Modern Biopolymer Science: Bridging the Divide between Fundamental Treatise and Industrial Application

Industrialists developing new food and pharmaceutical products face the challenge of innovation in an increasingly competitive market that must consider incredient cost, product added-value, expectations of a healthy life-style, improved sensory impact, controlled delivery of active compounds and last, but not lease, product stability. While much work has been done to explore, understand, and address these issues, a gap has emerged between recent advances in fundamental knowledge and its direct application to product situations with a growing need for scientific input.Modern Biopolymer Science matches science to application by first acknowledging the differing viewpoints between those working with low-solids and those working with high-solids, and then sharing the expertise of those two camps under a unified framework of materials science.

• Real-world utilisation of fundamental science to achieve breakthroughs in product development
• Includes a wide range of related aspects of low and high-solids systems for foods and pharmaceuticals
• Covers more than bio-olymer science in foods by including biopolymer interactions with bioactive compounds, issues of importance in drug delivery and medicinal chemistry

• Language: English
• Publisher: Academic Press
• Release date: Jul 21, 2009
• ISBN: 9780080921143

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Table of Contents

Cover image

Copyright

Contributors

Preface

CHAPTER 1. Biopolymer Network Assembly

1.1. Biopolymer networks and gels

1.2. Rheological characterization of biopolymer gels

1.3. Theoretical aspects

1.4. Conclusions

CHAPTER 2. Gelation

2.1. Introduction

2.2. Modeling gel networks and their rheological behavior

2.3. Molecular mechanisms causing aggregation/gelation

2.4. Gel structure type

2.5. Gel texture: oral processing, rheology/fracture, microstructure and sensory analysis

2.6. Concluding remarks and future challenges

CHAPTER 3. Antifreeze Proteins

3.1. Antifreeze proteins

3.2. Afp properties

3.3. AFP Mechanism of function

3.4. Applications of AFP

CHAPTER 4. Biopolymers in Food Emulsions

4.1. Introduction

4.2. Emulsion science and technology terminology

4.3. Emulsion droplet characteristics

4.4. Production of food emulsions

4.5. Emulsion stability

4.6. Physicochemical properties of food emulsions

4.7. Biopolymer emulsifiers

4.8. Biopolymer texture modifiers

4.9. Conclusions

CHAPTER 5. Functional Interactions in Gelling Biopolymer Mixtures

5.1. Introduction

5.2. Applicability of polymer blending laws to biphasic networks

5.3. Phase composition

5.4. Blending law analyses of gelatin–calcium pectinate co-gels

5.5. Co-gelation of whey protein isolate (wpi) with crosslinked starch

5.6. Associative interactions

5.7. Segregative interactions in single-phase mixtures

5.8. Current understanding and future challenges

CHAPTER 6. Effect of Processing on Biopolymer Interactions

6.1. Introduction

6.2. Fluid/sheared gels

6.3. Water-in-water emulsions

6.4. Processing inside people

6.5. The future

CHAPTER 7. Unified Application of the Materials-Science Approach to the Structural Properties of Biopolymer Co-Gels throughout the Industrially Relevant Level of Solids

7.1. Introduction and overview of product development concerns that necessitated work in phase-separated biopolymer gels

7.2. Experimental methods of pinpointing phase-separation phenomena in mixed gels

7.3. Utilization of reaction kinetics to identify phase-separation phenomena in biopolymer mixtures

7.4. Quantitative analysis of the structural properties of binary composite gels

7.5. Bridging the divide between the low- and high-solid analyses in binary co-gels

7.6. Molecular dynamics of bioactive compounds in a high-solids carbohydrate matrix

7.7. Structural properties of non-aqueous systems used in controlled topical delivery

7.8. Concluding remarks

CHAPTER 8. Mapping the Different States of Food Components Using State Diagrams

8.1. Introduction

8.2. Glass transition

8.3. Glass formation

8.4. Determination of glass transition

8.5. Water plasticization and plasticizers

8.6. Glass transition and water activity

8.7. Mechanical properties and relaxations

8.8. Stiffness

8.9. Collapse phenomena

8.10. Stickiness and caking

8.11. Glass transitions in frozen foods

8.12. Crystallization and recrystallization

8.13. State diagrams and stability

CHAPTER 9. Structural Advances in the Understanding of Carbohydrate Glasses

9.1. Carbohydrate phase behavior in the prediction of food and pharmaceutical stability

9.2. Effects of water on the structure of carbohydrate glasses

9.3. Molecular packing in glassy carbohydrates

9.4. Structural aspects of the aging of carbohydrate glasses

9.5. Dynamic properties close to the glass transition

9.6. Technological implications

9.7. Conclusions and perspectives

CHAPTER 10. Biopolymer Films and Composite Coatings

10.1. Introduction

10.2. Mechanisms of film formation

10.3. Obtaining a well-matched coating

10.4. Film-application stages and methods for testing films

10.5. Selecting biopolymers for specific applications

10.6. Edible protective films

10.7. Novel products

10.8. Non-food gum coatings

10.9. Next generation of edible films

CHAPTER 11. Protein + Polysaccharide Coacervates and Complexes

11.1. Introduction

11.2. Historical background

11.3. Structures formed during protein + polysaccharide associative phase separation

11.4. Protein + Polysaccharide associative phase separation kinetics

11.5. Internal structure of coacervates and interpolymeric complexes

11.6. Parameters affecting protein + polysaccharide attractive electrostatic interaction

11.7. Functional properties and potential applications of protein +polysaccharide complexes and coacervates

11.8. Main limitations for the use of coacervates and complexes in food applications and encapsulation

11.9. Perspectives

CHAPTER 12. Single Molecule Techniques

12.1. Atomic force microscopy

12.2. Surface forces

12.3. Conclusions

CHAPTER 13. Dietary Fiber

13.1. Recent developments in dietary fiber research

13.2. Technological properties of dietary fiber

13.3. Dietary fiber products: Chemistry, functional properties and applications in foods

13.4. Concluding remarks

CHAPTER 14. Resistant Starch in Vitro and in Vivo

14.1. Introduction

14.2. Measurement of resistant starch

14.3. Health benefits of RS

14.4. Effect of processing on resistant starch formation in foods

14.5. Model studies of isolated starches

14.6. Molecular and microstructural organization of resistant starches

14.7. Concluding remarks

CHAPTER 15. Glycemic Response Reduction in Processed Food Products

15.1. Introduction

15.2. Processing and carbohydrate digestibility

15.3. The effect of extrusion parameters and processing on food quality

15.4. Manipulating the glycemic impact of extruded snack products

15.5. The link between slowly digestible and rapidly digestible carbohydrates and the glycemic impact of processed foods

15.6. Use of dietary fiber in manipulating starch digestibility

15.7. Conclusion

CHAPTER 16. Biopolymers in Controlled-Release Delivery Systems

16.1. Introduction

16.2. Drug loading and release

16.3. Modeling diffusion

16.4. Higuchian model

16.5. Swelling

16.6. Temperature-sensitive hydrogels

16.7. Equilibrium swelling and the Flory-Rehner theory

16.8. Approaches to cross-linking

16.9. Glutaraldehyde

16.10. Genipin

16.11. Quinones and phenols

16.12. Polyelectrolyte cross-linking and complexes

16.13. Polymer–drug interactions

16.14. Collagen

16.15. Gelatin

16.16. Chitin and chitosan

16.17. Celluloses

16.18. Alginates

16.19. Summary

CHAPTER 17. Amyloid Fibrils – Self-Assembling Proteins

17.1. Introduction to protein misfolding and fibril formation

17.2. Amyloid formation, nature and disease

17.3. Why is there such a great interest in amyloid fibrils?

17.4. Amyloid fibrils in nature

17.5. Protein folding and misfolding in the cell

17.6. Amyloid formation and biotechnology

17.7. Fibril formation pathways

17.8. Analytical techniques to study amyloid formation

17.9. Techniques for studying amyloid fibril formation

17.10. Detection of amyloid fibrils

17.11. Alternative models to the cross β structure

17.12. Fibril formation kinetics

17.13. Conditions that promote fibril formation

17.14. Taking lessons from nature

17.15. Nanotubes and nanowires

17.16. Fibrillar gels

17.17. Future innovations?

17.18. Conclusions

CHAPTER 18. Hydrocolloids and Medicinal Chemistry Applications

18.1. Drug delivery

18.2. Tissue engineering

18.3. Future horizons

Index

Copyright

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Contributors

Anthony R. Bird

Commonwealth Scientific and Industrial Research Organisation, Food Futures National Research Flagship, and CSIRO Human Nutrition, Adelaide, Australia

Charles Stephen Brennan

Hollings Faculty, Manchester Metropolitan University, Manchester, UK

Margaret Anne Brennan

Institute of Food, Nutrition and Human Health, Massey University, Palmerston North, New Zealand

Sarah L. Buckley

Highton, Australia

Allan H. Clark

Pharmaceutical Science Division, King‘s College London, London, UK

Phil W. Cox

School of Engineering-Chemical Engineering, University of Birmingham, Edgbaston, UK

Steve W. Cui

Guelph Research Food Centre, Agriculture and Agri-Food Canada, Guelph, Canada

David E. Dunstan

Chemical & Biomolecular Engineering, University of Melbourne, Victoria, Australia

E. Allen Foegeding

Department of Food Science, North Carolina State University, Raleigh, USA

Michael J. Gidley

Centre for Nutrition & Food Sciences, University of Queensland, Brisbane, Australia

Liam M. Grover

School of Chemical Engineering, University of Birmingham, Edgbaston, UK

Victoria A. Hughes

Chemical & Biomolecular Engineering, University of Melbourne, Victoria, Australia

Stefan Kasapis

School of Applied Sciences, RMIT University, Melbourne, Australia

Sandra I. Laneuville

Dairy Research Centre STELA and Institute of Nutraceutical and Functional Foods INAF, Laval University, Quebec, Canada

Peter J. Lillford

CNAP-Department of Biology, The University of York, York, UK

Erik van der Linden

Agrotechnology and Food Sciences Group, Wageningen University, Wageningen, The Netherlands

Amparo Lopez-Rubio

Australian Nuclear Science and Technology Organisation, Bragg Institute, Menai, Australia

David Julian McClements

Department of Food Science, University of Massachussets Amherst, Amherst, USA

Edwin R. Morris

Department of Food & Nutritional Sciences, University College Cork, Ireland

Vic J. Morris

Institute of Food Research, Colney, UK

Ian T. Norton

School of Engineering-Chemical Engineering, University of Birmingham, Edgbaston, UK

Amos Nussinovitch

Faculty of Agricultural, Food and Environmental Quality Sciences, The Hebrew University of Jerusalem, Rehovot, Israel

Kunal Pal

Department of Chemistry and Biology, Ryerson University, Toronto, Canada

Allan T. Paulson

Department of Chemistry and Biology, Ryerson University, Toronto, Canada

Keisha Roberts

Guelph Research Food Centre, Agriculture and Agri-Food Canada, Guelph, Canada

Yrjö H. Roos

Department of Food & Nutritional Sciences, University College Cork, Ireland

Simon B. Ross-Murphy

Pharmaceutical Science Division, King's College London, London, UK

Dérick Rousseau

School of Nutrition, Ryerson University, Toronto, Canada

Ashok K. Shrestha

Centre for Nutrition & Food Sciences, University of Queensland, St. Lucia, Australia

Alan M. Smith

School of Chemical Engineering, University of Birmingham, Edgbaston, UK

Fotios Spyropoulos

School of Engineering-Chemical Engineering, University of Birmingham, Edgbaston, UK

Sylvie L. Turgeon

Dairy Research Centre STELA and Institute of Nutraceutical and Functional Foods INAF, Laval University, Quebec, Canada

Johan B. Ubbink

Nestle Research Centre Switzerland, Savigny, Switzerland

Preface

It has been a while since a book was put together to address the issues of the physics and chemistry of biopolymers in industrial formulations, including concise treatments of the relation between biopolymer functionality and their conformation, structure, and interactions. In these intervening years, some materials and concepts came to prominence while other ones have changed in their appeal or application. As ever, the industrialist is faced with the challenge of innovation in an increasingly competitive market in terms of ingredient cost, product added-value, expectations of a healthy life-style, improved sensory impact, controlled delivery of bioactive compounds and, last but not least, product stability. Proteins, polysaccharides and their co-solutes remain the basic tools of achieving the required properties in product formulations, and much has been said about the apparent properties of these ingredients in relation to their practical use. There is also an ever increasing literature on the physicochemical behaviour of well-characterised biopolymer systems based on the molecular physics of glassy materials, the fundamentals of gelation, and component interactions in the bulk and at interfaces. It appears, however, that a gap has emerged between the recent advances in fundamental knowledge and the direct application to product situations with a growing need for scientific input.

The above statement does not detract from the pioneering work of the forefathers in the field who developed the origins of biopolymer science. For example, there is no question that the pioneering work on conformational transitions and gelation, the idea of phase separation into water in emulsions, the development of physicochemical understanding that lead to the concept of fluid gels and the application of the glass transition temperature to dehydrated and partially frozen biomaterials has resulted not only in academic progress but in several healthy and novel products in the market place. Thus the first phase of the scientific quest for developing comprehensive knowledge at both the theoretical and applied levels of functional properties in basic preparations and systems has largely been accomplished. It is clear, though, that the future lies in the utilization of this understanding in both established and novel foodstuffs, and non-food materials (e.g. pharmaceuticals) with their multifaceted challenges. A clear pathway for processing, preservation and innovation is developing which is particularly important if progress is to be made in the preparation of indulgent yet healthy foods which are stable, for example, in distribution and storage. This requires a multi-scale engineering approach in which material properties and microstructure, hence the product performance are designed by careful selection of ingredients and processes. Examples of this can be found in the pioneering work on fat replacement and the reliance on the phenomenon of glass transition to rationalise the structural stability and mouthfeel of a complex embodiment.

Within this context of matching science to application, one feels compelled to note that a dividing line has emerged, which is quite rigorous, with researchers in the structure-function relationships of biopolymers opting to address issues largely in either high or low-solid systems. This divide is becoming more and more pronounced, as scientists working in the high-solid regime are increasingly inspired by the apparently universal molecular physics of glassy materials, which may or may not consider much of the chemical detail at the vicinity of the glass transition temperature. By comparison, their colleagues working on low-solid systems are shifting their focus from the relatively universal structure-function relationships of biopolymers in solution to the much more specific ones involving multi-scale assembly, complexation and molecular interactions. Sharing the expertise of the two camps under the unified framework of the materials science approach is a prerequisite to ensuring fully functional solutions to contemporary needs, spanning the full range of relevant time-, length- and concentration scales. This effort may prove to be the beginning of a modernized biopolymer science that, one the one hand, utilizes and further develops fundamental insights from molecular physics and the advanced synthetic polymer research as a source of inspiration for contemporary bio-related applications. On the other hand, such modernized science should be able to forward novel concepts dealing with the specific and often intricate problems of biopolymer science, such as the strong tendency for macromolecular hydrogen bonding, thus serving as an inspiration for related polymer advances and industrial applications. Sincere thanks are due to all our friends and colleagues whose outstanding contributions within their specialized areas made this a very worthwhile undertaking.

Stefan Kasapis, Ian T. Norton and Johan B. Ubbink

CHAPTER 1. Biopolymer Network Assembly

Measurement and Theory

Allan H. Clark and Simon B. Ross-Murphy

King's College London, Franklin-Wilkins Building, 150 Stamford Street, London, UK

Abstract

This chapter describes modern methods, both experimental and theoretical for the measurement of biopolymer network self-assembly from the solution to the gel state. We concentrate on rheological methods which embrace the analysis of kinetic ‘cure’ curves for the growth of gel modulus with respect to time, especially those involving oscillatory shear (‘mechanical spectroscopy’).

The theoretical methods include discussion of combinatoric, branching (‘cascade’) theory, and other modern methods involving percolation and fractal geometry analyses. We continue to emphasize the advantage of collecting data in a consistent and unified manner and extracting the most sensitive parameters, including the gelation time, t gel and the time and frequency independent gel modulus, G inf. Indeed we stress that the detailed analyses discussed can only be carried out reliably when experimental parameters such as these are obtained.

A number of biopolymer systems can self-assemble to form networks and gels and the assembly can occur by a variety of mechanisms. In this chapter we consider the nature of biopolymer gels and networks, the kinetics of assembly, and their characterization by rheological methods. The necessary theory to explain, for example, the complexities of gelation kinetics is then described in some detail. Before reaching this, we discuss the nature of network assembly, and the character of gels and their gelation.

1.1. Biopolymer networks and gels

1.1.1. Gels Versus Thickeners

1.1.1.1. What is a Polymer Network?

Polymer networks are molecular-based systems, whose network structure depends upon covalent or non-covalent interactions between macromolecules. The interactions can be simple covalent cross-links, or more complex junction zone or particulate-type interactions. Figure 1.1 illustrates different types of polymer network. Solvent swollen polymer networks are commonly known as gels – un-swollen networks are important for synthetic polymer systems, but are less relevant for biopolymers. Here, where the solvent is water or electrolyte, we can also introduce the term ‘hydrogel’.

1.1.1.2. What is a Gel?

We have already defined a gel above as a swollen polymer network, but unfortunately, one of the major issues in chapters such as the present one is that the term ‘gel’ means very different things to different audiences. In this respect, the widely cited 1926 definition by Dorothy Jordan Lloyd, that ‘the colloidal condition, the gel, is one which is easier to recognize than to define’ (Jordan Lloyd, 1926) is quite unhelpful, since it implies that a gel is whatever the observer thinks it is. Consequently we commonly see such products described as shower gels and pain release or topical gels.

Neither of these classes of systems follows a rheological definition such as that of the late John Ferry, in his classic monograph (Ferry, 1980). He suggests that a gel is a swollen polymeric system showing no steady-state flow; in other words if subjected to simple steady shear deformation it will fracture or rupture. Clearly neither shower nor topical gels follows this rule; indeed if they did, they would not be useful as products. In fact, commercial shower gels, for example, are simply highly viscous fluids formed by the entanglement of (often rod-like) micelles. For more rigorous definitions, at this stage it is necessary to introduce some common terminology.

Most modern rheological experiments on gelation (see below) employ oscillatory shear. In the simplest form of this, a small sinusoidal strain wave of frequency ω (typically 10 −3–10 s −1) is applied to the top surface of a gelling system (most likely constrained between parallel metal discs) and the resultant stress transmitted through the sample is measured. In general the stress and strain waves differ in both phase and amplitude, but using phase resolution, it is easy to extract the in-phase and 90 o out-of-phase components. Then G′ is the storage modulus given as the ratio of in-phase stress divided by strain, and G″ is the loss modulus, the ratio of 90 o out-of-phase stress to strain. There are other relationships between these and common experimentally determined parameters, as we describe later, but for now we are interested only in the storage – sometimes called elastic component – of the modulus, G′. For a perfect, so-called Hookean elastic material, such as a steel rod, G′ is effectively independent of the oscillatory frequency. The constancy of G′ with respect to frequency is then a useful definition of a solid.

One rheological definition of a gel is therefore a system that shows ‘a plateau in the real part of the complex modulus’ – G′ – ‘extending over an appreciable window of frequencies … they are … viscoelastic solids’ (Burchard and Ross-Murphy, 1990). A slightly later definition accepts this, but extends it and the Ferry definition by identifying a gel as a soft, solid or solid-like material, which consists of two or more components, one of which is a liquid, present in substantial quantity (Almdal et al., 1993). They therefore follow Ferry in accepting substantially swollen polymer networks as gels. However, according to them, a gel must also show a flat mechanical spectrum in an oscillatory shear experiment. In other words it should show a value of G′ which exhibits a pronounced plateau extending to times of the order of seconds, and a G′′ which is considerably smaller than the storage modulus in this region.

1.1.1.3. ‘Viscosifiers’

One of the problems in this area follows directly from the overuse of the term gel – as we outlined above, many viscous fluids are also described as gels or hydrogels. These include biopolymer solutions, whose properties are determined all but exclusively by entanglements of long chains, in this area typically represented by solutions of the galactomannan guar. These are analogous to solutions of common synthetic polymers in organic solvents, where entanglements involve reptation of chains (Doi and Edwards, 1986). Rheologically there are also a number of so-called structured liquids – which can suspend particles and appear solid-like – typically formed from liquid crystalline polymers or micellar solutions – and usefully exemplified in the present context by ordered solutions of the microbial polysaccharide xanthan (Richardson and Ross-Murphy, 1987b). To confuse matters, these have been referred to, in the past, including by one of the present authors as ‘weak gels’ (Ross-Murphy and Shatwell, 1993). We now reject this term totally, both because of its anthropomorphic connotation, and for its lack of precision – since they can show steady-state flow – in terms of the Ferry definition above.

1.1.1.4. Viscoelastic Solids vs. Viscoelastic Liquids

What then is the main difference between solids and liquids? It is the existence of an equilibrium modulus, i.e. a finite value of G′ even as the time of measurement becomes very long (or the oscillatory frequency tends to zero), usually referred to simply as the equilibrium shear modulus G. This means that a gel has (at least one) infinite relaxation time. Of course such a definition is partly philosophical, since given infinite time, all systems show flow, and in any case, most biopolymer gels will tend to degrade, not least by microbial action. However, this remains an important distinction, and in subsequent pages we regard biopolymer networks and gels as viscoelastic solids, and non-gelled systems, included pre-gelled solutions, ‘sols’, as viscoelastic liquids.

1.1.2. Brief History of Gels

1.1.2.1. Flory Types 1–4

Historically the term gel follows from the Latin gelatus ‘frozen, immobile’, and gelatin, produced by partial hydrolysis of collagen from, e.g. pigs, cattle or fish was probably recognized by early man. Gelatin has certainly been used in photography for almost 150 years, although this is, of course, a shrinking market.

In 1974, Flory (Flory, 1974) proposed a classification of gels based on the following:

1. Well-ordered lamellar structure, including gel mesophases.

2. Covalent polymeric networks; completely disordered.

3. Polymer networks formed through physical aggregation, predominantly disordered, but with regions of local order.

4. Particular, disordered structures.

In the present chapter, although we will not discuss specific systems in much depth, type 3 gels are represented by ‘cold set’ gelatins, and type 4 gels are represented by denatured protein systems. Type 2 systems are archetypal polymer gels. These are made up, at least formally, by cross-linking simpler linear polymers into networks, and their mechanical properties, such as elasticity, reflect this macroscopic structure.

1.1.2.2. Structural Implications

The structural implications of the above should be clear – gels will be formed whenever a super-molecular structure is formed, and Figure 1.1 illustrates the underlying organization of type 2, 3 and 4 gels. Of course this is highly idealized; for example if the solvent is ‘poor’, gel collapse is seen. Examples of each of these classes include the rubber-like arterial protein elastin – type 2; many of the gels formed from marine-sourced polysaccharides such as the carrageenans and alginates, as well as gelatin, type 3; and the globular protein gels formed by heating and/or changing pH, without substantial unfolding, type 4.

Of course, Figure 1.1 is highly idealized and the nature of network strands can vary substantially. For example, for the polysaccharide gels, such as the carrageenans, the classic Rees model of partial double helix formation (Morris et al., 1980) has been challenged by both small-angle X-ray scattering (SAXS) and atomic force microscopy (AFM) measurements, and it now seems likely that aggregation of junction zones and intertwining of pre-formed fibrils are additional contributory factors. This is certainly an on-going controversy, but one outside the remit of this chapter, except for its implications for the kinetic processes occurring during gelation. There are similar variations for protein gels too. When heated close to the isoelectric point, a coarse and random coagulate network is commonly formed but heating many globular proteins above their unfolding temperatures under acid conditions – say at pH 2 – results in fibrillar structures (Stading et al., 1992) that, at least at the nano-length scale, resemble the amyloid structures seen in a number of critical diseases such as Alzheimer's (Gosal, 2002, Gosal et al., 2002 and Dobson, 2003). This is now a very active area of research, but the subject of a separate chapter in this volume (Hughes and Dunstan, 2009).

1.2. Rheological characterization of biopolymer gels

1.2.1. Traditional Methods for Gel Characterization

A number of more traditional techniques have been used for gel measurements. They often have a major advantage in their low cost, compared to commercial apparatus. On the debit side, the actual strain deformation is sometimes unknown or, at best, requires calibration. Nowadays these approaches are less commonly employed, as almost all labs possess at least one oscillating rheometer, but they still have some advantages – not least from the financial viewpoint.

1.2.1.1. Falling Ball

This is one of the simplest and cheapest methods but, given a few precautions, it can still prove useful. In its simplest form, a magnet is used to raise a small metal sphere within a tube containing gelling material, and then the time taken to fall a fixed distance is registered (Richardson and Ross-Murphy, 1981). Clearly as gelation proceeds from the sol state, the rate of fall decreases, and eventually the sphere does not move any more. For low modulus systems there are potential problems since the sphere may locally rupture the gel and cut a channel through it – so-called ‘tunneling’ – and in this limit the method is more akin to a large deformation or failure method. The converse method of monitoring the fall of a sphere above a melting gel (or a series of such samples at different concentrations) is very commonly used to determine ‘melting temperatures’ (Eldridge and Ferry, 1954 and Takahashi, 1972), but again care must be taken to ensure that true melting is involved rather than localized pre-melt tunneling.

1.2.1.2. Oscillatory Microsphere

The microsphere rheometer is just the oscillatory analogue of a falling ball system. A small magnetic sphere is placed into the sample and using external AC and DC coils, the sphere can be positioned and made to oscillate with the frequency of the AC supply. The maximum deformation can be observed with a traveling microscope, or alternatively tracked, for example, using a position-sensitive detector array. A number of different designs have been published and used for measurements on systems including agarose and gelatin gels, and mucous glycoproteins (King, 1979 and Adam et al., 1984). The major limitation is that the measurement is very localized, so that again for some systems local rupture and tunneling can occur and then the modulus determined may not be representative of the whole system.

1.2.1.3. U-tube Rheometer

In this very simple assembly, originally designed by Ward and Saunders in the early 1950s for work on gelatin, the gel is allowed to set in a simple U-tube manometer, one arm of which is attached to an air line of known pressure, the other free to the air. Both may be observed with a traveling microscope. The air pressure exerts a compression stress in the sample (stress and pressure both have units of force/area), and the deformation of the sample can be measured from the differential heights of the manometer arms. The static (equilibrium Young's modulus) can be calculated directly using the analogue of Poiseuille's equation for capillary flow (Arenaz and Lozano, 1998).

As well as cheapness, this apparatus has the advantage that it becomes more sensitive for low modulus systems, since the deformation observed will be larger. However, in view of this, great care must be taken that the deformation induced is still in the linear region. The method has recently been extended for use with gels which synerese, by roughening the inner glass surfaces and by using an oscillatory set up (Arenaz et al., 1998; Xu and Raphaelides, 2005).

1.2.2. Modern Experimental Methods Employing Oscillatory Shear

Nowadays the vast majority of physical measurements on gels are made using oscillatory shear rheometry (Ferry, 1980, Ross-Murphy, 1994 and Kavanagh and Ross-Murphy, 1998). This is because rheometers are far cheaper and ‘user friendly’ than used to be the case. However, by the same token, some published data are poor and, just as seriously, the degree of understanding does not always appear to have kept pace with the rate of data collection. One of the major objectives of succeeding sections is to try to modify this situation.

The essential features of a typical rheometer for studying biopolymer systems consists of a vertically mounted motor (which can drive either steadily in one direction or can oscillate). In a controlled stress machine, this is usually attached to the upper fixture. A stress is produced, for example by applying a computer-generated voltage to a DC motor, and the strain induced in the sample can be measured using an optical encoder or radial position transducers attached to the driven member. In a controlled strain instrument, a position-controlled motor, which can be driven from above or below, is attached to one fixture, and opposed to this is a transducer housing with torque and in some cases, normal force transducers. Figure 1.2 represents a typical controlled stress instrument. The sample geometry can be changed from, e.g. Couette, to cone/plate and disc/plate, and the sample temperature controlled. Such a general description covers most of the commercial constant strain rate instruments (e.g. those produced under the names of TA Instruments, ARES series) and controlled stress rheometers (e.g. Malvern Bohlin, TA Instruments Carrimed, Rheologica, Anton Paar). In recent years the latter have begun to dominate the market, since they are intrinsically cheaper to construct, and they can provide good specifications at lower cost. Most claim to be usable in a servo-controlled (feedback) controlled strain mode, and are widely used in this mode. However, there are limitations here, as discussed in detail below.

Controlled stress instruments are ideal for time domain experiments, i.e. measuring creep, whereby a small fixed stress is applied to a gelled sample and the strain (‘creep’) is monitored over time (Higgs and Ross-Murphy, 1990). The time domain constant strain analogue of the creep experiment is stress relaxation. In this, a fixed deformation is quickly applied to the sample and then held constant. The decrease in induced stress with time is monitored. Few such measurements have been discussed for biopolymer systems and nowadays practically all modern instruments appear to be used predominantly in the oscillatory mode.

1.2.2.1. Mechanical Spectroscopy

We have already introduced the storage and loss moduli, G′ and G″, but there are a number of other commonly used rheological parameters, and all are interrelated (Ferry, 1980 and Ross-Murphy, 1994).

For example, G ∗, the complex modulus is given by:

(1.1)

and the ratio:

(1.2)

In the early days of oscillatory rheometry the phase angle, δ, was an experimentally observed parameter; nowadays instruments tend to hide the experimental measurables, the phase angle and the amplitude ratio, from the user.

Finally the complex viscosity, η ∗, is given by:

(1.3)

with ω the oscillatory shear (radial) frequency; here ω is just 2π x the frequency in Hertz. Of course, oscillatory measurements can also be made in tension/compression, leading to alternative parameters, such as E′ and E″, etc. However, for biopolymer gels and networks, this is relatively uncommon, and so we do not discuss these further.

1.2.2.1.1. Controlled Strain Versus Controlled Stress

We mentioned above that the majority of modern instruments are now of the controlled stress type. However most usually still generate results in the controlled strain form, that is as the modulus components, G′ and G″. Strictly speaking, since stress is applied and the strain is measured, then results should be reported as the components of complex compliance J′ and J″. However, most of the instruments circumvent this by applying a stress, measuring the strain, but in a servo- or feedback mode, so that it appears that they are indeed controlling the strain. For many applications and systems this is acceptable, but for systems very close to gelation, it is certainly not ideal. This is because there is no sure way of controlling the feedback when the system just changes from solution (sol) to gel, and yet at the same time guaranteeing that the strain remains very low. For such systems there is a further advantage in a genuine controlled strain technique, in that the mechanical driving head and the measurement transducer are completely separate assemblies – the only link between them is the test sample and geometry.

1.2.2.1.2. Time Independent Systems

Below we describe a typical experimental regime to collect the data in a form that is appropriate for an exploration of the kinetic assembly of biopolymer networks. However, since the overall outcome usually involves the conversion of a biopolymer solution (sol) to a viscoelastic solid (gel) it is useful to first understand the so-called mechanical spectra of these two systems, and their dependence on the experimental variables of oscillatory frequency, shear strain deformation (or shear stress, bearing in mind the caveats above) and temperature.

1.2.2.2. Frequency and Strain Dependence

1.2.2.2.1. Biopolymer Solutions

The mechanical spectrum of a liquid has the general form illustrated in Figure 1.3. At low frequencies (note the double log scale) G″ is greater than G′ but as the oscillatory frequency increases, G′ increases more rapidly than G″ (with a slope ~ 2 in the log–log representation, compared to a slope of 1 for G″) and at some frequency there is a ‘cross-over’. After this both G′ and G″ become much less frequency-dependent – we enter the so-called rubbery plateau region.

Whether or not the cross-over region is reached in the frequency window of conventional oscillatory measurements depends upon the biopolymer concentration, relative molecular mass (MW), and chain flexibility. For example for a typical high MW viscosifier such as guar, the G″– G′ cross-over may occur for concentrations of say 2–3% w/w (Richardson and Ross-Murphy, 1987a), whereas for a more flexible and lower MW biopolymer such as gelatin above its gel melting temperature, the concentration required may be above 25% w/w, and therefore essentially outside the experimentally interesting range.

At the same time, the mechanical spectrum measured will be essentially independent of the amount of shear strain, out to say 100% ‘strain units’ (i.e. a strain, in terms of the geometry of deformation, of 1). Rheologists may express this by saying that the linear viscoelastic (LV) strain extends out to ca. 100%.

1.2.2.2.2. Biopolymer Gels

The mechanical spectrum of a viscoelastic solid will, as we already mentioned in the discussion of the equilibrium modulus, have a finite G′, with a value usually well above (say 5–50 x) that of G″, at all frequencies, as illustrated in Figure 1.4 (Clark and Ross-Murphy, 1987; te Nijenhuis, 1997; Kavanagh et al., 1998; Kavanagh, 1998). In this respect it shows some similarities with the plateau region of the solution mentioned above – such a plateau has been referred to, somewhat imprecisely, as gel-like, for exactly this reason.

The strain-dependent behavior for biopolymer gels is more difficult to generalize, although the LV strain is rarely as great as 100% (some gelatin gels may be the exception here), and may be extremely low – say 0.1% as less. At values just greater than the LV strain, G′ and G″ may show an apparent increase with strain. This is, of course, largely an artefact of the experiment, since G′ and G″ are only defined within the LV region. This is then followed by a dramatic decrease, caused by failure – either by rupture or fracture, sometimes macroscopic – as often failure occurs at the geometry interface, especially if measuring in a disc plate (parallel plate) configuration.

1.2.2.3. Temperature Dependence

In this chapter we are not particularly interested in the temperature dependence of time-independent systems, since we are essentially concerned with the processes of self-assembly. However, in the study of synthetic polymer solutions and melts, this is of course of great importance. Again, although it has little to do with the formation of gel networks, many biopolymer gels do show so-called ‘glassy’ behavior at high enough frequencies or low enough temperatures, and the study of gels under these conditions, perhaps induced by measuring in highly viscous low MW solvents such as saturated sucrose, is a very active area of interest. This is discussed in further detail elsewhere in this book.

What the above does suggest, of course, is the well-known effect in polymer materials science, that high frequencies and low temperatures may be regarded as equivalent. This is the basis of the principle of time–temperature superposition (TTS). This is applied, for example, in the characterization of low-water gels, as mentioned above. Very often it works well, but caution should always be applied. The glass transition itself is related to polymer free volume, and temperature discontinuities in said free volume should make the approach invalid. If we are to follow the principles outlined by Ferry (Ferry, 1980) – one of the co-devisers of the method, and its strongest protagonist – then TTS should never be applied within 50°C of a phase transition within the system. For biopolymer gels, this should eliminate all TTS approaches from –50°C to 150°C – i.e. more than the whole regime of potential interest. In fact TTS can work well within this region, but should not be relied upon.

1.2.2.4. Time-Dependent Systems

1.2.2.4.1. The Kinetic Gelation Experiment

Clearly if we are, by some physical method (say heating), converting a biopolymer solution to a biopolymer gel, we will change the initial sol mechanical spectrum (Figure 1.3) to the gel spectrum (Figure 1.4). In a typical experiment, following the progress of gelation using mechanical spectroscopy, the oscillatory frequency is kept constant – and ca. 1Hz (6.28 rad s −1) – for convenience many workers use a frequency of 10 rad s −1 – and the strain is maintained constant and low – say typically 10% or less. The choice of frequency is always a compromise – we need a high enough value that a single frequency measurement does not take too long – so we can collect enough data – but not so high that instrumental artefacts begin to appear. In our experience these can be seen quite commonly for frequencies > say 30 rad s −1.

The temperature regime employed must also be carefully controlled, whether for heat-set, e.g. globular protein or cold-set, e.g. gelatin, gellan or carrageenan gels. A very common approach, not least because the instrument manufacturers supply it as an option, is to use a temperature ramp – say heating from 25°C to 75°C at 1°C per minute. The problem with this is, of course, that no serious study can be made of the kinetics of assembly, when the time-dependent assembly is convoluted with the change in temperature. Unfortunately many published data do employ such a heating ramp approach. Although an isothermal temperature profile can be difficult to achieve, modern Peltier heating systems are usually very fast to heat, cool and re-equilibrate. Originally these were only available on controlled stress instruments, but that limitation has now been overcome.

1.2.2.4.2. Gelation Time Measurement

Before considering the different approaches to the determination of say gelation time, we consider the expected self-assembly time profile. If we consider the equilibrium gel modulus, the ideal profile is seen in Figure 1.5a. Initially there is no response, but then G rises very rapidly, even on a log scale, at or just after the gelation time, before reaching a final asymptotic level, and the behavior illustrated is a simple consequence of the positive order kinetics of self-assembly (cross-linking) and the requirement for a minimum number of cross-links per ‘chain’ at the gel point. We note that some phenomenological models have neglected the pre-gel behavior, and simply fitted the G (>0) versus t behavior to an n-order kinetic model. From the data-fitting viewpoint, this is quite acceptable, providing it is appreciated that the underlying physics of self-assembly has been perverted.

The above scenario is, of course, complicated by the consideration that what is being evaluated by the instrument is not G, but G′ and G″. Both of these are finite even for a solution, although the respective moduli values may be very low. However, because of the finite frequency effect, and the contribution of non-ideal network assembly contributions, both G′ and G″ will tend to rise before the true gelation point, and something akin to Figure 1.5b is usually seen. The flattening off of G″ is not something predicted from theory, indeed some would expect a pronounced maximum in G″ after gelation, but this is rarely seen, except for some low concentration gelatin gels. This asymptotic level G″ behavior has been associated with the ‘stiffness’ of the network strands.

1.2.2.4.3. Extrapolation Methods for t gel

Accepting for now that there is a definitive gelation time, t gel, and that it is an important parameter, how best do we establish its value? There are a number of essentially empirical approaches, and we discuss some of these here (Clark et al., 1987; Kavanagh et al., 1998).

1. t gel is the time when G′ becomes greater than some pre-defined threshold value. This is a useful approach, albeit that the choice of the threshold value is obviously arbitrary. It needs to be in the very fast G′ increase region, but above the noise level. This, in turn, will depend upon the instrument and the system being investigated, but is typically in the range 1–10 Pa.

2. By linear back extrapolation of the G′ versus time to a pre-defined level. This can work well, but again is fairly arbitrary; normally the level here will be lower than that employed in method 1.

3. When there is a cross-over in G′, G″ – i.e. where G′ becomes greater than G″. This is a very common empirical approach in the external literature, and has been used for many years, for example when studying the setting of acrylic paints. However, this cross-over time will depend upon frequency, and so is really just as subjective as the previous methods. It also has limitations – for example some globular protein solutions act like charged colloids, so although G′ is low, it is always above G″. This means a cross-over will never be seen.

The different approaches above may appear somewhat capricious, but in practice, they can all be useful even if, typically, they give t gel’s which differ by say 50% or more. As we see later, the real interest is in investigating, and understanding, the concentration dependence of t gel. Here the concentration exponent can be such that (on the required log scale) a 50% difference in t gel is actually not so significant. The important thing is that, somehow or other, a value for t gel can be established, since this is an important parameter in subsequent kinetic modeling.

1.2.2.4.4. Chambon-Winter Method and Applicability in Biopolymer Self-Assembly

Some 20 years ago, Winter and Chambon (Chambon and Winter, 1985 and Winter and Chambon, 1986) suggested an elegant method, which despite its testing experimental requirements seemed to furnish a more absolute criterion for determining t gel, as that time when, in the mechanical spectrum of a gelling system, both (log) G′ and G″ versus log (ω), first show the same power law exponent. As we mentioned above, for a sol well before gelation, G″ and G′ will be proportional respectively to the oscillatory frequency ω, and to ω ². Well after the gel point, for a viscoelastic solid (gel), both G′ and G″ become essentially independent of frequency. In other words, the respective slopes of a log (G′, G″) versus log (ω) ‘frequency sweep’ will change from (2,1) to (0,0).

In the Winter approach (Winter et al., 1986) the time of congruency of slope, n, where

(1.4)

corresponds exactly to the gelation time, t gel (Figure 1.6).

The Winter-Chambon method for determining the gel point has become so popular that it has almost replaced more classical definitions, i.e. that the gel point is the conversion (or the corresponding time) when the average molecular weight (relative molecular mass) M w becomes infinite (in other words where the system first develops an infinite relaxation time) and it is now assumed, almost without reflection, that the two must be identical. However it might be more reasonable to say that there are so-called dynamic and static gel points, the former measured in a viscoelastic experiment, and the latter in an equilibrium (e.g. light scattering determination of M w) experiment (Trappe et al., 1992). In practice the two may be very close, but despite much effort it has not been proven (nor may it be possible to prove) that they are actually identical. This reflects the fact that there are always problems in making mechanical measurement on critical gels, connected with the strain dependence, the long relaxation times involved, and also the effect of entanglements, as we discuss below. This may be simplified by new techniques such as particle tracking, although these also tend to measure at high strains and frequencies.

The precise value of the slope n where congruency occurs can be calculated from a number of theories and is usually around 0.7 (Chambon et al., 1985; Winter et al., 1986; te Nijenhuis, 1997). In practice it is found that the experimentally observed congruent slope lies somewhere between 0.1 and 0.9, depending upon the precise system. This range is, unfortunately, very close to the extremes, viz. 0 and 1, given by the G″ slopes. This suggests that congruency of slope may not, of itself, be sufficient to identify the gelation point/time. This is a point that is sometimes misunderstood – if the exponent n is close to 0, the spectrum is that of Figure 1.3– which is well past, and has little to do with, the critical gelation profile described in the Winter-Chambon method.

1.2.2.4.5. Range of Viscoelastic Linearity

Yet another aspect of gel time measurement, and arguably one of greater significance, is the effect of finite strain on the tenuous mechanical system close to gelation (Ross-Murphy, 2005). In performing the kinetic gelation experiment it is usual practice to employ the smallest strain consistent with obtaining reliable data. In principle this can be checked to be within the linear viscoelastic region both before and after gelation by stopping the experiment and performing a so-called strain sweep. However, as we discussed above, many experiments are performed using controlled stress instruments in their pseudo-controlled strain mode, and such instruments do have more problems measuring a gelling system where the properties are changing quite rapidly within the oscillatory cycle, than when using the ‘controlled’ strain mode. This is because one might expect that the linear viscoelastic strain of the gelling system, rather than being constant, would tend to change during the gelation process, and would be a minimum just at the gel point (Rodd et al., 2001). The overall conclusion would appear to be that, even where, for biopolymer self-assembly systems, a definitive Winter-Chambon ‘gel point’ has been established, it may be impossible to equate this precisely with t gel.

1.2.2.4.6. G inf and the Equilibrium Modulus

For modeling the kinetics of network assembly, another parameter, in addition to t gel, is of value. This is, of course, the gel modulus. However, which gel modulus? As we have already hinted, what is really required is the equilibrium modulus, G; what we have measured is G′ (and G″), after a particular time, and at a preset frequency and strain. Assuming for the moment that the strain is sufficiently low, that the system is linearly viscoelastic, we still have the implicit effect of time and frequency. Strictly speaking we need to extrapolate to zero frequency, and very long (nominally infinite) set up (or ‘cure’) times. Of course, if the system is well into the gel state, G′ will be very largely independent of frequency (although this needs to be checked), so all that is required is to somehow extrapolate the values obtained during the cure experiment to infinite time, to obtain the parameter we have called G inf (Kavanagh et al., 1998, 2000).

One approach is simply to appeal to empiricism, and we have found, in practice (Kavanagh et al., 2000) that the form

(1.5)

where t is the time in seconds, B is an empirical parameter, and G inf is the required value of G′ at infinite time. As can be seen in Figure 1.7, this form reproduces satisfactorily both the asymptotic limit as t → ∞ and the required behavior (technically a logarithmic singularity) that log G′ → −∞, as t → t gel. Further thought reduces this to the simple case where we plot log(G′) versus 1/t, and find the intercept on the 1/t = 0 axis; this seems to be a valuable aid (Clark et al., 2001).

Two comments are worth noting here. First, very few workers make (or even appreciate the significance of) this extrapolation, and just assume G′ (after say 100 minutes of cure) is the same as G inf. This is risky, because only in retrospect can we judge the validity (or otherwise) of this. Second, for some systems, and the archetypal example is gelatin, it is almost impossible to make such an extrapolation anyway, because G′ never levels off. Instead gelatin gels enter an apparent ‘log phase’ of modulus growth, which is assumed to be due to formation of new structures, perhaps via the slow kinetic processes associated with the cis proline ‘flip’ (Djabourov et al., 1985 and Busnel et al., 1989). Since this chapter is concerned with generic issues, we mention this, but do not discuss it further.

1.3. Theoretical aspects

If we take a series of measurements of G inf for different initial biopolymer concentrations, and plot in the form log(G inf) versus log(C), several features are immediately apparent (Clark and Ross-Murphy, 1985; Clark et al., 1987); in fact, as we shall see, they already reflect the underlying ‘percolation’ type assembly behavior, and discussion of this behavior leads us naturally into the theoretical part of this chapter. As Figure 1.8 shows, at high concentrations, there is an apparent (power law) dependence of log(G inf) versus log(C), whereas at lower concentrations, log(G inf) shows increasingly pronounced curvature, and at a particular concentration, appears to vanish. This is the so-called critical gel concentration, here denoted, C 0.

1.3.1. Critical Gel Concentration

That a biopolymer gel does have a critical gel concentration seems, on the surface, obvious. It is well appreciated that there needs to be a certain concentration of the biopolymer present before a contiguous gel can be formed. (Self-evident though this may appear to be, it is not a feature of, for example, the fractal gel model we discuss later.)

What then is the significance of C 0, and what does it tell us about the mechanisms of self-assembly? The answer to this, and the corresponding behavior of the gelation time, is the subject of much of the remaining discussion. In real terms, values of C 0 vary from < 0.05% for some microbial polysaccharides to > 10% for certain more particulate gel systems. However, despite this, the form of the scaled log G versus log (C) or better still, from the generality viewpoint, versus log (C/C 0) curve, remains the same.

1.3.2. Gelation Time

We have already discussed methods for the determination of the gelation time, t gel, and its significance in the experimental context. If, for example, we repeat the type of plot seen in Figure 1.8, but instead of plotting log(G inf) we plot log(1/t gel), the experimental data seem to follow essentially the same behavior, with a power law behavior at high concentrations, and a logarithmic singularity, at the same concentration, and consequently with the same C 0. Many years ago, we conjectured that the shapes of these two curves were essentially congruent (Richardson et al., 1981), and in practice this often works quite well. However like many such practical conjectures, theory shows that this should not really be the case. Indeed, as we discuss in detail below, fitting modulus concentration and (reciprocal) gel time concentration data simultaneously can be extremely testing, and usually requires sophisticated multistage kinetic modeling (Clark et al., 2001).

1.3.3. Kinetic Modeling

1.3.3.1. Flory-Stockmayer (FS) Model

The basic model for gelation is that of non-linear or random step-growth polymerization (or, to use old terminology, polycondensation) which goes back to the classical work of Flory and Stockmayer in the 1940s on covalently formed, irreversible, networks (Flory, 1941, Flory, 1942, Stockmayer, 1943 and Stockmayer and Zimm, 1984). This model, which in today's terms we would describe as percolation on an infinite dimension, tree-like or Bethe lattice (Gordon and Ross-Murphy, 1975, Stauffer et al., 1982 and Stauffer, 1985) has proved of enormous value even though it neglects many features, such as the pre-gel formation of intramolecular links (cycles), which ‘wastes’ cross-links (Gordon and Scantlebury, 1968). Indeed it is fair to say that gelation in the absence of cross-link wastage reactions is well understood in terms of the Flory-Stockmayer theory and the gel point can be used as a reference point for the consideration of the effects of intramolecular reaction.

In the FS model, at the gel point, the species of infinite molar mass has a tree-like structure permeating through the whole reaction mixture. The critical conversion occurs when there is a non-zero probability that a randomly chosen chain continues to infinity. Given the previously mentioned random reaction (or equal reactivity) of like functional groups or sites, the gel point, and properties relating to the gelling system, may be predicted quite generally, in terms of the parameter α, representing the proportion of reacted groups, and the gel point conversion, α c. For example many properties can be related back directly to the ratio α/α c,, although very close to α/α c – in the so-called critical region – critical fluctuations need to be taken into account. (The extent of this critical region is governed by criteria such as that of Ginzburg; evaluation of the extent of the critical domain remains the realm of the theorist but a practical guide is that the upper limit is say 10 −2 <= (α/α c) – 1 <= 10 −1. This has relevance later, when we discuss the critical region in more detail.)

The original Flory-Stockmayer model was developed to describe the formation of polymer networks in the absence of a solvent, either through the condensation reaction of monomeric species or the cross-linking of pre-formed polymer chains. As Stockmayer demonstrated, this model can be developed in kinetic terms (Stockmayer, 1943) through a second-order differential equation for the change of α in terms of the fraction of unreacted sites (1 – α), i.e.

(1.6)

subject to the initial conditions α = 0 at t = 0. This allows the degree of reaction α to be specified as a function of time and properties of the network calculated from it. For example, one version of the model (Dobson and Gordon, 1965 and Gordon and Ross-Murphy, 1975) tells us that the gel modulus is given by:

(1.7)

Here N e, which is a function of the degree of conversion α, is the number of elastically active network chains (EANCs) per biopolymer chain. The parameter a is the so-called rubber theory ‘front factor’, RT is the usual gas constant term, and V mol is volume per mole of biopolymer chains. N e is zero before the gel point, so G = 0 until the gel point, which occurs (Flory, 1941) at:

(1.8)

where f is the so-called network functionality, the number of functional groups or attachment sites available per primary molecule to form cross-links. By substituting the appropriate expression from Eq. (1.6) into Eq. (1.7), albeit ignoring the various rate and other constants, we obtain G as a function of time.

Figure 1.9 illustrates this result with the dependence of both α and G on time plotted on logarithmic time axes (Ross-Murphy, 2005). At first glance, in the linear time axis plot, both α and G look to have a quite similar pseudo-rectilinear relationship, albeit that there is a small lag time, the gelation time, t gel, in the G (t) behavior.

1.3.4. Random Branching in Solution

The formation of highly solvated networks which is characteristic of biopolymer self-assembly in solution requires development of the Flory-Stockmayer model in a number of directions. A principal change is that the kinetic equations determining the reaction extent must now be written in terms of the concentrations of reacting functionalities and in some cases cross-link reversibility must be anticipated. Where cross-linking is believed to occur irreversibly there is no longer the possibility of ignoring wastage reactions such as the formation of cycles as, in solution, the absence of these would lead to some form of network collapse: i.e. the formation of homogeneous gels with a finite critical concentration would be impossible. Extension of the Flory-Stockmayer approach along these lines has been described in past literature by Gordon and co-workers (Gordon, 1962, Gordon and Scantlebury, 1964, Gordon and Scantlebury, 1966 and Gordon and Scantlebury, 1968; Gordon et al., 1975; Dusek et al., 1978) and their model is described in outline below. In this description their elegant mathematical approach using probability generating functions is adopted, an approach sometimes referred to as branching or cascade theory.

The model starts by assuming the presence in solution of a concentration C of molecular species each of molecular weight M and each bearing f equivalent reactive functional groups or sites available for bonding. Assuming a totally random cross-linking process which includes also the possibility of intramolecular cross-linking within aggregates, i.e. cyclization, the state of cross-linking of the system can be specified at any time by the link probability generating function (lpgf):

(1.9)

The subscript 0 of F 0( θαθσθω) indicates the zeroth or root generation of the tree, θα, θσ and θω are so-called ‘dummy variables’ and P ij is the probability of