Table of Contents
Related Titles
Title Page
Copyright
List of Contributors
Introduction
What is Product Design and Engineering?
Why This Book?
References
Chapter 1: Rheology of Disperse Systems
1.1 Introduction
1.2 Basics of Rheology
1.3 Experimental Methods of Rheology
1.4 Rheology of Colloidal Suspensions
1.5 Rheology of Emulsions
References
Chapter 2: Rheology of Cosmetic Emulsions
2.1 Introduction
2.2 Chemistry of Cosmetic Emulsions
2.3 Rheological Measurements
2.4 Dynamic Mechanical Tests (Oscillation)
References
Chapter 3: Rheology Modifiers, Thickeners, and Gels
3.1 Introduction
3.2 Classification of Thickeners and Gels
3.3 Definition of a “Gel”
3.4 Rheological Behavior of a “Gel”
3.5 Classification of Gels
3.6 Particulate Gels
3.7 Rheology Modifiers Based on Surfactant Systems
References
Chapter 4: Use of Rheological Measurements for Assessment and Prediction of the Long-Term Assessment of Creaming and Sedimentation
4.1 Introduction
4.2 Accelerated Tests and Their Limitations
4.3 Application of High Gravity (g) Force
4.4 Rheological Techniques for Prediction of Sedimentation or Creaming
4.5 Separation of Formulation (“Syneresis”)
4.6 Examples of Correlation of Sedimentation or Creaming with Residual (Zero Shear) Viscosity
4.7 Assessment and Prediction of Flocculation Using Rheological Techniques
4.8 Examples of Application of Rheology for Assessment and Prediction of Flocculation
4.9 Assessment and Prediction of Emulsion Coalescence Using Rheological Techniques
References
Chapter 5: Prediction of Thermophysical Properties of Liquid Formulated Products
5.1 Introduction
5.2 Classification of Products, Properties and Models
5.3 Pure Compound Property Modeling
5.4 Functional Bulk Property Modeling – Mixture Properties
5.5 Functional Compound Properties in Mixtures – Modeling
5.6 Performance Related Property Modeling
5.7 Software Tools
5.8 Conclusions
Appendix 5.A: Overview of the M&G GC+ Method
Appendix 5.B: Prediction of the UNIFAC Group Interaction Parameters
References
Chapter 6: Sources of Thermophysical Properties for Efficient Usein Product Design
6.1 Introduction
6.2 Overview of the Important Thermophysical Properties for Phase Equilibria Calculations
6.3 Reliable Sources of Thermophysical Data
6.4 Examples of Databases for Thermophysical Properties
6.5 Special Case and Challenge: Data of Complex Solutions
6.6 Examples of Databases with Properties of Electrolyte Solutions
6.7 Properties of New Component Classes: Ionic Liquids and Hyperbranched Polymers
References
Chapter 7: Current Trends in Ionic Liquid Research
7.1 Introduction
7.2 Ionic Liquids as Acido-Basic Media
7.3 Binary Mixtures of Ionic Liquids: Properties and Applications
7.4 Nanoporous Materials from Ionothermal Synthesis
7.5 Catalytic Hydrogenation Reactions in Ionic Liquids
7.6 Concluding Remarks
Acknowledgements
References
Chapter 8: Gelling of Plant Based Proteins
8.1 Introduction – Overview of Plant Proteins in Industry
8.2 Structure and Formation of Protein Gels
8.3 Factors Determining Physical Properties of Protein Gels
8.4 Evaluating Gelation of Proteins
8.5 Gelation of Proteins Derived from Plants
8.6 Protein Gels in Product Application
8.7 Future Prospects and Challenges
References
Chapter 9: Enzymatically Texturized Plant Proteins for the Food Industry
9.1 Introduction
9.2 Reactions Catalyzed by MTG
9.3 Current Sources of MTG
9.4 Need for Novel Sources of MTG
9.5 Vegetable Proteins Suitable for Crosslinking with MTG
9.6 Strategies to Modify and Improve Protein Sources for MTG Crosslinking
9.7 Applications of MTG in Processing Food Products Containing Vegetable Protein
9.8 Applications of MTG Crosslinked Leguminous Proteins in Food Models and Realistic Food Products
9.9 Safety of MTG and Isopeptide Bonds in Crosslinked Plant Proteins
9.10 Conclusions
References
Chapter 10: Design of Skin Care Products
10.1 Product Design
10.2 Skin Care
10.3 Emulsions
10.4 Structure of a Skin Care Cream
10.5 Essential Active Substances from a Medical Point of View
10.6 Penetration into the Skin
10.7 Targeted Product Design in the Course of Development
10.8 Production of Skin Care Products
10.9 Bottles for Cosmetic Creams
10.10 Design of all Elements
References
Chapter 11: Emulsion Gels in Foods
11.1 Introduction
11.2 Food Emulsions
11.3 Creating a Food Emulsion
11.4 Applications of Gel-Like Type Emulsions
11.5 Final Considerations
References
Index
Related Titles
Bröckel, U., Meier, W., Wagner, G. (eds.)
Product Design and Engineering
Best Practices
2007
ISBN: 978-3-527-31529-1}
Rähse, W.
Industrial Product Design of Solids and Liquids
A Practical Guide
2014
ISBN: 978-3-527-33335-6
Jameel, F., Hershenson, S. (eds.)
Formulation and Process Development Strategies for Manufacturing Biopharmaceuticals
2010
ISBN: 978-1-118-12473-4
Norton, I. (ed.)
Practical Food Rheology - An Interpretive Approach
2011
ISBN: 978-1-405-19978-0
Tadros, T.F.
Rheology of Dispersions
Principles and Applications
2010
ISBN: 978-3-527-32003-5
The Editors
Prof. Dr.-Ing Ulrich Bröckel
HS Trier
Umwelt-Campus Birkenfeld
Campus Allee 9916
55761 Birkenfeld
Germany
Dr. Willi Meier
DECHEMA e.V.
Theodor-Heuss-Allee 25
60486 Frankfurt
Germany
Dr.-Ing Gerhard Wagner
Global R&D Director DSM Biotechnology Center
Alexander Fleminglaan 1
2613 AX Delft
The Netherlands
and
Dianastrasse 12
4310 Rheinfelden
Switzerland
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List of Contributors
Product design is, in principle, a term that describes a very broad variety of designs, ranging from industry design of goods, for example cars, measurement instruments and furniture, to scientific and technological product design. Scientific and technological product design can be subdivided into four main clusters:
All four clusters of product design have in common a need for a multidisciplinary approach based on biology, chemistry and physics combined with engineering skills and sciences; furthermore, disciplines like nutrition, pharmacy and ergonomics and form design are crucial to ensure a suitable product. Product design is more of an iterative than a linear process; the understanding of the different disciplines is therefore important at the different steps of product design depending of the scale of the product design. Figure 1 shows how the disciplines impact on the physical properties relevant to product qualities within the different size ranges.
Figure 1 Product design – disciplines and scales.
In the first two volumes of this book series we focused mainly on basic technologies and solids. The very important aspect of liquid-like or gel-type applications have been excluded so far. In this third volume of our book series we will, therefore, elaborate these applications in more detail.
Product design has become a growing field of interest during recent years. The reasons for this are manifold. Looking at the markets of the chemical engineering field today, we observe well-established and quite saturated markets. Breakthrough innovation with brand new products is difficult to accomplish. Product design is, therefore, crucial to improve the following properties of a product:
With product design, existing products can be optimized, improved and positioned in the market, prolonging their life cycle or differentiating and making them applicable for new markets.
Product design has been seen as a paradigm shift in process design away from the unit operation concept to a new interdisciplinary thinking [9], speeding up the development process. But, obviously, there is still a long way to go before enough knowledge and basic understanding of the complex interactions of multicomponent systems is gathered in order to calculate the composition and the processing conditions in order to predict the performance of a new formulation using a computer.
As mentioned in the introduction of the first two volumes of this series, product design is much more than a buzz word and is of the highest importance for related industries. For example, Aspirin® as a highly developed pharmaceutical bulk chemical is available in new formulations every few years [10]. The same is true for formulated and encapsulated carotenoid products, which also show a continuous development in their product form and composition. In addition, beverages, like soft drinks, energy drinks and coffee-based beverages or washing powder, laundry products or detergents are subject to on-going development, owing to customers expecting an improved taste or a better performance of the new product. These kinds of improvements ensure the lifecycle management of products.
Several company strategies clearly state the importance of product design and engineering for the future of process industries. Customer demands have to be recognized and turned into products with the help of well-established processes and technologies. Fulfilling customer needs will automatically lead the business to new applications and new markets were real growth is created.
For such examples we elaborated in Volumes I and II different technologies, raw materials and additives. One area we were not able to investigate in more detail, even though we touched on emulsion technologies, is liquid and gel-type applications, which are very important for the broad field of, among other industries, the life science industry.
Volume I describes the basics and fundamentals of chemical engineering that are essential for product design and engineering. This enabled us to give an overview of the basic knowledge and related activities. The second volume describes recent applications that turn the technologies described in Volume I into customer oriented products. Volume II shows some examples of these new products with an introductory chapter on product design fundamentals. The superior behavior of, for example, coffee, aspirin and carotenoid products is crucial. The taste of coffee, the bioavailability of aspirin and carotenoid products and the UV absorption of polymers can be adjusted. Product design offers opportunities to change, to adopt, and to improve products.
The intention of this, the third volume of this series is to discuss in more detail the basics of rheology and how product design is carried out in liquid and gel-type applications. Differentiation and product design is essential for raw materials and additives. The behavior of, for example, starches and gelatins is designed and changed by product design to give a specific texture and/or performance of the final product.
The structure of the third volume of this series is illustrated in Figure 2. The fundamentals are the basics in rheology, which are important for describing and quantifying the properties of gels or pastes.
Figure 2 Structure and content of the third volume.
This volume starts with the essential chapter entitled “Rheology of Disperse Systems” (Chapter 1) while Chapter 3 gives an insight view into “Rheology Modifiers, Thickeners, and Gels.” The stability of dispersions is crucial for the shelf life of customer related products. This problem is discussed in the chapter on the “Use of Rheological Techniques for Assessment and Prediction of Stability of Dispersions (Suspensions and Emulsions)” (Chapter 4). The specific chapter “Rheology of Cosmetic Emulsions” (Chapter 2) focuses on products closer to cosmetic daily life application. A theoretical approach can be found in the chapters on the “Prediction of Thermophysical Properties of Liquid Products” (Chapter 5) and “Sources of Thermophysical Properties for the Efficient Use in Product Design” (Chapter 6). A more forward-looking contribution is “Trends in Ionic Liquid Research” (Chapter 7). The possibilities in terms of modifying the viscosity of liquid formulations are given in “Gelling of Plant Based Proteins” (Chapter 8) and “Enzymatic Texturized Plant Proteins for the Food Industry” (Chapter 9). Examples of some important applications are given in the “Design of Skin Care Products” (Chapter 10) and “Emulsion Gels in Foods” (Chapter 11).
A complete prediction of product formulation based on scientific knowledge is not possible given the current state of the art. Very specific and often still empirical knowledge and specific trials in the laboratories are still needed to ensure the design of a product. Product design and engineering it is a very multifaceted area. Nevertheless, this book aims to give a good overview of the different fields of technologies and successful instances for liquid and gel-type applications.
As product design will become increasingly important in the near future the teaching of students in this field should be intensified given that:
It is our intention to contribute with this book series to the on-going improvements in technologies, fundamentals and discussions in the community about teaching product design. Finally, without the highly qualified contributions of the persons most important for this book, our authors, this volume would still only be a nice idea.
1. Sutton, A. (2008) Product development of probiotics as biological drugs. Clin. Infect. Dis., 46, 128–132.
2. Merino, S.T. and Cherry, J. (2007) Progress and challenges in enzyme development for biomass utilization. Adv. Biochem. Eng. Biotechnol., 108, 95–120.
3. Rastall, R.A. (2007) Novel Enzyme Technology for Food Applications, Woodhead Press, Cambridge. ISBN: 9781420043969.
4. Cussler, E.L. and Moggridge, G.D. (2011) Chemical Product Design, 2nd edn, Cambridge University Press, Cambridge. ISBN: 978-0-521-16822-9.
5. Calorie Control Council (2013) Polyols Information Source http://www.polyol.org/ (accessed 8 March 2013).
6. Stummerer, S. and Hablesreiter, M. (2010) Food Design XL, Springer, New York. ISBN: 978-3-211-99230-2.
7. Ortega-Rivas, E. et al. (2005) Food Powders, Springer, New York. ISBN: 978-0-306-47806-2.
8. Norton, I.T. et al. (2013) Formulation Engineering of Foods, Wiley-Blackwell. ISBN: 10: 0470672900.
9. Costa, R. et al. (2006) Chemical product engineering: an emerging paradigm within chemical engineering. AIChE J., 52 (6), 1976–1986.
10. Bayer HealthCare LLC (2009) Bayer Group Aspirin http://www.aspirin.com/scripts/pages/en/aspirin_history/index.php (accessed 8 March 2013).
The rheology of disperse systems is an important processing parameter. Being able to characterize and manipulate the flow behavior of dispersions one can ensure their optimal performance. Waterborne automotive coatings, for example, should exhibit a distinct low-shear viscosity necessary to provide good leveling but to avoid sagging at the same time. Then, a strong degree of shear thinning is needed to guarantee good pump- and sprayability. The rheological properties of dispersions, especially at high solids content, are complex and strongly dependent on the applied forces and flow kinematics. Adding particles does not simply increase the viscosity of the liquid as a result of the hydrodynamic disturbance of the flow; it also can be a reason for deviation from Newtonian behavior, including shear rate dependent viscosity, elasticity, and time-dependent rheological behavior or even the occurrence of an apparent yield stress. In colloidal systems particle interactions play a crucial role. Depending on whether attractive or repulsive interactions dominate, the particles can form different structures that determine the rheological behavior of the material. In the case of attractive particle interactions loose flocs with fractal structure can be formed, immobilizing part of the continuous phase. This leads to a larger effective particle volume fraction and, correspondingly, to an increase in viscosity. Above a critical volume fraction a sample-spanning network forms, which results in a highly elastic, gel-like behavior, and an apparent yield stress. Shear-induced breakup and recovery of floc structure leads to thixotropic behavior. Electrostatic or steric repulsion between particles defines an excluded volume that is not accessible by other particles. This corresponds to an increase in effective volume fraction and accordingly to an increase in viscosity. Crystalline or gel-like states occur at particle concentrations lower than the maximum packing fraction.
Characterization of the microstructure and flow properties of dispersions is essential for understanding and controlling their rheological behavior. In this chapter we first introduce methods and techniques for standard rheological tests and then characterize the rheology of hard sphere, repulsive, and attractive particles. The effect of particle size distribution on the rheology of highly concentrated dispersions and the shear thickening phenomenon will be discussed with respect to the influence of colloidal interactions on these phenomena. Finally, typical features of emulsion rheology will be discussed with special emphasis on the distinct differences between dispersion and emulsion rheology.
According to its definition, rheology is the science of the deformation and flow of matter. The rheological behavior of materials can be regarded as being between two extremes: Newtonian viscous fluids, typically liquids consisting of small molecules, and Hookean elastic solids, like, for example, rubber. However, most real materials exhibit mechanical behavior with both viscous and elastic characteristics. Such materials are termed viscoelastic. Before considering the more complex viscoelastic behavior, let us first elucidate the flow properties of ideally viscous and ideally elastic materials.
Isaac Newton first introduced the notion of viscosity as a constant of proportionality between the force per unit area (shear stress) required to produce a steady simple shear flow and the resulting velocity gradient in the direction perpendicular to the flow direction (shear rate):
1.1 
where σ = F/A is the shear stress, η the viscosity, and the 
is the shear rate. Here A is the surface area of the sheared fluid volume on which the shear force F is acting and h is the height of the volume element over which the fluid layer velocity v varies from its minimum to its maximum value. A fluid that obeys this linear relation is called Newtonian, which means that its viscosity is independent of shear rate for the shear rates applied. Glycerin, water, and mineral oils are typical examples of Newtonian liquids. Newtonian behavior is also characterized by constant viscosity with respect to the time of shearing and an immediate relaxation of the shear stress after cessation of flow. Furthermore, the viscosities measured in different flow kinematics are always proportional to one another.
Materials such as dispersions, emulsions, and polymer solutions often exhibit flow properties distinctly different from Newtonian behavior and the viscosity decreases or increases with increasing shear rate, which is referred to a shear thinning and shear thickening, respectively. Figure 1.1a,b shows the general shape of the curves representing the variation of viscosity as a function of shear rate and the corresponding graphs of shear stress as a function of shear rate.
Figure 1.1 Typical flow curves for Newtonian, shear thinning and shear thickening (dilatant) fluids: (a) shear stress as a function of shear rate; (b) viscosity as a function of shear rate.
Materials with a yield stress behave as solids at rest and start to flow only when the applied external forces overcome the internal structural forces. Soft matter, such as, for example, dispersions or emulsions, does not exhibit a yield stress in this strict sense. Instead, these materials often show a drastic change of viscosity by orders of magnitude within a narrow shear stress range and this is usually termed an “apparent” yield stress (Figure 1.2a,b). Dispersions with attractive interactions, such as emulsions and foams, clay suspensions, and ketchup, are typical examples of materials with an apparent yield stress. Note that there are various methods for yield stress determination and the measured value may differ depending on the method and instrument used.
Figure 1.2 Flow curve of a material with an apparent yield stress σy: (a) shear stress as a function of shear rate; (b) viscosity as a function of shear stress.
The flow history of a material should also be taken into account when making predictions of the flow behavior. Two important phenomena related to the time-dependent flow behavior are thixotropy and rheopexy. For materials showing thixotropic behavior the viscosity gradually decreases with time under constant shear rate or shear stress followed by a gradual structural recovery when the stress is removed. The thixotropic behavior can be identified by measuring the shear stress as a function of increasing and decreasing shear rate. Figure 1.3 shows a hysteresis typical for a thixotropic fluid. Examples of thixotropic materials include coating formulations, ketchup, and concentrated dispersions in the two-phase region (Section 1.4.1.1). The term rheopexy is defined as shear-thickening followed by a gradual structural recovery when the shearing is stopped. Tadros pointed out that rheopexy should not be confused with anti-thixotropy, which is the time dependent shear thickening [1]. However, rheopectic materials are not very common and will not be discussed here.
Figure 1.3 Flow curve of a thixotropic material.
So far we have considered the flow behavior of viscous fluids in terms of Newton's law and a nonlinear change of viscosity with applied stress that can occur either instantaneously or over a long period of time. At the other extreme is the ideal elastic behavior of solids, which can be described by Hooke's law of elasticity:
1.2 
where γ is the shear deformation (also termed strain) and G is the shear modulus characterizing the rigidity of a material. The shear modulus of an ideal elastic solid is independent of the shear stress and duration of the shear load. As soon as a deformation is applied a constant corresponding stress occurs instantaneously. In viscoelastic materials stress relaxes gradually over time at constant deformation and eventually vanishes for viscoelastic liquids. When the stress relaxation is proportional to the strain we are talking about the so-called linear viscoelastic regime. Above a critical strain the apparent shear modulus becomes strain dependent. This is the so-called nonlinear viscoelastic regime. The linear viscoelastic material properties are in general very sensitive to microstructural changes and interactions in complex fluids.
A dynamic test or small amplitude oscillatory shear (SAOS) test is the most widely used rheological measurement to investigate the linear viscoelastic behavior of a fluid, since it has a superior accuracy compared to step strain or step stress experiments. When a sinusoidal oscillatory shear strain is applied with amplitude γ0 and angular frequency ω the deformation γ(t) can be written as:
1.3 
where t denotes the time. The shear rate is the time derivative of the shear strain and then reads as follows:
1.4 
A linear viscoelastic fluid responds with a sinusoidal course of shear stress σ(t) with amplitude σ0 and angular frequency ω, but is phase shifted by an angle δ compared to the imposed strain:
1.5 
Depending on material behavior, the phase shift angle δ occurs between 0° and 90°. For ideal elastic materials the phase shift disappears, that is, δ = 0, while for ideal viscous liquids δ = 90°. The shear modulus can be written in complex form:
1.6 
with the storage modulus G′ and loss modulus G′′. G′ is a measure of the energy stored by the material during a cycle of deformation and represents the elastic behavior of the material, while G′′ is a measure of the energy dissipated or lost as heat during the shear cycle and represents the viscous behavior of the material. The terms G′ and G′′ can be expressed as sine and cosine function of the phase shift angle δ:
1.7 
1.8 
Hence the tangent of the phase shift δ represents the ratio of loss and storage modulus:
1.9 
Analogous to the complex shear modulus we can define a complex viscosity η*:
1.10 
with:
1.11 
The viscoelastic properties of a fluid can be characterized by oscillatory measurements, performing amplitude- and frequency-sweep. The oscillatory test of an unknown sample should begin with an amplitude sweep, that is, variation of the amplitude at constant frequency. Up to a limiting strain γc the structure of the tested fluid remains stable and G′ as well as G′′ is independent of the strain amplitude. The linear viscoelastic range may depend on the angular frequency ω; often, γc decreases weakly with increasing frequency.
Frequency sweeps are used to examine the time-dependent material response. For this purpose the frequency is varied using constant amplitude within the linear viscoelastic range. At an appropriately high angular frequency ω, that is, short-term behavior, the samples show an increased rigidity and hence G′ > G′′. At lower frequencies (long-term behavior) stress can relax via long-range reorganization of the microstructure and the viscous behavior dominates and, correspondingly, G′′ > G′.
Rheometers can be categorized according to the flow type in which material properties are investigated: simple shear and extensional flow. Shear rheometers can be divided into rotational rheometers, in which the shear is generated between fixed and moving solid surfaces, and pressure driven like the capillary rheometer, in which the shear is generated by a pressure difference along the channel through which the material flows. Extensional rheometers are far less developed than shear rheometers because of the difficulties in generating homogeneous extensional flows, especially for liquids with low viscosity. Many different experimental techniques have been developed to characterize the elongational properties of fluids and predict their processing and application behavior, including converging channel flow [2], opposed jets [3], filament stretching [4], and capillary breakup [5, 6] techniques. However, knowledge about the extensional rheology of complex fluids like dispersions and emulsions is still very limited.
Rotational instruments are used to characterize materials in steady or oscillatory shear flow. Basically there are two different modes of flow: controlled shear rate and controlled shear stress. Three types of measuring systems are commonly used in modern rotational rheometry, namely, concentric cylinder, parallel plate, and cone-and-plate. Typical shear rates that can be measured with rotational rheometers are in the range 10−3 to 103 s−1.
As shown in Figure 1.4a, a cylinder measuring system consists of an outer cylinder (cup) and an inner cylinder (bob). There are two modes of operation depending on whether the cup or the bob is rotating. The Searle method corresponds to a rotating bob and stationary cup, while in the Couette mode the cup is set in motion and the bob is fixed. The gap between the two concentric cylinders should be small enough so that the sample confined in the gap experiences a constant shear rate. This requirement is fulfilled and the gap is classified as “narrow” when the ratio of the inner to the outer cylinder radius is greater than 0.97.
Figure 1.4 Schematic representation of (a) concentric cylinder, (b) parallel-plate, and (c) cone-and-plate measuring system.
When the bob is rotating at an angular velocity ω the shear rate is given by:
1.12 
where Ri and Ra are the radii of the bob and the cup, respectively. If the torque measured on the bob is Md, the shear stress σ in the sample is given by:
1.13 
where L is the effective immersed length of the bob.
Having the shear rate 
and shear stress σ, the sample viscosity η can be calculated according to Equation 1.1 For these calculations we ignore any end effects, which are actually likely to occur as a result of the different shearing conditions in the liquid covering the ends of the cylinders. To minimize the end effect the ratio of the length L to the gap between cylinders is maintained at greater than 100 and the shape of the bottom of the bob is designed as a cone with an angle α, which is chosen so that the shear rate in the bottom area matches that in the narrow gap between the concentric cylinders.
The concentric cylinder measuring system is especially suitable for low-viscous liquids, since it can be designed to offer a large shear area and at high shear rates the sample is not expelled from the gap. Other advantages of this geometry are that sample evaporation is of minor relevance since the surface area is small compared to the sample volume, the temperature can be easily controlled due to the large contact area, and even if suspensions exhibit sedimentation and particle concentration varies along the vertical direction the measured viscosity is a good approximation of the true value.
The parallel plate geometry is shown in Figure 1.4b. The sample, confined within the gap of height H between the two parallel plates, is sheared by the rotation of one of the plates at angular velocity ω. Thereby, the circumferential velocity v depends on the distance from the plate at rest h and the distance r from the rotational axis:
1.14 
and thus:
1.15 
The shear rate 
at constant ω is not constant within the gap. Typically, the calculations and analysis of rheological results in parallel-plate measuring systems are related to the maximum shear rate value at the rim of the plate (r = Rp). The shear rate can be varied over a wide range by changing the gap height H and the angular velocity ω.
The shear stress σ is a function of the shear rate 
, which is not constant within the gap. Thus, to relate the shear stress to the total torque an expression for the 
dependence is necessary. For Newtonian liquids the shear stress depends linearly on the shear rate and can be expressed as follows:
1.16 
This expression is called the apparent shear stress. For non-Newtonian fluids Giesekus and Langer [7] developed a simple approximate single point method to correct the shear rate data, based on the idea that the true and apparent shear stress must be equal at some position near the wall. It was found that this occurs at the position where r/Rp = 0.76 and this holds for a wide range of liquids.
The parallel-plate measuring system allows for measurements of suspensions with large particles by using large gap heights. On the other hand, by operating at small gaps the viscosity can be obtained at relatively high shear rates. Small gaps also allow for a reduction of errors due to edge effects and secondary flows. Wall slip effects can be corrected by performing measurements at different gap heights. Rough plates are often used to minimize wall slip effects. Note that for sedimenting suspensions the viscosity is systematically underestimated since the upper rotating plate moves on a fluid layer with reduced particle loading.
A cone-and-plate geometry is shown schematically in Figure 1.4c. The sample is contained between a rotating flat cone and a stationary plate. Note that the apex of the cone is cut off to avoid friction between the rotating cone and the lower plate. The gap angle ϕ is usually between 0.3° and 6° and the cone radius Rp is between 10 and 30 mm. The gap h increases linearly with the distance r from the rotation axis:
1.17 
The circumferential velocity v also increases with increasing distance r:
1.18 
Hence the shear rate is constant within the entire gap and does not depend on the radius r:
1.19 
The shear stress is related to the torque Md on the cone:
1.20 
A great advantage of the cone-and-plate geometry is that the shear rate remains constant und thus provides homogenous shear conditions in the entire shear gap. The limited maximum particle size of the investigated sample, difficulties with avoiding solvent evaporation, and temperature gradients in the sample as well as concentration gradients due to sedimentation are typical disadvantages of the cone-and-plate measuring system.
Figure 1.5 shows a schematic diagram of a piston driven capillary rheometer. A piston drives the sample to flow at constant flow rate from a reservoir through a straight capillary tube of length L. Generally, capillaries with circular (radius R) or rectangular (width B and height H) cross-sections are used. The measured pressure drop Δp along the capillary and the flow rate Q are used to evaluate the shear stress, shear rate, and, correspondingly, viscosity of the sample.
Figure 1.5 Schematic representation of a controlled flow rate capillary rheometer.
Pressure driven flows through a capillary have a maximum velocity at the center and maximum shear rate at the wall of the capillary, that is, the deformation is essentially inhomogeneous. Assuming Newtonian behavior and fully developed, incompressible, laminar, steady flow, the apparent wall shear stress σa in a circular capillary with radius R is related to the pressure drop Δp by:
1.21 
and the apparent or Newtonian shear rate at the wall can be calculated on the basis of measured flow rate according to:
1.22 
Therefore, we can evaluate the viscosity in terms of an apparent viscosity based on Newton's postulate (Equation 1.1).
To obtain the true shear rate in the case of non-Newtonian fluids the Weissenberg–Rabinowitch correction [8] for non-parabolic velocity profiles should be taken into account. A simpler method to determine the true shear rate has been developed by Giesekus and Langer [7] as well as Schümmer and Worthoff [9]. Their single point method is based on the idea that the true and apparent shear rate must be equivalent at a certain radial position near the wall and thus the true shear rate 
is given simply by:
1.23 
Note that this approximation does no differ significantly from the Weissenberg–Rabinowitch correction for weakly shear thinning fluids.
Other possible sources of error in capillary flow experiments are entrance effects, slippage at the capillary wall, and viscous heating effects. Furthermore, the pressure drop Δp is difficult to measure directly in the capillary. Therefore, it is usually detected by a transducer mounted above the entrance of the capillary. Hence, the measured pressure includes not only the pressure loss due to the laminar flow in the die but also the entrance pressure loss due to rearrangement of the streamlines at the entrance and the exit of the capillary. Bagley [10] proposed a correction that accounts for these additional pressure losses but for practical purposes it is sufficient to use a single capillary die with sufficiently large L/R ratio, typically L/R ≥ 60 [8].
For highly concentrated suspensions wall slip effects, due to shear induced particle migration (only for very large particles), and specific particle–wall interactions have to be considered. If the slip velocity is directly proportional to the applied stress it is possible to correct the apparent wall shear rate according to the procedure developed by Mooney [11], which compares the flow curves determined with dies of different radii but similar L/R.
The major advantage of the capillary rheometer is that the flow properties of fluids can be characterized under high shear conditions (up to 
= 106 s−1) and process-relevant temperatures (up to 400 °C). Another advantage is that the capillary flow is closed and has no free surface so that edge effects, solvent evaporation, and other problems that trouble rotational rheometry can be avoided.
The flow behavior of colloidal (often also termed Brownian) dispersions is controlled by the balance between hydrodynamic and thermodynamic interactions as well as Brownian particle motion. Thermodynamic interactions mainly include electrostatic and steric repulsion and van der Waals attraction. The relative importance of individual forces can be assessed on the basis of dimensionless groups, which can be used to scale rheological data. In this section we first consider dispersions of Brownian hard sphere particles and elucidate the effect of particle volume fraction, size, and shape of particles on dispersion rheology. Then, we take into account the effect of repulsive and attractive interactions on the microstructure of suspensions and its corresponding rheological response. Special attention will be paid to the rheological behavior of concentrated dispersions.
Hard-sphere dispersions are idealized model systems where no thermodynamic or colloidal particle–particle interactions are present unless these particles come into contact. In that sense, they represent the first step from ideal gases towards real fluids. Even such simple systems can show complex rheological behavior. The parameters controlling dispersion rheology will be discussed below.
Hard-sphere dispersions exist in the liquid, crystalline, or glassy state depending on the particle volume fraction similar to the temperature-dependent phase transition of atomic or molecular systems. Figure 1.6 demonstrates schematically the hard-sphere phase diagram in terms of particle volume fraction ϕ, constructed by means of light diffraction measurements [12]. At a low volume fraction ϕ particles can diffuse freely and there is no long-range ordering in particle position, that is, the dispersion is in the fluid state, while with increasing concentration above ϕ = 0.50 crystalline and liquid phases coexist in equilibrium and the fraction of crystalline phase increases until the sample is fully crystalline at ϕ = 0.55. With further increasing particle volume fraction, crystallization becomes slower due to reduced particle mobility. At a critical volume fraction ϕ = 0.58 particle mobility is so strongly reduced that no ordered structure can be formed and the dispersion remains in the disordered glassy (immobile) state. Crystalline ordering only occurs if all particles are of equal size, otherwise disordered gel-like structures form at ϕ > 0.5.
Figure 1.6 Hard-sphere phase-diagram constructed from light diffraction measurements [12].
The phase states of hard sphere dispersions are reflected in their characteristic flow curves. Figure 1.7 demonstrates the general features of the shear rate dependence of viscosity at various particle concentrations. At volume fractions up to ϕ = 0.50 the dispersion is in the liquid state and a low-shear Newtonian plateau is observed for the viscosity. The low-shear viscosity, as well as the shear thinning, increases with increasing particle volume fraction ϕ. In the two-phase region colloidal hard-sphere dispersions may show thixotropic behavior (see the curves in Figure 1.7 at ϕ = 0.52), due to the shear induced destruction and subsequent recovery of sample structure, associated with coexisting liquid and crystalline domains. The degree of thixotropy, if any, depends on the measuring conditions. For a particle volume fraction of ϕ ≥ 0.55 dispersions are in the crystalline or gel-like state and show shear thinning behavior in the whole shear rate range investigated. On the other hand, thixotropy vanishes since no long-range particle rearrangements are possible due to the dense particle packing.
Figure 1.7 Viscosity versus shear rate for hard sphere dispersions at various volume fractions ϕ. The downward and upward arrows indicate the viscosity measurement with increasing and consequent decreasing shear rate, respectively.
Viscosity in the low shear Newtonian plateau, referred to as zero-shear viscosity η0, depends only on the total volume occupied by the particles and is independent of particle size. The solvent viscosity ηs always acts as a constant pre-factor, and in the following we will focus on the relative viscosity ηr = η/ηs. Various models describing the volume fraction dependence of the zero-shear viscosity have been proposed. The classical model of Einstein [13, 14] for infinitely dilute, non-interacting hard spheres showed that single particles increase the viscosity of the dispersion medium as a linear function of the volume fraction ϕ according to the equation:
1.24 
The Einstein equation applies to ϕ < 0.01, assuring that the flow around a particle does not influence the velocity field of any other particle. At higher particle concentration the hydrodynamic interactions between particles become important and higher-order terms in ϕ have to be considered. The effect of two-sphere hydrodynamic interactions on the suspension viscosity was calculated by Batchelor [15]:
1.25 
This equation is validated to ϕ < 0.1. For higher particle concentrations multi-particle interactions become imperative and a prediction of viscosity from first principles is still lacking. Numerous phenomenological equations have been introduced to correlate the viscosity of concentrated dispersions to the particle volume fraction. Krieger and Dougherty [16] proposed a semi-empirical equation for the concentration dependence of the viscosity:
1.26 
where ϕmax is the maximum packing fraction or the volume fraction at which the zero shear viscosity diverges. This equation reduces to the Einstein relation (Equation 1.24) at low particle concentration. Quemada [17] suggested another phenomenological model to predict the ηr(ϕ) dependence:
1.27 
This model suits best as ϕ → ϕmax. Figure 1.8 shows the volume fraction dependence of relative viscosity, according to the models described above.
Figure 1.8 Schematic representation of the volume fraction dependence of relative viscosity ηr according to the Einstein, Batchelor, Krieger–Dougherty, and Quemada models.
The absolute value for the maximum packing fraction ϕmax is determined by the packing geometry, which depends on the particle shape and particle size distribution but not on particle size. The volume fraction at maximum packing has been calculated by theoretical models and different ϕmax values have been found depending on the type of packing. The ϕmax value for hard spheres is often taken as 0.64 [18], which is the value associated with random close packing. However, experiments on colloidal hard sphere dispersions have shown that zero-shear viscosity diverges at the volume fraction of the colloidal glass transition ϕg = 0.58 [19–22]. Above ϕg, particle diffusion is restricted to small “cages” formed by the nearest neighbors; correspondingly, the long-time self-diffusion coefficient decreases to zero and the viscosity diverges. The latter two quantities are related to each other by the generalized Stokes–Einstein equation:
1.28 
Let us now consider the shear rate dependence of dispersion viscosity in the liquid state. The transition from low shear to high shear plateau referred to as the shear-thinning region depends on the balance between Brownian and hydrodynamic forces. The Péclet number Pe is a useful dimensionless quantity to express the relative importance of these two contributions:
1.29 
where a is the particle size, kBT is the thermal energy, and D0 = D (ϕ → 0) is the diffusion coefficient.
The Péclet number is often called the dimensionless shear rate; equivalently, the dimensionless shear stress σr can be expressed as follows:
1.30 