material. Two types of stress can act on a surface: normal stress and shear stress. Normal stress acts perpendicular to the surface whereas shear stress acts parallel to the surface (Figure 2.7). Normal stress is further classified as tensile and compressive depending on the directions of force and unit normal vector of the surface. In tension these two vectors are in the same direction (angle 0°) while in compression they are in opposite directions (angle 180°). In simple shear the force is applied tangentially to the surface as shown in Figure 2.8. The solid lines indicate the original shape of the element. The deformation of the element is such that there is a change in the shape but not in the volume of the element. The tangential force divided by the area (of the x–y plane) it acts on gives the shear stress, denoted by σxy = τ: τ=

Ft A

(2.9)

In fact, stress and strain at a point in a material are tensorial quantities having nine components as shown in Figure 2.9.

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Normal force, F

Unit normal vector, n Tangential force, Ft Area, A

F

F

n

n

n

Ft

Tension:

Shear:

Compression:

FIGURE 2.7 Normal (tension and compression) and shear stresses acting on a surface and the unit normal vector. y

Surface area

A d

q

Ft

A

q

h

z

FIGURE 2.8 A cubic element undergoing simple shear due to tangential force, Ft.

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x

y σyy; εyy σyx; εyx σxy; εxy

σyz; εyz σzy; εzy σzx; εzx

σxx; εxx σxz; εxz

σzz; εzz

x

z

FIGURE 2.9 General state of stress acting on a cubic element (all stresses have positive sense). The corresponding strains are also indicated.

When stress calculation is based on the initial cross-sectional area of a specimen it is known as engineering stress, σ: σ≡

F Ao

(2.10)

Since the cross-sectional area of a test piece is changing continuously during a large deformation test, the engineering stress may not precisely represent the state of stress in the material. Thus, stress calculation based on the current cross-sectional area, known as true stress, σt is used more commonly: σt ≡

F A

(2.11)

The true stress and engineering stress are related to each other by the following expression: σ t = σ(1 + ε ) = σ ⋅ exp(ε H )

(2.12)

Here, exp(εH) = 1 + εH + εH2 /2!… Therefore, for small strains the true and engineering stresses are essentially the same (i.e., σt .σ). Equation (2.12) is valid

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for both in tension and compression as long as the magnitude of ε is used with appropriate sign (i.e., plus (+) for tension and minus (–) for compression). We should point out that true stress is smaller in compression and larger in tension than the corresponding engineering stress. After defining strain and stress, it is useful to demonstrate with an example why it is better to use stress–strain relations rather than force–deformation relations. For this, let us consider force–deformation graphs of three cylindrical specimens of the same material but different dimensions undergoing a tensile deformation (Figure 2.10A) (Riley et al., 1995). At first sight we cannot guess that all three curves describe the same material behavior. In the next figure (Figure 2.10B) the same data are replotted as stress vs. deformation. Here, we see that two of the previous graphs overlay, but the third one still appears to be different. Finally, we plot the data as the stress vs. strain (Figure 2.10C) to see that all three previously separate curves in fact form a single graph. This example illustrates that stress and strain are better parameters to use in evaluating and classifying the response of materials to applied forces.

STRAIN RATE The third important rheological variable, “time,” is introduced in the measurement of strain rate. The concept of strain rate is necessary to describe flow behavior of materials. In flow situations, since the strain will attain very large values with increasing time, it is preferred to discuss material behavior in terms of stress–strain rate rather than stress–strain. The strain rate is simply the time derivative of strain. For instance, strain rate in compression, and strain rate in simple shear, are given by: ε˙ H =

(dL L) = dt

1 dL Vz = L dt L

(2.13)

and γ˙ =

1 dδ Vx = h dt h

(2.14)

where, Vz = axial velocity and Vx = velocity of the moving plate.

FUNDAMENTAL METHODS UNIAXIAL COMPRESSION Uniaxial compression is the most popular test for determining rheological properties of foods, including cheese. This test is popular probably because it is easy to execute and there is no need for sample gripping (Luyten et al., 1992). Nearly all compression tests on cheese are done using one of the versatile instruments commonly referred to as Universal Testing Machine (UTM) (Figure 2.11). The UTM provides precise control of deformation while accurately measuring force.

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Force (kN)

9 8

No: 1

7

No: 2

6

No: 3

A

5 4 3 2 1 0 0

0.05

0.1

0.15

Deformation (mm) 500 No: 1

Stress (MPa)

400

B

No: 2 No: 3

300 200 100 0 0

0.05

0.1

0.15

Deformation (mm) 500 C

No: 1

Stress (MPa)

400

No: 2 No: 3

300 200 100 0 0

0.1

0.2

0.3

0.4

0.5

Strain (%)

FIGURE 2.10 Diagrams for accounting the height and cross-sectional area effects of three specimens (No. 1, No. 2, and No. 3) of same material but different dimensions. No. 1: area = 10 mm2, length = 30 mm; No. 2: area = 10 mm2, length = 60 mm; No. 3: area = 20 mm2, length = 30 mm. (After Riley et al., 1995.)

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Load cell

Moving crosshead

Specimen

FIGURE 2.11 Schematic view of a universal testing machine (UTM).

TABLE 2.1 Some Major Manufacturers of Universal Testing Machines (UTMs) Company

Web Site

ATS Applied Test Systems Instron Lloyd Instruments M&L Testing MTS Shimadzu Stable Micro Systems Tinius Olsen Thwing-Albert United Testing Systems Zwick

http://www.atspa.com http://www.instron.com http://www.lloyd-instruments.co.uk http://mltest.com http://www.mts.com http://www.shimadzu.com http://www.stablemicrosystems.com http://www.tiniusolsen.com http://www.thwingalbert.com http://www.tensiletest.com http://www.zwick.com

UTMs can be used to conduct compression as well as tension, bending, and shear tests. A number of companies (Table 2.1) are making computer-controlled UTMs with many useful features for operating the machine and acquiring, storing, analyzing, and reporting of the data. The UTMs are designed for various materials such as metals, concrete, ceramics, papers, polymers, and foods.

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F F: Force Vz: Deformation rate Upper Platen (moving) Cheese specimen

Vz

Lower Platen (stationary) BEFORE COMPRESSION

Without friction (homogeneous deformation)

With friction (inhomogeneous deformation)

AFTER COMPRESSION

AFTER COMPRESSION

FIGURE 2.12 Schematic drawing of uniaxial compression test with and without frictional effects. proportional limit for A

Elastic limit for A

lB

proportional limit for B

m

at

er

ia

al A teri ma

Stress

Elastic limit for B

Slope A > Slope B (A is stiffer than B) Slope gives Young’s modulus Strain

FIGURE 2.13 Linear elastic constants to be obtained from stress–strain curve.

A typical arrangement for a compression test is shown in Figure 2.12. In this test, a specimen of known shape and size is placed between two parallel rigid plates of a UTM, and often the upper plate is moved downward at a constant (crosshead) speed (i.e., constant deformation rate) while recording the force as a function of time. The resulting force–time data pairs are converted into corresponding stress and strain values from which other rheological quantities such as Young’s modulus or modulus of elasticity can be calculated (Figure 2.13). A higher value of modulus of elasticity

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Specimen with ends not parallel

Force

Force

Specimen with parallel ends

Deformation

Deformation Tail

FIGURE 2.14 Effect of shape of test piece on the force–deformation diagram. The “tail” at the beginning of the curve on right is due to specimen ends not being parallel. (After Bourne, 1982.)

corresponds to a higher material stiffness, as shown schematically in Figure 2.13 for materials A and B. It is important to note that since Young’s modulus is calculated from the initial part of the stress–strain data the test piece must have perfectly flat and parallel ends for accurate determination of this property (Figure 2.14). For highly nonlinear stress–strain curves, the Young’s modulus is sometimes expressed as the 5% strain secant modulus (Charalambides et al., 1995). Since food materials are viscoelastic and the stress–strain curve is nonlinear, the existence of a true elastic limit, as seen in Hookean solids, is questionable. Therefore, to maintain the purity of the term modulus of elasticity, representing rigidity or stiffness of a material, Mohsenin and Mittal (1977) proposed the term “modulus of deformability” to be used when dealing with stress–strain curves of food materials. The point at which the linearity between stress and strain ceases to exist is called the proportional limit (Figure 2.13). Said differently, the Hooke’s law is applicable only up to the proportional limit of the material. On the other hand, the elastic limit is the greatest stress, which a material is capable of sustaining without any permanent strain remaining upon release of the stress (ASTM, 1995). Thus, if a stress is applied to a specimen and then removed, the specimen will return to its original shape and size as long as the stress did not exceed the elastic limit. In general, the elastic limit is greater than the proportional limit, but for some materials they are difficult to distinguish. In addition to the proportional limit and elastic limit, several other significant parameters, particularly for engineering materials, can be determined from the stress–strain curves such as yield point or yield strength, ultimate strength, resilience, toughness, etc. Each of these parameters is described in Table 2.2 and its calculation illustrated in Figures 2.15–2.18. In uniaxial compression tests on cheese, mostly cylindrical specimens are preferred, except few cases where cubic samples are used. It must be noted that sharp corners in cubic samples are prone to stress concentrations. The tools to

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TABLE 2.2 Description of Some Commonly Encountered Terms in Analysis of Stress–Strain Curves of Engineering Materials Parameter

Description

Young’s modulus or modulus of elasticity

It is a measure of material’s resistance to axial deformation. It represents the stiffness of the material to an applied load. The larger the stiffness, the higher the force or stress needed to cause a given deformation or strain. Its value is calculated as the slope of the stress–strain curve in the linear section. In shear loading it is called Rigidity Modulus. See Figure 2.13. It is defined as the slope of a line drawn tangent to the stress–strain curve at a particular stress (or strain) level. Thus, the tangent modulus can have different values according to the point at which it is calculated. When this point falls within the linear part of the stress–strain curve, the tangent modulus is equal to the Young’s modulus. In general, the tangent modulus describes the stiffness of a material in the plastic region. See Figure 2.15. It is the slope of a line connecting the origin of the stress–strain curve and any point (e.g., 5% strain) on the curve. It is therefore true that the secant modulus takes different values depending upon the strain at which it is evaluated. The secant modulus also describes the stiffness of a material in the inelastic region of the stress–strain diagram. See Figure 2.15. It is the highest stress at which stress is directly proportional to strain. Hooke’s law applies up to the proportional limit. The proportional limit also marks the start of nonlinearity in the stress–strain curve. See Figure 2.13. It is the maximum stress the material can sustain without any measurable permanent strain remaining upon the full release of load. Thus, the material will return to its original shape/size when the stress is removed. To determine the elastic limit, one conducts a cumbersome incremental loading–unloading test procedure until a permanent (or plastic) deformation is detected. See Figure 2.13. A small increase in stress above the elastic limit results in a relatively large increase in strain. The specimen is permanently deformed even if the load is reduced to zero. The stress that causes this yielding is termed the yield stress or yield strength. Some materials exhibit a distinct yield point, but many others do not have a welldefined yield point. Therefore, it is common practice to define a yield strength using a procedure called the offset method. In this method, most commonly a 0.2% strain offset is applied. For that, a line parallel to the initial straight-line portion of the stress–strain curve is drawn starting from point 0.002 (or 0.2%) on the strain axis. The point where this line intersects the stress–strain curve is taken as the yield strength. The 0.2% strain offset is arbitrarily chosen and it can be different. Note that the offset method requires a linear portion in the stress–strain curve. For materials showing no linear portion it is practical to define the yield strength as the stress to produce a certain strain (e.g., 0.5%). See Figure 2.16. It is the highest stress reached in the stress–strain curve before fracture. The important point is that the ultimate strength is based on the original cross-sectional area of the test piece. If the specimen develops necking (e.g., ductile materials), then the engineering stress will decrease with further increase in strain, but the true stress will continue to increase until fracture. See Figure 2.16. It is the amount of energy absorbed by a material in the elastic range. Its value is obtained from the area under the stress–strain curve up to the elastic limit of a material. Materials with a high-yield stress and a low modulus of elasticity will have good resilience. See Figure 2.17. It is the amount of energy absorbed by a material until fracture (or the amount of work per unit volume). Its value is equal to the total area under the stress–stain diagram. The larger the area, the tougher the material. See Figure 2.17.

Tangent modulus

Secant modulus

Proportional limit

Elastic limit

Yield point/yield strength

Ultimate strength

Resilience

Toughness (or modulus of toughness)

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TABLE 2.2 (continued) Description of Some Commonly Encountered Terms in Analysis of Stress–Strain Curves of Engineering Materials Parameter

Description

Brittle material

It is a material that fractures before undergoing little or no plastic deformation. See Figure 2.18. It is a material that exhibits yield point and undergoes significant plastic deformation before failure. The usual measures of ductility are the engineering strain at fracture and the reduction of area (particularly for tension) at fracture. See Figure 2.18.

Ductile material

Slope = Young’s modulus

Stress

Slope = Tangent modulus at selected location (e.g., at 8% strain)

Slope = Secant modulus at selected location (e.g., at 5% strain)

5% Strain

8%

FIGURE 2.15 Different ways of obtaining modulus from stress–strain relationship.

Stress

Ultimate strength Offset (e.g., 0.2%) yield strength

0.2%

Fracture stress

Strain

FIGURE 2.16 Schematic illustration of yield strength, ultimate strength, and fracture stress on the stress–strain curve.

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Fracture

Stress

A

Proportional limit

Modulus of resilience

Strain

B

Stress

Fracture

Modulus of toughness

Strain

FIGURE 2.17 Description of modulus of resilience (A) and modulus of toughness (B).

prepare cylindrical cheese specimens typically include a cork borer and a wire cutter (Figure 2.19). The cork borer is useful for cutting out uniform cylinders of certain diameter from a cheese block, and the wire cutter is good to obtain a certain height (or length) from the cheese cylinders. The success of a compression test depends largely on the quality and accuracy of the test specimen. It is therefore important that the samples maintain true cylindrical shape with perfect parallel end faces. This is easier said than done, especially when preparing cylindrical cheese specimens. Van Vliet and Peleg (1991) made a number of recommendations to ensure proper sample preparation and, consequently, correct experimental data. They state that after boring or cutting an elastic material the shape and dimensions of the resulting test piece are different from that of the borer or cutter used to produce it. A similar concern is expressed in ASTM Standard

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Brittle material

Stress

Fracture

Ductile material

Strain

FIGURE 2.18 Schematic drawing of stress–strain curves for brittle and ductile materials.

(1996) for sample preparation in compression testing of rubber, where the suggestion is to use a cutting tool (Figure 2.20) that is larger in diameter than the specimen to allow for cutting pressure. This hollow cutting tool is rotated in a drill press and lubricated with soapy water to obtain a smooth-cut surface. Luyten (1988) used a similar borer shown in Figure 2.21 for cutting cylindrical specimens of Gouda cheese without compression and erosion at the sides of the specimens. This borer has an inner diameter slightly larger than the diameter of the cutting part. Cheese boring or cutting should be carried out as slowly as possible. Other recommendations for proper cheese sample preparation are listed in Table 2.3. A wide range of test conditions has been used in uniaxial compression of cheese (Ak and Gunasekaran, 1992). A brief list of experimental conditions is presented in Table 2.4. Since most food products are viscoelastic, and therefore strain-rate sensitive, they can produce significantly different responses according to the crosshead speeds involved. For instance, it is shown that White Stilton cheese is firmer or less firm than Gouda cheese depending on the compression speed and the degree of compression (Shama and Sherman, 1973). It is considered crucial to know the conditions prevailing (e.g., strain rates) during sensory evaluation in order to select the optimum conditions for instrumental measurements (Sherman, 1975; Voisey, 1975). Bourne (1977) remarked that the rate of compression of solid foods in the mouth varies widely with many factors such as variations in chewing speeds from person to person, length of stroke of the jaws, type of food, etc. It is reported that the maximum rate of jaw movement during chewing ranges from 15 mm/s to 30 mm/s, with males chewing at a faster rate than females (Langley and Marshall, 1993). We shall note, as an exception, that for UF-Feta cheese it is not necessary to run uniaxial compression testing at the chewing rate to get a good correlation with sensory evaluation, provided that the crosshead speed is greater than 50 mm/min (Wium et al., 1997).

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FIGURE 2.19 Tools that can be used for sample preparation for compression testing of cheese. A: a cork-borer machine (from www.hmc-hsi.com, with permission); B: set of cork borers and sharpener (from www.boekelsci.com, with permission); C: table-top wire cheese cutter (from www.ashtongreen.com); D: cheese specimens for compression (after Ak, 1993).

Obviously, accurate control of test temperature is extremely important as physical properties of cheeses (and foods in general) are greatly affected by temperature changes. It is equally important to make sure that the specimen and the platens are maintained at the same temperature during a compression test. The choice of specimen dimensions requires special care. Buckling may occur if the aspect ratio (ratio of sample length or height, L to its diameter, D) is relatively large (e.g., L/D > 2). On the other hand, if L/D ratio is small (e.g., L/D < 1) the test results may be greatly affected by the specimen end conditions (e.g., friction effects). For instance, Chu and Peleg (1985) examined apparent deformability modulus, (determined as the engineering stress divided by engineering strain at 20% deformation) and failure conditions (i.e., failure stress and failure strain) of potato, bologna sausage, and process American cheese as a function of height-to-diameter ratio in the range 0.12 ≤ L/D ≤ 1.00. The flat (i.e., lower L/D ratio) specimens exhibited higher apparent deformability modulus (i.e., higher stiffness). This is particularly

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ASTM Cutting Tool

DTOOL

DTOOL = 28.804 mm (+0.000/−0.025) DSPECIMEN = 28.6 ± 0.1 mm LSPECIMEN = 12.5 ± 0.5 mm

FIGURE 2.20 The ASTM cutting tool for cutting cylindrical compression test specimens. (Per ASTM D638.) (After ASTM, 1996. With permission.) 17 to 21 mm

14.7 mm

14-mm diameter Cheese sample Cheese borer

Cheese block

14 mm

FIGURE 2.21 Cork borer for cutting cylindrical cheese specimens. (After Luyten, 1988; van Vliet and Peleg, 1991.)

the case when there is considerable friction between the specimen ends and the machine plates (e.g., when plates are coated with emery cloth). For instance, the apparent deformability modulus of the cheese sample with L/D = 0.12 is about 3.5 times greater than that for the sample with L/D = 1.0. Culioli and Sherman (1976) examined the effect of contact surface conditions on force–compression behavior of Gouda cheese at crosshead speeds of 2.5, 10, and 50 cm/min and L/D ratio of 1.0. At any of three crosshead speeds the level of friction

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TABLE 2.3 Recommendations for Cheese Sample Preparation for Uniaxial Compression Tests If it exists, remove hard rind of cheese Prepare specimens when they are cold (refrigeration temperature) Take sample from the same location in a cheese (e.g., center) and in the same direction Take samples from locations that are sufficiently away from each other since fracture may occur in the cheese loaf during previous sampling Cut specimens as slowly as possible, preferably using a motorized borer and cutting tool Lubricate all surfaces of the borer with mineral oil to minimize distortion during cutting out Use a tightly stretched, thin wire cutter, lubricated with mineral oil Measure actual dimensions of each specimen before testing Make sure that the size of specimen is large compared to the size of the heterogeneity Source: After van Vliet and Peleg, 1991.

at contact surfaces (i.e., with emery paper or with oil) exerted no influence on the force–compression behavior until 40% compression. At fracture point the force was lower when an emery paper was used at the interface than when the specimen ends were lubricated with oil. However, when the true stress is plotted against percent compression, the stress was greater when an emery paper was used at the interface than when the specimen ends were lubricated with oil. On the other hand, Luyten et al. (1992) found no clear effect of using emery paper or oil at the specimen–machine interface on the fracture stress of Gouda cheese. Moreover, they recalculated stress–strain curves from the data of Culioli and Sherman for emery paper and oil, and found no difference between them. The effect of aspect ratio on the fracture stress of young Gouda cheese is depicted in Figure 2.22. In this figure, data for different friction conditions (i.e., normal plates, lubricated, and emery paper) are pooled together since no clear effect of friction on the fracture stress is reported (Luyten et al., 1992). It is seen that the fracture stress of this cheese shows a tendency to become constant beyond the L/D ratio of 1.5. Although the uniaxial compression test appears simple in principle and practice (e.g., no need to grip the sample), the data analysis is complicated by the effect of friction between the specimen and the testing machine platens. Friction influences not only the magnitude of the force for compression but also the appearance of the compressed specimens (Culioli and Sherman, 1976). Barreling due to friction (Figure 2.12) is an indication of nonhomogeneous deformation. During compression the cheese specimen is to move relative to the platens and thus the force required to achieve a certain level of compression depends also on the friction. Therefore, stresses in the presence of friction (i.e., shear-plus compression) are always greater than in the absence of friction (i.e., shear-free compression). The common practice of reducing the friction is to lubricate the sample–platen interfaces with low-viscosity oil. It has been shown that type of lubricating oil can have a significant impact on the measurements (Charalambides et al., 1995). An alternative way of accounting for friction is to bond the sample to the platens using adhesives such as cyanoacrylate (Casiraghi et al., 1985). Charalambides et al. (2001) described a method based on the Cooke and Larke procedure to account for friction.

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TABLE 2.4 Experimental Conditions Used in Uniaxial Compression Testing of Cheese Cheese Type Cheddar Leicester Brie UF-Feta Immature Cheddar, Cheshire, Leicester Mozzarella cheese analogs Camembert

Crosshead speed (mm/min)

Temperature (°C)

Sample Diameter, D (mm)

Sample Height, L (mm)

Aspect ratio: L/D

2.5, 6.4, 12.7, 25.4, 50.8, 127 25, 50, 250, 500, 1000 33.3

22

19

19, 29

1, 1.53

Room, 23–26, 30–37 5, 20

25

25

1

15

19.5

1.3

100, 200, 300, 400

13

15.3

15.3

1

Mineral oil; emery sheet Smooth hydrophobic paper Low-viscosity oil

5–1000

0–40

28.5–29.5

30

~1

Machine plates

20

20

23

20

0.87

Machine plates

10

20

13.8

10

0.72

Machine plates

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Surface Condition Mineral oil

Reference Ak and Gunasekaran (1992) Vernon-Carter and Sherman (1978) Molander et al. (1990) Wium and Qvist (1997) Dickinson and Goulding (1980)

Yang and Taranto (1982) Mpagana and Hardy (1986)

Emmentaler

5, 20, 80

15

16.4

17.5

1.07

Gouda

5, 10, 50

20–21

7.5–35

0.3–3.5

Gouda Mild Cheddar, Sharp Cheddar, Monterey Jack Process American cheese Mozzarella, Cheddar, Processed cheese spread Gouda Gruyere, processed Mozzarella

0.1–500 10

20 4

10, 20, 25 (also cubes of 10 and 20 mm) 15 20

20–30 7, 10, 13, 20

1.33–2 0.35–1

5

Room

10–21

2.5–10

0.12–1

0.5, 5, 50

7, 22

57

20, 30, 40

0.35–0.70

10 3.6, 5.8, 7.9, 10.8, 14.4

20 room

20 20

20 5, 8, 11, 15, 20

1 0.25–1

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Paraffin oil Mineral oil; emery paper Machine plates Machine plates

Machine plates; emery cloth (No.120) Paraffin oil; bonding with cyanoacrylate; machine plates Paraffin oil Machine plates; synthetic grease

Rohm and Lederer (1992) Culioli and Sherman (1976) Luyten et al. (1991a) Charalambides et al. (1995)

Chu and Peleg (1985)

Casiraghi et al. (1985)

Rohm et al. (1997) Charalambides et al. (2001)

Fracture stress (kPa)

200

150

100

50

0 0.0

0.5

1.0

1.5

2.0

2.5

Aspect ratio

FIGURE 2.22 Effect of aspect ratio (height-to-diameter ratio) on fracture stress of Gouda cheese in compression. (After Luyten et al., 1992.)

According to this procedure, measurements are made on specimens with a constant diameter and various heights. Then, results are plotted as true stress against 1/H for constant values of strain. The intercept of the resulting line (i.e., at 1/H = 0) gives the correction to be applied to the stress for that strain level. For uniaxial compression, Hencky strain can be written in terms of the deformation rate (or crosshead speed), Vz as below:  L (t )   Lo − ∆L   Lo − Vz t  ε H = ln   = ln   = ln    Lo   Lo   Lo 

(2.15)

where, L = current sample height, Lo = initial sample height, ∆L = deformation (= Vz t), and t = time. Since L ≤ Lo in uniaxial compression, εH will have a negative value, an appropriate sign for compressive strains. Most often the right-hand-side of Equation (2.15) is multiplied with –1 to make the resulting strain values positive for common use. In lubricated compression a cylindrical specimen of radius Ro and height Lo is deformed into a cylinder of radius R and height L. From the assumption of constantvolume deformation we can obtain radius at any time from the following relation: L  R = Ro  o   L

1/ 2

(2.16)

The true stress for lubricated compression is calculated from: σt =

[

F(t ) F(t ) L(t ) F(t ) Lo − Vz t = = A(t ) Ao Lo Ao Lo

]

(2.17)

where, F(t) = applied force at any time, A(t) = cross-sectional area at any time, Ao = initial cross-sectional area, Lo = initial length.

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In bonded compression, the cross-sectional area in contact with the compression platens remains constant, and the stress in bonded compression is given as (Casiraghi et al., 1985): σB =

F (t ) π Ro2

(2.18)

The stress is then corrected, σBC, for the shape changes using the following equation: σ BC =

σB R2 (1 + o 2 ) 2L

(2.19)

This corrected stress equation is shown to be effective to bring results in bonded compression into agreement with those in lubricated compression up to a strain level of 0.37 for Cheddar, 1.4 for Mozzarella, and 0.8 for processed cheese spread, where strain is defined as ∆L/L. This definition of strain is used since it relates directly to the extent of bulging (i.e., δo). For the bonded sample the relation is given by the following equation (Christianson et al., 1985): δo =

3 ∆L R 4 o L

(2.20)

Kamyab et al. (1998) and Charalambides et al. (2001) analyzed the uniaxial compression test with friction between the sample and the compression platens, which is quantified by the coefficient of friction. They provided a scheme that enables calculation of true stress–Hencky strain curve from uniaxial compression data influenced by friction. The resulting analytical equation in its simpler form is given as: FL µL  = σt + σt  o  ∀o 3 L

3/ 2

 Do  L   o

(2.21)

where, ∀o = initial sample volume (i.e., πRo2Lo), µ = coefficient of friction, and Do = initial sample diameter. The results from compression tests are analyzed by plotting [FL/∀o] as a function of [Do/Lo] at fixed values of [Lo/L]. The intercept gives the true stress, σt, and the slope can be used to calculate the coefficient of friction, µ. Thus, using this procedure one can construct true stress–Hencky strain curves and determine variation of µ with strain. More parameters, such as the modulus of deformability and fracture energy per volume or toughness per volume, can be extracted from stress–strain curves from uniaxial compression. To facilitate parameter calculations, Ak and Gunasekaran (1992) suggested using polynomial expressions to describe the stress–strain curves as:

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N

σ t = a1ε H + a2 ε 2H + a3ε 3H + ... =

∑a ε

i i H

(2.22)

i =1

where, i denotes an index for the coefficients (ai) and a power for the Hencky-strain term (εΗi ). The coefficients (ai) can readily be determined using a curve-fitting procedure. The modulus of deformability, ED, can be obtained from Equation 2.22 using the following definition:  dσ  ED =  t  = a1  dε H  ε H →0

(2.23)

The coefficient of the first term in Equation 2.22 becomes equal to the modulus of deformability, and this is why it is necessary to apply a constraint so that it has non-negative values in curve-fitting procedure. As an alternative approach, we shall mention that Wium et al. (1997) determined the deformability modulus of UF-Feta cheese as the maximum slope in the range 0 ≤ εΗ ≤ 0.05. The peak strain, εf, which may sometimes correspond to fracture strain, can be estimated by locating the strain at which the slope becomes zero. That is: dσ t =0= dε H

N

∑a ε

i −1 i H

(2.24)

i =1

The strain that makes the slope zero (or nearly zero with some tolerance) can be determined using a root-finding procedure with computation software. Once the peak strain is computed, then its value can be inserted back into Equation 2.22 to determine the corresponding peak stress, σf , which may sometimes correspond to fracture stress. Of course, in some cases, it may be easier to obtain the fracture stress and fracture strain values directly from the experimental data without using Equation 2.24. However, for cases where a distinct peak is not readily discernible, use of Equation 2.24 is a practical approach. It is important to restate that in compression tests the fracture usually starts in the interior of the specimen and often before the maximum stress is reached (Luyten et al., 1991a). One can obtain fracture work per unit volume (W) (or the total energy per unit volume, or modulus of toughness) from the area under the stress–strain curve (Figure 2.17): εf

W=

∫ σ dε t

H

(2.25)

0

The calculation of fracture work becomes simpler with the substitution of Equation 2.22 for stress in Equation 2.25. With this approach we can also easily

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True stress (kPa)

70

A

60 50 40 30 20 10 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Hencky strain (-) 250 B Slope (kPa)

200 150 100 50 0 −50 0.00

0.50 Hencky strain (-)

1.00

1.50

FIGURE 2.23 Example showing extracting various parameters from stress–strain data (symbols) by fitting a polynomial equation ( σ = 214ε H − 695ε H2 + 1342ε H2 − 1158ε H4 + 355ε H5 ) to the true  dσ 

stress–Hencky strain curve (line) (A). From the plot of slope   vs. Hencky strain (B), the  dε H  modulus of deformability (= slope when Hencky strain approaches zero = 214 kPa) and fracture strain (εH,f = 0.95), and fracture stress (σf = 58 kPa) are calculated at zero slope. Integrating the true stress–Hencky strain polynomial equation between the limits εH = 0 and εH = 0.95, according to Equation (2.25), yields the value of fracture work or modulus of toughness as 35 kJ/m3. (After Ak and Gunasekaran, 1992.)

obtain the work up to any given strain or deformation by simply changing the upper limit of the integral. For instance, if we insert the proportional limit for the upper limit of integral the resulting area is termed the modulus of resilience and is the work done per unit volume to reach the proportional limit (Figure 2.17). The modulus of toughness is a measure of the ductility of a material. The larger the modulus of toughness relative to the modulus of resilience, the more ductile is the material (Fletcher, 1985). In Figure 2.23 we present a numerical example where the mechanical parameters mentioned above are extracted from the experimental data. As can be seen in Figure 2.23, there are more than one minimum and maximum points in the slope vs. strain curve. It is not clear if these points relate to some structural changes during the compression test.

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F

∆L

L0 L

∆D/2

D0

∆D/2

D

FIGURE 2.24 When a material is compressed by ∆L (from original length Lo to final L), its diameter increases by ∆D (from original Do to final D). The associated axial and lateral strains are used to calculate the Poisson’s ratio. (Equations 26a and 26b.)

Another important rheological parameter is the Poisson’s ratio (Figure 2.24). When a specimen of length Lo is compressed to final length L it experiences a concomitant increase in diameter from original value of Do to final D. That is, the imposed axial strain brings about a lateral strain. The Poisson’s ratio is the ratio of lateral strain to axial strain as given below: ν=−

ε ( D − D0 ) / Do lateral strain = − lateral = − axial strain ε axial ( L − Lo ) / Lo

(2.26a)

or, in terms of Hencky strains, it is given as:

ν=−

ε H lateral ε H axial

ln  D D    O =−  ln  L L   0 

(2.26b)

The negative sign indicates that the lateral dimensions decrease as the axial dimensions increase. It also makes ν a positive number since the lateral and axial strains are of opposite sign. The Poisson’s ratio is a material property and is based on the observation that when a material is subjected to an axial force, let us say tension, it will not only elongate but it will also contract laterally (Riley et al., 1995).

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FIGURE 2.25 Picture of compressed Mozzarella samples taken parallel and perpendicular to the fiber direction. (After Ak, 1993.)

Equations 26a and 26b are applicable only to homogeneous isotropic materials; that is, materials with the same properties in all directions. It assumes that the lateral expansion in uniaxial compression and lateral contraction in uniaxial tension is uniform in the radial direction. This may not be the case for some cheeses. For instance, Ak and Gunasekaran (1997) demonstrated that the kneading and stretching of the curd in hot water results in a Mozzarella cheese with anisotropic tensile and compressive properties. Anisotropy refers to the fact that material behavior is dependent on the direction in which stress is applied or on the direction in which sampling is done. As shown in Figure 2.25, the postdeformation appearances of compressed specimens taken parallel and perpendicular to the fiber orientation are clearly different. Thus, Poisson’s ratio calculations based on the lateral expansion of these two cases would certainly produce different values. For some cheeses, anisotropic mechanical properties may also be of commercial importance. For Gruyère de Comté, a Swiss-type hard cheese, it is reported that anisotropic rheological properties are important in the formation of eyes and slits (Grappin et al., 1993). The resistance to wire cutting of Cheddar cheese (4 months old) from different manufacturers is reported to vary with the cutting direction (Ney, 1985). Although allowable range of Poisson’s ratio is from –1 to 0.5, its value generally varies from 0 (totally compressible) to 0.5 (incompressible). For most metals it has a value between 0.25 and 0.35 (Riley et al., 1995). Rubber has a Poisson’s ratio about 0.5, making it nearly incompressible. On the other hand, cork has a Poisson’s ratio close to zero, which makes it a good bottle stopper. An axially loaded cork will not show a lateral expansion, thus making its insertion into a bottle easy. A negative value for Poisson’s ratio has been reported for polymeric and metallic foam structures (Lakes, 1987; Friis et al., 1988). Regarding cheese, experimental

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results indicate that the Poisson’s ratios of Cheddar and Gouda cheeses vary between 0.40 and 0.45; that is 20% and 10% lower than the theoretical value of 0.5 for incompressible materials (Calzada and Peleg, 1978; Luyten et al., 1991b; Rohm et al., 1997). The Poisson’s ratio is related to the elastic constants such as Young’s modulus (E), shear modulus (G), and bulk modulus (K) by the following formulas (Findley et al., 1989): E = 2(1 + ν)G K=

E 3(1 − 2V )

(2.27)

Bulk modulus describes the change in volume in response to hydrostatic pressure (i.e., equal pressures in all directions). From Equation 2.27, for an incompressible (ν = 0.5) linear elastic solid we can compute E = 3G and K = ∞.

UNIAXIAL TENSION As far as the direction of applied stress is concerned, uniaxial tension is simply the opposite of uniaxial compression. However, a more fundamental difference between tension and compression tests is in the strain rate. When a specimen is deformed at a constant speed, the strain rate decreases in tension but increases in compression. Various features of different fundamental methods are listed in Table 2.5. Uniaxial tension tests are considered not suitable for routine measurements since they are more difficult to execute due to lengthy sample preparation and difficulty of gripping (Luyten et al., 1992). Specially designed grips are often necessary in order to eliminate slippage and breakage of sample in the grips. Grip surfaces can be scored or serrated to enable better holding. It is generally assumed that the grip assembly and the specimen ends are nearly rigid and all of the deformation is taking place in the gage section of the specimen. A large number of grips are commercially available for different purposes and materials (Figure 2.26). The existence of many sophisticated grips designed for particular materials is sufficient to show that the tensile test is difficult to perform even for engineering materials. Therefore, it is not a commonly performed test for cheese, particularly at temperatures above melting point of fat in the cheese. The test specimen used in tensile testing may have either a circular or a rectangular cross-section. The latter shape is more suitable for cheese. The ends of tensile specimens are generally enlarged to provide extra area for gripping and to prevent sample prematurely breaking at the grips. A typical tension specimen described in ASTM Standards for plastics is illustrated in Figure 2.27. The specimen must be aligned as perfectly as possible with the direction of stretching so that the long axis of the test specimen will coincide with the direction of the grip assembly. Strictly speaking, data from only those tests that produce failure in the gage length should be used to obtain material properties. In practice, specimens may fail near the grips where there is stress concentration. Therefore, a small notch can be made in the central part of the test piece to ensure the location where the fracture will start (Luyten, 1988) (Figure 2.28).

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TABLE 2.5 Advantages and Drawbacks of Various Test Methods Used for Cheese Test Method I.

Test Type

Uniaxial Constant compression rate

Advantages

a) Difficult to obtain specimens with flat and perfectly parallel ends b) Friction affects the property calculations c) The strain rate increases during a test if a constant crosshead speed is applied (typical of Universal Testing Machines used in food studies) d) Rheological parameters depend on specimen size as a result of friction, varying strain rate, and the inhomogeneities in the cheese e) Start of fracture is often inside the test piece and does not correspond to the maximum stress in the stress–strain curve f) Assumptions of constant volume and perfect cylinder shape during deformation may not hold g) Test piece undergoes inhomogeneous straining Lubricated a) Biaxial extensional viscosity a) Difficult to separate elastic contribution squeeze may be determined relatively from viscous contribution to material’s flow easily when cheese response specimen is compressed b) Difficult to completely eliminate friction between lubricated plates c) Equations will not apply unless the b) Deformation rate can easily assumption of perfect cylinder shape holds be varied to obtain biaxial extensional viscosity as a function of strain rate Constant a) Constant force tests (creep a) Stress decreases during the test if the force tests) can be executed easily plate diameter is greater than specimen for long time scales provided diameter that necessary precautions are b) Strain rate varies during the test taken to avoid physical and c) Drawbacks mentioned above for the chemical changes constant rate case applies here as well b) The Young’s modulus and except that related to strain rate the compliance can be determined c) An apparent viscosity can be calculated from the so-called “secondary stage” where the strain rate is nearly constant d) Relevant to hole (eye) formation in cheese and sagging of cheese under its own mass

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a) Easy to prepare samples and to perform the test b) Young’s modulus can be obtained c) Fracture stress and strain can be obtained d) Easy to vary deformation rate e) Small samples with less chance of containing undesired inhomogeneities

Drawbacks

TABLE 2.5 (continued) Advantages and Drawbacks of Various Test Methods Used for Cheese Test Method

Test Type

II. Uniaxial tension

Constant rate or force

III. Bending

Constant rate or force

IV. Cutting (wire or wedge)

Constant rate

Advantages

Drawbacks

a) Fundamental rheological and fracture properties can be determined, such as Young’s modulus, fracture stress, fracture strain, toughness b) Friction effect is not present c) Fracture initiation and propagation can be controlled using notched samples a) Easy to perform the test b) No need to fix the specimen to an apparatus c) Fracture can be observed mostly on the outside (tension side) of the specimen d) Test imitates closely the sensory evaluation of cheese by graders a) b) c) d) e)

a) It is often difficult to grip the sample, thus requires specially designed grips to hold samples b) Fracture may occur at the grips, which is avoided using special sample shapes (e.g., dog-bone shape) c) Test piece must be long compared to other dimensions for deformation to be homogeneous and for reliable stress–strain calculations. d) Strain rate decreases during a tension test a) Test can be used only for cheese of some rigidity and fairly short texture b) Large test pieces increase the possibility of containing an undesired inhomogeneity c) Length of samples must be much larger than the other dimensions, which is sometimes not practical d) Deformation is far from being homogeneous as it varies from a compressive strain to a no strain at neutral axis, and to a tensile strain at the outside Easy to execute and no need a) Only fracture energy is determined from to clamp sample this test, and other tests are to be carried Small test piece out to obtain other rheological and fracture Useful for determination of properties fracture energy of cheese b) Additional cracks may be formed due to Similar to biting food with inhomogeneities in the structure of brittle teeth materials Fracture is in tension c) Friction between the wedge and the specimen may contribute to the measured force, which can be reduced by lubricating the wedge

Source: After van Vliet, 1991a; Luyten et al., 1992.

Parameter calculations in uniaxial tension are similar to those given earlier for uniaxial compression. Strain and stress in uniaxial tension can be calculated using the following expressions, respectively:  Lo + Vz t   L( t )   L o + ∆L  ε H = ln   = ln   = ln   = ln(1 + ε)  Lo   Lo   Lo 

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(2.28)

FIGURE 2.26 Various designs of commercial grips for tensile tests. (From www.itinscale.com/ grips.htm; www.dillon.fm/grip.htm#clevis; www.cscforce.com/gripping.htm. With permission.)

σt =

[

F(t ) F(t ) L(t ) F(t ) Lo + Vz t = = A(t ) Ao Lo Ao Lo

]

(2.29)

Here, Lo and L are initial and final gage lengths instead of total specimen lengths. In stress calculation it is assumed that the volume of specimen remains constant during extension (i.e., A(t) L(t) = Ao Lo). In strain calculation it is assumed that the deformation as a result of the crosshead movement is taking place in the gage length of the specimen. If this assumption is of suspect, one way to obtain strain values is

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D G W

R

WO

L LO

Recommended tensile specimen dimensions (in mm) for samples of the thickness (mm) listed below (after Swallowe, 1999). Sample thickness

Dimension W L WO (min) LO (min) G D R

7 to 14 19 57 29 246 50 115 76

4 to 7 13 57 19 165 50 115 76

<4 3.18 9.53 9.53 63.5 7.62 25.4 12.7

Tolerance ±0.5 ±0.5 +6.4 (3.18 for <4) No max ±0.25 ±5 ±1

FIGURE 2.27 Tensile specimen with recommended dimensions. Gage length

Notch

FIGURE 2.28 Tensile specimens with notches.

to draw marker lines on the specimen before testing and take photographs or videotape of the specimen during the deformation (Figure 2.29). Using the distance between the lines one can calculate the real strain values. Of course, more advanced techniques involving video recording and image analysis would yield quicker and more accurate results. There is an alternative technique to measure tensile properties while actually performing the test in compression. This technique is called the diametral compression (also known as the Brazilian test, indirect tension test, or compression splitting test), which is simpler to execute than the uniaxial tension test. It is a well-established method to measure the tensile strength of brittle materials such as concrete and ceramics (Fahad, 1996). Its application to determine the tensile strength of rice grains

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FIGURE 2.29 Pictures of a deforming cheese specimen at different times in the uniaxial horizontal extension experiment. (After Ak, 1993.)

has also been reported (Kamst et al., 1999). The diametral compression test is based on the fact that tensile stresses develop when a circular disc is compressed between two diametrically opposite faces (Fahad, 1996). For a specimen in the form of a right-circular cylinder of diameter D and thickness t undergoing diametral compression (Figure 2.30) the tensile strength value σΤ is calculated from: σT =

2F πDt

(2.30)

where, F is the applied force. This equation is derived for a Hookean solid for which the stress is proportional to the strain. Equation 2.30 is strictly valid for samples with a thickness-to-diameter ratio of 0.25 ≤ t/D ≤ 0.5 (Newton et al., 2000).

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F

R

t σT

FIGURE 2.30 Schematic drawing of diametral compression test (F = compressive force; R = sample radius; t = sample thickness; σT = tensile stress developed in the material).

BENDING TEST In sensory evaluation of cheese the panelist takes the ends of a cheese plug by the forefingers and thumbs of both hands and bends the plug slowly into a semicircle to observe when the sample breaks, as well as the nature of the break. This sensory evaluation sort of mimics the bending test. The bending test can be readily practiced with a rubber eraser or with a piece of string cheese. The major advantage of the bending test is that there is no need to fix the specimen to the measuring instrument. In this respect the bending test is easy to conduct, especially for brittle and other hard-to-grip materials. Typically, a cylindrical or rectangular cross-section sample is laid horizontally on a support with two (blunt or rounded) supporting edges, and the sample is pushed down at the center of the specimen (Figure 2.31) by means of a blunt (or rounded) plunger attached to the crosshead of a UTM. There are three points at which the material comes in contact with the test device. Thus, this test is also known as the three-point-bending test. The part of the sample on the side of the plunger experiences compressive stress (and compressive strain), and the opposite side experiences tensile stress (and tensile strain). The plane of transition from compressive to tensile stress (and strain) is known as the neutral axis. Fracture can often be observed on the side of the sample experiencing tensile stress (and strain). For a proper bending test, the test piece must have a large ratio of length to diameter or thickness, which is hard to obtain with cheese (van Vliet, 1991a). According to Luyten (1988), for Gouda cheese, this ratio should be 3.33 or greater

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F Plunger b (width) Neutral axis

h (height or thickness) -Side view-

-Front viewL F/2

F/2 compression

neutral

tension

FIGURE 2.31 Schematic drawing of three-point bending test.

in order to obtain reproducible results. Large specimen size implies higher probability of presence of inhomogeneities in the specimen (see Table 4.1 for various possible inhomogeneities in cheese). Luyten (1988) further noted that the bending test is good for mature and acid cheese with low fracture strain (i.e., short consistency), but not suitable for young cheese with high fracture strain where the test piece can slide from the supports before it fractures. For soft and deformable materials, penetration of the plunger used for bending complicates the situation as it introduces a significant compressive stress (Luyten, 1988). It can generally be stated that the bending test is most suitable for studying brittle materials. The equation for a linear elastic solid giving the bending stress can generally be expressed in the following form: σ=

Mc I

(2.31)

where, M is the bending moment, c the vertical distance between the neutral axis and the point at which the stress is sought, and I the area moment of the crosssection of the beam. The maximum deformation occurs at the center of the beam where the load is applied. We can write the following relations for stress and strain. Specimen with a Rectangular Cross-Section Maximum stress is given by: σ max = ±

3FL 2 b h2

(2.32)

where, F is the applied force, L the span or the distance between supports, b the width of test piece, and h the height or thickness of test piece.

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The corresponding strain at the base of the beam is given by:

ε=

6 h ∆y L2

(2.33)

where, ∆y is the (maximum) deflection at the center of the beam. We note that for the test piece with rectangular cross-section the maximum compressive stress (Equation 2.32 with negative sign) occurs at the top of the beam, whereas the maximum tensile (Equation 2.32 with positive sign) stress occurs at the bottom of the beam. The inherent assumption is that the elastic moduli of the material are the same in tension and compression. Specimen with a Circular Cross-Section Maximum stress is given by:

σ max =

8FL π D3

(2.34)

The corresponding strain is given by:

ε=

6 D ∆y L2

(2.35)

All of the equations given for bending are valid only for small strains and linear elastic materials (Luyten, 1988).

TORSION TEST Torsion test is applied by twisting a specimen about its longitudinal axis. Specimens used in torsion tests are usually circular in cross-section (Figure 2.32). The test piece can be fixed to the apparatus by gluing; for instance, by using cyanoacrylate adhesive. The torsion test produces pure shear, and hence does not change the specimen volume. Therefore, it is ideally suited for materials that may exude some of their contents (e.g., moisture, fat, etc.) under applied force. For a homogeneous and isotropic material exhibiting linearly elastic behavior (i.e., Hooke’s law applicable, τ = Gγ) the equations to calculate stress and strain are given below. Maximum shear stress at the surface of test piece:

τ max =

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2 Tmax π R3

(2.36)

γ T γ

θ R

r

L τ

R

r

FIGURE 2.32 Schematic drawing of torsion test (L = specimen length; R = specimen radius; T = torque applied during test; θ = angle of twist; γ = shear strain).

Maximum shear strain at the surface of test piece: γ max = R

θ L

(2.37)

where, Tmax is the torque on the surface, R the radius of the specimen, θ the total angle of twist in radians, and L the length of the specimen. The equations given above apply to solid, circular materials. Montejano et al. (1983) used capstan-shaped (i.e., narrow mid-section and enlarged ends) specimens to minimize undesirable stress concentrations at the locations where the twisting moments are applied (Figure 2.33). Since the diameter of the specimen is not uniform, a geometric correction factor, K, is to be applied to Equations 2.36 and 2.37 for calculating shear stress and shear strain (Hamann, 1983). The maximum shear stress is given by: τ max = K

2 Tmax 3 π Rmin

(2.38)

where, Rmin, is the specimen minimum radius, K the constant depending on the sample geometry. K is given by (Lanier, 2000): 1/ 2 2      31 +  Rmin + 1        Rc     K=  1/ 2    Rmin    4 1 + 2 + 1   Rc         

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(2.39)

D T

Rmin

Rc

T

FIGURE 2.33 Schematic drawing of a capstan-shaped test piece for torsion test (D = specimen diameter at ends; Rmin minimum radius at the specimen center, Rc = radius of curvature; T = torque applied during test). (After Montejano et al., 1983.)

and the maximum shear strain is obtained from: γ max =

2Kθ 3 Q π Rmin

(2.40)

Q is given by:

Q=

4 π

zo

∫ (R 0

[

min

dz

(

+ Rc ) − Rc2 − z 2

)

]

1/ 2 4

(2.41)

where, z varies from 0 (center of the groove) to zo (boundary of the groove and end section). For instance, Montejano et al. (1983) used cylindrical gel specimens with L = 28.7 mm, zo = 6.4 mm, D = 18.6 mm, Rmin = 5 mm, and Rc = 9.4 mm. Using these numerical values and Equations 2.39 and 2.41 we find that K = 1.08 and Q = 8.32 × 106 m–3.

VANE METHOD The vane geometry has been developed to eliminate slip effects frequently observed in yield stress measurements with the rotational viscometers (Barnes and Nguyen, 2001). It is simple but effective means of directly measuring the yield stress (Barnes and Nguyen, 2001; Nguyen and Boger, 1983; Nguyen and Boger, 1985). In a recent review, Barnes and Nguyen (2001) discuss the utility of the vane technique in measuring various rheological quantities such as (a) modulus of linear elastic solids and viscosity of Newtonian liquids; (b) yield stress; and (c) flow-curves of nonNewtonian fluids. In food rheology, the vane geometry is primarily used for direct measurement of yield stress (Table 2.6). © 2003 by CRC Press LLC

TABLE 2.6 Experimental Conditions in Food-Related Studies Using the Vane Method Vane Spindle Test Material Frozen ice cream Spreadable foods

Property Measured

Height (mm)

18 10

Stirred yogurt

Equilibrium stress

45

38 20 28 36 67.5

Set yogurt Processed cheese analogs Processed and natural cheese Cream cheese

Yield stress Yield stress

35 6

40–60 5.5

6–10

15–20

10b

20 25b 35 20 24 28 32

Protein foams

Yield stress Yield stress

Diameter (mm)

Fracture stress and equilibrium stress Yield stress and strain Yield stress

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10

Instrument Used Lab-made Brookfield HBTDV-I & 5X HBTDV-I Physica Rheolab MC100 Haake VT 550 Brookfield HBTDV-I Haake VT 550

Number of Blades

Thickness of Blades (mm)

Rotational Speed (rpm)

4

0.1

1

4

0.80

0.5

4

NAa

NA

6 4

NA NA

NA NA

4

NA

0.028–5

4

0.7

0.5

4

NA

0.3

Haake VT 550

Brookfield DV-I 25xLVTDV

References Briggs et al., 1996 Daubert et al., 1998

Geraghty and Butler, 1999 Dimonte et al., 1998 Mleko and Foegeding, 2000 Truong and Daubert, 2001 Breidinger and Steffe, 2001 Pernell et al., 2000

TABLE 2.6 (continued) Experimental Conditions in Food-Related Studies Using the Vane Method Vane Spindle Test Material

Property Measured

Molten chocolate

Yield stress

Food dispersions

Yield stress

a b c d

NA = Not Available Recommended dimensions CR = Controlled rate mode CS = Controlled stress mode

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Diameter (mm)

Height (mm)

15 20 25 40 (CRc mode) 21 (CSd mode)

40 55 70 60 (CR mode) 30 (CS mode)

Instrument Used Brabender Rheotron Haake RV2 Deer Rheometer III

Number of Blades

Thickness of Blades (mm)

4

0.55

6 (CR mode) 4 (CS mode)

NA NA

Rotational Speed (rpm) 0.064 0.120 0.224 0.4

References Wilson et al., 1993

Yoo et al., 1995

Although the existence of a yield stress as a physical reality has been debated in the literature (Barnes, 1999), the utility of this rheological parameter is recognized in food processing (Campanella and Peleg, 1987a; Steffe, 1996). Contributing to the discussion on yield stress, Schurz (1992) mentioned the critical role of this parameter in polymer processing and suggested the use of the term apparent yield stress as a resolution of the debate. Astarita (1990) stated that whether yield stress is or is not an engineering reality depends on the problem in consideration. Thus, from a processing point of view, the concept of yield stress is useful as long as the Deborah number (i.e., the ratio of a material’s characteristic relaxation time to the characteristic process time) is large (Zhu et al., 2001). Many reasons can be advanced to support usefulness of the concept of yield stress in food rheology. For instance, (a) for the short time scales encountered in food processing, consumption, and handling activities, a viscoplastic or elastoplastic material may demonstrate solid-like behavior (i.e., not flowing within the time available under a given stress); (b) rheological equations with a yield-stress term (e.g., Bingham, Casson, Herschel-Bulkley models) are successfully used in modeling behavior of several foods; (c) yield stress is considered a key parameter for quality control and evaluation of many products (e.g., ketchup, cream cheese, mayonnaise, various spreads) (Steffe, 1996). These commercially important products are formulated to display yield stress. The vane geometry is similar to the concentric cylinder system, except that the inner cylinder (i.e., the bob) is replaced by a vane spindle. The vane spindle is simply an attachment adapted to fit an existing rotational rheometer or viscometer (e.g., Brookefield viscometer) (Nijman and Chakrabarti, 1997). A vane consists of a number (2 to 8) of thin blades arranged at equal angles around a slender central shaft. A schematic drawing of a typical four-bladed vane rotor is presented in Figure 2.34. Several examples and dimensions of vane rotors used are listed in Table 2.6. The key assumptions in vane rheometry are that the shearing stress is uniform over the virtual cylindrical surface described by the outer edges of the blades, the material trapped between the blades of the vane is rotating as a rigid body, and there are no secondary flows between the blades (Barnes and Nguyen, 2001). These assumptions are valid if the vane consists of four or more blades and rotates at low speeds. There are experimental results on various nonfood and food systems such as shaving cream (Zhang et al., 1998), oil-in-water emulsions (Yoshimura et al., 1987), and applesauce (Qiu and Rao, 1988), as well as simulation data (Yan and James, 1997) to support validity of assumptions under stated conditions. In addition to eliminating the wall slip, the vane geometry offers other advantages over conventional rotational techniques (e.g., concentric cylinder): the sample preparation is simple and gentle, which enables measurements on weak materials inserting the vane spindle into the sample causing little disturbance to the material structure. This is important, particularly when working with thixotropic systems and delicate structures such as foams (Zhang et al., 1998). Multi-phase systems (e.g., suspensions, emulsions) have a tendency to form a low-viscosity, particle-depleted layer adjacent to the shearing surfaces of the traditional viscometer geometries. The velocity gradient in the low-viscosity layer at a fixed shear stress is larger than that in the bulk material, which then results in apparent slip. However, the vane geometry avoids

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Ω

H H

D

D

FIGURE 2.34 The vane rotor with four blades (H = vane height; D = vane diameter; Ω = angular velocity).

the wall-slip problem by using the test material contained between the vane blades as the virtual shearing surface. In a way, the vane acts like a solid cylinder (i.e., bob) without the wall-slip complications. It is also worth noting that the slip at the outer cylindrical boundary can be eliminated by inserting a slender gauze basket inside the outer cylinder (Barnes and Nguyen, 2001) or lining the inner wall with aluminum foil (Yoshimura et al., 1987). Another advantage of the vane method is that original product containers can be used as sample holders while measuring with the vane spindle. There is no need for a narrow gap (in contrast to concentric cylinder geometry), and the vane is less susceptible to artifacts arising from the presence of large particles. Breidinger and Steffe (2001) recommend the vane dimensions of 10 mm in length by 25 mm in diameter after considering the size of most commercial cream-cheese containers and maximum torque capacity of rotational viscometers. Nguyen and Boger (1983) studied the effect of rotational speed on the yield stress of bauxite residue suspensions (red mud) over a range of speeds from 0.1 rpm to 256 rpm. Their results, replotted in Figure 2.35, indicate a practically constant yield stress between 0.1 rpm to 8 rpm, followed by a rising yield stress with

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400

Yield stress (Pa)

300

200

100

0 0.1

1

10

100

1000

Rotational speed (rpm)

FIGURE 2.35 Effect of vane rotational speed on yield stress of red mud. Vane geometry: H/D = 1.923. (After Nguyen and Boger, 1983.)

increasing rotational speed. However, Nguyen and Boger (1983) used the lowest available speed of 0.1 rpm to minimize any unforeseen errors. As noticed in Table 2.6, in studies on foods, the vane is often rotated at a constant speed less than 1 rpm. There is experimental evidence indicating the effect of rotational speed on the yield stress values of molten chocolates and food dispersions (Wilson et al., 1993; Qiu and Rao, 1988) and TiO2 suspensions (Liddell and Boger, 1996). High vane speeds, particularly in low-viscosity liquids, are risky since secondary flows may develop between the blades (Barnes, 1999). A typical torque-time response for a material having a yield stress is illustrated in Figure 2.36. The initial elastic region is followed by a slightly curved part before reaching the peak torque. After a distinct maximum torque, a decline (rapid or

Torque

Tmax

tmax

Time

FIGURE 2.36 Typical torque-time response from the vane in a rate-controlled mode. Maximum torque (Tmax) occurs after a certain time (tmax) of rotation of the vane.

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Ω

Ω

Z1

H

D Z2 DT (a)

(b)

FIGURE 2.37 Schematic diagram of vane system (diameter, D; height, H, and container diameter, DT; Ω = angular velocity) in different positions: (a) immersed in the sample (Z1 and Z2 height of sample above and below the vane, respectively), (b) top surface in level with the sample.

gradual depending upon conditions) is observed towards an equilibrium torque value. In general, the peak torque is the only parameter derived from these curves to compute the yield stress. The peak torque is, of course, readily and unequivocally determined in comparison to other points of possible interest (e.g., departure from linearity, equilibrium level). When the vane blades are entirely immersed in a sample, as shown in Figure 2.37a, the following equation is used to calculate the yield stress (τy) from the measured maximum torque and vane dimensions:  2T  τ y =  max3  πD 

 H + 1  D 3 

−1

(2.42)

where Tmax is the maximum torque, D the vane diameter, H the vane height. When the top of the vane blades are aligned even with the top surface of the sample (Figure 2.37b), the stress contribution from the material above the vane is eliminated, and Equation 2.42 takes the form:  2T  τ y =  max3  πD 

H + 1  D 6 

−1

(2.43)

The following geometrical ratios have been proposed for accurate measurements with the vane method: H/D < 3.5; DT/D > 2.0; Z1/D > 1.0; Z2/D > 0.5 (Nguyen and Boger, 1985). Here, DT is the diameter of the container, Z1 and Z2 are height of © 2003 by CRC Press LLC

material above and below the vane, respectively. Typically, the blades are made of stainless steel with the thickness less than 1 mm. Recently, a new instrument called “slotted-plate device” has been developed to directly measure static yield stresses of suspensions (Zhu et al., 2001). The slottedplate device is reported to be more reliable for evaluating smaller yield stresses and avoids possible secondary flows between the blades and nonuniform stress distribution along a virtual cylindrical surface — the key assumptions in the vane geometry. The success with the vane method has resulted in new applications and designs of this geometry such as oscillatory testing and texture analysis (Junus and Briggs, 2001) and hand-held versions of the vane instrument (Keener et al., 1999).

STRESS-RELAXATION TEST One of the fundamental tests to study viscoelastic response is stress relaxation. This test can be performed in (uniaxial) tension, compression, shear, bending, torsion, etc. When a constant strain is applied to a viscoelastic material isothermally, the stress necessary to maintain that strain is not constant but decreases with time. Hence, the decrease of stress at constant strain is called stress relaxation. Two kinds of relaxation experiments can be conducted: stress relaxation after a sudden step strain, which is often applied to solids; and stress relaxation following a cessation of steady flow, which is often applied to liquids (Figure 2.38) (Dealy, 1995; Ferry, 1980; Whorlow, 1980). Stress-relaxation response permits rapid characterization of material behavior as shown in Figure 2.39. When a step strain is applied to ideal elastic solid, a finite and constant stress will be reached. Ideal elastic solids store all the energy charged during the straining step and would expend this energy upon removal of stress to return to its original size and shape. In a way, ideal elastic solids possess a perfect memory of the initial state. Thus, the same stress should be kept on the specimen at

Strain

Cessation of flow

Step strain

Time

FIGURE 2.38 Two types of stress relaxation test: step strain for solids and cessation of steady flow for liquids.

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Strain

Constant strain input

Time

Stress

Ideal elastic response

Stress

Time

Ideal viscous response

Time 2

Stress

Linear viscoelastic response

3

4

1

Time ε0

ε0

ε0

1

2

3

4

σ1=0

σ2

σ3

σ4

σ2 > σ3 > σ4

FIGURE 2.39 Step-strain input (A) and stress relaxation response of ideal elastic material (B), ideal viscous material (C), and viscoelastic material (D). The stress and strain on the viscoelastic material is schematically depicted at different times during the test at the bottom.

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all times for strain to remain constant. On the other hand, for ideal viscous liquids the stress decays to zero immediately after the cessation of strain application. Ideal viscous liquids do not store any energy and have no memory of the initial state. However, various materials, including foods, are viscoelastic and exhibit an intermediate response where stress relaxes at a finite rate characterized by the relaxation time (Peleg, 1987). Viscoelastic materials can be considered as materials with “fading memory.” If a viscoelastic material relaxes to zero stress within a certain period (e.g., experimental time) it is further characterized as “viscoelastic liquid.” In contrast, a viscoelastic material is considered as “viscoelastic solid” if a finite stress remains unrelaxed (i.e., residual stress) after a sufficiently long time. With most foods, however, the “sufficiently long time” is on the order of few minutes due to chemical, enzymatic, and physical changes that foods normally experience. The residual stress after an arbitrary time for test duration (e.g., 10 min) is suggested as a quantitative measure of the degree of “solidity” of foods (Peleg, 1987). The relaxation experiment can be viewed as composed of two steps: the straining step and the relaxation step. Ideally, the straining step is instantaneous, but in reality it takes finite time. The time it takes to apply the step strain is called the rise time. The rise time depends upon the capability of the instrument used and the magnitude of strain. For instance, if the highest crosshead speed of a UTM machine were 1000 mm/min, then it would require 0.09 s to apply 10% deformation on a sample of 15 mm height. Since stress relaxation of a viscoelastic material is affected by the history of deformation, the time taken for the straining step is important (Meissner, 1978). Accurate stress-relaxation tests require the rise time of the applied strain to be short in comparison with the relaxation times to be measured. With the advanced rheometers it is possible to apply a step strain within few seconds or milliseconds (e.g., 20 ms to 1000 ms) (Lauger and Huck, 2002). Obviously, for a proper test the applied strain, and consequently the resulting stress, should be lower than the corresponding fracture value. Although in nonfood applications stress-relaxation tests can be continued for a long time, the test duration for foods is limited (on the order of minutes, e.g., 10 min or less) because degradation of sample may occur before the test is completed as a result of physical changes (e.g., moisture exchange with environment), microbial activity, and chemical and biochemical changes (e.g., enzymatic browning in fruits, oxidation in oil-containing foods) (Peleg, 1987). For a linear viscoelastic material subjected to an instantaneous constant strain (εo),* the initial stress will be proportional to the applied strain and will decrease with time (Figure 2.40). By linearity it is meant that when the applied strain is multiplied by any factor (e.g., doubling), the stress it produces also changes by the same factor (e.g., doubling). The rate of stress decay is quantified by a material characteristic time known as relaxation time, λ. The relaxation time in practice is defined as the time for stress to decay to about 37% of the initial level. However, single-relaxation time is often insufficient to fully describe the relaxation curve of most food materials. A better representation of relaxation curve is possible by using more than one relaxation time or, ideally, by using a continuous relaxation-time spectrum (Peleg and Normand, 1983). * The symbol ε hereafter denotes Hencky strain unless stated otherwise.

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Stress

Straining stage Relaxation stage

σ0

0.37σ0

Rise time

Relaxation time, λ

Time

FIGURE 2.40 In actual relaxation test, the strain is applied over a finite time (rise time). The peak stress at the end of the straining stage is the inital stress (σo). Time taken for the stress to decay to 0.37σo during the relaxation stage is the relaxation time, λ.

For linear viscoelastic materials the stress decay with time t can be described, in tension or compression, by the following equation: σ(t ) = E (t ) ε o (2.44)

or E (t ) =

σ (t ) εo

where, the function E(t) is called the relaxation modulus. The relaxation modulus represents the change in stress per unit of applied strain and is a material property. For linear elastic solids the E(t) = E, the Young’s modulus. In shear configuration the corresponding equations are given as: τ( t ) = G( t ) γ o (2.45)

or G( t ) =

τ( t ) γo

where, τ(t) is the shear stress, G(t) the shear stress relaxation modulus, and γ the applied constant shear strain.

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o

Analysis of Relaxation Behavior Stress relaxation is a basic test providing information on the viscoelastic character of materials rapidly. Although in principle relaxation test can be done in any configuration (i.e., compression, tension, shear, torsion, bending, etc) the most common one in cheese studies is compression. To determine linear viscoelastic region of a material in relaxation, a series of relaxation curves is obtained by sequentially increasing the applied strain. When the resulting data is replotted in terms of modulus vs. time, the curves within the linear viscoelastic region will overlap. The strain level at which the curve does not overlap indicates that the linear viscoelastic region is exceeded. Alternately, the linear range of the isochronal — the plot of stress against strain at a specific time — will indicate the extent of strain level over which the material response can be considered linear (Ak and Gunasekaran, 2001). The process of obtaining isochronal plots is illustrated in Figure 2.41. The data obtained at two constant strains (ε1 and ε2) are represented in Figures 2.41A and 2.41B. From these, data points (for, e.g., a, b, c, d in Figure 2.41A and 2.41B) are gathered at different times (e.g., t1 and t2). Then the corresponding σ(t) vs. ε plot is constructed for each of the times at which the stress response is measured (Figure 2.41C). The strain value at which the isochronal begins to deviate from linearity (indicated by dotted line in Figure 2.41C) is the upper limit of the liner viscoelastic region for the material. The hatched region in Figure 2.41C indicates the nonlinear range of the material studied. The mechanical model most suitable for quantification of relaxation behavior of foods and a variety of polymeric materials has traditionally been the generalized Maxwell model (Figure 2.42) with a discrete number of elements (Peleg and Normand, 1983):

σ (t ) = E(t ) = Eo + εo



n



t 

∑ E exp − λ   i

i =1

(2.46)

i

where, Eo is the modulus of the single spring (λ = ∞) in parallel to Maxwell elements in Figure 2.42, t the time, Ei the modulus of each Maxwell element, and λi the relaxation time of each Maxwell element. It must be mentioned that for a true viscoelastic liquid the first term (i.e., Eo) will be zero and the material will eventually relax completely. For linear viscoelastic behavior, the relaxation parameters are a function of time only. However, for nonlinear viscoelastic behavior, the relaxation parameters will be a function of time as well as imposed strain and strain history. An alternative model to describe relaxation and creep curves of viscoelastic solids is suggested by Peleg (1979, 1980): t = k1 + k2 t Y (t )

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(2.47)

ε2 ε1 t

t

σ1(t)

σ2(t) c a

d

b

t1

t

t2

t1

t2

(a)

t

(b)

σ σ(t1)

Linear

c σ(t2)

a d Nonlinear b ε1

ε2

ε (c)

FIGURE 2.41 Plotting isochronals to determine linear viscoelastic range from stress relaxation data. (A) relaxation experiment at applied strain ε1; (B) relaxation experiment at applied strain ε2; and (C) isochronals plotted using data points a, b, c, and d from A and B at times t1 and t2 . (After Ak and Gunasekaran, 2001.)

and Y (t ) =

σ o − σ (t ) σo

(2.48)

where, σo is the initial stress and σ(t) the decaying stress. This linearization makes the calculation of model parameters easy as the slope gives k2 and the intercept gives the k1 . As it is also true for the parameters of the generalized Maxwell model,

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E0

E1

h1

E2

h2

E3

h3

En

hn

FIGURE 2.42 Generalized Maxwell element for stress relaxation (E0, E2, E3, … En are spring stiffnesses; η1, η2, η3, … ηn are dashpot viscosities).

the dependency of the constants k1 and k2 on the applied strain is an indication of nonlinear viscoelasticity related to the structural modifications that occur during deformation. Peleg (1980) further stated that 1/k1 represents the initial decay rate, while 1/k2 represents asymptotic level of Y(t) when t→∞.

CREEP TEST As with stress relaxation, a creep test can be performed in different configurations (i.e., compression, tension, shear, torsion, etc). In an isothermal creep test, a constant stress is applied to the material, and the resultant strain is recorded as a function of time (Figure 2.43). In an actual test the stress application is not instantaneous but can be rapid such as by dropping the weight on the specimen. Analysis of Creep Behavior In a creep test a constant step-stress is applied to a material and the resulting deformation or strain is measured as a function of time. The distinction between constant stress and constant force is necessary, especially for highly deformable foods, because of the progressive change in the cross-sectional area of the specimen. Hence, a constant force (i.e., dead weight) results in a progressively increasing stress in uniaxial tension and decreasing stress in uniaxial compression (Purkayastha et al., 1985). Although in principle creep tests can be done in any configuration (i.e., compression, tension, shear, torsion, bending, etc.), the most common one in cheese studies is compression. For linear materials, the time-dependent compliance, D(t), is given by (Findley et al., 1989): D(t ) ≡ ε(t ) σ o

(2.49)

where, ε(t) is the tensile or compressive strain. Symbol J(t) is used to represent the shear creep compliance, that is, J(t) = γ (t)/τo, where γ (t) is the shear strain, and

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Load removed Constant stress application

A

Stress

σo

0 B

t1

Time

t1

Time

Elastic solid response

Strain

εo quid

li cous

onse

resp

Vis

0

Recovery

Creep C 4

εi

Strain

3

5 2 Permanent strain

εi 1 t1

0 W

1

Time

Linear viscoelastic response

2

W

3

W W

4

5

FIGURE 2.43 Typical creep–recovery test. (A) application of instantaneous and constant stress (σo); (B) strain response of elastic solid and viscous liquid; (C) strain response of viscoelastic material. The application and removal of load (W) is shown at the bottom at various times along the creep–recovery curve for the viscoelastic material.

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D(t) Creep Curve ∑Di

Delayed creep Steady state flow slope = 1/η

Do

Instantaneous compliance

Time

FIGURE 2.44 Typical creep compliance D(t) vs. time response of a viscoelastic material comprises of instantaneous compliance, delayed creep, and steady state flow (of viscosity η).

σ3

Strain

σ1 σ3 > σ2 > σ1

Time (a)

σ3 Compliance, D (t)

σ2

σ2 & σ1

Time (b)

FIGURE 2.45 Determining linear viscoelastic region by creep tests. A. Creep curves at different stress levels σ1, σ2, and σ3; B. Compliance D(t) vs. time for the corresponding creep curves overlap within the linear viscoelastic region (σ1 to σ2). Stress level σ3 is outside of linear viscoelastic region. (After Anon, 2002.)

τo the applied shear stress. The objective of creep tests is to determine material properties D(t) and J(t) from the experimental strain vs. time data (Figure 2.44). To determine linear viscoelastic region of a material in creep, a series of creep curves is obtained by sequentially increasing the applied stress (Figure 2.45). When the resulting data is replotted in terms of creep compliance vs. time the curves within the linear viscoelastic region will overlap. A typical creep compliance curve is shown schematically in Figure 2.44. Unless the applied force or stress is carefully selected the test terminates with the failure of the specimen, especially in tension. Quantification of creep behavior of foods and a variety of biological and polymeric materials has been traditionally based on the © 2003 by CRC Press LLC

E2

E3

En η1

E0

η2

η3

ηn

FIGURE 2.46 Generalized Voigt-Kelvin element for creep (E0, E2, E3, … En are spring stiffnesses; η1, η2, η3, … ηn are dashpot viscosities).

generalized Kelvin-Voigt model (Figure 2.46) with a discrete number of elements (Purkayastha et al., 1984): ε (t ) t = D(t ) = Do + + σo η1



n



t 

∑ D 1 − exp − τ   i

i =2

(2.50)

i

where, Do is the instantaneous compliance (= 1/Eo), t the time, η1 the Newtonian viscosity while 1/η1 being the slope of the linear portion of the creep curve after sufficiently long time, Di the delayed compliance of each Kelvin-Voigt element (= 1/Ei), and τi the retardation time of each Kelvin-Voigt element. The last term in Equation 2.50 is called the creep function and denoted by ψ(t) (Purkayastha et al., 1984). It must be mentioned that for a true viscoelastic solid material the second term (i.e., t/η1) will be zero, and the material will eventually reach an equilibrium creep compliance. Equation 2.50 also reveals that a typical creep curve is composed of three components, as illustrated in Figure 2.44. For linear viscoelastic behaviors the creep parameters are a function of time only. However, for nonlinear viscoelastic behaviors the creep parameters will be a function of time as well as imposed stress and stress history. As for the relaxation case, the Peleg model (Purkayastha et al., 1984) can be used to linearize and represent the creep behavior of foods. For creep data, the Peleg model is used to represent the creep function using constants k1″ and k2″ as: ψ (t ) = D(t ) − Do −

t t = " η1 k1 + k2" t

(2.51)

SHEAR RHEOMETRY Polymer and food-processing applications involve a wide range of shear rates as shown in Figure 2.47. Thus, various rheometry measurements based on different geometries are essential and complementary to each other.

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Oscillatory/Vibrational Rotational Elongational Capillary Polymer Processing Food Processing 10−6

10−3

10−2

10−1

100

101

102

103

104

105

Shear rate (s−1)

FIGURE 2.47 Approximate shear rate ranges for different rheometry measurements and those involved in polymer processing and food processing applications. (After Riande et al., 2000.)

Rheological measurements based on shear flow are conveniently divided into two groups: (a) drag flows in which shear is generated between a moving and a fixed solid surface, and (b) pressure-driven flows in which shear is generated by a pressure difference over a closed channel (Macosko, 1994). Examples of shear-flow geometries belonging to the first group include sliding plates, concentric cylinders, parallel disks, and cone and plate. Examples of shear-flow geometries belonging to the second group include capillary or Poiseuille flow, slit flow, and axial annulus flow. The working equations for some of these measurement techniques are presented here. Interested readers are referred to other sources (Collyer and Clegg, 1988; Macosko, 1994; Steffe, 1996) for detailed discussions. The measurement systems described below can be used to conduct a variety of tests (e.g., steady shear, dynamic, relaxation, creep). In some tests, one system may be preferred over the others due to the shear-rate range or other advantages it offers. For instance, parallel-plate geometry is often preferred for measuring viscoelastic properties of solid cheese (e.g., relaxation modulus, creep compliance, dynamic moduli) since the sample handling is easier and the sensitivity to gap setting is less as compared to the cone-and-plate geometry. There are several companies manufacturing highly advanced rheometers and viscometers that will satisfy the measurement needs of researchers. Barnes et al. (1999) summarized the history of commercial viscometry and rheometry. Web sites of various companies offering rheological instruments are given in Table 2.7 for readers to have quick access.

SLIDING-PLATES GEOMETRY The schematic drawing of the sliding-plate geometry is shown in Figure 2.48. This relatively simple arrangement is generally used in defining shear viscosity. The

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TABLE 2.7 Some Major Manufacturers of Rheometers and Viscometers Company

Web Site

Alpha Technologies ATS RheoSystems Bohlin Instruments Brookfield Camtel Dynisco Polymer Test GBC Scientific Equipment Pty Ltd. Goettfert Inc. Haake Infra Scientific Kaltec Scientific Paar Physica Porpoise Viscometers Pressure Profile Systems, Inc. Reologica Instruments AB Research Equipment (London) Ltd. Rheometric Scientific TA Instruments Vilastic Scientific Inc.

http://www.alpha-technologies.com/ http://www.atsrheosystems.com http://www.bohlin.com/ http://www.brookfieldengineering.com/ http://www.camtel.co.uk/ http://www.dynisco.com/ http://www.gbcsci.com/ http://www.goettfert.com http://www.thermo.com/ http://www.infra.uk.com/ http://www.kaltecsci.com/ http://www.physica.de/ http://www.porpoise.co.uk/ http://www.pressure-profile.com/ http://www.reologica.se/ http://www.research-equipment.com/ http://www.rheosci.com/ http://www.tainst.com/ http://www.vilastic.com/

Source: The Society of Rheology Web site, http://www.rheology.org/sor.

Moving plate ∆X

y

Fx,Vx sample

H

sample

x Stationary plate

L Width of plate = W; Area of plate, A = L.W

FIGURE 2.48 Description of sliding plate geometry. Application of shear force Fx moves the top plate by ∆X at a velocity Vx.

sliding-plate rheometer can be operated in either strain-controlled or stress-controlled mode. The shear strain (γ), shear rate (γ˙), and shear stress (τ) can be calculated from the following equations (Dealy and Wissbrun, 1989; Macosko, 1994): γ=

∆X Vx t = H H

(2.52)

Vx H

(2.53)

γ˙ =

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τ=

Fx F = x A LW

(2.54)

where, L is the plate length, W the plate width, H the sample thickness, ∆X the sliding-plate displacement, Vx the velocity of sliding plate, Fx the shear force, and t the time. The shear viscosity is then computed from η = τ/ γ˙. The assumptions involved in the sliding-plates geometry include: (a) negligible inertial and edge effects to establish homogenous, simple shear flow, and (b) L and W are much greater than H, and H is as small as possible. The principal advantage of the sliding-plate geometry is that it is ideally suited for studying nonlinear viscoelasticity. The relative advantages and disadvantages of different rheometer geometries for studying nonlinear viscoelasticity are listed in Table 6.1. Further discussion on the theory and application of the sliding-plate rheometer for studying nonlinear viscoelasticity of cheese is presented in Chapter 6.

CONCENTRIC-CYLINDERS GEOMETRY The schematic views of different concentric-cylinder geometries are shown in Figure 2.49. The concentric-cylinder geometry has long been used in commercial viscometers. The concentric-cylinder system consists of an inner cylinder (called “bob”) positioned inside an outer cylinder (called “cup”). The sample is contained in the annular gap between the “infinitely” long bob and cup. In some cases the cup is rotated at a steady angular velocity while the bob is kept stationary, and in others, the bob is rotated at a constant angular velocity and the cup is fixed. The concentric-cylinder system is typically used with low-viscosity materials and mobile suspensions. The double-gap or double-Couette geometry (Figure 2.49) offers greater sensitivity than the other concentric-cylinder systems at low shear rates and viscosities as a result of its larger surface area. If sample drying (or skin formation) is likely to be an issue, which is a common problem in working with low-fat cheeses, it is better to use a solvent trap with the measuring system or alternatively a low-viscosity oil can be used as a barrier provided that the oil used does not interact with the sample to alter the sample properties. Working equations for shear strain γ, shear strain rate γ˙, and shear stress τ are given as (Macosko, 1994): Shear strain:

γ=

Ωi t R R (for narrow gaps; that is κ = i ≥ 0.99) Ro − Ri Ro (2.55)

and R=

( Ro + Ri )

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2

Bob

Cup

Wi

Wi

Ri

L Ro

L z

r

a

Lb

Lb

sample

Standard concentric cylinder

Coaxial cylinder with conic base

Wi

Wi

trapped air

Mooney cell (recessed bottom cylinder)

Double gap (double Couette)

FIGURE 2.49 Different concentric-cylinder measurement systems.

Shear strain rate: γ˙ ( Ri ) ≅ γ˙ ( Ro ) =

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Ωi R 2 Ωi = Ro − Ri 1 − κ 2

(for κ > 0.99)

and γ˙ ( Ri ) =

(

2 Ωi

n 1 − κ2/n

)

and γ˙ ( Ro ) =

(

−2 Ω i

n 1 − κ −2 / n

)

(for 0.5 < κ < 0.99) (2.56)

where n=

d ln Mi d ln Ω i

and

R=

Ro + Ri 2

Shear stress: τ( Ri ) =

Mi 2 π Ri2 L

(2.57)

where, Ri is the radius of bob, Ro the radius of cylinder, Mi the torque on bob and Ωi the angular velocity of bob, L the height of bob. Other alternative designs that are generally used to minimize end effects or make possible to account for the end effects are also depicted in Figure 2.49. The shear stress for the coaxial cylinder with conic base is given by: τ( Ri ) =

Mi 2 πR 2 L + 2 πR3  i 3 i  

(2.58)

With the Mooney cell or recessed bob the air is trapped underneath the bob and contributes practically no torque to the overall response. As can be seen from Equation (2.56), the shear rate changes across the gap for a wide-gap viscometer. This is a serious concern when using concentric-cylinder geometry with concentrated suspensions. Rotating the bob in a concentrated suspension causes particles to migrate away from higher shear-rate regions near the bob to lower shear-rate regions near to the cup (Abbott et al., 1991).

CONE-AND-PLATE GEOMETRY Sketches of different cone-and-plate (C&P) geometries are shown in Figure 2.50. The C&P system consists of a rotating (stationary) cone and a stationary (rotating) plate with a sample contained between them. The apex of the cone is essentially in contact with the plate. As seen in the following working equations the main advantage of using the C&P geometry is that the shear rate is approximately constant (i.e., independent of radial position) throughout the sample provided that the cone angle does not exceed a few degrees (e.g., ≤ 4°) (Macosko, 1994). This feature of constant shear rate is the reason why C&P geometry is particularly useful for studying nonNewtonian behavior.

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Ω, φ

θ

α

sample

Rt r

R

R

Normal cone-and-plate system

Truncated cone-and-plate system

R hc Rd

Double cone system

Extended cone-and-plate system Cone-and-dish system

FIGURE 2.50 Different cone-and-plate measurement systems.

Shear strain: γ=

φ tan α

(2.59)

where φ is the angle of rotation and α the cone angle. It is true that for a small cone angle tan(α) ≅ α*, for which the shear strain becomes: γ=

φ α

(2.60)

It is seen that for a given cone angle the strain is homogeneous and independent of position in the sample. Shear rate: dγ dφ 1 Ω = γ˙ = = dt dt sin α sin α

(2.61)

For small cone angles the shear rate can be simplified to:

* Maclaurin series for tan(α) is given as: tan(α) = α + (α3/3) + (2α5/15) + … where α in radian unit (Thomas and Finney, 1988).

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γ˙ =

Ω α

(2.62)

where, Ω = specified angular velocity. This equation shows that a uniform shear rate is realized experimentally with cone-and-plate system with small cone angles. According to the calculations of Adams and Lodge, reported in Lodge (1964), the errors involved in shear rate approximation by Equation 2.62 are 0.02, 0.18, 0.50, and 2% for cone angles of 1°, 3°, 5°, and 10°, respectively, for a material with a constant viscosity. Shear stress: τ=

3M 2 π R3

(2.63)

where, M = torque and R = radius of plate. This equation indicates that proper loading of the specimen is vital so that a full contact of the cone with the specimen is established since the torque measurement (and, consequently, the stress calculation) depends on R3. The C&P geometry is typically configured with cones having angles less than 4°. The cone angle shall be chosen with care since, for instance, for large cone angles the shear rate across the gap will vary; on the other hand, for the small cone angles there is higher chance for errors due to gap settings. For instance, for a cone angle of 10° the variation in shear rate across the gap is 3%, and the resulting error in calculated viscosity of Newtonian liquids is 2%. However, since the cone angles typically used is 4° or less the resulting error is quite small (Dealy, 1982). The C&P geometry cannot accommodate materials that contain particles since the particles can be subjected to grinding action near the tip of the cone. Therefore, quite often the tip of the cone is slightly truncated to allow measurements on particulate fluids. Cones are often slightly truncated, as shown in Figure 2.50, by removing the tip of the cone to make them more robust measurement tools. Errors due to cone truncation are generally negligible since the radius of truncation Rt is much smaller than R. The maximum error in torque associated with truncated cones can be calculated from the following equation:  R3 − Rt3  Maximum error = 1 −  100 R3  

(2.64)

For instance, if Rt = 0.2R then the maximum theoretical error in torque is 0.8%. Although C&P geometry is very simple and useful there are cases in which it shall not be preferred: (a) it is not recommended when conducting temperature sweeps unless the rheometer is equipped with an automatic system for thermal expansion compensation; and (b) it is not recommended for testing samples with particulate materials. The particles can jam the cone apex, giving erroneous data. Moreover, cone-and-plate system is not suitable for materials with a high concentration of solids, as the solids become expelled from the gap under high shear rates.

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In the extended C&P geometry the apex of the cone does not touch the plate, instead there is a finite distance between the apex and the plate, denoted by hc in Figure 2.50. The shear rate for this case is determined from (Powell, 1988): γ˙ ec =

RΩ hc + R tan α

(2.65)

It is sometimes essential to replace the flat plate with a dish to contain liquid materials as seen in Figure 2.50. Truncated and extended cones can also be utilized with a dish in place of a flat plate. Furthermore, double cone (or biconical) sensors (Figure 2.50) have been developed and employed for measuring very low viscosity liquids with small sample volume. This geometry eliminates the free surface and variation of its shape with rotational speed, and minimizes the sample exposure to the environment. On the other hand, it introduces a new type of edge effect (Dealy, 1982).

PARALLEL-PLATE GEOMETRY A schematic drawing of the parallel-plate geometry is shown in Figure 2.51. The parallel-plate system consists of a rotating (stationary) upper plate and a stationary (rotating) lower plate separated by the sample to be tested. Although similar in many ways to C&P system, the major difference between the parallel-plate and C&P systems is that the shear rate in the parallel-plate system is not constant but varies across the sample. Thus, if the objective is to subject the entire sample to a uniform shear, then the parallel-plate geometry is not appropriate. On the other hand, parallelplate geometry is not as sensitive to the gap-setting errors as the cone-and-plate system (Macosko, 1994). The working equations for the parallel-plate geometry is given as follows: Shear strain: γ=

φr h

(2.66)

It is clear that shear strain is not homogeneous and varies with radial position r. Shear rate at the edge (at r = R): γ˙ R =

R Ω h

(2.67)

It is seen that the shear rate can be varied in two ways: (a) by changing the rotational speed, Ω, and (b) by changing the gap between plates or sample thickness, h. Shear stress: τR =

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M 2 π R3

 d ln M  3 +  d ln γ˙ R  

(2.68)

Ω, φ

sample h R

Sample thickness or gap setting

R h

Rdish

FIGURE 2.51 Parallel-plate measurement systems.

where, M = torque on the rotating plate. The derivative term in the brackets makes the shear stress and viscosity calculations in the parallel-plate geometry more involved and difficult than those in cone-and-plate geometry. The accurate evaluation of the derivative term requires sufficient amount of torque versus edge shear rate data. For a Newtonian liquid the derivative term is equal to 1.0, and the equation reduces to: τR =

2M π R3

(2.69)

When the lower plate is replaced with a dish to contain the liquid sample there will be additional torque contribution caused by the increased frictional drag of the dish. Vrentas et al. (1991) presented an analysis of the dish effect (referred to as reservoir effect in their paper) for the flow of a Newtonian fluid in a parallel-plate rotational viscometer. According to their results, provided that the ratio of radius of

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dish, Rdish, to radius of upper plate, R, is greater than 1.10, the following equation can be used to calculate the torque: M=

Mmeasured h 1 + 1.9 R

(2.70)

The torque, M, from this equation can be used in Equation 2.69 to calculate shear stress τR for a Newtonian fluid.

CAPILLARY RHEOMETRY Capillary tube rheometry is a well-established technique for studying shear properties of materials. It has been applied to study viscosity of cheese (Smith et al., 1980), butter (Shukla and Rizvi, 1995), and many other food materials (Halliday and Smith, 1995; Sharma et al., 1993; White et al., 1993). Capillary viscometers are often used in laboratories and as an on-line instrument in process industries to measure viscosity (Roberts, 2001). The capillary rheometer consists of a small tube through which an incompressible fluid is forced to undergo steady axial laminar flow, either by means of an imposed pressure or a piston moving at a constant speed (Figure 2.52). The capillary rheometers can also be designed to have several capillary sections of different diameters in series so that non-Newtonian fluids can be characterized in a single pass of fluid (White et al., 1993). The quantities normally measured are the volumetric flow rate, Q, and the driving pressure, Pdriving. When a moving piston generates the flow, the driving pressure is related to the piston force (Fpiston) and reservoir radius (Rreservoir) as follows (Dealy and Wissbrun, 1989): Pdriving =

F piston 2 π Rreservoir

(2.71)

The important assumptions made in the analysis of capillary flow are (Macosko, 1994): (a) fully developed, steady, isothermal, laminar flow; and (b) fluid velocity is zero at wall — that is no slip at the wall. The total pressure drop (∆P) for flow of fluid from a reservoir, through a capillary and out to the ambient pressure consists of two components (Dealy and Wissbrun, 1989)*: Pdriving − Pambient = ∆P = ∆Pend + ∆Pcapillary

(2.72)

where, ∆Pend = excess pressure loss due to the entrance and exit flow (i.e., ∆Pend = ∆Pentrance + ∆Pexit), and Pambient = ambient pressure. The components of total pressure drop are schematically illustrated in Figure 2.53. * Although ∆ means “final-initial,” which makes ∆P a negative value; for convenience we consider the term ∆P as “Phigher – Plower” to make it a positive quantity.

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W (Dead weight) or Vz (Constant velocity) z Plunger r Rr

Sample Reservoir or barrel section

R

Capillary section

L

Q (Volumetric flow rate)

FIGURE 2.52 Schematic drawing of a piston-driven capillary rheometer.

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∆P

∆Pentrance

∆Pcapillary

∆Pexit In

Out

Reservoir section

Capillary section

FIGURE 2.53 The pressure profile in different sections of a capillary rheometer.

The working equations for capillary rheometry are given as (Macosko, 1994): Wall shear stress:

τw =

1 ∆Pcapillary 1 ∆P 1 ∆P − ∆Pend = = 2 ( L / R) 2 [( L / R) + CB ] 2 ( L / R)

(2.73)

where, CB (= ∆Pend /(2τw) is the Bagley correction, which takes into account the pressure losses in the entrance and exit of the capillary. The Bagley correction is either applied to capillary length-to-radius (L/R) term or to the pressure term, as written in Equation 2.73 The Bagley correction procedure involves measuring the pressure drop for a number of capillaries having different lengths (thus, different L/R ratios) at selected values of apparent wall shear rate. It is common practice to use at least three tubes of the same diameter but different lengths. The magnitudes of end corrections are determined from Bagley plots as shown schematically in Figure 2.54. It may be necessary to apply corrections to the measured volumetric flow rates (Qm ) if there is wall slippage. Wall slip is to be suspected when plots of τw vs. apparent shear rate at wall γ˙aw , (see below) for capillaries with the same L/R ratio but different diameters do not fall on a single curve. In accounting for slip, the slope of [Qm/(πR3τw)] versus [1/R2 ] is taken as the corrected slip coefficient, βc, and this parameter is used in the following equation to calculate the corrected volumetric flow rate (Qc): Qc = Qm − β c π R τ w

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(2.74)

∆P

. γaw3

. γaw2

. γaw1

∆Pend3 ∆Pend2 ∆Pend1

e3

e2

e1

L /R

FIGURE 2.54 Bagley plot for pressure corrections for the end effects. The measured pressure drop (∆P) for different length-to-radius ratio (L/R) of the pipe is obtained for different wall shear rates (γ˙ aw1, γ˙ aw2, γ˙aw3). The correction factors (∆Pend1, ∆Pend2, ∆P end3) are obtained from the intercepts.

Apparent or Newtonian shear rate at the wall is given by: γ˙ aw =

4 Qc π R3

(2.75)

For a non-Newtonian liquid the shear stress at the wall τw is unchanged while the shear rate at the wall is calculated from Weissenberg-Rabinowitsch-Mooney equation (Macosko, 1994):  3 1 d ln Qc  γ˙ w = γ˙ aw  +   4 4 d ln Pc 

(2.76)

The term in square brackets is called the Rabinowitsch correction. The slope (dlnQc/dlnPc) is equal to 1.0 for Newtonian fluids, and to (1/n) for power-law fluids with n being the flow-behavior index. Once shear rate and shear stress are known at the same location we can then calculate shear viscosity (η = τw /γ˙w) and construct either the flow curve (τw vs. γ˙w) or viscosity curve (η vs. γ˙w). The capillary rheometer can be operated in two modes: (a) controlled volume or displacement mode, where Q is controlled and ∆P is measured, and (b) controlled pressure mode, where ∆P is controlled and volumetric flow rate (Q) is measured. The first mode can be realized using a Universal Testing Machine with constant crosshead speeds. The second mode can be realized either by applying a dead weight on the plunger or by using gas pressure to move the plunger.

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EXTENSIONAL RHEOMETRY Extensional or elongational rheometry is a relatively new area of active research when compared to shear rheometry, and began to receive increased attention around 1970. (Macosko, 1994; Doraiswamy, 2002; Barnes et al., 1989). Its development stems from the observation that several industrially important polymer processing operations such as extrusion, molding, fiber spinning, calendaring, blowing, coating, and foam production involve significant extensional deformation in addition to shear deformation (Cogswell, 1981; Macosko, 1994; Baird, 1999). Currently, commercial elongational rheometers are available to measure extensional properties of polymer melts (Schulze et al., 2001; Meissner and Hostettler, 1994). Moreover, the filamentstretching rheometer has been developed to measure extensional properties of mobile polymer solutions (Tirtaatmadja and Sridhar, 1993; Sridhar, 2000). Extensional flows are more sensitive to variations in molecular structure of a polymeric material, and thus offer a powerful means of polymer characterization (Münstedt et al., 1998). It is possible for polymers to have identical shear flow properties while exhibiting extremely different extensional flow properties. Only for deformations that are either very small or very slow, the theory of linear viscoelasticity provides relationships between material functions determined using various kinds of deformations (Dealy and Wissbrun, 1989). For instance, the following limiting relation between extensional and shear properties is established (Barnes et al., 1989; Dealy, 1995): ηE (ε˙ ) ε˙ →0 = 3η( γ˙ ) γ˙ →0

(2.77)

where, ηE is the tensile or extensional viscosity and η is the shear viscosity. For Newtonian fluids ηE = 3η for all values of strain rates. This relationship is named as Trouton ratio, TR, defined as (Jones et al., 1987): TR =

η E (ε˙ ) η γ˙ = 3ε˙

(

)

(2.78)

For calculating TR, the shear viscosity should be evaluated at a shear rate numerically equal to 3ε˙ . Trouton ratio is exactly 3 for inelastic flows, and any departure from the value of 3 is associated with viscoelastic effects (Jones et al., 1987; Barnes et al., 1989). It is clear that when the relation given in Equation 2.77 is valid there is no need to make extensional tests since the extensional viscosity can be calculated from the shear viscosity function determined at small and slow shearing experiment (Dealy and Wissbrun, 1989). However, for large and rapid deformations the relation given in Equation 2.77 is not valid, except at small strains, and therefore it is essential to make extensional measurements. There are several kinds of extensional-flow geometries such as uniaxial extension, squeezing flow, sheet stretching, fiber spinning, bubble collapse, stagnation flows, and entrance flows (Macosko, 1994). Direct extensional-flow measurements have an advantage over shear measurements in that the measurement does not involve

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a sample–instrument interface, and thus there is no slip problem to consider (Cogswell, 1981). Some of these extensional methods are already applied to cheese with some success, as described in Chapter 9. Here we limit our attention to lubricated squeezing flow (LSF) technique, which is the most popular and promising method of studying extensional properties of (melted) cheese.

LUBRICATED SQUEEZING FLOW Squeeze flow or squeezing flow is often used to determine flow properties of highly viscous materials such as polymer melts and semiliquid or semisolid food products (e.g., cream cheese, peanut butter, melted cheese, tomato paste, butter, dough, etc). A schematic diagram of the squeezing-flow geometry is shown in Figure 2.55. Although it is a simple test to perform, the analysis of squeezing flow may not be straightforward, particularly if there is friction between the specimen ends and compression plates (i.e., unlubricated squeezing flow). In unlubricated squeezing flow, if a significant shearing component is present it alters the pattern of outward flow of material between the plates. Thus, it is necessary to lubricate the sample–platen contact surfaces in order to eliminate the shear in the sample and obtain purely shear-free or biaxial extensional flow. Lubricated or unlubricated squeezing flow can be conducted in constant volume or constant area configurations depicted in Figure 2.55. The material is squeezed out between two parallel plates at either controlled force (or stress) or controlled speed (or strain rate). Quite often the upper plate is moving at a constant speed while the lower plate is stationary. The squeezing-flow configuration represents one of the few cases where specimen loading and cleaning of equipment are fairly easy. The simple geometry of the lubricated squeezing flow (LSF) makes it also convenient for performing stress relaxation or creep experiments. Replacing the lower plate with a shallow dish results in a new configuration named as “imperfect squeezing flow” (Lee and Peleg, 1992) (Figure 2.55). LSF has been developed and used first by Chatraei and Macosko (1981) to measure biaxial extensional viscosity of polydimethyl siloxane and polyisobutylene melts under constant stress. The LSF technique was introduced to food rheology in the mid-1980s (Casiraghi et al., 1985) and has since been applied to various kinds of food products (Campanella et al., 1987; Campanella and Peleg, 1987b; Hoffner et al., 1997; Shukla et al., 1995; Huang and Kokini, 1993; Bagley et al., 1990; Corradini et al., 2000; Suwonsichon and Peleg, 1999; Wang et al., 1998; Ak and Gunasekaran, 1995). Recently, Campanella and Peleg (2002) reviewed LSF applications to semiliquid foods. A good example of application of LSF for cheese is the UW Meltmeter, the cheese meltability measurement device developed by Wang et al. (1998). In this, the fat melting from cheese at high temperature self-lubricates the compression plates, making it an ideal test method. The UW Meltmeter is discussed in detail in Chapter 8. The LSF geometry and velocity profile in the specimen and the lubricant layer are depicted in Figure 2.56. In an ideal situation, the lubricant undergoes shear deformation, and the sample undergoes biaxial extension. The working equations for LSF method are given as follows (Chatraei and Macosko, 1981; Macosko, 1994):

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R

R

Sample

H

r(t) Unlubricated constant volume squeezing flow

Unlubricated constant area squeezing flow

R

R

Sample

H

r(t) Lubricated constant volume squeezing flow

Lubricated constant area squeezing flow

R Sample Imperfect squeezing flow = Constant Load, W or = Constant Velocity, Vz

FIGURE 2.55 Configurations of unlubricated squeezing flow, lubricated squeezing flow, and imperfect squeezing flow tests.

Axial Hencky strain:  H ε H = ln   Ho 

(2.79)

Axial strain rate: ε˙ H =

1 dH Vz = H dt H

(2.80)

where Vz is the velocity in the vertical direction (e.g., crosshead speed of a universal testing machine).

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Upper plate (moving) δ/2

Lubricant layer

The ideal velocity profile

H

Specimen

Lubricant layer

δ/2

*Shear in the lubricant *Extension in the sample

Lower plate (fixed)

FIGURE 2.56 Lubricant and sample velocity profiles in ideal lubricated squeezing flow (the sample thickness H is much greater than the total lubricant layer thickness δ). (After Papanastasiou et al., 1986.)

Radial or biaxial strain:  R 1  H ε B = ln  = − ln  2  Ho   Ro 

(2.81)

where, the constant volume assumption, (R/R0) = (H0/H)1/2, is applied. Radial or biaxial strain rate: ε˙ B = −

1 dH 2 H dt

(2.82)

It is assumed that at any moment the lubricant film thickness, δ, is smaller than the specimen thickness, H, and therefore, H + δ ≅ H. This is justified at the start of the test, but may be questionable at later stages when the specimen thickness becomes small at large strains (e.g., when εΒ = 2.0, H/Ho = 0.018). Biaxial (compressive) stress when the gap is fully filled with sample: σB =

F π R2

(2.83)

In this case Rspeciman ≥ Rplate = R. Biaxial (compressive) stress when the gap is partially filled with sample: σB =

F π r(t) 2

In this case r(t) represents the instantaneous radius of the specimen.

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(2.84)

Biaxial stress growth coefficient (Dealy, 1995): η+B (t, ε˙ B ) ≡

σB ε˙ B

(2.85)

Biaxial extensional or elongational viscosity:

[

]

η B (ε˙ B ) = lim η+B (t, ε˙ B )

t →∞

(2.86)

For Newtonian fluids the biaxial elongational viscosity is six times the shear viscosity, ηB = 6η. One important consideration in LSF, which is often not addressed in food-related studies, is the loss of effective lubrication, which limits the maximum achievable strain. Macosko (1994) mentions that the strain in LSF is limited to 1.0–1.5 because of the loss of effective lubrication. It is experimentally demonstrated that lubrication is maintained up to a higher total strain if the lubricant has a higher zero-shear viscosity (Chatraei and Macosko, 1981). The optimum ratio of the zero-shear viscosities of the sample to the lubricant is reported to range from 500 Pa.s to 1000 Pa.s (Papanastasiou et al., 1986; Soskey and Winter, 1985). The criterion for good lubrication is given as (Macosko, 1994): 2δ η L R 2 < < 20 H ηS δ 2

(2.87)

where ηS is the sample viscosity, δ the lubricant thickness, H the sample thickness, ηL the lubricant viscosity, and R the plate radius.

EQUATIONS

FOR

DIFFERENT FLUIDS

IN

LUBRICATING SQUEEZING FLOW

Analytical solutions for lubricated-squeezing flow of Newtonian and non-Newtonian fluids are given in various publications in the rheology literature. We present below these equations describing the specimen thickness as a function of time under constant load, or the load as a function of time under constant velocity for Newtonian and non-Newtonian fluids. (1) Newtonian fluids (a) Flow under constant load–constant volume (Lee and Peleg, 1989): 1 1 W t = + H (t ) Ho 3ηΛ

(2.88)

where, W is the constant load, η the Newtonian viscosity, Λ the specimen volume, and t the time.

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(b) Flow under constant load–constant area:  Wt  H (t ) = Ho exp − π 3 R 2 η  

(2.89)

where, R is the radius of the plate (= radius of the sample). (c) Flow under constant velocity–constant volume: 3ηΛ V [H(t)]2 z

F (t ) =

(2.90)

where, Vz is the squeezing speed. (d) Flow under constant velocity–constant area: F (t ) =

3π η R 2 Vz H(t)

(2.91)

(2) Power-law fluids (a) Flow under constant load–constant volume (Campanella and Peleg, 1987b): 1/ n     t  W   1  1 = +    H (t )  Ho1 / n n  n2+1 K 3 Λ    

n

(2.92)

where, K is the consistency index, and n the flow behavior index. (b) Flow under constant velocity–constant volume: F (t ) = 3

n +1 2

ΛK

(Vz )n

(2.93)

[ H (t )]n+1

(c) Flow under constant load–constant area: 1/ n    W  t  H (t ) = Ho exp  − n+1   2 2  3 π R K  

(2.94)

(d) Flow under constant velocity–constant area: F (t ) = 3

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n +1 2

 V  πKR  z   H (t )  2

n

(2.95)

(3) Herschel-Bulkley fluids (a) Flow under constant load–constant volume (Ak and Gunasekaran, 2000):   1 1 −n 1 3 τo Λ  ,1 + , +  1 / n  Hypg2 F1 , n H (t ) W  n n  ( H (t ) W )  1/ n   1 1   −n 1 3 τo Λ  1   ,1 + , t  = − ( n +1)  1 / n Hypg2 F1 , n Ho W   ( Ho W )  K Λ 3  n n  

(2.96)

where, Hypg2F1[a,b,c,z] is the hypergeometric function 2F1[a,b;c;z] (Mathematica 3.0, Wolfram Research; Andrews, 1992). The validity of this hypergeometric solution, Equation 2.96, is verified by the fact that it reduces correctly to Equation 2.88 for n = 1 and τo = 0, and to Equation 2.92 for τo = 0 (Ak and Gunasekaran, 2000). (b) Flow under constant velocity–constant volume: V   Λ  F (t ) = 3 τ o + K  3 z   H(t)  H (t ) 

n

  

(2.97)

(c) Flow under constant load–constant area:   W − πR2 3 τ  1 / n  o H (t ) = Ho exp − n +1  t 2   πR 3 K  

(2.98)

(d) Flow under constant velocity–constant area: n  V    F (t ) = 3 π R 2  τ o + K  3 z    H (t )   

(2.99)

REFERENCES Abbott, J. et al. 1991. Experimental observations of particle migration in concentrated suspensions: Couette flow. Journal of Rheology 35(5):773–795. Ak, M.M. 1993. Rheological Measurements on Mozzarella Cheese. University of WisconsinMadison, Ph.D. thesis. Ak, M.M. and S. Gunasekaran. 1992. Stress–strain curve analysis of Cheddar cheese under uniaxial compression. Journal of Food Science 57(5):1078–1081. Ak, M.M. and S. Gunasekaran. 1995. Evaluating rheological properties of Mozzarella cheese by the squeezing flow method. Journal of Texture Studies 26:695–711.

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Ak, M.M. and S. Gunasekaran. 1997. Anisotropy in tensile properties of Mozzarella cheese. Journal of Food Science 62(5):1031–1033. Ak, M.M. and S. Gunasekaran. 2000. Simulation of lubricated squeezing flow of a HerschelBulkley fluid under constant force. Applied Rheology 10(6):274–279. Ak, M.M. and S. Gunasekaran. 2001. Linear viscoelastic methods, in Nondestructive Food Evaluation: Techniques to Analyze Properties and Quality, S. Gunasekaran, Ed., pp 287–334. New York: Marcel Dekker, Inc. Andrews, L.C. 1992. Special Functions of Mathematics for Engineers. New York: McGraw-Hill, Inc. Anon. 2002. Determining the linear viscoelastic region in polymers. Rheology Application Notes: RS-23 (TA Instruments), http://www.tainst.com. Astarita, G. 1990. Letter to the Editor: The engineering reality of the yield stress. Journal of Rheology 34(2):275–277. ASTM. 1995. Standard test method for tensile properties of plastics (Metric). 59–67. ASTM. 1996. Standard test methods for rubber properties in compression. 113–116. Bagley, E., D. Christianson, and D. Trebacz. 1990. The computation of viscosity and relaxation time of doughs from biaxial extension data. Journal of Texture Studies 21:339–354. Baird, D. 1999. The role of extensional rheology in polymer processing. Korea-Australia Rheology Journal 11(4):305–311. Barnes, H.A. 1999. The yield stress — a review or ‘παντα ρει — everything flows? Journal of Non-Newtonian Fluid Mechanics 81:133–178. Barnes H.A., J.F. Hutton, and K. Walters. 1989. An Introduction to Rheology. Amsterdam: Elsevier Science Publishers B.V. Barnes, H.A., H. Schimanski, and D. Bell. 1999. 30 Years of progress in viscometers and rheometers. Applied Rheology 9(2):69–76. Barnes, H.A. and Q.D. Nguyen. 2001. Rotating vane rheometry — a review. Journal of Non-Newtonian Fluid Mechanics 98(1):1–14. Bourne, M.C. 1977. Compression rates in the mouth. Journal of Texture Studies 8:373–376. Bourne, M.C. 1982. Food Texture and Viscosity: Concept and Measurement. New York: Academic Press. Breidinger, S.L. and J.F. Steffe. 2001. Texture map of cream cheese. Journal of Food Science 66(3):453–456. Briggs, J.L., J.F. Steffe, and Z. Ustunol. 1996. Vane method to evaluate the yield stress of frozen ice cream. Journal of Dairy Science 79(4):527–531. Calzada, J.F. and M. Peleg, 1978. Mechanical interpretation of compressive stress-strain relationships of solid foods. Journal of Food Science 43:1087–1092. Campanella, O.H. and M. Peleg, 1987a. Determination of the yield stress of semi-liquid foods from squeezing flow data. Journal of Food Science 52(1):214–215, 217. Campanella, O.H. and M. Peleg, 1987b. Squeezing flow viscosimetry of peanut butter. Journal of Food Science 52(1):180–184. Campanella, O.H., L. Popplewell, J. Rosenau and M. Peleg, 1987. Elongational viscosity measurements of melting American process cheese. Journal of Food Science 52(5):1249–1251. Campanella, O.H. and M. Peleg, 2002. Squeezing flow viscometry for nonelastic semiliquid foods — Theory and applications. Critical Reviews in Food Science & Nutrition 42(3):241–264. Casiraghi, E.M., E.B. Bagley, and D.D. Christianson. 1985. Behavior of Mozzarella, Cheddar and processed cheese spread in lubricated and bonded uniaxial compression. Journal of Texture Studies 16:281–301. Charalambides, M.N. et al. 2001. The analysis of the frictional effect on stress–strain data from uniaxial compression of cheese. Journal of Materials Science 36(9):2313–2321.

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Hamann, D.D. 1983. Structural failure in solid foods, in Physical Properties of Foods, M. Peleg and E.B. Bagley, Eds., pp 351–383. Westport, CT: AVI Publishing Company, Inc. Hoffner, B., C. Gerhards, and M. Peleg, 1997. Imperfect lubricated squeezing flow viscometry for foods. Rheologica Acta 36:686–693. Huang, H. and J. Kokini. 1993. Measurement of biaxial extensional viscosity of wheat flour doughs. Journal of Rheology 37(5):879–891. Jones, D.M., K. Walters, and P.R. Williams. 1987. On the extensional viscosity of mobile polymer solutions. Rheologica Acta 26:20–30. Junus, S. and J.L. Briggs. 2001. Vane sensor system in small strain oscillatory testing. Applied Rheology 11(5):264–270. Kamst, G.F. et al. 1999. A new method for the measurement of the tensile strength of rice grains by using the diametral compression test. Journal of Food Engineering 40:227–232. Kamyab, I., S. Chakrabarti, and J. G. Williams. 1998. Cutting cheese with wire. Journal of Materials Science 33(11):2763–2770. Keener, K.M., C.R. Daubert, and T.A. Glenn. 1999. Development and evaluation of a hand vane device for rapid quality measurement during food processing. Abstracts, IFT Annual Meeting, Chicago, IL. Konstance, R.P. and V.H. Holsinger. 1992. Development of rheological test methods for cheese. Food Technology 46:105–109. Lakes, R.S. 1987. Foam structure with a negative Poisson’s ratio. Science 235:1038–1040. Langley, K.R. and R.J. Marshall. 1993. Jaw movement during mastication of fibrous and nonfibrous composite foods by adult subjects. Journal of Texture Studies 24:11–25. Lanier, T.C. 2000. Measurement of fracture of solid and semi-solid foods with the Hamann torsion gelometer, in Proceedings of the 2nd International Symposium on Food Rheology and Structure, P. Fischer, I. Marti, and E.J. Windhab, Eds., pp 121–125. Zürich, Switzerland. Lauger, J. and S. Huck. 2002. Real controlled stress and controlled strain experiments with the same rheometer. www.physica.de. Lee, S. and M. Peleg, 1989. Squeezing flow of a double layered array of two Newtonian liquids. Chemical Engineering Science 44(12):2979–2986. Lee, S. and M. Peleg, 1992. Imperfect squeezing flow viscosimetry with a wide plate and a shallow container. Journal of Texture Studies 23:267–278. Liddell, P.V. and D.V. Boger. 1996. Yield stress measurements with the vane. Journal of Non-Newtonian Fluid Mechanics 63:235–261. Lodge A.S. 1964. Elastic Solids: An Introductory Vector Treatment of Finite-Strain Polymer Rheology. New York: Academic Press. Luyten H. 1988. The Rheological and Fracture Properties of Gouda Cheese. Wageningen Agricultural University, The Netherlands, Ph.D. thesis. Luyten, H., T. van Vliet, and P. Walstra. 1991a. Characterization of the consistency of Gouda cheese: Fracture properties. Netherlands Milk and Dairy Journal 45:55–80. Luyten, H., T. van Vliet, and P. Walstra. 1991b. Characterization of the consistency of Gouda cheese: Rheological properties. Netherlands Milk and Dairy Journal 45:33–53. Luyten, H., T. van Vliet, and P. Walstra. 1992. Comparison of various methods to evaluate fracture phenomena in food materials. Journal of Texture Studies 23:245–266. Macosko C.W. 1994. Rheology: Principles, Measurements, and Applications. New York: VCH Publishers, Inc. Meissner, J. 1978. Combined constant strain rate and stress relaxation test for linear viscoelastic studies. Journal of Polymer Science 16:915–919.

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Meissner, J. and J. Hostettler. 1994. A new elongational rheometer for polymer melts and other highly viscoelastic liquids. Rheologica Acta 33:1–21. Mitchell, J.R. 1984. Rheological Techniques, in Food Analysis: Principles and Techniques Volume 1. Physical Characterization, D.W. Gruenwedel and J. R. Whitaker, Eds., pp 151–220. New York: Marcel Dekker, Inc. Mleko, S. and E.A. Foegeding. 2000. Physical properties of rennet casein gels and processed cheese analogs containing whey proteins. Milchwissenschaft 55(9):513–516. Mohsenin, N. and J. Mittal. 1977. Use of rheological terms and correlation of compatible measurements in food texture research. Journal of Texture Studies 8:395–408. Molander, E., K.R. Kristiansen and H. Werner. 1990. Instrumental and sensoric measurement of Brie texture. Milchwissenschaft 45(9):589–593. Montejano, J.G., H.H. Hamann, and T.C. Lanier. 1983. Final strengths and rheological changes during processing of thermally induced fish muscle gels. Journal of Rheology 27(6):557–579. Mpagana, M. and J. Hardy. 1986. Effect of salting on some rheological properties of fresh Camembert cheese as measured by uniaxial compression. Milchwissenschaft 41(4):210–213. Münstedt, H., S. Kurzbeck, and L. Egersdörfer. 1998. Influence of molecular structure on rheological properties of polyethylenes. II. Elongational behavior. Rheologica Acta 37:21–29. Newton, J.M., I. Haririan, and F. Podczeck. 2000. The influence of punch curvature on the mechanical properties of compacted powders. Powder Technology 107:79–83. Ney, K.H. 1985. Rheology of foods. Anisotropy in Cheddar cheese. Gordian 85(9):172, 174. Nguyen, Q.D. and D.V. Boger. 1983. Yield stress measurement for concentrated suspensions. Journal of Rheology 27(4):321–349. Nguyen, Q.D. and D.V. Boger. 1985. Direct yield stress measurement with the vane method. Journal of Rheology 29(3):335–347. Nijman, J. and S. Chakrabarti. 1997. A rotational rheometer for material characterization by the rheologist and the nonrheologist. American Laboratory 29(16):17–18, 20. Papanastasiou, A., C. Macosko, and L. Scriven. 1986. Analysis of lubricated squeezing flow. International Journal of Numerical Methods in Fluids 6:816–839. Peleg, M. 1979. Characterization of the stress relaxation curves of solid foods. Journal of Food Science 44(1):277–281. Peleg, M. 1980. Linearization of relaxation and creep curves of solid biological materials. Journal of Rheology 24(4):451–463. Peleg, M. 1984. A note on the various strain measures at large compressive deformations. Journal of Texture Studies 15:317–326. Peleg, M. 1987. The basics of solid foods rheology, in Food Texture: Instrumental and Sensory Measurement, H.R. Moskowitz, Ed., pp 3–33. New York: Marcel Dekker. Peleg, M. and M.D. Normand. 1983. Comparison of two methods for stress relaxation data presentation of solid foods. Rheologica Acta 22:108–113. Pernell, C.W., E.A. Foegeding, and C.R. Daubert. 2000. Measurement of the yield stress of protein foams by vane rheometry. Journal of Food Science 65(1):110–114. Powell, R.L. 1988. Rotational viscometry, in Rheological Measurement, A.A. Collyer and D.W. Clegg, Eds., pp 247–296. London: Elsevier Applied Science. Purkayastha, S. et al. 1985. A computer aided characterization of the compressive creep behavior of potato and cheddar cheese. Journal of Food Science 50:45–50, 55. Purkayastha, S., M. Peleg, and M. Normand. 1984. Presentation of the creep curves of solid biological materials by a simplified mathematical version of the generalized KelvinVoigt model. Rheologica Acta 23:556–563.

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Qiu, C.G. and M.A. Rao. 1988. Role of pulp content and particle size in yield stress of apple sauce. Journal of Food Science 53(4):1165–1170. Reiner, M. 1964. The Deborah number. Physics Today 17(1):62. Riande E. et al. 2000. Polymer Viscoelasticity — Stress and Strain in Practice. New York: Marcel Dekker, Inc. Riley W.F., L.D. Sturges, and D.H. Morris. 1995. Statics and Mechanics of Materials: An Integrated Approach. New York: John Wiley & Sons. Roberts, I. 2001. In-line and on-line rheology measurement, in Instrumentation and Sensors for the Food Industry, E. Kress-Rogers and C.J.B. Brimelow, Eds., pp 403–422. Cambridge: Woodhead Publishing Ltd. Rohm, H., D. Jaros, and M. deHaan. 1997. A video-based method for determination of average stress–strain relations in uniaxial compression of selected foods. Journal of Texture Studies 28:245–255. Rohm, H. and H. Lederer. 1992. Uniaxial compression of Swiss-type cheese at different strain rates. International Dairy Journal 2:331–343. Schulze, J. et al. 2001. A comparison of extensional viscosity measurements from various RME rheometers. Rheologica Acta 40:457–466. Schurz, J. 1992. Letter to the editor: A yield value in a true solution. Journal of Rheology 36(7):1319–1321. Shama, F. and P. Sherman. 1973. Evaluation of some textural properties of foods with the Instron universal testing machine. Journal of Texture Studies 4:344–352. Sharma, N., M.A. Hanna, and Y.R. Chen. 1993. Flow behavior of wheat flour-water dough using a capillary rheometer. I. Effect of capillary geometry. Cereal Chemistry 70(1):59–63. Sherman, P. 1975. Factors influencing the instrumental and sensory evaluation of food emulsions, in Theory, Determination and Control of Physical Properties of Food Materials, C.-K. Rha, Ed., pp 251–266. Dordrecht-Holland: D. Reidel. Shoemaker, C., J. Lewis, and M. Tamura. 1987. Instrumentation for rheological measurements of food. Food Technology (3):80–84. Shukla, A. and S. Rizvi. 1995. Measurement of flowability of butter by capillary rheometry. Journal of Texture Studies 26:299–311. Shukla, A., S. Rizvi, and J. Bartsch. 1995. Rheological characterization of butter using lubricated squeezing flow. Journal of Texture Studies 26:313–323. Smith, C., J. Rosenau, and M. Peleg, 1980. Evaluation of the flowability of melted Mozzarella cheese by capillary rheometry. Journal of Food Science 45:1142–1145. Soskey, P. and H. Winter. 1985. Equibiaxial extension of two polymer melts: polystyrene and low density polyethylene. Journal of Rheology 29(5):493–517. Sridhar, T. 2000. From rheometry to rheology. Korea-Australia Rheology Journal 12(1):39–53. Steffe J.F. 1996. Rheological Methods in Food Process Engineering. Michigan: Freeman Press. Suwonsichon, T. and M. Peleg, 1999. Imperfect squeezing flow viscometry of mustards with suspended particles. Journal of Food Engineering 39:217–226. Swallowe, G.M. 1999. Tensile and Compressive Testing, in Mechanical Properties and Testing of Polymers: An A-Z Reference, G.M. Swallowe, Ed., pp 242–243. AH Dordrecht, The Netherlands: Kluwer Academic Publishers. Thomas, G.B. Jr. and R.L. Finney. 1988. Calculus and Analytic Geometry. 7th edition. New York: Addison-Wesley Publishing Company. Tirtaatmadja, V. and T. Sridhar. 1993. A filament stretching device for measurement of extensional viscosity. Journal of Rheology 37(6):1081–1102. Truong, V.D. and C.R. Daubert. 2001. Textural characterization of cheeses using vane rheometry and torsion analysis. Journal of Food Science 66(5):716–721.

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3

Uniaxial Testing of Cheese

Uniaxial testing is the most popular configuration for evaluating mechanical and rheological properties of cheeses. In this chapter, we attempt to summarize the extensive literature on different cheeses. The available data are diverse and are often collected by various testing modes, which can be grouped under the word “uniaxial.” Uniaxial testing is also the most popular method for instrumental evaluation of cheese texture (see Chapter 7). One way to handle such diverse literature on properties of different cheeses would be to present the available data according to the type of cheese. However, we organized the information according to the specific test method (e.g., uniaxial compression, tension, relaxation, etc.), since effects of many experimental factors (e.g., deformation rate, cheese age, etc.) on rheological properties of different cheeses bear some similarities. Rigorous analysis of the literature data is seriously hampered by the lack of standardization in terms of sample preparation, measurement conditions, parameter evaluations, and reporting style. It may not be feasible to specify particular requirements for each of these issues since the objectives of measurements can be totally different (e.g., quality control, correlations with sensory results, etc.), and the prevailing conditions under which the tests are made may also vary. However, minimum requirements in data reporting may be (and perhaps should be) universally agreed upon, which would greatly facilitate comparison of results from different sources. Masi (1987) reported a major attempt to improve the comparability of results from different laboratories and to identify the most suitable measurement conditions for generating reproducible mechanical data on cheese. Even in this collaborative work the participating laboratories used different experimental conditions (i.e., sample size and shape, test temperature, sample handling, sample age at testing, number of samples tested, etc.). The recommendations based on the results of this extensive study are given in Table 3.1 for cheese, as well as general recommendations for reporting compression results. Some of these suggestions are no longer relevant due to advances in the instrumentation. For instance, the recommended crosshead speeds are naturally based on the characteristics of uniaxial testing machines available at that time and the capabilities of the strip-chart recorders, which are practically obsolete now. A group at the International Dairy Federation (IDF) also formed a working team (IDF E703) to develop standards for testing and reporting uniaxial test results on cheese. Though a draft was prepared, it was never officially published.* Nevertheless, the special bulletin of IDF (1991) includes expert opinions on important issues related to cheese rheology and texture measurements, and recommendations on test methods. * Philip Watkinson, Fonterra Research Center, New Zealand, personal communication.

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TABLE 3.1A Recommendations for Compression Testing of Cheese Measurement Conditions

Data to Be Reported

a. Shape: vertical cylinder or prism b. Size: aspect ratio > 1 (aspect ratio = sample height/sample width or diameter) c. Crosshead speed: 5 cm/min and two other speeds in the range 0–20 cm/min d. Boundaries: Both bonded and lubricated platens

a. Rupturea (or yield) stress b. Rupture (or yield) strain c. Elastic modulus d. Rupture (or yield) work per unit of volume e. Failure mode

a

If the cheese does not show rupture or yield, then the stress and the work per unit of volume should correspond to 80% deformation. Source: After Masi, 1987. With permission.

TABLE 3.1B Information to be Included in Compression-Test Reports Contextual Information

Measurement Conditions

Results

a. Detailed description of sample b. Sample dimensions c. Sample history (including preparation and conditioning)

a. Temperature of environment b. Relative humidity of environment c. Sample temperature d. Interface between sample and compression surfaces I. Material II. Roughness III. Dimensions e. Compression rate f. Machine details Accuracy of (1) compression rate, and (2) force measurement g. Initial position of crosshead in relation to the sample h. Response time

a. All original and derived results b. Complete force–deformation curve

Source: After McKenna, 1987. With permission.

The theory of fracture mechanics and the methods specially designed to study fracture properties of materials are discussed in Chapter 4. Fracture properties reported in this chapter are those that are often routinely determined even though the primary objective of the research is not to study fracture behavior. For instance, work-to-fracture (or area under stress–strain curve) values are given in this chapter, whereas the specific fracture energy data are reported in Chapter 4.

UNIAXIAL COMPRESSION MEASUREMENTS Mechanical properties commonly determined from uniaxial compression tests on cheese include modulus of deformability ED, fracture stress σf , fracture strain εf , and work to fracture Wf . All these variables are defined in Chapter 2 (see Figure 2.23). © 2003 by CRC Press LLC

The numerical results for these properties extracted from many publications on different cheeses are listed in Tables 3.2 to 3.5. In the following sections, we will frequently refer to these compilations. For viscoelastic materials, Mohsenin and Mittal (1977) suggested the term “modulus of deformability” instead of Young’s modulus, which is reserved for engineering materials obeying the Hooke’s law. The modulus of deformability is the slope of the “initial linear” part of the stress–strain curve. There are several methods for getting a representative value of the slope. One of these methods is to fit the stress–strain data to a polynomial equation (see Chapter 2, Equation 2.22) and to determine properties from the resulting fit equation (Ak and Gunasekaran, 1992). The other approach is to take the maximum slope within the strain range from 0 to 0.05 (Wium et al., 1997). Another method is to take the slope at a particular strain level (e.g., 5%) as the modulus of deformability (i.e., secant modulus described in Chapter 2). Linear regression on the data pertaining to the initial part of the stress–strain curve is yet another option to calculate the modulus of deformability. Watkinson and Jackson (1999) suggested a new procedure to calculate the modulus of deformability using the gradient of the inflection at the lowest strain in a stress vs. strain curve. This new procedure was used to calculate modulus of deformability ED for three cheeses and compared with the results from three alternative procedures, including the simple polynomial fitting suggested by Ak and Gunasekaran (1992). As shown in Table 3.6, the ranking of ED for each cheese was the same for each procedure, and the relative magnitude of ED for each cheese along with the coefficient of variation was similar for each procedure. Watkinson and Jackson (1999) discussed some special features of their procedure. The usual practice in uniaxial compression of foods is to run the crosshead at constant speed, since Universal Testing Machines (UTMs) machines are normally designed to do that. It is quite unusual, but perhaps highly necessary, to see compression tests made on cheese at constant true strain rate. The true strain rate continuously increases in uniaxial compression at a constant crosshead speed as the specimen height decreases, and this has a great effect in uniaxial compression of foods (Peleg, 1977a; 1977b). We are not aware of any such studies on cheese except that of Jaros and Rohm (1994). A simple device has been described earlier by Luton et al. (1974) for use with Instron testing machines to produce a constant true strain rate in compression or tension tests. Jaros and Rohm (1994) described a method to conduct uniaxial compression tests at constant strain rate using an Instron testing machine. Based on the analysis of rheological data on 136 Swiss cheese samples, it was shown that stress and strain at fracture are significantly lower in constant strain rate compression than in constant speed compression, due to the differences in strain history of the two modes. The modulus of deformability, obtained at 0.04 strain level, is, however, not affected by the test setup. The authors reported the following equations relating fracture stress σf and fracture strain εf from the two modes of deformation: σ f (constant strain rate) = 1.27 σ f (constant speed)0.917

(3.1)

ε f (constant strain rate) = 1.01 ε f (constant speed)0.867

(3.2)

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TABLE 3.2 Values of Modulus of Deformability from Uniaxial Testing of Several Cheesesa

Cheese Variety Apericube processed Bel, France Appenzell Arzua-Ulloa Spanish soft cheese — type I Arzua-Ulloa Spanish soft cheese — type II Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Caciocavallo Caerphilly Camembert 1-d before brining Camembert 1-d after brining Camembert 8-d-old Camembert 15-d-old Camembert 22-d-old Camembert 29-d-old Cheddar Cheddar Cheddar 20 days Cheddar 8 weeks old Cheddar 64 weeks old Cracker Barrel Tasty: Kraft, USA Danbo cheese with 45% fat Danish Feta Double Gloucester Double Gloucester Edam Edam Emmental Emmental Emmentaler 4 months Emmentaler 4 months Emmentaler 4 months Emmentaler 4 months English Mature Cheddar Galbanino Galbanino Galbanino Galbanino Garrotxa-type goat milk cheese Gouda Gouda

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Initial Strain Rate s–1b

Modulus kPa

— — 46 46 110 220 330 440 4.2 — 5.56 5.56 5.56 5.56 5.56 5.56 — — 1.4–110 32 32 — 833 — — — — — — — 4.76 19 76.2 19 — 0.556 11 28 56 500 — —

70 22 84 44 229 221 225 176 687 8 357 429 749 1070 251 0 48 180 242 640 290 580 189 470 1000 850 ~500 290 18 470 139 182 234 182 890 600 1000 1400 1800 343 405 390

Ref. Agrawal et al., 1997 Prentice et al., 1993 Almena et al., 1998 Almena et al., 1998 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Masi and Addeo, 1986 Prentice et al., 1993 Schlesser et al., 1992 Schlesser et al., 1992 Schlesser et al., 1992 Schlesser et al., 1992 Schlesser et al., 1992 Schlesser et al., 1992 Prentice et al., 1993 Prentice et al., 1993 Ak and Gunasekaran, 1992 Hort and Le Grys, 2001 Hort and Le Grys, 2001 Agrawal et al., 1997 Madsen and Ardö, 2001 Agrawal et al., 1997 Prentice et al., 1993 Agrawal et al., 1997 Prentice et al., 1993 Agrawal et al., 1997 Prentice et al., 1993 Agrawal et al., 1997 Rohm and Lederer, 1992 Rohm and Lederer, 1992 Rohm and Lederer, 1992 Rohm et al., 1992 Agrawal et al., 1997 Masi, 1989 Masi, 1989 Masi, 1989 Masi, 1989 Saldo et al., 2000 Prentice et al., 1993 Prentice et al., 1993

TABLE 3.2 (continued) Values of Modulus of Deformability from Uniaxial Testing of Several Cheesesa

Cheese Variety Gouda Gruyère Gruyère Gruyère-type strong cohesion Gruyère-type strong cohesion — Tension Gruyère-type weak cohesion Gruyère-type weak cohesion — Tension Jarlsberg La Serena 2 days old, with lactic starter La Serena 2 days old, without lactic starter La Serena 60 days old, with lactic starter La Serena 60 days old, without lactic starter Lancashire Lancashire Leicester Mahon >150 days Mahon <60 days Mahon from 60 to 150 days Mild Cheddar Mild Cheddar 37 days old Mild Cheddar 182 days old Mild Cheddar — Tension Montasio Monterey Jack 46 days old Monterey Jack 185 days old Mozzarella Mozzarella Mozzarella Münster Parmesan Parmesan Parmigiano Reggiano 12 months old Parmigiano Reggiano 18 months old Parmigiano Reggiano 28 months old Pecorino Romano Process cheese loaf, light Process cheese loaf, regular Process cheese slice, American 1 Process cheese slice, American 1 — Tension Process cheese slice, American 2 Process cheese slice, American 2 — Tension

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Initial Strain Rate s–1b

Modulus kPa

Ref.

— — — 55.5 11 55.5 11 — — — — — — — — 8.33 8.33 8.33 3 8.3–24 8.3–24 3 — 8.3–24 8.3–24 — — 4.2 — — — 42 42 42 — 3 3 3 3 3 3

410 690 77 479 111 468 93 350 61 32 348 104 ~1250 710 650 2487 497 1426 349 1176 1050 747 470 1008 652 15 150 49 6 1980 2260 1340 1530 2280 2840 154 107 318 374 138 162

Agrawal et al., 1997 Agrawal et al., 1997 Prentice et al., 1993 Pesenti and Luginbühl, 1999 Pesenti and Luginbühl, 1999 Pesenti and Luginbühl, 1999 Pesenti and Luginbühl, 1999 Agrawal et al., 1997 Medina et al., 1991 Medina et al., 1991 Medina et al., 1991 Medina et al., 1991 Prentice et al., 1993 Agrawal et al., 1997 Agrawal et al., 1997 Benedito et al., 2000 Benedito et al., 2000 Benedito et al., 2000 Kamyab et al., 1998 Charalambides et al., 1995 Charalambides et al., 1995 Kamyab et al., 1998 Prentice et al., 1993 Charalambides et al., 1995 Charalambides et al., 1995 Prentice et al., 1993 Agrawal et al., 1997 Masi and Addeo, 1986 Prentice et al., 1993 Prentice et al., 1993 Agrawal et al., 1997 Noel et al., 1996 Noel et al., 1996 Noel et al., 1996 Prentice et al., 1993 Kamyab et al., 1998 Kamyab et al., 1998 Kamyab et al., 1998 Kamyab et al., 1998 Kamyab et al., 1998 Kamyab et al., 1998

TABLE 3.2 (continued) Values of Modulus of Deformability from Uniaxial Testing of Several Cheesesa

Cheese Variety Processed cheese analogs high fat, high moisture Processed cheese analogs high fat, high moisture Processed cheese analogs high fat, high moisture Processed cheese analogs low fat, low moisture Processed cheese analogs low fat, low moisture Processed cheese analogs low fat, low moisture Provolone Provolone Raclette Red Brick UF-Feta 8–10 weeks Red Brick UF-Feta 8–10 weeks Red Brick UF-Feta 8–10 weeks Red Brick UF-Feta 8–10 weeks Reduced Fat Cheddar Mainland, New Zealand Sbrinz Sharp Cheddar Sharp Cheddar 1 month old Sharp Cheddar 6 months old Sharp Cheddar — Tension Silano Smoked Cheddar King Island, Australia String Swiss Appenzeller type — rapeseed-added diet Swiss Appenzeller type — regular diet Teleme 1 month old Tilsit Tin UF-Feta 8–10 weeks Tin UF-Feta 8–10 weeks Tin UF-Feta 8–10 weeks Tin UF-Feta 8–10 weeks Vorarlberger Bergkase White Cheshire a b

Initial Strain Rate s–1b

Modulus kPa

5.56

27

Marshall, 1990

55.6

48

Marshall, 1990

556

50

Marshall, 1990

5.56

223

Marshall, 1990

55.6

231

Marshall, 1990

556

276

Marshall, 1990

— 4.2 — 110 220 330 440 — — 3 8.3–24 8.3–24 3 4.2 — 11 17 17 36

240 559 210 301 256 260 190 940 195 198 1580 1938 220 137 990 100 401 490 175

— 110 220 330 440 16.7 —

30 465 387 404 407 600 470

Prentice et al., 1993 Masi and Addeo, 1986 Agrawal et al., 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Agrawal et al., 1997 Prentice et al., 1993 Kamyab et al., 1998 Charalambides et al., 1995 Charalambides et al., 1995 Kamyab et al., 1998 Masi and Addeo 1986 Agrawal et al., 1997 Taneya et al., 1992 Jaros et al., 2001 Jaros et al., 2001 Raphaelides and Antoniou, 1996 Prentice et al., 1993 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Jaros and Rohm, 1997 Agrawal et al., 1997

Ref.

Data are from uniaxial compression test unless indicated otherwise by a remark next to the variety name. Values in this column are already multiplied by a factor of 1000 to avoid small numbers.

© 2003 by CRC Press LLC

TABLE 3.3 Values of Strain at Fracture from Uniaxial Testing of Several Cheesesa

Cheese Variety

Initial Strain Rate s–1b

Peak Strain

American Light — Shear American — Shear Appenzell Arzua-Ulloa Spanish soft cheese — type I Arzua-Ulloa Spanish soft cheese — type II Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Caciocavallo Caerphilly Cheddar Cheddar Cheddar Cheddar Cheddar Cheddar Cheddar 225 d old Cheddar 8 weeks old Cheddar 64 weeks old Cheddar 5-mo-old, full-fat: 32% Cheddar 5-mo-old, low-fat: 5% Cheshire Danbo Double Gloucester Edam Emmental Emmental 16 weeks Emmental 16 weeks Emmental 16 weeks Emmental 7-d-old Emmental 28-d-old Emmental 70-d old Emmental 112-d-old Emmentaler 4 months Emmentaler 4 months Emmentaler 4 months Emmentaler 4-mo-old Garrotxa-type goat milk cheese Gouda Gouda Gouda 1 week, pH = 4.94 Gouda 1 week, pH = 4.94

261 261 — 46 46 110 220 330 440 — — — — — 1.4–110 11 110 29 32 32 440 440 — 833 — — — 16.7 16.7 16.7 16.7 16.7 16.7 16.7 4.8 19 76.2 19 500 — — 2.8 28

1.05 1.01 0.63 0.37 0.71 0.31 0.32 0.34 0.34 0.80 0.17 0.20 0.21 0.21 0.81–0.91 0.85 0.97 0.37 0.73 0.27 0.42 0.13 0.33 1.10 0.24 0.63 1.05 1.19 1.21 1.27 1.47 1.63 1.43 1.25 1.08 1.17 1.15 1.12 0.24 0.72 0.37 0.55 0.60

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Ref. Gwartney et al. 2002 Gwartney et al., 2002 Prentice et al., 1993 Almena et al., 1998 Almena et al., 1998 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Masi and Addeo, 1986 Prentice et al., 1993 Prentice et al., 1993 Prentice et al., 1993 Prentice et al., 1993 Ak and Gunasekaran, 1992 Rosenau et al., 1978 Rosenau et al., 1978 Fenelon and Guinee, 2000 Hort and Le Grys, 2001 Hort and Le Grys, 2001 Kucukoner et al., 1998 Kucukoner et al., 1998 Prentice et al., 1993 Madsen and Ardö, 2001 Prentice et al., 1993 Prentice et al., 1993 Prentice et al., 1993 Rohm et al., 1996 Rohm et al., 1996 Rohm et al., 1996 Jaros et al., 1997 Jaros et al., 1997 Jaros et al., 1997 Jaros et al., 1997 Rohm and Lederer, 1992 Rohm and Lederer, 1992 Rohm and Lederer, 1992 Rohm et al., 1992 Saldo et al., 2000 Prentice et al., 1993 Prentice et al., 1993 Luyten, 1988 Luyten, 1988

TABLE 3.3 (continued) Values of Strain at Fracture from Uniaxial Testing of Several Cheesesa

Cheese Variety Gouda 1 week, pH = 4.94 Gouda 2 week, pH = 5.24 Gouda 2 week, pH = 5.24 Gouda 2 week, pH = 5.24 Gouda 2 week, pH = 5.24 Gouda 6 months Gouda 6 months Gouda 6 months Gouda 6 months Gruyère Gruyère-type strong cohesion Gruyère-type strong cohesion — Tension Gruyère-type weak cohesion Gruyère-type weak cohesion — Tension Lancashire Leicester Mild Cheddar — A Reduced Fat — Shear Mild Cheddar — A — Shear Mild Cheddar — B Light — Shear Mild Cheddar — B — Shear Montasio Monterey Jack Light — Shear Monterey Jack — Shear Mozzarella Mozzarella Münster Parmesan Parmigiano Reggiano 12 months old Parmigiano Reggiano 18 months old Parmigiano Reggiano 28 months old Pecorino Romano Processed American Processed American Processed cheese analogs high fat, high moisture Processed cheese analogs high fat, high moisture Processed cheese analogs high fat, high moisture Processed cheese analogs low fat, low moisture Processed cheese analogs low fat, low moisture

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Initial Strain Rate s–1b

Peak Strain

140 0.67 2.7 50 250 0.28 2.8 28 170 — 55.5 11 55.5 11 — — 261 261 261 261 — 261 261 — — — — 42 42 42 — 11 110 5.56

0.61 1.48 1.39 1.31 1.07 0.29 0.31 0.30 0.32 0.51 0.75 1.30 0.73 0.46 0.20 0.30 1.28 1.13 1.47 0.95 0.58 1.36 1.41 0.55 1.08 0.10 0.14 0.22 0.18 0.12 0.25 0.86 0.79 1.20

Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Prentice et al., 1993 Pesenti and Luginbühl, Pesenti and Luginbühl, Pesenti and Luginbühl, Pesenti and Luginbühl, Prentice et al., 1993 Prentice et al., 1993 Gwartney et al., 2002 Gwartney et al., 2002 Gwartney et al., 2002 Gwartney et al., 2002 Prentice et al., 1993 Gwartney et al., 2002 Gwartney et al., 2002 Prentice et al., 1993 Masi and Addeo, 1986 Prentice et al., 1993 Prentice et al., 1993 Noel et al., 1996 Noel et al., 1996 Noel et al., 1996 Prentice et al., 1993 Rosenau et al., 1978 Rosenau et al.,1978 Marshall, 1990

55.6

1.43

Marshall, 1990

556

1.47

Marshall, 1990

5.56

0.92

Marshall, 1990

55.6

1.02

Marshall, 1990

Ref.

1999 1999 1999 1999

TABLE 3.3 (continued) Values of Strain at Fracture from Uniaxial Testing of Several Cheesesa

Cheese Variety Processed cheese analogs low fat, low moisture Provolone Provolone Red Brick UF-Feta 8–10 weeks Red Brick UF-Feta 8–10 weeks Red Brick UF-Feta 8–10 weeks Red Brick UF-Feta 8–10 weeks Sbrinz Sharp Cheddar Light — Shear Sharp Cheddar — Shear Silano Tilsit Tin UF-Feta 8–10 weeks Tin UF-Feta 8-10 weeks Tin UF-Feta 8-10 weeks Tin UF-Feta 8-10 weeks Tybo Argentino up to 114 days Vorarlberger Bergkase a b

Initial Strain Rate s–1b

Peak Strain

556

1.02

Marshall, 1990

— — 110 220 330 440 — 261 261 — — 110 220 330 440 42 16.7

0.53 0.80 0.30 0.33 0.34 0.35 0.41 1.26 0.94 0.99 0.78 0.21 0.20 0.20 0.22 1.20 0.55

Prentice et al., 1993 Masi and Addeo, 1986 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Prentice et al., 1993 Gwartney et al., 2002 Gwartney et al., 2002 Masi and Addeo, 1986 Prentice et al., 1993 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Bertola et al., 1992 Jaros and Rohm, 1997

Ref.

Data are from uniaxial compression test unless indicated otherwise by a remark next to the variety name. Values in this column are already multiplied by a factor of 1000 to avoid small numbers.

The constant strain rate compression has also been applied to biopolymer gels (Rohm, et al., 1995). Prentice (1992) discussed main issues related to the measurement and data interpretation. Prentice (1992)* also listed the mechanical properties for a number of cheeses, which are included in Tables 3.2 to 3.5. These properties are zero-strain modulus, Eo, which we can take as modulus of deformability, peak stress, and peak strain, which we can identify, as often done by many authors, as fracture stress and fracture strain. The zero-strain modulus is based on the following expression: σ = Eo ε − c ε 2

(3.3)

Where, σ is the true stress, ε is the Hencky strain below some critical level, and c is a constant. * The mechanical properties presented in Table 8.1 of Prentice (1992) were, however, different than those reported by the same author and coworkers in a later publication (Prentice et al., 1993). Since our calculations based on data from the few original papers agreed better with the results of Prentice et al. (1993), these values are included in our Tables 3.2 to 3.5.

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TABLE 3.4 Values of Stress at Fracture from Uniaxial Testing of Several Cheesesa

Cheese Variety

Initial Strain Rate s–1b

Peak Stress kPa

American Light — Shear American — Shear Appenzell Arzua-Ulloa Spanish soft cheese — type I Arzua-Ulloa Spanish soft cheese — type II Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Caciocavallo Caerphilly Cheddar Cheddar Cheddar Cheddar Cheddar Cheddar Cheddar 32.5% fat, 225 d old Cheddar 6.3% fat, 225 d old Cheddar 64 weeks old Cheddar 8 weeks old Cheddar 20-d-old Cheddar 20-d-old Cheddar 20-d-old Cheddar 20-d-old Cheddar 20-d-old Cheddar 20-d-old Cheddar 20-d-old Cheddar 20-d-old Cheddar 20-d-old Cheddar 20-d-old Cheddar 20-d-old Cheddar 20-d-old Cheddar 5-mo-old, full-fat: 32% Cheddar 5-mo-old, low-fat: 5% Cheshire Danbo Double Gloucester Edam Emmental Emmental 16 weeks Emmental 16 weeks Emmental 16 weeks

261 261 — 46 46 110 220 330 440 4.2 — — — — — 11.1 110 29 29 32 32 1.4 2.2 3.7 5.6 7.2 11 14.5 22.2 29.2 44.6 72.9 111 440 440 — 833 — — — 16.7 16.7 16.7

38 29 7 39 220 20 22 23 22 383 64 8 44 23 108 44 59 220 740 30 60 39 45 50 57 50 68 59 72 62 82 75 100 94 525 44 94 94 146 12 165 126 91

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Ref. Gwartney et al., 2002 Gwartney et al., 2002 Prentice et al., 1993 Almena et al., 1998 Almena et al., 1998 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Masi and Addeo, 1986 Prentice et al., 1993 Prentice et al., 1993 Prentice et al., 1993 Prentice et al., 1993 Prentice et al., 1993 Rosenau et al., 1978 Rosenau et al., 1978 Fenelon and Guinee, 2000 Fenelon and Guinee, 2000 Hort and Le Grys, 2001 Hort and Le Grys 2001 Ak and Gunasekaran 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Kucukoner et al., 1998 Kucukoner et al., 1998 Prentice et al., 1993 Madsen and Ardö, 2001 Prentice et al., 1993 Prentice et al., 1993 Prentice et al., 1993 Rohm et al., 1996 Rohm et al., 1996 Rohm et al., 1996

TABLE 3.4 (continued) Values of Stress at Fracture from Uniaxial Testing of Several Cheesesa

Cheese Variety

Initial Strain Rate s–1b

Peak Stress kPa

Ref.

Emmental 112-d-old Emmental 28-d-old Emmental 70-d-old Emmental 7-day-old Emmentaler 4 months Emmentaler 4 months Emmentaler 4 months Emmentaler 4-mo-old Garrotxa-type goat milk cheese Gouda Gouda Gouda 1 week, pH = 4.94 Gouda 1 week, pH = 4.94 Gouda 1 week, pH = 4.94 Gouda 2 week, pH = 5.24 Gouda 2 week, pH = 5.24 Gouda 2 week, pH = 5.24 Gouda 2 week, pH = 5.24 Gouda 6 months Gouda 6 months Gouda 6 months Gouda 6 months Gruyère Gruyère-type strong cohesion Gruyère-type strong cohesion — Tension Gruyère-type weak cohesion Gruyère-type weak cohesion — Tension Lancashire Leicester Leicester Mild Cheddar 37 days old Mild Cheddar 182 days old Mild Cheddar — A Reduced Fat — Shear Mild Cheddar — A — Shear Mild Cheddar — B Light — Shear Mild Cheddar — B — Shear Montasio Monterey Jack 46 days old Monterey Jack 185 days old Monterey Jack Light — Shear Monterey Jack — Shear Mozzarella Mozzarella

16.7 16.7 16.7 16.7 4.8 19 76.2 19 500 — — 2.8 28 140 0.67 2.7 50 250 0.28 2.8 28 170 — 55.5 11.1 55.5 11.1 — — — 8.3–24 8.3–24 261 261 261 261 — 8.3–24 8.3–24 261 261 — 4.2

170 265 205 280 78 95 121 96 65 69 68 27 41 66 35 55 108 182 42 64 82 108 15 1338 70 909 32 87 48 50 170 140 43 37 50 33 125 110 80 50 25 3 147

Jaros et al., 1997 Jaros et al., 1997 Jaros et al., 1997 Jaros et al., 1997 Rohm and Lederer, 1992 Rohm and Lederer, 1992 Rohm and Lederer, 1992 Rohm et al., 1992 Saldo et al., 2000 Prentice et al., 1993 Prentice et al., 1993 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Luyten, 1988 Prentice et al., 1993 Pesenti and Luginbühl, 1999 Pesenti and Luginbühl, 1999 Pesenti and Luginbühl, 1999 Pesenti and Luginbühl, 1999 Prentice et al., 1993 Prentice et al., 1993 Prentice et al., 1993 Charalambides et al., 1995 Charalambides et al., 1995 Gwartney et al., 2002 Gwartney et al., 2002 Gwartney et al., 2002 Gwartney et al., 2002 Prentice et al., 1993 Charalambides et al., 1995 Charalambides et al., 1995 Gwartney et al., 2002 Gwartney et al., 2002 Prentice et al., 1993 Masi and Addeo, 1986

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TABLE 3.4 (continued) Values of Stress at Fracture from Uniaxial Testing of Several Cheesesa

Cheese Variety Münster Parmesan Parmigiano Reggiano 12 months old Parmigiano Reggiano 18 months old Parmigiano Reggiano 28 months old Pecorino Romano Processed American Processed American Processed cheese analogs high fat, high moisture Processed cheese analogs high fat, high moisture Processed cheese analogs high fat, high moisture Processed cheese analogs low fat, low moisture Processed cheese analogs low fat, low moisture Processed cheese analogs low fat, low moisture Provolone Provolone Red Brick UF-Feta 8–10 weeks Red Brick UF-Feta 8–10 weeks Red Brick UF-Feta 8–10 weeks Red Brick UF-Feta 8–10 weeks Sbrinz Sharp Cheddar 1 month old Sharp Cheddar 6 months old Sharp Cheddar Light — Shear Sharp Cheddar — Shear Silano Swiss Appenzeller type — rapeseed diet Swiss Appenzeller type — regular diet Tilsit Tin UF-Feta 8–10 weeks Tin UF-Feta 8–10 weeks Tin UF-Feta 8–10 weeks Tin UF-Feta 8–10 weeks Vorarlberger Bergkase a b

Initial Strain Rate s–1b

Peak Stress kPa

— — 42 42 42 — 11.1 110 5.56

3 112 211 199 186 187 20 31 48

Prentice et al., 1993 Prentice et al., 1993 Noel et al., 1996 Noel et al., 1996 Noel et al., 1996 Prentice et al., 1993 Rosenau et al., 1978 Rosenau et al., 1978 Marshall, 1990

55.6

79

Marshall, 1990

556

102

Marshall, 1990

5.56

368

Marshall, 1990

55.6

424

Marshall, 1990

556

453

Marshall,1990

— 4.2 110 220 330 440 — 8.3–24 8.3–24 261 261 4.2 17 17 — 110 220 330 440 17

57 314 25 27 27 27 22 120 155 51 44 157 104 135 7 33 39 40 46 123

Prentice et al., 1993 Masi and Addeo, 1986 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Prentice et al., 1993 Charalambides et al., 1995 Charalambides et al., 1995 Gwartney et al., 2002 Gwartney et al., 2002 Masi and Addeo, 1986 Jaros et al., 2001 Jaros et al., 2001 Prentice et al., 1993 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Jaros and Rohm, 1997

Ref.

Data are from uniaxial compression test unless indicated otherwise by a remark next to the variety name. Values in this column are already multiplied by a factor of 1000 to avoid small numbers.

© 2003 by CRC Press LLC

TABLE 3.5 Values of Work-to-Fracture from Uniaxial Testing of Several Cheesesa

Cheese Variety Arzua-Ulloa Spanish soft cheese — type I Arzua-Ulloa Spanish soft cheese — type II Blue Brick UF-Feta Blue Brick UF-Feta Blue Brick UF-Feta Blue Brick UF-Feta Caciocavallo Cheddar 8 weeks old Cheddar 64 weeks old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Cheddar cheese 20-d-old Danbo Garrotxa-type goat milk cheese Gruyere-type strong cohesion Gruyere-type strong cohesion — Tension Gruyere-type weak cohesion Gruyere-type weak cohesion — Tension Mozzarella Parmigiano Reggiano 12 months old Parmigiano Reggiano 18 months old Parmigiano Reggiano 28 months old Provolone Red Brick UF-Feta Red Brick UF-Feta Red Brick UF-Feta Red Brick UF-Feta Silano Tin UF-Feta Tin UF-Feta Tin UF-Feta Tin UF-Feta a b c

Initial Strain Rate s–1b

Toughness kJ/m3c

46 46

7.1 31.7

Almena et al., 1998 Almena et al., 1998

110 220 330 440 4.2 32 32 1.4 2.2 3.7 5.6 7.2 11 14.5 22.2 29.2 44.6 72.9 111 833 500 55.5 11.1 55.5 11.1 4.2 42 42 42 4.2 110 220 330 440 4.2 110 220 330 440

4.4 5.2 5.5 5.5 8 42.3 7 23 27 26 32 34 42 30 35 40 47 51 61 61 6.5 210 57 152 8 2 30.6 23.8 18.4 6.4 5.5 6.4 6.7 6.7 3.3 5 5.4 5.5 6.2

Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist,1997 Wium and Qvist, 1997 Masi and Addeo, 1986 Hort and Le Grys, 2001 Hort and Le Grys, 2001 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Ak and Gunasekaran, 1992 Madsen and Ardö, 2001 Saldo et al., 2000 Pesenti and Luginbühl, 1999 Pesenti and Luginbühl, 1999 Pesenti and Luginbühl, 1999 Pesenti and Luginbühl, 1999 Masi and Addeo, 1986 Noel et al., 1996 Noel et al., 1996 Noel et al., 1996 Masi and Addeo, 1986 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Masi and Addeo, 1986 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997

Ref.

Data are from uniaxial compression test unless indicated otherwise by a remark next to the variety name. Values in this column are already multiplied by a factor of 1000 to avoid small numbers. Obtained from area under the stress–strain curve up to peak location.

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TABLE 3.6 Comparison of Modulus of Deformability Values Obtained Through Different Procedures for Three Cheeses at 20°C Procedure

Standard Cheddar (18-week-old)

Granular Cheddar (34-week-old)

Standard Mozzarella (7-week-old)

A B C D

1210 (10)a 1150 (11) 934 (13) 996 (13)

958 (8) 946 (10) 694 (3) 685 (4)

144 (21) 133 (27) 115 (21) 93 (22)

A: Gradient at inflection (Inf.) method. B: Gradient at strain range Inf./2 to 1.5 Inf. C: Gradient at strain range 0 to 0.04. D: Linear term of the polynomial fit (Equation 2.22, Chapter 2). a

Numbers in parentheses are coefficients of variation (100*Std.Dev./Mean).

Source: After Watkinson and Jackson, 1999.

As stated in Chapter 2, it is not sure at which stage and where the fracture actually starts in compression testing of cheese (Luyten et al., 1991b). Nevertheless, in the literature it is common to identify the maximum point on the true stress–Hencky strain curve as the fracture point in order to extract fracture stress and fracture strain data. Fracture properties determined by this practical method are likely to be related to the actual stress and strain at fracture. Luginbühl (1996) contested the use of true stress–Hencky strain curve for determination of fracture parameters. He stated that the true stress calculation at large deformation is based on incorrect assumptions. Two of these assumptions are the constancy of volume and the retention of cylindrical shape. He also stated that the shift of the apparent fracture strain towards smaller values is merely due to a mathematical effect. He advocated that the fracture parameters of hard and semihard cheeses should be determined from the coordinates of the apparent fracture point in the engineering stress–engineering strain curve. If need arises, Equations 2.7 and 2.12 can be used to convert fracture parameters from one set of coordinates to the other. Following the criticisms of Luginbühl (1996), Rohm et al. (1997) conducted compression tests on various foods, including Gouda cheese, where the contact surfaces were generously lubricated with paraffin oil. The compression tests were recorded, and the resulting video frames were analyzed using image-processing software. Their results showed that the shape of Gouda cheese specimen remained nearly cylindrical even at a relative deformation of 60% or Hencky strain of 0.92. Furthermore, for Gouda cheese, the true stress calculated by Equation 2.17, was shown to be in close agreement with the average stress based on the actual crosssectional area obtained from the video image analysis. In addition to the previous findings of other researchers (Culioli and Sherman, 1976; Calzada and Peleg, 1978; Luyten et al., 1991a), these results confirmed the assumptions of constant volume

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and shape retention during uniaxial compression with lubrication, at least for Gouda cheese. Taneya et al. (1992) calculated, using Equation 2.27, Poisson’s ratio of 0.493 for string cheese, signifying that it is practically incompressible, and thus the constant volume assumption is valid. Moreover, Brennan and Bourne (1994) and Charalambides et al. (2001) presented photographic evidence illustrating that cylindrical shapes in lubricated compression of mild Provolone, Gruyere, and Mozzarella cheeses were maintained. On the other hand, compression between molar teeth in the mouth followed the nonlubricated pattern (i.e., barreling), although some lubrication was provided by the saliva (Brennan and Bourne, 1994). For truly elastic solids or Hookean solids, the Young’s modulus will not be affected by the deformation rate. Cheese, however, is a viscoelastic material, and, consequently, the rate of loading or deformation is expected to have a significant impact on its rheological properties. Many researchers have actually observed the rate effect for a variety of cheeses (Shama and Sherman, 1973; Culioli and Sherman, 1976; Vernon-Carter and Sherman, 1978; Dickinson and Goulding, 1980). The results of Shama and Sherman (1973) on double Gloucester cheese clearly demonstrate the effect of deformation rate on cheese properties. The modulus of deformability and the stress at fracture are usually more influenced by the deformation rate than the strain at fracture. For instance, the fracture stress of 20-day-old Cheddar cheese almost doubled upon increasing the initial strain rate (i.e., crosshead speed/original specimen height) by fiftyfold during the lubricated compression; meanwhile, only a small change in the fracture strain was observed (Ak and Gunasekaran, 1992). In a similar fashion, the modulus of deformability for 4-month-old Emmentaler cheese increased by 1.7 times when the initial strain rate was increased by sixteenfold. For this change in the rate of deformation the fracture strain increased only by 1.07 times (Rohm and Lederer, 1992). A quick look at the data in Tables 3.2 to 3.5 will confirm the effect of deformation rate on the mechanical properties of various cheeses. Generally speaking, as the deformation rate of deformation is increased, the stress at fracture and the modulus of deformability increase, whereas the strain at fracture may increase, decrease, or remain unchanged. The common explanation for the rate effect is that at higher deformation rates the viscoelastic cheese has less time to relax some of the stress during the loading stage and therefore attains higher values of stress. In other words, Deborah number, which is the ratio of the characteristic time of the material to the time-scale of the experiment, determines the way a viscoelastic material responds to mechanical perturbations. The rate dependence of stress–strain relationship as a result of viscoelasticity of cheese can be represented by a power-law relationship: σ = Cε˙ n or ln σ = ln C + n ln ε˙

(3.4)

Where, C is a constant and n is the exponent signifying the relative contributions of viscous and elastic components. The n value is zero for an ideally elastic material (i.e., no viscous part) and increases with the extent of the viscous contribution. According to the analysis of Luyten et al. (1991a) the relation between n and the tan δ at the same time scale is given by:

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TABLE 3.7 Values for the Exponent n in Equation 3.4 for Different Types of Cheese

Cheese

Range of Initial Strain Rate s–1a

n

Ref.

Double Gloucester Gouda Cream cheese American Cheddar Cheddar Leicester Cheshire Cheddar Processed cheese Provolone Gouda-type 1 Cheddar Emmentaler Tin Feta Red Brick Feta Blue Brick Feta

20 fold 20–300 2–500 2–500 2–500 3–600 3–600 3–600 2–400 0.2–400 4–200 0.2–200 1.4–110 4.8–76 110–440 110–440 110–440

0.23 0.19 0.18 0.14 0.22 0.22 0.21 0.14 0.19 0.15 0.19 0.15–0.20 0.19 0.16 0.21 0.06 0.09

Luyten et al., (1991a) Luyten et al., (1991a) Luyten et al., (1991a) Luyten et al., (1991a) Luyten et al., (1991a) Luyten et al., (1991a) Luyten et al., (1991a) Luyten et al., (1991a) Luyten et al., (1991a) Luyten et al., (1991a) Luyten et al., (1991a) Luyten et al., (1991a) Ak and Gunasekaran, (1992) Rohm and Lederer, (1992) Wium and Qvist, (1997) Wium and Qvist, (1997) Wium and Qvist, (1997)

a

Values in this column are already multiplied with a factor of 1000 to avoid small numbers.

tan δ =

π n 2

(3.5)

This equation forms a bridge between nonlinear property n from the static experiments and the linear property tan δ from the dynamic experiments (see Chapter 5). The n values for a variety of cheeses are listed in Table 3.7. Luyten and van Vliet (1996) offered two mechanisms to explain the relation between fracture strain and rate of deformation or time-scale of deformation. The first mechanism is related to energy dissipation due to viscous flow. As the viscouslike character of a material increases with decreasing deformation rate (or increasing time-scale of deformation), the energy dissipation due to viscous flow becomes relatively more important (e.g., in cheese with high water content). Therefore, the amount of elastically-stored energy available for fracture at a given deformation (see Chapter 4) decreases so that fracture will not take place before the cheese is deformed further. Hence, the fracture strain will attain a larger value. Accordingly, a truly viscous liquid will not fracture at all, and a truly elastic solid will store all mechanical energy until fracture. The second mechanism is related to another dissipative effect in cheese, which is internal friction due to relative motion of different components. The energy dissipation through friction, however, increases with increasing rate of

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deformation. Thus, this mechanism lowers more of the energy available for fracture at higher deformation rates, resulting in a larger fracture strain. There may also be cheeses for which the fracture strain seems independent of the deformation rate due to the balance between these two mechanisms. The instrumental texture measurements often seek to obtain high correlations with the sensory evaluations, so that the sensory panel can be replaced with instrumental tests (Bourne, 1982; Peleg, 1983). The instrumental texture determinations are discussed in Chapter 7. It is usually hoped that when instrumental tests are made under conditions similar to those prevailing during sensory evaluations, one can obtain good correlations. There are, however, many theoretical and practical issues that need to be addressed in order to obtain satisfactory and accurate correlations (Peleg, 1983). In addition, poor correlations may sometimes stem from other reasons; for instance, the difference in fracture mechanisms of Cheddar cheese and Cheshire cheese in the mouth and in compression testing (Green et al., 1985). Often the deformation rates employed in instrumental measurements are smaller than the rates estimated during the sensory evaluations (e.g., chewing rate). This discrepancy has long been considered to exert a negative effect on the correlation between instrumental and sensory measurements. However, a recent study on UF-Feta cheese indicated that the correlation between oral firmness and stress at fracture was independent of deformation rate (Wium et al., 1997). Whether this is a rule or exception is yet to be established with other cheeses. For instance, the effects of deformation level and deformation rate on the maximum compression force, which is related to sensory hardness, were recently investigated for 26 commercial cheeses (Xiong et al., 2002). The highest correlation was obtained at a combination of compression level between 70 and 90% and deformation rate of 1.0 mm/s, which was not the highest rate used in the study (i.e., 10 mm/s). As for many other cheese varieties, the compression force increased with the initial strain rate in log-linear fashion (Figure 3.1).

Maximum compression force (N)

400 Deli-Provolone Farmer’s

300

Edam Gouda

200

100 0 0.01

0.1

1

Initial strain rate (1/s)

FIGURE 3.1 Relationship between maximum compression force and strain rate in uniaxial compression of various cheeses. (After Xiong et al., 2002.)

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TENSION

PARALLEL PERPENDICULAR

CHEESE BLOCK

COMPRESSION Parallel

Perpendicular

FIGURE 3.2 Sample preparations for parallel and perpendicular orientations.

STRUCTURE AND COMPOSITION EFFECTS When cylindrical samples of cheese with oriented fibrous structure are evaluated by uniaxial compression, according to Prentice et al. (1993), the values of Young’s modulus and maximum stress may be the same whether they are deformed along or across the direction of the internal fibers. Prentice et al. (1993) commented that during uniaxial compression of cheese cylinders the stresses are distributed radially throughout the sample such that stresses will be both along and across the direction of the internal fibers. Quantitative results for the effect of sampling direction on compression properties of Mozzarella cheese have been presented in Ak (1993). The schematic drawing of compression test is given in Figure 3.2, and the experimental conditions are given in Table 3.8. The deformability modulus ED of samples with perpendicular orientation was initially higher than those of parallel orientation. As the cheese matured, the difference in ED values first diminished at day 14, and later reversed (Figure 3.3). Examining the compression forces at three deformation levels (i.e., 25%, 50%, 75%), we see no definite trend emerging (Table 3.9) (Ak, 1993). Nevertheless, especially at 50% and 75% deformations, the force was usually higher when the fibers in the sample were perpendicular to the compression direction. This is consistent with the findings of Cervantes et al. (1983) with Mozzarella cheese, where force at 50%

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TABLE 3.8 Details of Experimental Conditions in Uniaxial Compression of Mozzarella Cheese with Different Sampling Directions Explanation Cheese Ambient temperature Specimen temperature

Crosshead speed Storage time Compression level Replication

Commercial low-moisture, part-skim Mozzarella. Tests were conducted at room temperature (~23°C). Specimens were thermally conditioned in an incubator at 15°C. This temperature was selected to reduce complications that may arise from melting of cheese fat. It was measured that the specimen temperature increased at most by 0.5°C during 40 s of testing It was set to 50 mm/min in order to reduce the time a specimen was exposed to ambient conditions during testing. Tests were made 7, 14, 21, and 28 days after the production date. Samples stored at refrigeration temperature 25, 50, and 75% compression level was applied separately on different days Sixteen specimens were tested for each orientation at each storage time

Source: After Ak, 1993.

170 Deformability modulus (kPa)

Parallel Perpendicular 150

130

110

90 7

14

21

28

Storage time (day)

FIGURE 3.3 Effect of aging on deformability modulus of cheese with parallel and perpendicular orientations. (After Ak, 1993.)

compression was used for comparison. On the other hand, Pompei et al. (1987) examined whether the anisotropy of Provolone cheese, due to stretching of hot curd, affected its rheological behavior. For that purpose they applied force in direction parallel and normal to the run of the macroscopic fibers. The hardness, energy, and cohesiveness values from the texture profile analysis (TPA, see Chapter 7) tests did not reveal sufficient evidence to claim the existence of anisotropic behavior in Provolone cheese of 5–6 mo old.

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TABLE 3.9 Force Values at Different Compression Levels for Samples Taken Parallel and Perpendicular to the Fiber Orientation in Mozzarella Cheese During Refrigerated Storage Mean Compressive Force (N) 25% Compression Cheese Age (day) 7 14 21 28 a

50% Compression

Parallel

Perpendicular

Sig. Levela

8.1 12.3 10.6 10.2

8.3 11.9 9.4 9.1

0.73 0.36 0.009 0.005

75% Compression

Parallel

Perpendicular

Sig. Level

Parallel

Perpendicular

Sig. Level

21.2 29.2 25.0 26.0

23.5 31.1 25.1 23.7

0.013 0.050 0.93 0.027

51.5 65.1 51.4 57.7

61.1 73.7 57.8 54.0

0.0001 0.003 0.015 0.124

Significance level

Source: After Ak, 1993.

TABLE 3.10 Effect of Sampling Direction on Compression Properties of Gruyère de Comté Cheese Rheological Parameter Modulus (kPa) Fracture stress (kPa) Fracture strain (%)

Axial Compression

Perpendicular Compression

Significance Level

367 184 43

429 142 38

** ** **

Note: ** = significant p<0.01; comparison of the means by t-test. Source: After Grappin et al., 1993. With permission.

For some cheeses (e.g., Gruyère de Comté, a Swiss-type, hard cheese), anisotropy in the rheological properties is of commercial importance. As an example, we cite the study of Grappin et al. (1993), where significantly different rheological properties were reported for samples taken according to the direction of pressing (axial) or perpendicular to it (Table 3.10). The authors established that the formation of slits favors large differences in stress and strain values at fracture for axial and perpendicular directions. They further established clearly that the difference in fracture strains between the two directions is chiefly responsible for the formation of slits — a quality defect that downgrades commercial value of Comté cheese (Grappin et al., 1993). It is known that rheological properties of cheeses vary with the composition (i.e., the amounts and states of constituents, namely casein, fat, water, salt, pH, etc.) and the degree of maturation. However, it may not be easy to establish sound relations between composition and rheological properties of the cheeses. The reason for this, © 2003 by CRC Press LLC

Penetration stress (kPa)

1600 y = 73 x − 1256 R2 = 0.76 1200

800

400

0 10

20

30

40

Protein content (%)

FIGURE 3.4 Relationship between penetration force and protein content of 11 different cheeses. (After Chen et al., 1979.)

as already acknowledged by many researchers (Creamer and Olson, 1982; Luyten et al., 1982; Marshall, 1990; Wium et al., 1998), is the difficulty of making cheese that differs only in one constituent (e.g., fat content), while keeping others (e.g., water content, salt content, pH, etc.) constant. In some cases, statistical methods are used to analyze the simultaneous variations in more than one variable. Rohm et al. (1992) evaluated chief compositional and maturation parameters affecting uniaxial compression properties of Swiss-type cheese (4-month-old Emmental) by multiple regression analysis. The mean values for rheological and fracture parameters of this work are presented in Tables 3.2 to 3.5. Casein is the main constituent in cheese that builds the structure and gives, when intact, the elastic or solid character. Based on the data of Chen et al. (1979), the relation between penetration stress (required pushing the plunger, D = 0.64 cm, into the sample for 1.91 cm) and protein content of different cheeses is shown in Figure 3.4. Based on the same study (excluding processed Cheddar cheese), Prentice (1992) concluded that about 25%* of the cheese by weight must be casein in order to provide a rigid framework. A similar procedure is applied to the compression data taken from Lee et al. (1978), and the resulting plot is given in Figure 3.5. It is seen that the compression stress at 80% deformation increases with the protein content of different cheeses (protein values taken from Bassette and Acosta, 1988). In the study by Chen et al. (1979) the penetration tests were made at 12.2 to 13.3°C, while