UvA-DARE is a service provided by the library of the University of Amsterdam (http://dare.uva.nl)

UvA-DARE (Digital Academic Repository)

Understanding the rheology of yield stress materials

Paredes Rojas, J.F.

Link to publication

Citation for published version (APA):Paredes Rojas, J. F. (2013). Understanding the rheology of yield stress materials.

General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s),other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).

Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, statingyour reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Askthe Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam,The Netherlands. You will be contacted as soon as possible.

Download date: 15 Apr 2020

1

Introduction

1.1 Yield stress materials

Materials can be classified in a very general way into elastic solids and Newtonian

liquids. The former is a material that will deform when a force is applied to it,

but returns to its initial shape after the force is removed. For the latter material,

there is a constant shear viscosity (η) that represents its mechanical properties;

however, this is temperature and pressure dependent [1].

In addition, many everyday materials are neither elastic solids nor Newtonian

fluids. For example, there are shear-thinning fluids [2–12] for which the shear

viscosity decreases with increasing deformation rate, and shear-thickening fluids

[2, 3, 7, 8, 13–19] for which the viscosity increases with increasing deformation rate.

In addition to these, there are materials that respond elastically to small applied

stresses (σ) but flow once a threshold stress is exceeded [20]; this threshold is called

the yield stress (σy) and these type of materials are called yield stress materials.

Classical examples of yield stress materials are paints, foams, wet cement, cleansing

creams, mayonnaise and toothpaste.

Yield stress materials are widely used in industrial, pharmaceutical and cosmetic

applications. Therefore, the physical and flow properties of this type of materials

are important for their formulation and handling [21]. In this sense, it is extremely

1

Chapter 1

important to know the yield stress, as this is the stress at which the material starts

to flow.

The classical description of yield stress materials was initiated by Bingham [22],

who captured the phenomenological behavior of these in the following way:

σ = σy + f(γ) if σ > σy

and

γ = 0 if σ ≤ σy (1.1)

where σ is the shear stress, σy the yield stress and f(γ) some function of the

deformation or shear rate γ. Thus, if σ < σy there is no flow, and if σ ≥ σy the

stress is a monotonically increasing function of γ [23].

The most commonly used model for describing yield stress materials is the Herschel-

Bulkley model [24], which presupposes a well-defined yield stress:

σ = σy +K · γn (1.2)

whereK and n are adjustable model parameters. After the yield stress is overcome,

if n = 1 the material flows with a constant viscosity, if n < 1 the viscosity decreases

with γ (shear-thinning behavior), and if n > 1 the viscosity increases with γ (shear-

thickening behavior).

1.1.1 Problems when dealing with yield stress materials

The main problem encountered by scientists and engineers dealing with yield stress

materials is that the yield stress of a given material is very difficult to determine

[23, 25–28]. Indeed, it has been demonstrated that variations of more than one

order of magnitude can be obtained depending on the way the yield stress is

measured [29–33].

Because it is necessary to define a yield stress value for use in engineering design

and operation of processes, it was proposed to work with two yield stresses—one

static and one dynamic—or even with a range of yield stresses (see Mujudumdar

et. al. [26]): the static yield stress would be the stress above which the material

2

Introduction

Figure 1.1: Yield stress materials: Silicone oil-in-water emulsion (a), Castoroil-in-water emulsion (b), Commercial foam (c), Commercial hair gel (d).

turns from a solid state to a liquid one, while the dynamic yield stress is the stress

where the material turns from a liquid state to a solid one [34].

The second problem is that when yield stress materials are made to flow, most of

the time shear banding occurs: part of the material flows but another part remains

quiescent.

In addition to the above-mentioned problems, it has been shown that yield stress

materials can exhibit thixotropy [34, 35], i.e., the shear viscosity decreases con-

tinuously over time when a sample that has been previously at rest is made to

flow.

Yield stress materials that have been widely studied are emulsions, foams and

gels, because of their wide applicability; therefore, I use these materials as model

systems for studying the flow behavior of yield stress materials (Figure 1.1).

3

Chapter 1

1.2 Emulsions

An emulsion is defined, according to the International Union of Pure and Applied

Chemistry IUPAC, as “a dispersion of droplets of one liquid in another one with

which it is incompletely miscible” [36]. The dispersed liquid is normally called the

internal or dispersed phase, and it is in the form of spherical droplets with a diame-

ter that varies between 1-100 µm (in some special cases, between 0.1-500 µm [37]).

The liquid in which the droplets are dispersed is normally called the continuous

phase. Examples of emulsions are mayonnaise, margarine, milk, coffee whiteners,

cosmetic and pharmaceutical creams and cortisol cream.

Emulsions are thermodynamically unstable, due to the large positive interfacial

free energy of the dispersed system, which decreases with the reduction of the

interfacial area [38]. Therefore, the dispersed droplets eventually coalesce, leading

to separation of the constituent liquids.

Drop coalescence is controlled by adding emulsifiers, normally surfactants; these

are amphiphilic materials that adsorb at the drop interface and produce some

interdrop repulsion according to a variety of static and dynamic phenomena [39].

In addition, certain solid nanoparticles and polymers can be used as emulsifiers, as

they also adsorb at the liquid-liquid interface, reducing drop coalescence [40–43].

1.2.1 Emulsions and yield stress

The flow properties of emulsions are determined by several variables, such as

viscosity of the liquids, type and concentration of emulsifier, temperature and

drop size of the dispersed droplets. A parameter of paramount importance is the

dispersed phase content, normally called internal volume fraction (φ) given by the

ratio of the volume of the dispersed phase to the total volume of the emulsion.

When φ is higher than a critical value (φc), the dispersed droplets jam and a

yield stress appears, due to the short-range repulsive forces between the droplets

[34, 44, 45]. Thus, for φ > φc emulsions will not flow unless the yield stress is

overcome. The value of φc ≈ 0.64, but it is highly dependent on the polydispersity

of the emulsion [46–48]. In addition, it has been reported that by adding clays

or by increasing the concentration of surfactant, it is possible to induce attractive

4

Introduction

forces between the droplets, which leads to thixotropy [34, 35, 49, 50], making the

determination of the yield stress complicated.

On the one hand, after emulsions with φ > φc are made to flow, they behave like

shear-thinning fluids, and the viscosity decreases with increasing shear rate. On

the other hand, emulsions with φ < φc behave either like Newtonian liquids at low

shear rates or like shear-thinning fluids at high shear rates [45].

1.3 Foams compared to emulsions

Foam is defined, according to the International Union of Pure and Applied Chem-

istry, as “a dispersion in which a large proportion of gas by volume in the form

of gas bubbles, is dispersed in a liquid solid or gel. The diameter of the bubbles is

usually larger than 1 µm” [36].

The mechanical behavior of foams is similar in many aspects to that of emulsions.

At small stresses, foams and emulsions behave as viscoelastic solids, while above

the yield stress they flow [1, 47]. For foams, the yield stress is also given by the

repulsive interactions between the bubbles [51].

1.4 Carbopol

Carbopol are crossed linked acrylic acid resins that form small “sponges” when

dispersed in water. These sponges swell in water under neutral pH, so that con-

centrations of carbopol higher than approximately 0.035 %wt are sufficient for the

particles to jam; thus a yield stress arises [52]. People refer to carbopol suspen-

sions as carbopol ‘gels’; however, when carbopol shows yield stress behavior, it is

not a gel in the sense that there is no percolating network of polymers connected

by chemical bonds, but rather a concentrated suspension of sponge-like particles

[25].

Emulsions, foams and carbopol ‘gels’ behave in a similar way in that they exhibit a

yield stress when the amount of dispersed phase is high enough. In addition, they

do not exhibit thixotropic behavior, which makes them excellent model systems

for studying the flow properties of yield stress materials.

5

Chapter 1

1.5 Scope of this thesis

This thesis focuses on the study of the flow properties of yield stress materials.

To this aim, I use model systems such as emulsions, foams and carbopol ‘gels’,

together with rheological measurements and flow visualization techniques. This

thesis is divided as follows:

• Chapter 2 : in this chapter I describe the experimental techniques and the

preparation protocols of the materials studied in this thesis.

• Chapter 3 : this chapter compares a ‘simple’ yield stress material with a

thixotropic yield stress material. It is shown that for a ‘simple’ yield stress

material the yield stress is a well defined value, while for the thixotropic one

the yield stress is ill-defined.

• Chapter 4 : in this chapter I show the different types of shear banding

that can occur in both a ‘simple’ and a thixotropic emulsion. For both

of them, shear banding manifests if the shear stress is inhomogeneous or

due to wall slip; in addition, the thixotropic emulsion exhibits shear banding

when sheared below a critical shear rate value.

• Chapter 5 : in this chapter I use a transparent emulsion to show that wall

slip can be controlled by modifying the wettability of the shearing walls.

This enables the disentanglement of two effects leading to gap-dependent

viscosities: wall slip and cooperative effects.

• Chapter 6 : in this chapter I use emulsions with different internal volume frac-

tions to show that the entire mechanical behavior below and above jamming

can be understood in a way analogous to a phase transition. In addition, a

two-state model is proposed to describe the transition between jammed and

unjammed states.

• Chapter 7 : in this chapter I compare different methods for measuring the

yield stress of emulsions, foams and carbopol gels, providing a physical ex-

planation of what each of these methods measure.

• Chapter 8 : in this chapter I show partial results of topics that deserve more

research, such as rheological behavior of mayonnaise and margarine, and

6

Introduction

stored stress in yield stress materials. In addition, I suggest experiments,

with the aim of improving our current understanding regarding the mechan-

ical behavior of yield stress materials.

References

[1] R. G. Larson, The Structure and Rheology of Complex Fluids. Oxford Uni-

versity Press, Inc., 1999.

[2] C. E. Chaffey and I. Wagstaff, “Shear thinning and thickening rheology. 2.

volume fraction and size dispersion of particles,” J. Colloid Interface Sci.,

vol. 59, pp. 63–75, 1977.

[3] I. Wagstaff and C. E. Chaffey, “Shear thinning and thickening rheology. 1.

concentrated acrylic dispersions,” J. Colloid Interface Sci., vol. 59, pp. 53–62,

1977.

[4] M. M. Cross, “Relation between viscoelasticity and shear-thinning behaviour

in liquids,” Rheol. Acta, vol. 18, pp. 609–614, 1979.

[5] S. T. Milner, “Relating the shear-thinning curve te the molecular weight

distribution in linear polymer melts,” J. Rheol., vol. 40, pp. 303–315, 1996.

[6] S. M. Al-Zahrani, “A generalized rheological model for shear thinning fluids,”

J. Petro. Sci. Eng., vol. 17, pp. 211–215, 1997.

[7] L. Bergstrom, “Shear thinning and shear thickening of concentrated ceramic

suspensions,” Colloid Surf. A, vol. 133, pp. 151–155, 1998.

[8] M. C. Newstein, H. Wang, N. P. Balsara, A. A. Lefebvre, Y. Shnidman,

H. Watanabe, K. Osaki, T. Shikata, H. Niwa, and Y. Morishima, “Microstruc-

tural changes in a colloidal liquid in the shear thinning and shear thickening

regimes,” J. Chem. Phys., vol. 111, pp. 4827–4838, 1999.

[9] G. P. Roberts, H. A. Barnes, and P. Carew, “Modelling the flow behaviour of

very shear-thinning liquids,” Chem. Eng. Sci., vol. 56, pp. 5617–5623, 2001.

7

Chapter 1

[10] M. P. Escudier, I. W. Gouldson, A. S. Pereira, F. T. Pinho, and R. J. Poole,

“On the reproducibility of the rheology of shear-thinning liquids,” J. Non-

Newtonian Fluid Mech., vol. 97, pp. 99–124, 2001.

[11] S. Forster, M. Konrad, and P. Lindner, “Shear thinning and orientational

ordering of wormlike micelles,” Phys. Rev. Lett., vol. 94, p. 017803, 2005.

[12] P. T. Reardon, S. Feng, A. L. Graham, V. Chawla, R. S. Admuthe, and

J. Abbott, “Shear-thinning of polydisperse suspensions,” J. Phys. D: Appl.

Phys., vol. 41, p. 115408, 2008.

[13] A. Fall, N. Huang, F. Bertrand, G. Ovarlez, and D. Bonn, “Shear thickening of

cornstarch suspensions as a reentrant jamming transition,” Phys. Rev. Lett.,

vol. 100, p. 018301, 2008.

[14] W. J. Frith, P. d’Haene, R. Buscall, and J. Mewis, “Shear thickening in model

suspensions of sterically stabilized particles,” J. Rheol., vol. 40, pp. 531–548,

1996.

[15] J. Bender and N. J. Wagner, “Reversible shear thickening in monodisperse

and bidisperse colloidal dispersions,” J. Rheol., vol. 40, pp. 899–916, 1996.

[16] S. R. Raghavan and S. A. Khan, “Shear-thickening response of fumed silica

suspensions under steady and oscillatory shear,” J. Colloid Interface Sci.,

vol. 185, pp. 57–67, 1997.

[17] R. L. Hoffman, “Explanations for the cause of shear thickening in concen-

trated collidal suspensions,” J. Rheol., vol. 42, pp. 111–123, 1998.

[18] Y. T. Hu, P. Boltenhagen, and D. J. Pine, “Shear thickening in low-

concentration solutions of wormlike micelles. i. direct visualization of transient

behavior and phase transitions,” J. Rheol., vol. 42, pp. 1185–1208, 1998.

[19] Y. T. Hu, P. Boltenhagen, E. Matthys, and D. J. Pine, “Shear thickening in

low-concentration solutions of wormlike micelles. ii. slip, fracture, and stabil-

ity of the shear-induced phase,” J. Rheol., vol. 42, pp. 1209–1226, 1998.

[20] A. Fall, F. Bertrand, G. Ovarlez, and D. Bonn, “Yield stress and shear band-

ing in granular suspensions,” Phys. Rev. Lett., vol. 103, p. 178301, 2009.

[21] P. Coussot, Q. D. Nguyen, H. T. Huynh, and D. Bonn, “Avalanche behavior

in yield stress fluids,” Phys. Rev. Lett., vol. 88, p. 175501, 2002.

8

Introduction

[22] E. C. Bingham, Fluidity and Plasticity. McGraw-Hill, New York, 1922.

[23] D. Bonn and M. M. Denn, “Yield stress fluids slowly yield to analysis,” Sci-

ence, vol. 324, pp. 1401–1402, 2009.

[24] W. Herschel and R. Bulkley, “Measurement of consistency as applied to

rubber-benzene solutions,” Proc. Am. Assoc. Test Mater., vol. 26, pp. 621–

633, 1926.

[25] P. Møller, A. Fall, V. Chikkadi, D. Derks, and D. Bonn, “An attempt to

categorize yield stress fluid behavior,” Phil. Trans. Roy. Soc. A, vol. 367,

pp. 5139–5155, 2009.

[26] A. Mujumdar, A. N. Beris, and A. B. Metzner, “Transient phenomena in

thixotropic systems,” J. Non-Newtonian Fluid Mech., vol. 102, pp. 157–178,

2002.

[27] D. C. H. Cheng, “Yield stress: A time-dependent property and how to mea-

sure it,” Rheol. Acta, vol. 25, pp. 542–554, 1986.

[28] F. Varnik, L. Bocquet, J.-L. Barrat, and L. Berthier, “Shear localization in a

model glass,” Phys. Rev. Lett., vol. 90, p. 095702, 2003.

[29] A. E. James, D. J. A. Williams, and P. R. Wiliams, “Direct measurement of

static yield properties of cohesive suspensions,” Rheol. Acta, vol. 26, pp. 437–

446, 1987.

[30] L. Zhu, N. Sun, K. Papadopoulos, and D. D. Kee, “A slotted plate device for

measuring static yield stress,” J. Rheol., vol. 45, p. 1105, 2001.

[31] Q. D. Nguyen and D. V. Boger, “Measuring the flow properties of yield stress

fluids,” Annu. Rev. Fluid Mech., vol. 24, pp. 47–88, 1992.

[32] H. A. Barnes, “The yield stress—a review or ′παντα ρει′— everything flows?,”

J. Non-Newtonian Fluid Mech, vol. 81, pp. 133–178, 1999.

[33] H. A. Barnes and Q. D. Nguyen, “Rotating vane rheometry—a review,” J.

Non-Newtonian Fluid Mech., vol. 98, pp. 1–14, 2001.

[34] A. Fall, J. Paredes, and D. Bonn, “Yielding and shear banding in soft glassy

materials,” Phys. Rev. Lett., vol. 105, p. 225502, 2010.

9

Chapter 1

[35] J. Paredes, N. Shahidzadeh-Bonn, and D. Bonn, “Shear banding in

thixotropic and normal emulsions,” J. Phys-Condens Mat, vol. 23, p. 284116,

2011.

[36] International Union of Pure and Applied Chemistry, Manual of Symbols and

Terminology for Physicochemical Quantities and Units, 1972.

[37] J.-L. Salager, “Cuaderno FIRP S210A Formulacion HLB, PIT, R de Winsor,”

1998.

[38] W. Kegel, Interfacial Properties and Phase Bahaviour of an Ionic Microemul-

sion System. PhD thesis, Universiteit Utrecht, Faculteit Scheikunde, 1993.

[39] M. Rosen, Surfactants and Interfacial Phenomena. J.Wiley, New York, 1978.

[40] N. Yan, M. Gray, and J. H. Masliyah, “On water-in-oil emulsions stabilized

by fine solids,” Colloids Surf. A, vol. 193, pp. 97–107, 2001.

[41] E. Vignati, R. Piazza, and T. Lockhart, “Pickering emulsions: Interfacial

tension, colloidal layer morphology, and trapped-particle Motion,” Langmuir,

vol. 19, p. 6650 6656, 2003.

[42] G. Romet-Lermonne, J. Daillant, P. Guenoun, J. Yang, D. Holley, and

J. Mays, “Oil-in-water microemulsions stabilized by charged diblock copoly-

mers,” J. Chem. Phys., vol. 122, p. 064703, 2005.

[43] L. Besnard, F. Marchal, J. F. Paredes, J. Daillant, N. Pantoustier, P. Per-

rin, and P. Guenoun, “Multiple emulsions controlled by stimuli-responsive

polymers,” Adv. Mater., vol. 25, pp. 2844–2848, 2013.

[44] T. G. Mason, J. Bibette, and D. A. Weitz, “Elasticity of compressed emul-

sions,” Phys. Rev. Lett., vol. 75, p. 2051, 1995.

[45] J. Paredes, M. A. J. Michels, and D. Bonn, “Rheology across the zero-

temperature jamming transition,” Phys. Rev. Lett., vol. 111, p. 015701, 2013.

[46] T. G. Mason, J. Bibette, and D. A. Weitz, “Yielding and flow of monodisperse

emulsions,” J. Colloid Interface Sci., vol. 179, pp. 439–448, 1996.

[47] F. Rouyer, S. Cohen-Addad, and R. Hohler, “Is the yield stress of aqueous

foam a well-defined quantity?,” Colloid. Surface. A, vol. 263, pp. 111–116,

2005.

10

Introduction

[48] A. Saint-Jalmes and D. J. Durian, “Vanishing elasticity for wet foams: Equiv-

alence with emulsions and role of polydispersity,” J. Rheol., vol. 43, pp. 1411–

1422, 1999.

[49] L. Becu, S. Manneville, and A. Colin, “Yielding and flow in adhesive and

nonadhesive concentrated emulsions,” Phys. Rev. Lett., vol. 96, p. 138302,

2006.

[50] A. Ragouilliaux, G. Ovarlez, N. Shahidzadeh-Bonn, B. Herzhaft, T. Palermo,

and P. Coussot, “Transition from a simple yield-stress fluid to a thixotropic

material,” Phys. Rev. E, vol. 76, p. 051408, Nov 2007.

[51] M. van Hecke, “Jamming of soft particles: geometry, mechanics, scaling and

isostaticity,” J. Phys-Condens. Mat., vol. 22, p. 033101, 2010.

[52] J. M. Piau, “Carbopol gels: Elastoviscoplastic and slippery glasses made of

individual swollen sponges. Meso- and macroscopic properties, constitutive

equations and scaling laws,” J. Non-Newtonian Fluid Mech, vol. 144, pp. 1–

29, 2007.

11