Shear Thickening in Concentrated Soft Sphere Colloidal Suspensions: A Shear Induced Phase Transition
RResearch Article
Shear Thickening in Concentrated Soft Sphere ColloidalSuspensions: A Shear Induced Phase Transition
Joachim Kaldasch, Bernhard Senge, and Jozua Laven Technische Universit¨at Berlin, K¨onigin-Luise-Strasse 22, 14195 Berlin, Germany Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands
Correspondence should be addressed to Joachim Kaldasch; [email protected] 26 September 2014; Accepted 17 December 2014Academic Editor: Brian J. EdwardsCopyright © 2015 Joachim Kaldasch et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.A model of shear thickening in dense suspensions of Brownian soft sphere colloidal particles is established. It suggests that shearthickening in soft sphere suspensions can be interpreted as a shear induced phase transition. Based on a Landau model of thecoagulation transition of stabilized colloidal particles, taking the coupling between order parameter fluctuations and the localstrain-field into account, the model suggests the occurrence of clusters of coagulated particles (subcritical bubbles) by applyinga continuous shear perturbation. The critical shear stress of shear thickening in soft sphere suspensions is derived while reversibleshear thickening and irreversible shear thickening have the same origin. The comparison of the theory with an experimentalinvestigation of electrically stabilized colloidal suspensions confirms the presented approach.
1. Introduction
Concentrated colloidal dispersions are of technological rel-evance for various industrial products such as paints, phar-maceuticals, cosmetics, lubricants, and food. They are oftenprocessed at high shear rates and stresses. In strong viscousflows, colloidal dispersions exhibit a unique transition relatedto an increase of the shear viscosity, termed shear thickening[1]. This effect makes it harder to pump suspensions and cancause equipment damage and failure in the production pro-cesses (flow blockage). Next to reversible shear thickening,also the irreversible aggregation of the dispersion after shearthickening has been reported known as irreversible shearthickening [2]. A fundamental understanding of the relationbetween particle and flow properties of dense colloidalsuspensions is required in order to provide clues minimizingundesired effects or exploiting useful applications of shearthickening [3].It has long been an issue to understand the underlyingmechanisms involved in shear thickening. A number oftheoretical attempts were made explaining shear thickening in sheared suspensions of Brownian particles where inertialeffects can be neglected (non-Brownian particles [4] anddeformable spheres like emulsions, etc. are not consideredhere). We have to distinguish between hard sphere sus-pensions with an interaction potential confined to the bareparticle (Born-) repulsion and soft sphere suspensions with acombination of van der Waals attraction and long-range elec-trostatic or steric repulsion in addition to the Born-repulsion.The dynamics of sheared hard sphere suspensions canbe studied in simulations. In dense suspensions, the particleinteraction is essentially determined by lubrication forces.When particles are pushed together along the compres-sion axis of a sheared suspension, they must overcomethe viscous drag forces between neighbouring particles inorder to move away from each other. Above a critical shearrate, particles stick together generating the formation ofshear induced clusters (hydroclusters) [5–8]. At high volumefractions, these transient touched particles induce jammingand an increase of the viscosity at a critical shear stress[9]. An alternative concept explaining shear thickening isrelated to an order-disorder transition. In this approach, an
Hindawi Publishing CorporationJournal of ermodynamicsVolume 2015, Article ID 153854, 10 pageshttp://dx.doi.org/10.1155/2015/153854
Journal of Thermodynamicsordered, layered structure of colloidal particles in a shearedsuspension becomes unstable above a critical shear rate[10–13]. Shear thickening occurs when lubrication forcesbetween neighbouring particles cause particles to rotate outof alignment of the sheared structure, destabilizing the flow.A more sophisticated approach is based on an ad hoc mode-coupling model. It describes the instability as a stress-inducedtransition into a jammed state [14–17]. Also suggested is thatdilation of confined suspensions may cause a rapid increaseof the viscosity termed discontinuous shear thickening [18].The presented theory is confined to the study of Browniansoft sphere suspensions. As long as the attractive interactioncan be neglected, soft spheres can be approximated as so-called effective hard spheres and the abovementioned mech-anisms for shear thickening in hard sphere suspensions mayapply. However, soft sphere particles have a much more com-plex interaction than hard spheres. With increasing volumefraction van der Waals attraction causes a coagulation of col-loidal particles, in particular when the repulsive stabilizationof the particles is small. For soft sphere suspensions, the hardsphere approach to shear thickening has to be extended. Anapproach aiming at understanding shear thickening in softsphere suspensions is an activation model [19–21]. It takesadvantage from the complex interaction potential and sug-gests that colloidal particles arranged along the compressionaxis of a sheared suspension may overcome the mutual repul-sion at a critical shear stress. As a result, the viscosity increaseswhen clusters of coagulated particles are formed. For a suffi-ciently strong attraction between particles, these clusters can-not be disrupted by the applied shear leading to irreversibleshear thickening. Otherwise, shear thickening is reversible.While the activation model is a microscopic approach,this paper presents an alternative mesoscopic approach toshear thickening in dense soft sphere suspensions. It suggeststhat shear thickening is related to a shear induced phasetransition. The transition is formulated in terms of a thermo-dynamic standard concept known as the Landau model [22,23]. The Landau model is a mean field theory originally devel-oped to understand symmetry breaking phase transitions. Itis a widely accepted theory utilized in particular to modelthe dynamics of fluids and polymers [24]. Here the theoryis used to describe the equilibrium coagulation transitionof soft spheres. Taking the coupling of density fluctuationsto the viscoelastic medium into account, it can be shownthat sheared dense suspensions induce clusters of coagulatedparticles in equivalence to the microscopic activation model.The paper is organized as follows. First, shear thickeningin suspensions of hard and soft spheres is discussed. Afterestablishing a hydrodynamic model for a dense suspensionand deriving a Landau model for the coagulation transition,the models are combined in a subcritical bubble theory topredict the occurrence of coagulated particle clusters. Theapplication of the subcritical bubble approach to sheareddense suspensions allows an estimation of the critical stressfor shear thickening. After comparing the model with anexperimental investigation, the paper ends with concludingremarks on the rheology of concentrated soft sphere suspen-sions in relation to their equilibrium phase diagram.
2. The Model
Hardspheres have an interaction potential that is zero when parti-cles do not overlap and infinite otherwise (Born-repulsion).The phase diagram depends on the volume fraction Φ of thebare particles determined by Φ = 𝜋𝑎 𝑁6𝑉 , (1)where 𝑎 is the particle radius and 𝑁 the number of particlesin the volume 𝑉 . As shown by simulations, monodispersehard sphere suspensions form a liquid phase and a (face-centred cubic) solid phase for volume fractions Φ > 0.54 in the absence of flow. However, even for a polydispersityof the particles > Φ > 0.58 , the relaxation time becomes largecompared to typical experimental time scales. The systemdoes not relax anymore. This jammed state is called a colloidalglass [25, 26]. Approaching the jamming volume fraction Φ 𝑗 ,the apparent shear viscosity of a hard sphere suspension 𝜂 HS diverges with 𝜂 HS (Φ) ∼ 1(Φ − Φ 𝑗 ) 𝛽 (2)while 𝛽 ≈ 2 is the critical exponent for low shear rates ̇𝛾 →0 [27]. The mode-coupling theory predicts that a dynamicglass transition occurs already at Φ ≈ 0.516 . It suggeststhat approaching the jamming volume fraction large densityfluctuations with glass-like dynamics occur in concentratedsuspensions. They determine the internal relaxation time of aconcentrated suspension and are related to the divergence ofthe viscosity approaching the jamming transition by 𝜏 HS ∼ 𝜂 HS (Φ) ∼ (Φ − Φ 𝑗 ) −𝛽 . (3)The flow properties of sheared dense suspensions areessentially governed by 𝜏 HS . If the applied shear rate is muchsmaller than the inverse relaxation time ̇𝛾𝜏 HS ≪ 1 , densityfluctuations disappear before they can be perturbed. Forthe case ̇𝛾𝜏 HS ≈ 1 , however, they are deformed by theconvective shear flow. As a consequence, density fluctuationsare compressed along the compression axis and stretchedalong the elongation axis of the sheared suspension whilerotating in time as schematically displayed in Figure 1. Thiseffect causes a decrease of the apparent viscosity of hardsphere suspensions termed shear thinning [28].The deformation of density fluctuations leads to the com-pression of particles arranged along the compression axis. Tomove away from each other, they must overcome the viscousdrag forces created by the small gaps between neighbouringparticles. This lubrication effect determines the characteristicseparation time 𝜏 𝑃 of a pair of particles. For shear rates ̇𝛾𝜏 𝑝 ≪1 , there is sufficient time for hard spheres to detach. However,for ̇𝛾𝜏 𝑝 ≫ 1 , particles arranged along the compressionaxis start to form transient clusters (so-called hydroclusters).Shear thickening occurs in this view when the applied shearournal of Thermodynamics 3 Elongation axisCompression axis 𝜎 > 0 𝜎 = 0V f xy Figure 1: Displayed is the deformation of a large density fluctuationof volume 𝑉 𝑓 in a simple shear geometry for an applied shear stress 𝜎 . rate is ̇𝛾 HS 𝐶 𝜏 𝑝 ≈ 1 [8]. Based on this approach, several relationshave been established characterizing the critical shear stress[2]. As a rule of thumb, the onset of shear thinking canbe estimated for Brownian colloids by the relation betweenBrownian and hydrodynamic forces governed by a Pecletnumber Pe = 6𝜋𝜂 𝑆 𝑎 ̇𝛾/𝑘 𝐵 𝑇 , where 𝑘 𝐵 is the Boltzmannconstant, 𝑇 the temperature, and 𝜂 𝑆 the Newtonian viscosityof the solvent. Shear thickening occurs for hard spheres whenthe shear stress 𝜎 increases thermal diffusion of the particleswhen Pe > 100 . Hence, the critical stress for the onset of shearthickening is of the order [2] 𝜎 HS 𝐶 = ̇𝛾 HS 𝐶 𝜂 𝑆 ≈ 50𝑘 𝐵 𝑇3𝜋𝑎 . (4)Shear thickening in hard sphere suspensions occurs as agradual increase of the viscosity with increasing shear rate(continuous shear thickening), since lubrication forces pro-hibit a direct contact of the particles. Discontinuous shearthickening, related to a rapid increase of the viscosity at thecritical shear rate ̇𝛾 HS 𝐶 , occurs only when the particles comesufficiently close such that lubrication breaks down and thesurface roughness of the particles comes into play [18]. The hard sphere model is anidealization of the colloidal particle interaction. In practicalapplications mainly soft sphere suspensions with electrostaticor steric repulsion are utilized. For convenience, we wantto confine our considerations to soft sphere suspensions ofelectrically stabilized monodisperse colloidal particles. TheDLVO theory states that the total two-particle interactionpotential can be expressed as the sum of the double-layerpotential 𝑈 el and van der Waals attraction 𝑈 vdW (note that forhigh volume fractions short range surface forces and also thesurface roughness may come into play not taken into accounthere; for more details, see [20]): 𝑈 (ℎ) = 𝑈 el (ℎ) + 𝑈 vdW (ℎ) , (5)where ℎ is the surface-to-surface distance. In numerous cases,a simple equation derived by Hogg et al. [29] was found to bea suitable approximation, which for our case can be expressedfor a constant surface potential by 𝑈 CP (ℎ) = 2𝜋𝜀 𝜀 𝑟 𝑎𝜁 ln (1 + 𝑒 −𝜅ℎ ) (6) Primaryminimum SecondaryminimumEffective hard spheres
U(h) hh h max Figure 2: Displayed are two interaction potentials
𝑈(ℎ) of electri-cally stabilized colloidal particles as a function of the surface-to-surface distance ℎ . The combination of electrostatic repulsion andvan der Waals attraction leads to the occurrence of a primary anda secondary potential minimum, while ℎ max indicates the potentialmaximum and ℎ the average mutual distance of the particles. Forhighly stabilized particles (effective hard spheres), the secondaryminimum disappears. and for a constant surface charge approach of the particles by 𝑈 CC (ℎ) = −2𝜋𝜀 𝜀 𝑟 𝑎𝜁 ln (1 − 𝑒 −𝜅ℎ ) , (7)where we approximated the surface potential by the 𝜁 -poten-tial. The parameters 𝜀 and 𝜀 𝑟 are the absolute and relativedielectric constants. The Debye reciprocal length 𝜅 is definedby 𝜅 = √ 2𝐶 𝑆 𝑁 𝐴 𝑍 𝑒 𝜀 𝜀 𝑟 𝑘 𝐵 𝑇 (8)while 𝑒 is the elementary electric charge, 𝑁 𝐴 the Avogadronumber, 𝑍 the ionic charge number, and 𝐶 𝑆 the salt concen-tration. The nonretarded van der Waals attraction betweentwo spheres can be described by 𝑈 vdW (ℎ) = − 𝐴𝑎12ℎ , (9)where the effective Hamaker constant 𝐴 is determined bythe dielectric constants of the solvent-particle combination.The corresponding interaction potential is schematicallydisplayed in Figure 2. It consists of a primary and a secondaryminimum due to van der Waals attraction and a repulsivepotential barrier caused by the electric double layer.While the hard sphere phase diagram is a function ofthe volume fraction, the phase diagram of soft spheres ismuch richer. The two-particle interaction potential dependson many variables characterizing the impact of attractive andrepulsive forces. We want to confine the discussion here totwo variables: the volume fraction Φ and the Debye screeningreciprocal length 𝜅 . The latter characterizes the electrostatic Journal of Thermodynamics Effective hard spheresPotential barrier Vapor SolidLiquidFlocculation Coagulation
F(Ψ) Ψ L = 0 Ψ s Ψ 𝜅Φ C Φ𝜎 = 𝜎 C Figure 3: Schematically displayed are the Landau free energies of a stabilized dense suspension in equilibrium near the liquid-solid(coagulation) transition (solid line) and at the critical shear stress 𝜎 = 𝜎 𝐶 (dashed line). Also displayed schematically is the equilibrium phasediagram of electrically stabilized colloidal suspensions as a function of the volume fraction Φ and Debye screening parameter 𝜅 [30–32]. Itshows the first order transition lines separating the colloidal vapor, liquid, and solid phase. interaction while an increasing 𝜅 indicates decreasing electro-static repulsion. The equilibrium phase diagram is schemat-ically displayed in Figure 3 [30–32]. The phase diagramof electrically stabilized monodisperse colloidal particlesseparates into colloidal vapour, liquid, and solid phase (theglass state arises for polydisperse particles). For high 𝜅 andlow Φ , Brownian particles with an attractive interaction forma vapour phase that undergoes a flocculation (gel) transitioninto a solid with increasing volume fraction. The binodal linesof this first order phase transition are displayed in Figure 3.With decreasing 𝜅 , the vapour phase becomes more stableand turns into a liquid phase. In this phase, Brownian par-ticles have to overcome an increasing electrostatic potentialbarrier. The transition into (fcc-) colloidal crystal occurs atvolume fraction Φ 𝐶 . This first order liquid-solid transitionis termed coagulation. At high volume fractions, however,suspensions do not exhibit a fast phase separation due to theslow dynamics of the bare (hard sphere) particles. Instead, theparticles are captured in metastable glassy states.The coagulation transition induced by Brownian motionat low volume fractions (also termed perikinetic coagulation)can be described as an activation process determined bythe frequency ] of bond-forming events. In equilibrium,particles may overcome their repulsion with the frequency ] by thermal excitations. The nucleation rate is governed for apair of colloidal particles by ] = ] exp (− 𝑈 𝐵 𝑘 𝐵 𝑇 ) , (10) where 𝑈 𝐵 is the repulsive energy barrier and ] the collisionfrequency. The potential barrier formed by the two-particleinteraction has the form 𝑈 𝐵 = 𝑈 (ℎ max ) − 𝑈 (ℎ ) (11)while ℎ is the equilibrium particle surface-to-surface dis-tance and ℎ max expresses the distance to the maximum of theinteraction potential (Figure 2).In sheared suspensions, colloidal particles arranged alongthe compression axis are pushed together. The key idea tounderstand shear thickening in soft sphere suspensions is torealize that these particles may also overcome their mutualrepulsion and form transient bonds. For a sheared suspen-sion, we have to correct the undisturbed activation modelequation (10) by taking a perturbation due to the appliedshear stress 𝜎 into account. For a pair of colloidal particlesarranged along the compression axis, the perturbed fre-quency can be written as [19] ] ∼ exp (− 𝑈 𝐵 − 𝜎𝑉 ∗ 𝑘 𝐵 𝑇 ) , (12)where the activation volume 𝑉 ∗ is of the order of the free vol-ume per particle: 𝑉 ∗ = 43 𝜋𝑎 Φ 𝐶 Φ . (13)The critical shear stress at which the frequency becomes amaximum is determined by 𝜎 𝐶 = 𝑈 𝐵 𝑉 ∗ = 𝜂 ̇𝛾 𝐶 . (14)ournal of Thermodynamics 5The critical stress is a function of the repulsive potentialbarrier and the free volume. It scales with 𝑎 −3 like (4) for hardspheres. The nucleation rate can be rewritten with (14) as ] ∼ exp (− 𝜎 𝐶 − 𝜎𝑘 𝐵 𝑇 ) . (15)The critical shear stress 𝜎 𝐶 can be also expressed as theproduct of a viscosity 𝜂 and a critical shear rate ̇𝛾 𝐶 . Sincethe critical shear stress is nearly independent of the volumefraction (see below), the corresponding critical shear rate isgoverned by the divergence of the viscosity due to lubricationforces with increasing volume fraction 𝜂(Φ) ∼ 𝜂 HS (Φ) . Thecritical shear rate for soft spheres is therefore determined by ̇𝛾 𝐶 (Φ) = 𝜎 𝐶 (Φ)𝜂 (Φ) ≈ 𝐶 𝑂 (Φ − Φ 𝐶 ) 𝛽 , (16)where 𝐶 𝑂 is a free parameter.For volume fractions Φ > Φ 𝑝 (where Φ 𝑝 is the percola-tion volume fraction), clusters of aggregated colloidal parti-cles may even span the entire suspension and form jammedforce chains along the compression axis of the sheared sus-pension. These force chains can cause dilation of the shearedsuspension at 𝜎 𝐶 , often observed in connection with discon-tinuous shear thickening [2]. Differently from the two-particle approach of the activationmodel, in this section a continuous model is established. Forthis purpose, we first determine the rheological properties ofdense suspensions in a hydrodynamic approach. A colloidalsuspension close to a phase transition can be described by aLandau model and is termed here as near-critical suspension.Taking advantage from these models, the critical shearstress for the occurrence of aggregated colloidal particles isobtained from a subcritical bubble approach. (1) The Hydrodynamic Model.
We want to specify the rheo-logical properties of a near-critical suspension by applying atwo-fluid model. The suspending medium of volume fraction , with density 𝜌 𝑠 and viscosity 𝜂 𝑆 , is treated as the firstfluid. The second fluid is a continuous medium formed by theensemble of colloidal particles of volume fraction Φ , density 𝜌 𝑝 , and viscosity 𝜂 𝑃 . The mass densities of the two fluids, 𝜌 𝑠 and 𝜌 𝑝 , are convected by the velocities v 𝑝 and v 𝑆 governed bythe conservation relations: 𝜕𝜌 𝑆 𝜕𝑡 = −∇ (𝜌 𝑆 v 𝑆 ) ; 𝜕𝜌 𝑃 𝜕𝑡 = −∇ (𝜌 𝑝 k 𝑝 ) . (17)The average velocity v is defined by v = 1𝜌 (𝜌 𝑆 v 𝑆 + 𝜌 𝑝 v 𝑝 ) , (18)where total density reads 𝜌 = 𝜌 𝑆 + 𝜌 𝑃 . (19) The equations of motion for the two fluids are (1 − Φ) 𝜌 𝑆 𝜕 v 𝑆 ( r , 𝑡)𝜕𝑡 = − ∇𝑝 𝑆 ( r , 𝑡) + ∇ 𝜎 𝑆 ( r , 𝑡) − Ξ w ,Φ𝜌 𝑃 𝜕 v 𝑃 ( r , 𝑡)𝜕𝑡 = − ∇𝑝 𝑃 ( r , 𝑡) + ∇ 𝜎 𝑃 ( r , 𝑡) + Ξ w , (20)where 𝑝 𝑃 , 𝑝 𝑆 and 𝜎 𝑝 , 𝜎 𝑆 are the pressure and viscousstress tensor of the colloidal particle and solvent medium,respectively. The velocity difference between the two fluids is w = k 𝑆 − k 𝑝 . (21)The friction coefficient can be approximated for near-criticalsuspensions by Ξ ∼ 6𝜋𝜂 𝑆 𝜉, (22)where 𝜉 is the correlation length of density fluctuationsformed by colloidal particles. Since 𝜉 is a large quantity fornear-critical suspensions, we confine the discussion here tothe limit of strong coupling w ≈ 0 . In this case, the continuityequation of the suspension has the usual form 𝜕𝜌 ( r , 𝑡)𝜕𝑡 = −∇ (𝜌 ( r , 𝑡) v ( r , 𝑡)) , (23)where the mean velocity is given by (18). However, the vis-cosity of dense suspensions diverges with increasing volumefraction as suggested by (2). It causes a viscoelastic responseof dense suspensions. This effect can be taken into account byapplying a Maxwell model for the momentum relation withthe constitutive equation [33] 𝜕𝜎 ( r , 𝑡)𝜕𝑡 + 1𝜏 𝜎 ( r , 𝑡) = 𝐺 𝜕𝛾 ( r , 𝑡)𝜕𝑡 (24)while 𝛾 is a shear deformation and 𝐺 an effective shearmodulus. The mechanical properties of a viscoelastic mediumdepend on the relaxation time 𝜏 given for a dense suspensionby (3). For Φ → 0 , the suspension behaves as a viscousfluid because 𝜏 → 0 . For large volume fractions, however,the relaxation time diverges ( 𝜏 → ∞ ) and the suspensionresponds as an elastic solid. (2) The Landau Model.
The phase diagram displayed inFigure 3 suggests that a first order liquid-solid coagulationtransition occurs in concentrated monodisperse soft spheresuspensions at rest. In order to describe this phase tran-sition, we take advantage from a Landau model [22]. Forthis purpose, an order parameter characterizing the densitydifference between the average density of the colloidal liquidphase 𝜌 and the local density can be defined by Ψ ( r , 𝑡) = 𝜌 ( r , 𝑡) − 𝜌 , (25)where the vector r indicates the spatial location. The spatiallyaveraged order parameter ⟨Ψ⟩ has the property to be zeroin the liquid phase and nonzero in the coagulated solidphase. In thermal equilibrium, the free energy density canbe established as a Taylor expansion of the order parameter Journal of Thermodynamicsaround the instability. Expanding the free energy up to theforth order in Ψ( r , 𝑡) , we obtain the standard Ginzburg-Landau free energy scaled by the thermal energy 𝑘 𝐵 𝑇 [23]: 𝐹 (Ψ) = 1𝑘 𝐵 𝑇 ∫ ( 𝜄2 |∇Ψ( r , 𝑡)| + 𝛼2 Ψ( r , 𝑡) + 𝜆3 Ψ ( r , 𝑡) + 𝜒4 Ψ ( r , 𝑡) ) 𝑑 𝑟. (26)The first term takes contributions due to spatial variations ofthe order parameter into account and 𝜄, 𝛼, 𝜆 , and 𝜒 are freeparameters. For simplicity, the first order coagulation transi-tion is treated as weakly first order such that 𝜄, 𝜒 > 0, 𝜆 < 0 ,and |𝜆| ≪ |𝛼|, |𝜒| . It implies that we confine our considera-tions to near-critical colloidal liquid with small 𝜅 (Figure 3).The free energy minimum of the spatially averaged orderparameter ⟨Ψ⟩ corresponds to ⟨Ψ⟩ ≅ {{{{{{{{{0 for 𝛼 ≥ 𝜆 − 𝛼𝜒 for 𝛼 < 𝜆
2𝜒 . (27)For 𝛼 ≫ 0 , the liquid state is stable, since the free energyhas a minimum at ⟨Ψ⟩ = Ψ 𝐿 = 0 . For 𝛼 ≪ 0 , the orderparameter becomes nonzero and describes the solid phasewith a higher stationary mean density ⟨Ψ⟩ , while Ψ ≈ 𝛼/𝜒 .The coagulation transition into a solid occurs when 𝛼(Φ, 𝜅) ≈0 . Hence, the parameter 𝛼 can be expanded near the liquid-solid transition as 𝛼 (Φ, 𝜅) ≅ 𝛼 (Φ − Φ 𝐶 ) (𝜅 𝐶 − 𝜅) , (28)where 𝜅 𝐶 indicates a critical Debye screening parameter and 𝛼 is a free parameter.Near-critical suspensions are characterized by large orderparameter fluctuations with a characteristic size that can beestimated by the correlation length 𝜉 ≈ (𝜕 𝐹/𝜕⟨Ψ⟩ ) −1/2 . (Fora second order transition, the correlation length diverges andthe fluctuations become scale invariant, such that no charac-teristic length exists. However, for first order transitions, thecorrelation length remains finite.) The time evolution of theorder parameter fluctuations is determined by [23] 𝑑Ψ ( r , 𝑡)𝑑𝑡 = −Γ∇ 𝛿𝐹 (Ψ)𝛿Ψ ( r , 𝑡) , (29)where Γ is a kinetic coefficient, 𝛿 indicates a variational der-ivation of the free energy, and we used that the order param-eter Ψ is conserved. For the lowest order contribution of theorder parameter, we obtain in the Fourier-space with wavevector k 𝜕Ψ ( k , 𝑡)𝜕𝑡 ≅ −Γ𝑘 (𝛼 + 𝑘 𝜄) Ψ ( k , 𝑡) . (30)The relaxation time of order parameter fluctuations becomes,therefore, 𝜏 ∼ 1Γ𝑘 𝛼 (Φ, 𝜅) . (31) Obviously the relaxation time of the density fluctuationsbecomes large for 𝛼(Φ, 𝜅) → 0 . This effect is known ascritical slowing-down.We have to keep in mind that in a dense suspensioninternal deformations relax slowly. This effect can be takeninto account by including a coupling between the orderparameter Ψ( r , 𝑡) and internal deformations of the straincomponent 𝛾( r , 𝑡) . The free energy equation (26) becomes 𝐹 (Ψ, 𝛾) ≅ 1𝑘 𝐵 𝑇 ∫ ( 𝜄2 |∇Ψ ( r , 𝑡)| + 𝛼2 Ψ ( r , 𝑡) + 𝜆3 Ψ( r , 𝑡) + 𝜒4 Ψ( r , 𝑡) + 𝐺2 𝛾 ( r , 𝑡) − 𝜃𝛾 ( r , 𝑡) Ψ ( r , 𝑡) ) 𝑑 𝑟, (32)where the fifths term is the contribution of a shear pertur-bation to the free energy. The last term expresses the lowestorder coupling between a shear deformation component 𝛾( r , 𝑡) and the order parameter Ψ( r , 𝑡) , with the coupling con-stant 𝜃 > 0 . The lowest order coupling must be proportionalto Ψ [34] (a shear deformation cannot generate a nonzeromean order parameter). In equilibrium, the magnitude of theaveraged strain component ⟨𝛾⟩ can be obtained from 𝜕𝐹 (Ψ, ⟨𝛾⟩)𝜕 ⟨𝛾⟩ = 𝐺 ⟨𝛾⟩ + 𝜃 ⟨Ψ⟩ = 0. (33)In the liquid phase, the mean order parameter is ⟨Ψ⟩ = 0 .As expected, the mean deformation disappears in a colloidalliquid at rest ⟨𝛾⟩ = 0 , since 𝐺 ̸= 0 . (3) The Subcritical Bubble Model. We want to study a near-critical dense soft-sphere suspension in a simple shear geom-etry perturbed by a uniform shear flow in the 𝑥 -direction: V 𝑥 = 𝑦 ̇𝛾 , where internal shear deformations have the form 𝛾 𝑥𝑦 = 𝜕𝑢 𝑥 /𝜕𝑦 (Figure 1). When a continuous shear flow isimposed to the suspension, the order parameter kinetics canbe described by a convection-diffusion equation of the form[23] 𝑑Ψ ( r , 𝑡)𝑑𝑡 + ∇ (Ψ ( r , 𝑡) V 𝑥 ( r , 𝑡)) = −Γ∇ 𝛿𝐹 (Ψ)𝛿Ψ ( r , 𝑡) . (34)It suggests that density fluctuations are convected duringtheir lifetime by the applied shear stress 𝜎 𝑥𝑦 as displayed inFigure 1. However, as suggested by the hydrodynamic model,a sheared dense suspension generates stationary internalshear deformations: 𝛾 𝑥𝑦 ( r , 𝑡) ≅ 𝐺 −1 𝜎 𝑥𝑦 ( r , 𝑡) . (35)That means that a considerable contribution of the appliedshear stress is not immediately dissipated but induces internaldeformations.The key idea of this consideration is to estimate the chancefor the occurrence of fluctuations of the solid (coagulated)phase within a sheared suspension in the liquid phase. Forournal of Thermodynamics 7this purpose, we take advantage from a subcritical bubbleapproach [34]. The subcritical bubble model is based on theidea that subcritical bubbles can be approximately describedin equilibrium as symmetric Gaussian-shaped spheres withradius 𝑅 . The bubble can be parameterized as [35] Ψ (𝑟) = Ψ
Core exp (− 𝑟 𝑅 ) (36)while the core value of the order parameter is Ψ Core ≈ Ψ 𝑆 . Thenucleation rate of subcritical bubbles of coagulated particlesof volume 𝑉(𝑅) generated in a sheared suspension can beestimated from ] ∼ exp (− 1𝑘 𝐵 𝑇 ∫
𝑉(𝑅)
𝐹 (𝑟, Ψ
Core , 𝛾 𝑥𝑦 ) 𝑑 𝑟) . (37)Since the initial shape of a subcritical bubble can be approxi-mately given by (36), we obtain for the lowest order in Ψ Core a nucleation rate: ] ∼ exp (− (𝛼 − 𝜃 ⟨𝛾 𝑥𝑦 ⟩) Ψ Core 𝑘 𝐵 𝑇 ) . (38)Because the mean strain-field in a sheared dense suspensionis not zero but of the order ⟨𝛾 𝑥𝑦 ⟩ ≅ 𝜎 𝑥𝑦 𝐺 , (39)the nucleation rate has a maximum at a critical shear stress: 𝜎 𝐶 = 𝐺𝛼𝜃 . (40)Therefore, the nucleation rate of subcritical bubbles consist-ing of a few coagulated particles can be written as ] ∼ exp (− 𝜎 𝐶 − 𝜎 𝑥𝑦 𝑘 𝐵 𝑇 ) . (41)The generation rate of subcritical bubbles is formally equiv-alent to the nucleation rate equation (15). It suggests thatthe critical shear stress related to the occurrence of smallsubcritical bubbles can be approximated by the two-particlecritical stress established by the activation model 𝜎 𝐶 ≈ 𝜎 𝐶 .Since subcritical bubbles (coagulated clusters) increase theviscosity, this transition is accompanied by shear thickening.For decreasing volume fractions Φ → 0 , the relaxation timevanishes: 𝜏 → 0 . As a consequence the shear thickeningeffect disappears in low concentrated suspensions becauseinternal deformations are very small ⟨𝛾⟩ ∼ ̇𝛾𝜏 ≈ 0 . Inthis case, the nucleation rate of coagulated particle clustersequation (38) disappears.The model gives also an explanation for the occurrenceof reversible and irreversible shear thickening (orthokineticcoagulation [36]). As displayed in Figure 3, the Landaufree energy of a suspension exhibiting irreversible shear thickening must have a stable minimum at Ψ 𝑆 ( at 𝜎 = 0) .Applying a shear stress, the potential barrier of the free energyeffectively decreases such that at 𝜎 𝐶 the nucleation rate issufficiently high to form subcritical bubbles of coagulatedparticles. Ceasing the shear stress, the coagulated structurecannot relax into the liquid state when the potential barrierincreases the thermal energy. In the case of reversible shearthickening, however, it is known that for any applied shearstress no coagulated structure is evident after ceasing thestress [37]. The free energy must have therefore an unstableminimum at Ψ 𝑆 ( at 𝜎 = 0) with a potential barrier thatis much less than the thermal energy. Therefore, thermalexcitations are sufficient to break up the bonded structure.Colloidal particles have in this case an effective hard sphereinteraction potential (Figure 2).Though van der Waals attraction diverges for ℎ → 0 such that particles should be always bonded in the primaryminimum ones, they are captured. The explanation for thethermal breakup of the bonds is given in [20]. The point isthat the potential maximum ℎ max for effective hard spheresis only a few Angstrom from the bare particle surface. Since ℎ max is of the order of the diameter of the atomic constituents,it can be expected that the surface roughness of the particlesprohibits a permanent bond.Note that, for 𝛼 < 0 , the colloidal particles are alreadybounded in the primary minimum and (40) suggests that 𝜎 𝐶 < 0 . That means that shear thickening disappears for acoagulated structure and the rheological properties are thoseof a coagulated solid. That this is the case has been shownexperimentally by Barnes [1]. Modifying the chemistry of astable suspension by adding a large amount of flocculatingagents shear thickening disappears.Rheological experiments indicate that shear thickeningin concentrated suspensions exhibits a viscosity hysteresisbetween increasing and decreasing shear rates. It means thatthe critical shear stress related to increasing shear rates ismuch higher than the critical stress associated with decreas-ing shear rates. The presented theory can explain this finding.If subcritical bubbles of particles with large attraction (high 𝜅 )are formed, their relaxation time is large. Decreasing the shearrate they remain in the bounded state even at the critical shearstress related to increasing shear rates. The model suggeststhat the hysteresis effect increases with increasing attractionbetween the particles, which is known from empirical inves-tigations of discontinuous shear thickening.
3. Comparison with Experimental Results
Both the activation approach and the presented shear inducedtransition model suggest the occurrence of shear thickeningat a critical shear stress 𝜎 𝐶 . In order to show the applicabilityof the theory, we want to consider an experimental investiga-tion performed by Maranzano and Wagner on nonaqueouselectrically stabilized suspensions [38]. They studied silicaparticles in THFFA (tetrahydrofurfuryl alcohol), a systemdesigned to minimize the van der Waals attraction. We wantto focus here only on samples denoted by HS600, becausefor these samples the flow curves have been published Journal of Thermodynamics 𝜎 C ( P a ) Φ ̇ 𝛾 C ( s − ) Figure 4: Experimental critical stresses: a function of the volume fraction of the sample HS600 (squares) investigated by Maranzano andWagner [38]. The solid line indicates the critical stress obtained from the activation model for a constant charge interaction potential and thedotted line with constant potential applying the experimental data 𝑎 = 316 nm, 𝜁 = 92.7 , 𝜅𝑎 = 87.1 , 𝑘 𝐵 = 1.38 ∗ 10 −23 J, 𝑒 = 1.6 ∗ 10 −19 As, 𝜀 𝑟 = 8.2 , 𝜀 = 8.854 ∗ 10 −12 F/m, 𝑧 = 1 , 𝑇 = 298 K, 𝐴 = 0.41 ∗ 10 −21 J, Φ 𝐶 = 0.6 , and 𝜂 THFFA = 0.005
Pas. Also displayed are the empiricalcritical shear rates at the onset of shear thickening (diamonds) of the same sample. The dashed line indicates a fit of (16). for a large number of volume fractions. The concentratedsuspensions were found to exhibit discontinuous reversibleshear thickening for volume fractions
Φ ≥ 0.55 in a constantstress sweep. Taking advantage from the experimentallycharacterized data of the colloidal particles, the critical shearstresses for constant potential (dotted line) and constantcharge (solid line) obtained from the activation model aredisplayed in Figure 4, together with the measured criticalshear stresses (squares). The experimental critical stresses fallinto the predicted range between the two lines, while they arecloser to the constant charge critical stress. (This result can beinterpreted as a consequence of the double layer dynamics ofthe electrostatic repulsion. There is obviously not sufficienttime at high critical shear rates to establish a constantpotential double layer differently from low critical shear ratesat high volume fractions.) Note that (4) suggests a criticalshear stress 𝜎 HS 𝐶 ≈ 0.6 Pa which is an order of magnitudebelow the experimental data. Also displayed in Figure 4 arethe experimental data of the critical shear rate of the sampleHS600 at the onset of shear thickening (diamonds). Thedashed line indicates a fit of (16) with a volume fraction Φ 𝐶 =0.6, 𝛽 = 2 , and 𝐶 𝑂 = 5∗10 . Note that the activation model isapplicable not only to concentrated electrically stabilized butalso to sterically stabilized colloidal suspensions [39].
4. Conclusion
The paper establishes a model that suggests the occurrenceof a shear induced phase transition in dense soft spheresuspensions accompanied with shear thickening. The theoryis based on the idea that near-critical suspensions containlarge density fluctuations. Because a concentrated suspensioncan be treated as a viscoelastic medium, the application of acontinuous shear flow generates internal deformations. Thecoupling of these deformations to the order parameter relatedto the liquid-solid transition induces subcritical bubbles ofthe solid phase in the stable liquid phase if a continuous shearperturbation is applied. Since subcritical bubbles consist of coagulated particles, their appearance causes an increase ofthe viscosity at a critical stress (shear thickening).The flow properties of sheared suspensions are governedby the interaction potential of the particles. This interactionis related to the location in the equilibrium phase diagram.Differently from shear thickening in hard sphere suspensions,the phase diagram of soft spheres depends on many parame-ters characterizing the mutual interaction. We want to confinethe discussion here to electrically stabilized monodispersesuspensions where the volume fraction Φ and the Debyescreening reciprocal length 𝜅 are treated as free parameters.Displayed schematically in Figure 5 are two paths throughthe equilibrium phase diagram of electrically stabilized sus-pensions and the expected rheological response in terms ofthe viscosity 𝜂(Φ, 𝜅) . The dashed line indicates a path withincreasing volume fraction Φ and constant 𝜅 . In this case, thecolloidal particles are stabilized by the electrostatic repulsionand the critical stress is nearly independent of the volumefraction. These suspensions are governed by the divergenceof the viscosity with increasing volume fraction caused bylubrication forces. As found experimentally, increasing thevolume fraction, the character of the transition turns intodiscontinuous shear thickening [40].The second path in Figure 5 is related to suspensions withan increasing Debye reciprocal length 𝜅 of the colloidal parti-cles and constant Φ . This path is absent in a hard sphere phasediagram. Starting in the liquid phase, the particles are highlystabilized (effective hard spheres) and continuous shearthickening occurs at relatively high shear rates. Increasing 𝜅 implies that the particle solvent combination approachesthe coagulation transition. This can be done, for example,by increasing the salt concentration. As a consequence,the electrostatic repulsion decreases and the critical stressdeclines. The model suggests that reversible shear thickeningturns into irreversible shear thickening with increasing 𝜅 .Finally shear thickening disappears in a stable coagulatedstructure as known from empirical investigations [41].For a quantitative comparison, the presented theory isapplied to experimental data of shear thickening in denseournal of Thermodynamics 9 Solid
Liquid ̇ 𝛾 C ̇ 𝛾 C ΦΦ ̇ 𝛾 ̇ 𝛾𝜂( ̇ 𝛾, Φ) 𝜂( ̇ 𝛾, 𝜅)𝜅 𝜅 Figure 5: Schematically displayed is the equilibrium phase diagram of electrically stabilized monodisperse colloidal suspensions. The modelsuggests that shear thickening in soft sphere suspensions depends on the position in the phase diagram. The expected viscosity is displayedfor two different paths as a function of the shear rate (logarithmic scaling). The dashed line in the phase diagram indicates a path with varyingvolume fraction and the dotted line a path with changing the Debye screening parameter 𝜅 . For comparison with empirical data, see, forexample, [40, 41]. electrically stabilized suspensions [38]. The good coinci-dence between the model predictions and the experimentallyobtained critical shear stresses confirms this approach. Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
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