- Email: [email protected]
Theory of rheology of colloidal dispersions
Current Opinion in Colloid & Interface Science 6 Ž2001. 484᎐488
Theory of rheology of colloidal dispersions Johan BergenholtzU Department of Chemistry, Physical Chemistry, Goteborg Uni¨ ersity, S-412 96 Goteborg, Sweden ¨ ¨ Received 14 August 2001; accepted 15 August 2001
Abstract This brief review contains a survey of recent literature on theory of rheology of colloidal dispersions. Areas of active research are highlighted, such as approximations for many body interactions in weakly sheared dispersions and the flow-induced microstructural distortions in more strongly sheared dispersions. The former approach seeks to capture the viscosity increase on increasing particle concentration, whereas the latter explains shear thinning and shear thickening. 䊚 2001 Elsevier Science Ltd. All rights reserved. PACS: 2.70.Dd; 83.80.Hj; 83.50.Ax; 83.10.-y Keywords: Rheology; Colloidal dispersions; Linear response; Shear thinning; Shear thickening; Microstructure; Normal stress
1. Introduction Predicting the rheological behavior of colloidal dispersions is a challenging task. Non-Newtonian effects appear because of interactions among the dispersed Žrigid, spherical. particles, and any predictive theory must accordingly consider the effect of colloidal forces on the spatial arrangement of the particles in the externally applied flow field. Because of the relatively small size of colloids, the large solid᎐liquid interfacial area dictates that the surface chemistry is important. In addition, the colloidal size range is one in which Žeffective. Brownian forces, direct interparticle forces, and viscous, hydrodynamic forces are all of comparable magnitude. Hence, the rheology of colloidal dispersions exhibits a rare diversity, the growing recognition of which attracts engineers, chemists and physicists alike. Just like in equilibrium, the central quantity of interest in the statistical mechanics of dispersions
under flow is the spatial distribution of particles, usually referred to as the microstructure. The microstructure rearranges to accommodate the applied flow and the colloidal-level forces and appropriate averages weighted by this non-equilibrium microstructure yield the sought-after macroscopic properties. As is now widely recognized, direct experimental probes of the microstructure Žfor which colloidal dispersions are exemplary systems for study. w1,2x and computer simulations w3,4 ,5x serve as critical tests of such microstructurally-based theories. A well known route to obtaining the microstructure is to solve a deterministic position-space Smoluchowski equation w6 ,7x. An alternative starting point is the Fokker᎐Planck equation w8x, where the distribution function contains an additional dependence on the particle momenta. Usually the former is preferred because the latter has to contend with the objection that the translational and angular momentum of a colloidal particle relax on the same time scale as hydrodynamic interactions propagate w6 ,7x. This brief review will focus on recent advances in formulating theory for the microstructural response 䢇
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Tel.: q46-31-772-1000; fax: 46-31-167-194. E-mail address: [email protected] ŽJ. Bergenholtz..
1359-0294r01r$ - see front matter 䊚 2001 Elsevier Science Ltd. All rights reserved PII: S 1 3 5 9 - 0 2 9 4 Ž 0 1 . 0 0 1 1 2 - 1
J. Bergenholtz r Current Opinion in Colloid & Interface Science 6 (2001) 484᎐488
to applied flows and rheological properties of model, spherical colloids in Newtonian solvents.
2. Survey of theory for low shear rates For low values of the dimensionless shear rate, Pe, the microstructure is linearly perturbed away from equilibrium by the flow. The main focus of theoreticians is to extend Batchelor’s rigorous theory to higher volume fractions Ž .. The most common route is to integrate the Smoluchowski equation over all but two particles, which leads to a hierarchy of coupled equations governing the sheared microstructure. Current work focuses on formulating closure relations that capture the effect of many particle correlations at low Pe w9᎐11 ,12,13x. An additional complication that requires special consideration is the presence of higher than two-particle hydrodynamic interactions. While these can be handled accurately for periodic particle arrangements w14x, the situation is far more difficult for disordered dispersions. The most fruitful effort at resolving this problem is presently the approximate Žand simple for hard spheres!. hydrodynamic functions constructed by Lionberger and Russel w10,11 ,15x, although some progress is being made towards handling three-particle hydrodynamics w16x. In a series of articles, Lionberger and Russel have isolated the many body thermodynamic w9᎐11 x and hydrodynamic interactions w10,11 ,15 x from one another, such that approximations can be judged in an objective manner. A generalization of the hypernetted-chain closure succeeds in describing the weakly sheared microstructure and the low-shear transport coefficients up to f 0.4 for a variety of interaction potentials. Furthermore, the work demonstrates the importance of hydrodynamic interactions at low and intermediate not just quantitatively, but also qualitatively in the high-frequency limit w15 ,17x. Dhont has similarly formulated a closure relation designed to capture the effect of long-range particle correlations in shear flows, which dominate near the gas᎐liquid spinodal. Recent applications of this theory include studies of the non-linear microstructural response to oscillatory flows w18 x as well as steadyroscillatory superposition flows w18 ,19x. Although, for critical correlations, it would suffice to include the far-field part of the hydrodynamic interactions in the Smoluchowski equation, this remains a future challenge; however, it is of vital importance to include hydrodynamics in the evaluation of the stress, as this produces a marked enhancement of the viscosity divergence in the critical region. Generally speaking, such inconsistencies in the treatment of hydrodynamics should be avoided ᎏ an argument 䢇
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䢇
䢇
䢇
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485
which has been placed on a more formal footing recently w13x ᎏ but are common-place and are of course borne out of a desire to derive a tractable theory. In the low-Pe limit the time-correlation route presents an equally rigorous starting point for constructing approximate theories. Since the response of the system to a weak external disturbance is the same as that to spontaneously occurring fluctuations at equilibrium, it follows that the shear viscosity is given by a time integration over a transverse stress auto-correlation function. The frequency Ž .-dependent dynamic viscosity is given by w20᎐22 x 䢇
U Ž . s ⬘⬁ q
1 k B TV
⬁
yi t
H0 dte
=² x y Ž 0 . x y Ž t .:eq s ⬘⬁ q
1 k B TV
⬁
yi t
H0 dte
⌬Ž t .
Ž1.
where V is the volume, k B T is the thermal energy, ⬘⬁ is the high-frequency viscosity, and ² ⭈ ⭈⭈ :eq denotes an equilibrium ensemble average. The microscopic shear stress x y is given in w20,22 x. This Green-Kubo formula is in effect a generalization of Batchelor’s result to arbitrary wit contains both the renormalized Brownian shear stress and a differential equation governing the ŽLaplace-transformed. fluctuating shear stress x; in fact, the fluctuating shear stress x y Ž t . is directly proportional to an N-particle generalization of the structural perturbation introduced by Batchelor. In addition, the time-correlation formalism leads to expressions for the high-frequency shear modulus, G⬘⬁ s ⌬Ž t ª 0., and the lowfrequency shear modulus, G⬘0 s ⌬Ž t ª ⬁.. The latter exists only when the shear relaxation function ⌬Ž t . does not vanish at long times, such as for colloidal crystals and Žideal. colloidal glasses w23x and gels w24x. So far the above expressions have found use in Stokesian Dynamics simulations w4 x and as the starting point for the idealized mode coupling theory ŽMCT. of linear viscoelasticity w20᎐22 x Žsee also Verberg et al. w25x.. While MCT produces the accurate predictions for high concentrations the previously mentioned closures seek w21,22 x, it comes at the expense of dispensing with the notion that the viscosity diverges at random close packing. Instead, MCT predicts a glass transition at a lower , the location of which varies in the phase diagram with the range and magnitude of the interaction potential w26x. At the Židealized. glass transition long-time structural relaxation ceases and an amorphous, structurally arrested solid is formed, which originates from the strongly correlated dynamics associated with the caging of particles w26x. Unfortunately, the microstructure is not 䢇
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J. Bergenholtz r Current Opinion in Colloid & Interface Science 6 (2001) 484᎐488
readily available from MCT and how to incorporate hydrodynamic interactions remains an open question.1 Additional work at low Pe examines the effect of pair-interactions on the viscosity of dilute dispersions, which yields the Huggins coefficient. Recent work has revisited the square-well and square-step interactions w27x and hard sphere mixtures w28x, and has produced predictions for electrostatic and van der Waals interactions w29x as well as interaction models of polymeric stabilization w30x Žneglecting, however, the influence of the polymer coats on the hydrodynamic interactions w17,31x..
3. Shear thinning At higher Pe the microstructural response to the flow field is no longer linear. Many studies leave the complicated many-body interactions aside and focus on the tractable pair-limit Žsee, however, w6 ,18,19x., which qualitatively reproduces much of the behavior seen in more concentrated dispersions. A standard perturbation approach to determining the microstructure in this case is hampered by the Smoluchowski equation being singularly perturbed by the flow field at small Pe. However, a regular perturbation expansion suffices for determining the microstructure up to O Ž Pe 2 ., which permits for calculations of the leading order normal stress differences w32x. Pursuit of higher order corrections necessitates the use of matched asymptotic expansions, which have been discussed by Brady and Vicic w32,33x. Numerical, non-perturbative methods capitalize on the microstructure being well captured by a low-order expansion in spherical harmonics w33᎐35x wup to O Ž1. values of Pe x and the relative ease by which hydrodynamics can be included. Aside from confirming the results of the perturbation theory, they demonstrate that the viscosity shear thins because of a decrease of the thermodynamic contribution to the viscosity, proportional to Pe 2 , which far outweighs the slight increase of the hydrodynamic component of the viscosity at low Pe ŽFig. 1.. The MCT has recently been extended to small, but 䢇䢇
Fig. 1. Variation of the O Ž b2 . viscosity coefficient with dimensionless shear rate Pe b s Ž b r a. 2 Pe for two effective hard-sphere radii, b r as 1.00001 f 1 and b r as 1.1. Colors distinguish between two different numerical methods for solving the two-particle Smoluchowski equation with simple shear flow w35x.
finite, Pe for hard-sphere colloidal dispersions w36 x. This has been accomplished in part by generalizing Eq. Ž1. to finite Pe, albeit in the absence of hydrodynamic interactions. The aim is to understand how a steady shear interacts with the structural relaxation just below and above the hard-sphere glass transition within the MCT formalism. The predictions of the new MCT w36 x are promising and rely on there being no shear ordering on application of a flow. The Newtonian plateau of the flow curves shifts to lower Pe as the glass state is approached Žfrom the liquid. and eventually vanishes at the glass transition. The glass shear melts, and in the process exhibits a shear thinning viscosity proportional to Pey1 , which signals the presence of a yield stress that is smaller than, but proportional to, G 0 . 䢇
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4. Strong flows The Pe4 1 limit of the Smoluchowski equation has been analyzed at the pair level by Brady and Morris w37 x. The no-flux boundary condition renders this limit singular Žcompare with Feke and Schowalter w38x sec. 3.2.. The shear flow forces particles together along the compressive axes which generates large concentration gradients such that the comparatively weak diffusion remains important in a boundary-layer near particle᎐particle contact. They extract the microstructure and rheology both with and without hydrodynamic interactions by a simple radial diffusion-convection balance, which captures the behavior qualitatively and even semi-quantitatively w35x. 䢇
Fuchs and Mayr w23x and Banchio and colleagues w21,22 䢇 x superficially include hydrodynamic interactions via the short-time particle diffusivity, which does not affect the long-time relaxation of the microstructure and leaves the glass transition location in the phase diagram unaltered. In w20x Žpair-wise additive, far-field. hydrodynamics are directly coupled to Žthe slow. concentration fluctuations, which would alter the dynamics at the glass transition and its location in the phase diagram. This is unfortunate as MCT has been so successful for concentrated hard-sphere dispersions w26x and intuition should perhaps warn us against letting hydrodynamics affect a glass transition. 1
J. Bergenholtz r Current Opinion in Colloid & Interface Science 6 (2001) 484᎐488
Moreover, their analysis connects the strongly sheared Brownian hard spheres to non-Brownian suspensions w39,40x, as the non-Newtonian behavior is predicted to vanish as Pey0 .22 w37 x Žat least for a planar extensional flow.. The most important conclusion is that the viscosity shear thickens because the build-up of closely spaced particle pairs generates large hydrodynamic stresses. The pair-limit has recently been analyzed numerically for higher Pe w35x than in w34,41x. The analysis is restricted to effective hard-sphere dispersions for which the minimum approach distance is 2 b whereas the physical diameter is 2 a. Fig. 1 shows the shear rate dependence of the O Ž 2b . term in the expansion for the relative viscosity, r s 1 q Ž5r2. q C Ž Pe,br a.2b wwith the volume fraction s Ž br a.y3 b x. As seen, the viscosity shear thins, irrespective of br a, owing to the decrease of thermodynamic contributions. For near hard-sphere interactions Ž br a f 1. the boundary-layer on the compressive axes combined with lubrication interactions create a shear thickening viscosity at higher Pe; however, pushing the boundary-layer to larger separation distances Ž br as 1.1., results in a high-shear Newtonian plateau because now the increasing hydrodynamic contribution is balanced by the decrease of the thermodynamic contribution. In addition, the first normal stress difference goes from positive to negative values as Pe increases, whereas the second normal stress difference remains negative throughout w35x. This analysis of the pair-limit is in full agreement with results from Stokesian Dynamics simulations on concentrated dispersions w3,4 ,5x and is in conflict with the notion that continuous shear thickening arises because of a break-down of shear-induced order w42x. Lubrication interactions cause hard spheres to rotate as doublets in shear flows, which disrupts any tendency to form ordered structures w4 x. Particles with longer-range repulsions, however, may order on shearing w4 x, which is the subject of recent theory w43,44x, although the focus here has not yet shifted away from hard spheres. The self-diffusivity and self-mobility are also altered by a shear flow, rendering these quantities anisotropic already at O Ž Pe ., which was initially studied in the pair limit in the absence of hydrodynamic interactions w45x. Subsequent work added hydrodynamics w46x and an analysis of the high-Pe limit w37 x. In light of the connections being established between marginally Brownian dispersions Ž Pe4 1. and non-colloidal systems w37 x, it now becomes important, also from a theoretical standpoint, to view colloidal dispersions in a broader context w47x. The development of theory of dispersion rheology, especially at high and Pe, will likely be aided by ex䢇
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ploration of theory in adjacent fields w39,40,48,49x, just as many of today’s theories have benefited from concepts and methods developed within simple liquid theory w20᎐22 ,25x. 䢇
References and recommended reading 䢇 䢇䢇
of special interest of outstanding interest
w1x Wagner NJ. Rheo-optics. Curr Opin Coll Int Sci 1998; 3:391᎐400. w2x Newstein MC, Wang H, Balsara NP et al. Microstructural changes in a colloidal liquid in the shear thinning and shear thickening regimes. J Chem Phys 1999;111:4827᎐4838. w3x Brady JF. Model hard-sphere dispersions: statistical mechanical theory, simulations, and experiments. Curr Opin Coll Int Sci 1996;1:472᎐480. w4x Foss DR, Brady JF. Structure, diffusion and rheology of 䢇 Brownian suspensions by Stokesian Dynamics simulation. J Fluid Mech 2000;407:167᎐200. Stokesian Dynamics results for the rheology and self-diffusivity of hard-sphere dispersions are reported for wide ranges of and Pe. No shear-induced ordering is observed. w5x Brady JF. Computer simulation of viscous suspensions. Chem Eng Sci 2001;56:2921᎐2926. w6x Dhont JKG. An introduction to dynamics of colloids. Amster䢇䢇 dam: Elsevier Science, 1996. Excellent introductory text on wide-ranging topics in colloidal dynamics. w7x Nagele G. On the dynamics and structure of charge-stabilized ¨ suspensions. Phys Rep 1996;272:215᎐372. w8x Jorquera H, Dahler JS. A kinetic theory of suspensions. ii. the steady flow of a hard-sphere suspension. J Chem Phys 1994;101:1392᎐1411. w9x Lionberger RA, Russel WB. Effectiveness of nonequilibrium closures for the many body forces in concentrated colloidal dispersions. J Chem Phys 1997;106:402᎐416. w10x Lionberger RA, Russel WB. A Smoluchowski theory with simple approximations for hydrodynamic interactions in concentrated dispersions. J Rheol 1997;41:399᎐425. w11x Lionberger RA, Russel WB. Microscopic theories of the 䢇 rheology of stable colloidal dispersions. Adv Chem Phys 2000;111:399᎐475. Review that also contains a number of new results, such as refinements of the approximate hydrodynamic functions in w10x and comparisons of non-equilibrium closure results with data on monodisperse and polydisperse hard spheres and polymerically stabilized spheres. w12x Szamel G. Nonequilibrium structure and rheology of concentrated colloidal suspensions: linear response. J Chem Phys 2001;114:8708᎐8717. w13x Wagner NJ. The Smoluchowski equation for colloidal suspensions developed and analyzed through the GENERIC formalism. J Non-Newtonian Fluid Mech 2001;96:177᎐201. w14x Hofman JMA, Clercx HJH, Schram PPJM. Effective viscosity of dense colloidal crystals. Phys Rev E 2000;62:8212᎐8233. w15x Lionberger RA, Russel WB. High frequency modulus of hard 䢇 sphere colloids. J Rheol 1994;38:1885᎐1908. Gives the high-frequency asymptotic behavior of G⬘⬁ for arbitrary of hard spheres in the absence of hydrodynamics, which can be used to judge Brownian Dynamics algorithms of hard spheres. Furthermore, the work identifies and analyzes boundary-layer formation at high oscillation frequencies. w16x Cichocki B, Ekiel-Jezewska ML, Wajnryb E. Lubrication corrections for three-particle contribution to short-time self-dif-
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fusion coefficients in colloidal dispersions. J Chem Phys 1999;111:3265᎐3273. w17x Elliott SL, Russel WB. High frequency shear modulus of polymerically stabilized colloids. J Rheol 1998;42:361᎐378. w18x Dhont JKG, Nagele G. Critical viscoelastic behavior of col¨ 䢇 loids. Phys Rev E 1998;58:7710᎐7732. Expressions for the thermodynamic part of the non-linear dynamic viscosity are derived and applied in a mean-field treatment of shear-induced long-range structural distortions close to the critical point. w19x Dhont JKG, Wagner NJ. Superposition rheology. Phys Rev E 2001;63:021406. w20x Nagele G, Bergenholtz J. Linear viscoelasticity of colloidal ¨ mixtures. J Chem Phys 1998;108:9893᎐9904. w21x Banchio AJ, Bergenholtz J, Nagele G. Rheology and dy¨ namics of colloidal suspensions. Phys Rev Lett 1999; 82:1792᎐1795. w22x Banchio AJ, Nagele G, Bergenholtz J. Viscoelasticity and ¨ 䢇 generalized Stokes-Einstein relations of colloidal dispersions. J Chem Phys 1999;111:8721᎐8740. Comprehensive presentation of MCT results for the linear viscoelastic behavior. Experimental data for shear relaxation times of hard spheres are captured by MCT at high concentrations. w23x Fuchs M, Mayr MR. Aspects of the dynamics of colloidal suspensions: further results of the mode-coupling theory of structural relaxation. Phys Rev E 1999;60:5742᎐5752. w24x Bergenholtz J, Fuchs M, Voigtmann Th. Colloidal gelation and non-ergodicity transitions. J Phys Condens Matter 2000;12:6575᎐6583. w25x Verberg R, de Schepper IM, Cohen EGD. Viscosity of colloidal suspensions. Phys Rev E 1997;55:3143᎐3158. w26x Gotze ¨ W. Recent tests of the mode-coupling theory for glassy dynamics. J Phys Condens Matter 1999;11:A1᎐A45. w27x Wajnryb E, Dahler JS. The viscosity of a moderately dense suspension of particles with piece-wise constant potentials of interaction. Physica A 1998;250:142᎐166. w28x Wajnryb E, Dahler JS. The viscosity of a moderately dense, polydisperse suspension of spherical particles. Physica A 1998;253:77᎐104. w29x Wajnryb E, Dahler JS. The viscosity of electrostatically stabilized dispersions of spherical colloid particles. J Coll Int Sci 1999;217:249᎐258. w30x Wajnryb E, Dahler JS. The viscosity of polymerically stabilized dispersions of spherical colloid particles. J Coll Int Sci 1999;217:259᎐268. w31x Potanin AA, Russel WB. Hydrodynamic interactions of particles with grafted polymer brushes and applications to rheology of colloidal dispersions. Phys Rev E 1995;52:730᎐737. w32x Brady JF, Vicic M. Normal stresses in colloidal dispersions. J Rheol 1995;39:545᎐566. w33x Vicic M. Rheology and microstructure of complex liquids: dispersions, emulsions and polymer solutions wPhD Thesisx. California: California Institute of Technology; 1999. w34x Lionberger RA. Shear thinning of colloidal dispersions. J Rheol 1998;42:843᎐863.
w35x Bergenholtz J, Brady JF, Vicic M. The non-Newtonian rheology of dilute colloidal suspensions. J Fluid Mech 2001, submitted. Numerical calculations of the pair-limit for effective hard-sphere dispersions over wide ranges of Pe, showing varying degrees of continuous shear thickening and sign changes of N1 at high Pe. w36x Fuchs M, Cates ME. Sheared dense colloidal suspensions Žto 䢇 be submitted.. Finds universal shear thinning behavior at the glass transition and establishes a firmer connection between yield stress and lowfrequency modulus. A simple, schematic model that qualitatively captures the cage-melting phenomenon is also presented. w37x Brady JF, Morris JF. Microstructure of strongly sheared 䢇 suspensions and its impact on rheology and diffusion. J Fluid Mech 1997;348:103᎐139. Presents a detailed analysis of the Peª ⬁ limit, which demonstrates that it is a singular limit and that it leads to boundary-layer formation. w38x Feke DL, Schowalter WR. The effect of Brownian diffusion on shear-induced coagulation of colloidal dispersions. J Fluid Mech 1983;133:17᎐35. w39x Wilson H, Davis RH. The viscosity of a dilute suspension of rough spheres. J Fluid Mech 2000;421:339᎐367. w40x Zarraga IE, Leighton DT. Normal stress and diffusion in a dilute suspension of hard spheres undergoing simple shear. Phys Fluids 2001;13:565᎐577. w41x Blawzdziewicz J, Szamel G. Structure and rheology of semidilute suspensions under shear. Phys Rev E 1993;48:4632᎐4636. w42x Hoffman RL. Explanation for the cause of shear thickening in concentrated colloidal dispersions. J Rheol 1998;42: 111᎐123. w43x Morin B, Ronis D. Disorder and order in sheared colloidal suspensions. ii. stochastic simulations. Phys Rev E 1999; 59:3100᎐3115. w44x Schwartz M. Shear-induced order in suspensions. Physica A 1999;269:395᎐402. w45x Blawzdziewicz J, Ekiel-Jezewska ML. How shear flow of a semidilute suspension modifies its self-mobility. Phys Rev E 1995;51:4704᎐4708. w46x Morris JF, Brady JF. Self-diffusion in sheared suspensions. J Fluid Mech 1996;312:223᎐252. Reports a result for the low-Pe, pair limit of the hard-sphere self-diffusivity that is different from that by Blawzdziewicz and Ekiel-Jezewska w45x. w47x Coussot P, Ancey C. Rheophysical classification of concentrated suspensions and granular pastes. Phys Rev E 1999;59:4445᎐4457. w48x Edwards SF, Grinev DV. The statistical-mechanical theory of stress transmission in granular materials. Physica A 1999;263:545᎐553. w49x Cates ME, Wittmer JP, Bouchaud J-P, Claudin P. Jamming and static stress transmission in granular materials. Chaos 1999;9:511᎐522.
Theory of rheology of colloidal dispersions Johan BergenholtzU Department of Chemistry, Physical Chemistry, Goteborg Uni¨ ersity, S-412 96 Goteborg, Sweden ¨ ¨ Received 14 August 2001; accepted 15 August 2001
Abstract This brief review contains a survey of recent literature on theory of rheology of colloidal dispersions. Areas of active research are highlighted, such as approximations for many body interactions in weakly sheared dispersions and the flow-induced microstructural distortions in more strongly sheared dispersions. The former approach seeks to capture the viscosity increase on increasing particle concentration, whereas the latter explains shear thinning and shear thickening. 䊚 2001 Elsevier Science Ltd. All rights reserved. PACS: 2.70.Dd; 83.80.Hj; 83.50.Ax; 83.10.-y Keywords: Rheology; Colloidal dispersions; Linear response; Shear thinning; Shear thickening; Microstructure; Normal stress
1. Introduction Predicting the rheological behavior of colloidal dispersions is a challenging task. Non-Newtonian effects appear because of interactions among the dispersed Žrigid, spherical. particles, and any predictive theory must accordingly consider the effect of colloidal forces on the spatial arrangement of the particles in the externally applied flow field. Because of the relatively small size of colloids, the large solid᎐liquid interfacial area dictates that the surface chemistry is important. In addition, the colloidal size range is one in which Žeffective. Brownian forces, direct interparticle forces, and viscous, hydrodynamic forces are all of comparable magnitude. Hence, the rheology of colloidal dispersions exhibits a rare diversity, the growing recognition of which attracts engineers, chemists and physicists alike. Just like in equilibrium, the central quantity of interest in the statistical mechanics of dispersions
under flow is the spatial distribution of particles, usually referred to as the microstructure. The microstructure rearranges to accommodate the applied flow and the colloidal-level forces and appropriate averages weighted by this non-equilibrium microstructure yield the sought-after macroscopic properties. As is now widely recognized, direct experimental probes of the microstructure Žfor which colloidal dispersions are exemplary systems for study. w1,2x and computer simulations w3,4 ,5x serve as critical tests of such microstructurally-based theories. A well known route to obtaining the microstructure is to solve a deterministic position-space Smoluchowski equation w6 ,7x. An alternative starting point is the Fokker᎐Planck equation w8x, where the distribution function contains an additional dependence on the particle momenta. Usually the former is preferred because the latter has to contend with the objection that the translational and angular momentum of a colloidal particle relax on the same time scale as hydrodynamic interactions propagate w6 ,7x. This brief review will focus on recent advances in formulating theory for the microstructural response 䢇
䢇䢇
䢇䢇
U
Tel.: q46-31-772-1000; fax: 46-31-167-194. E-mail address: [email protected] ŽJ. Bergenholtz..
1359-0294r01r$ - see front matter 䊚 2001 Elsevier Science Ltd. All rights reserved PII: S 1 3 5 9 - 0 2 9 4 Ž 0 1 . 0 0 1 1 2 - 1
J. Bergenholtz r Current Opinion in Colloid & Interface Science 6 (2001) 484᎐488
to applied flows and rheological properties of model, spherical colloids in Newtonian solvents.
2. Survey of theory for low shear rates For low values of the dimensionless shear rate, Pe, the microstructure is linearly perturbed away from equilibrium by the flow. The main focus of theoreticians is to extend Batchelor’s rigorous theory to higher volume fractions Ž .. The most common route is to integrate the Smoluchowski equation over all but two particles, which leads to a hierarchy of coupled equations governing the sheared microstructure. Current work focuses on formulating closure relations that capture the effect of many particle correlations at low Pe w9᎐11 ,12,13x. An additional complication that requires special consideration is the presence of higher than two-particle hydrodynamic interactions. While these can be handled accurately for periodic particle arrangements w14x, the situation is far more difficult for disordered dispersions. The most fruitful effort at resolving this problem is presently the approximate Žand simple for hard spheres!. hydrodynamic functions constructed by Lionberger and Russel w10,11 ,15x, although some progress is being made towards handling three-particle hydrodynamics w16x. In a series of articles, Lionberger and Russel have isolated the many body thermodynamic w9᎐11 x and hydrodynamic interactions w10,11 ,15 x from one another, such that approximations can be judged in an objective manner. A generalization of the hypernetted-chain closure succeeds in describing the weakly sheared microstructure and the low-shear transport coefficients up to f 0.4 for a variety of interaction potentials. Furthermore, the work demonstrates the importance of hydrodynamic interactions at low and intermediate not just quantitatively, but also qualitatively in the high-frequency limit w15 ,17x. Dhont has similarly formulated a closure relation designed to capture the effect of long-range particle correlations in shear flows, which dominate near the gas᎐liquid spinodal. Recent applications of this theory include studies of the non-linear microstructural response to oscillatory flows w18 x as well as steadyroscillatory superposition flows w18 ,19x. Although, for critical correlations, it would suffice to include the far-field part of the hydrodynamic interactions in the Smoluchowski equation, this remains a future challenge; however, it is of vital importance to include hydrodynamics in the evaluation of the stress, as this produces a marked enhancement of the viscosity divergence in the critical region. Generally speaking, such inconsistencies in the treatment of hydrodynamics should be avoided ᎏ an argument 䢇
䢇
䢇
䢇
䢇
䢇
䢇
䢇
485
which has been placed on a more formal footing recently w13x ᎏ but are common-place and are of course borne out of a desire to derive a tractable theory. In the low-Pe limit the time-correlation route presents an equally rigorous starting point for constructing approximate theories. Since the response of the system to a weak external disturbance is the same as that to spontaneously occurring fluctuations at equilibrium, it follows that the shear viscosity is given by a time integration over a transverse stress auto-correlation function. The frequency Ž .-dependent dynamic viscosity is given by w20᎐22 x 䢇
U Ž . s ⬘⬁ q
1 k B TV
⬁
yi t
H0 dte
=² x y Ž 0 . x y Ž t .:eq s ⬘⬁ q
1 k B TV
⬁
yi t
H0 dte
⌬Ž t .
Ž1.
where V is the volume, k B T is the thermal energy, ⬘⬁ is the high-frequency viscosity, and ² ⭈ ⭈⭈ :eq denotes an equilibrium ensemble average. The microscopic shear stress x y is given in w20,22 x. This Green-Kubo formula is in effect a generalization of Batchelor’s result to arbitrary wit contains both the renormalized Brownian shear stress and a differential equation governing the ŽLaplace-transformed. fluctuating shear stress x; in fact, the fluctuating shear stress x y Ž t . is directly proportional to an N-particle generalization of the structural perturbation introduced by Batchelor. In addition, the time-correlation formalism leads to expressions for the high-frequency shear modulus, G⬘⬁ s ⌬Ž t ª 0., and the lowfrequency shear modulus, G⬘0 s ⌬Ž t ª ⬁.. The latter exists only when the shear relaxation function ⌬Ž t . does not vanish at long times, such as for colloidal crystals and Žideal. colloidal glasses w23x and gels w24x. So far the above expressions have found use in Stokesian Dynamics simulations w4 x and as the starting point for the idealized mode coupling theory ŽMCT. of linear viscoelasticity w20᎐22 x Žsee also Verberg et al. w25x.. While MCT produces the accurate predictions for high concentrations the previously mentioned closures seek w21,22 x, it comes at the expense of dispensing with the notion that the viscosity diverges at random close packing. Instead, MCT predicts a glass transition at a lower , the location of which varies in the phase diagram with the range and magnitude of the interaction potential w26x. At the Židealized. glass transition long-time structural relaxation ceases and an amorphous, structurally arrested solid is formed, which originates from the strongly correlated dynamics associated with the caging of particles w26x. Unfortunately, the microstructure is not 䢇
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J. Bergenholtz r Current Opinion in Colloid & Interface Science 6 (2001) 484᎐488
readily available from MCT and how to incorporate hydrodynamic interactions remains an open question.1 Additional work at low Pe examines the effect of pair-interactions on the viscosity of dilute dispersions, which yields the Huggins coefficient. Recent work has revisited the square-well and square-step interactions w27x and hard sphere mixtures w28x, and has produced predictions for electrostatic and van der Waals interactions w29x as well as interaction models of polymeric stabilization w30x Žneglecting, however, the influence of the polymer coats on the hydrodynamic interactions w17,31x..
3. Shear thinning At higher Pe the microstructural response to the flow field is no longer linear. Many studies leave the complicated many-body interactions aside and focus on the tractable pair-limit Žsee, however, w6 ,18,19x., which qualitatively reproduces much of the behavior seen in more concentrated dispersions. A standard perturbation approach to determining the microstructure in this case is hampered by the Smoluchowski equation being singularly perturbed by the flow field at small Pe. However, a regular perturbation expansion suffices for determining the microstructure up to O Ž Pe 2 ., which permits for calculations of the leading order normal stress differences w32x. Pursuit of higher order corrections necessitates the use of matched asymptotic expansions, which have been discussed by Brady and Vicic w32,33x. Numerical, non-perturbative methods capitalize on the microstructure being well captured by a low-order expansion in spherical harmonics w33᎐35x wup to O Ž1. values of Pe x and the relative ease by which hydrodynamics can be included. Aside from confirming the results of the perturbation theory, they demonstrate that the viscosity shear thins because of a decrease of the thermodynamic contribution to the viscosity, proportional to Pe 2 , which far outweighs the slight increase of the hydrodynamic component of the viscosity at low Pe ŽFig. 1.. The MCT has recently been extended to small, but 䢇䢇
Fig. 1. Variation of the O Ž b2 . viscosity coefficient with dimensionless shear rate Pe b s Ž b r a. 2 Pe for two effective hard-sphere radii, b r as 1.00001 f 1 and b r as 1.1. Colors distinguish between two different numerical methods for solving the two-particle Smoluchowski equation with simple shear flow w35x.
finite, Pe for hard-sphere colloidal dispersions w36 x. This has been accomplished in part by generalizing Eq. Ž1. to finite Pe, albeit in the absence of hydrodynamic interactions. The aim is to understand how a steady shear interacts with the structural relaxation just below and above the hard-sphere glass transition within the MCT formalism. The predictions of the new MCT w36 x are promising and rely on there being no shear ordering on application of a flow. The Newtonian plateau of the flow curves shifts to lower Pe as the glass state is approached Žfrom the liquid. and eventually vanishes at the glass transition. The glass shear melts, and in the process exhibits a shear thinning viscosity proportional to Pey1 , which signals the presence of a yield stress that is smaller than, but proportional to, G 0 . 䢇
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4. Strong flows The Pe4 1 limit of the Smoluchowski equation has been analyzed at the pair level by Brady and Morris w37 x. The no-flux boundary condition renders this limit singular Žcompare with Feke and Schowalter w38x sec. 3.2.. The shear flow forces particles together along the compressive axes which generates large concentration gradients such that the comparatively weak diffusion remains important in a boundary-layer near particle᎐particle contact. They extract the microstructure and rheology both with and without hydrodynamic interactions by a simple radial diffusion-convection balance, which captures the behavior qualitatively and even semi-quantitatively w35x. 䢇
Fuchs and Mayr w23x and Banchio and colleagues w21,22 䢇 x superficially include hydrodynamic interactions via the short-time particle diffusivity, which does not affect the long-time relaxation of the microstructure and leaves the glass transition location in the phase diagram unaltered. In w20x Žpair-wise additive, far-field. hydrodynamics are directly coupled to Žthe slow. concentration fluctuations, which would alter the dynamics at the glass transition and its location in the phase diagram. This is unfortunate as MCT has been so successful for concentrated hard-sphere dispersions w26x and intuition should perhaps warn us against letting hydrodynamics affect a glass transition. 1
J. Bergenholtz r Current Opinion in Colloid & Interface Science 6 (2001) 484᎐488
Moreover, their analysis connects the strongly sheared Brownian hard spheres to non-Brownian suspensions w39,40x, as the non-Newtonian behavior is predicted to vanish as Pey0 .22 w37 x Žat least for a planar extensional flow.. The most important conclusion is that the viscosity shear thickens because the build-up of closely spaced particle pairs generates large hydrodynamic stresses. The pair-limit has recently been analyzed numerically for higher Pe w35x than in w34,41x. The analysis is restricted to effective hard-sphere dispersions for which the minimum approach distance is 2 b whereas the physical diameter is 2 a. Fig. 1 shows the shear rate dependence of the O Ž 2b . term in the expansion for the relative viscosity, r s 1 q Ž5r2. q C Ž Pe,br a.2b wwith the volume fraction s Ž br a.y3 b x. As seen, the viscosity shear thins, irrespective of br a, owing to the decrease of thermodynamic contributions. For near hard-sphere interactions Ž br a f 1. the boundary-layer on the compressive axes combined with lubrication interactions create a shear thickening viscosity at higher Pe; however, pushing the boundary-layer to larger separation distances Ž br as 1.1., results in a high-shear Newtonian plateau because now the increasing hydrodynamic contribution is balanced by the decrease of the thermodynamic contribution. In addition, the first normal stress difference goes from positive to negative values as Pe increases, whereas the second normal stress difference remains negative throughout w35x. This analysis of the pair-limit is in full agreement with results from Stokesian Dynamics simulations on concentrated dispersions w3,4 ,5x and is in conflict with the notion that continuous shear thickening arises because of a break-down of shear-induced order w42x. Lubrication interactions cause hard spheres to rotate as doublets in shear flows, which disrupts any tendency to form ordered structures w4 x. Particles with longer-range repulsions, however, may order on shearing w4 x, which is the subject of recent theory w43,44x, although the focus here has not yet shifted away from hard spheres. The self-diffusivity and self-mobility are also altered by a shear flow, rendering these quantities anisotropic already at O Ž Pe ., which was initially studied in the pair limit in the absence of hydrodynamic interactions w45x. Subsequent work added hydrodynamics w46x and an analysis of the high-Pe limit w37 x. In light of the connections being established between marginally Brownian dispersions Ž Pe4 1. and non-colloidal systems w37 x, it now becomes important, also from a theoretical standpoint, to view colloidal dispersions in a broader context w47x. The development of theory of dispersion rheology, especially at high and Pe, will likely be aided by ex䢇
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ploration of theory in adjacent fields w39,40,48,49x, just as many of today’s theories have benefited from concepts and methods developed within simple liquid theory w20᎐22 ,25x. 䢇
References and recommended reading 䢇 䢇䢇
of special interest of outstanding interest
w1x Wagner NJ. Rheo-optics. Curr Opin Coll Int Sci 1998; 3:391᎐400. w2x Newstein MC, Wang H, Balsara NP et al. Microstructural changes in a colloidal liquid in the shear thinning and shear thickening regimes. J Chem Phys 1999;111:4827᎐4838. w3x Brady JF. Model hard-sphere dispersions: statistical mechanical theory, simulations, and experiments. Curr Opin Coll Int Sci 1996;1:472᎐480. w4x Foss DR, Brady JF. Structure, diffusion and rheology of 䢇 Brownian suspensions by Stokesian Dynamics simulation. J Fluid Mech 2000;407:167᎐200. Stokesian Dynamics results for the rheology and self-diffusivity of hard-sphere dispersions are reported for wide ranges of and Pe. No shear-induced ordering is observed. w5x Brady JF. Computer simulation of viscous suspensions. Chem Eng Sci 2001;56:2921᎐2926. w6x Dhont JKG. An introduction to dynamics of colloids. Amster䢇䢇 dam: Elsevier Science, 1996. Excellent introductory text on wide-ranging topics in colloidal dynamics. w7x Nagele G. On the dynamics and structure of charge-stabilized ¨ suspensions. Phys Rep 1996;272:215᎐372. w8x Jorquera H, Dahler JS. A kinetic theory of suspensions. ii. the steady flow of a hard-sphere suspension. J Chem Phys 1994;101:1392᎐1411. w9x Lionberger RA, Russel WB. Effectiveness of nonequilibrium closures for the many body forces in concentrated colloidal dispersions. J Chem Phys 1997;106:402᎐416. w10x Lionberger RA, Russel WB. A Smoluchowski theory with simple approximations for hydrodynamic interactions in concentrated dispersions. J Rheol 1997;41:399᎐425. w11x Lionberger RA, Russel WB. Microscopic theories of the 䢇 rheology of stable colloidal dispersions. Adv Chem Phys 2000;111:399᎐475. Review that also contains a number of new results, such as refinements of the approximate hydrodynamic functions in w10x and comparisons of non-equilibrium closure results with data on monodisperse and polydisperse hard spheres and polymerically stabilized spheres. w12x Szamel G. Nonequilibrium structure and rheology of concentrated colloidal suspensions: linear response. J Chem Phys 2001;114:8708᎐8717. w13x Wagner NJ. The Smoluchowski equation for colloidal suspensions developed and analyzed through the GENERIC formalism. J Non-Newtonian Fluid Mech 2001;96:177᎐201. w14x Hofman JMA, Clercx HJH, Schram PPJM. Effective viscosity of dense colloidal crystals. Phys Rev E 2000;62:8212᎐8233. w15x Lionberger RA, Russel WB. High frequency modulus of hard 䢇 sphere colloids. J Rheol 1994;38:1885᎐1908. Gives the high-frequency asymptotic behavior of G⬘⬁ for arbitrary of hard spheres in the absence of hydrodynamics, which can be used to judge Brownian Dynamics algorithms of hard spheres. Furthermore, the work identifies and analyzes boundary-layer formation at high oscillation frequencies. w16x Cichocki B, Ekiel-Jezewska ML, Wajnryb E. Lubrication corrections for three-particle contribution to short-time self-dif-
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fusion coefficients in colloidal dispersions. J Chem Phys 1999;111:3265᎐3273. w17x Elliott SL, Russel WB. High frequency shear modulus of polymerically stabilized colloids. J Rheol 1998;42:361᎐378. w18x Dhont JKG, Nagele G. Critical viscoelastic behavior of col¨ 䢇 loids. Phys Rev E 1998;58:7710᎐7732. Expressions for the thermodynamic part of the non-linear dynamic viscosity are derived and applied in a mean-field treatment of shear-induced long-range structural distortions close to the critical point. w19x Dhont JKG, Wagner NJ. Superposition rheology. Phys Rev E 2001;63:021406. w20x Nagele G, Bergenholtz J. Linear viscoelasticity of colloidal ¨ mixtures. J Chem Phys 1998;108:9893᎐9904. w21x Banchio AJ, Bergenholtz J, Nagele G. Rheology and dy¨ namics of colloidal suspensions. Phys Rev Lett 1999; 82:1792᎐1795. w22x Banchio AJ, Nagele G, Bergenholtz J. Viscoelasticity and ¨ 䢇 generalized Stokes-Einstein relations of colloidal dispersions. J Chem Phys 1999;111:8721᎐8740. Comprehensive presentation of MCT results for the linear viscoelastic behavior. Experimental data for shear relaxation times of hard spheres are captured by MCT at high concentrations. w23x Fuchs M, Mayr MR. Aspects of the dynamics of colloidal suspensions: further results of the mode-coupling theory of structural relaxation. Phys Rev E 1999;60:5742᎐5752. w24x Bergenholtz J, Fuchs M, Voigtmann Th. Colloidal gelation and non-ergodicity transitions. J Phys Condens Matter 2000;12:6575᎐6583. w25x Verberg R, de Schepper IM, Cohen EGD. Viscosity of colloidal suspensions. Phys Rev E 1997;55:3143᎐3158. w26x Gotze ¨ W. Recent tests of the mode-coupling theory for glassy dynamics. J Phys Condens Matter 1999;11:A1᎐A45. w27x Wajnryb E, Dahler JS. The viscosity of a moderately dense suspension of particles with piece-wise constant potentials of interaction. Physica A 1998;250:142᎐166. w28x Wajnryb E, Dahler JS. The viscosity of a moderately dense, polydisperse suspension of spherical particles. Physica A 1998;253:77᎐104. w29x Wajnryb E, Dahler JS. The viscosity of electrostatically stabilized dispersions of spherical colloid particles. J Coll Int Sci 1999;217:249᎐258. w30x Wajnryb E, Dahler JS. The viscosity of polymerically stabilized dispersions of spherical colloid particles. J Coll Int Sci 1999;217:259᎐268. w31x Potanin AA, Russel WB. Hydrodynamic interactions of particles with grafted polymer brushes and applications to rheology of colloidal dispersions. Phys Rev E 1995;52:730᎐737. w32x Brady JF, Vicic M. Normal stresses in colloidal dispersions. J Rheol 1995;39:545᎐566. w33x Vicic M. Rheology and microstructure of complex liquids: dispersions, emulsions and polymer solutions wPhD Thesisx. California: California Institute of Technology; 1999. w34x Lionberger RA. Shear thinning of colloidal dispersions. J Rheol 1998;42:843᎐863.
w35x Bergenholtz J, Brady JF, Vicic M. The non-Newtonian rheology of dilute colloidal suspensions. J Fluid Mech 2001, submitted. Numerical calculations of the pair-limit for effective hard-sphere dispersions over wide ranges of Pe, showing varying degrees of continuous shear thickening and sign changes of N1 at high Pe. w36x Fuchs M, Cates ME. Sheared dense colloidal suspensions Žto 䢇 be submitted.. Finds universal shear thinning behavior at the glass transition and establishes a firmer connection between yield stress and lowfrequency modulus. A simple, schematic model that qualitatively captures the cage-melting phenomenon is also presented. w37x Brady JF, Morris JF. Microstructure of strongly sheared 䢇 suspensions and its impact on rheology and diffusion. J Fluid Mech 1997;348:103᎐139. Presents a detailed analysis of the Peª ⬁ limit, which demonstrates that it is a singular limit and that it leads to boundary-layer formation. w38x Feke DL, Schowalter WR. The effect of Brownian diffusion on shear-induced coagulation of colloidal dispersions. J Fluid Mech 1983;133:17᎐35. w39x Wilson H, Davis RH. The viscosity of a dilute suspension of rough spheres. J Fluid Mech 2000;421:339᎐367. w40x Zarraga IE, Leighton DT. Normal stress and diffusion in a dilute suspension of hard spheres undergoing simple shear. Phys Fluids 2001;13:565᎐577. w41x Blawzdziewicz J, Szamel G. Structure and rheology of semidilute suspensions under shear. Phys Rev E 1993;48:4632᎐4636. w42x Hoffman RL. Explanation for the cause of shear thickening in concentrated colloidal dispersions. J Rheol 1998;42: 111᎐123. w43x Morin B, Ronis D. Disorder and order in sheared colloidal suspensions. ii. stochastic simulations. Phys Rev E 1999; 59:3100᎐3115. w44x Schwartz M. Shear-induced order in suspensions. Physica A 1999;269:395᎐402. w45x Blawzdziewicz J, Ekiel-Jezewska ML. How shear flow of a semidilute suspension modifies its self-mobility. Phys Rev E 1995;51:4704᎐4708. w46x Morris JF, Brady JF. Self-diffusion in sheared suspensions. J Fluid Mech 1996;312:223᎐252. Reports a result for the low-Pe, pair limit of the hard-sphere self-diffusivity that is different from that by Blawzdziewicz and Ekiel-Jezewska w45x. w47x Coussot P, Ancey C. Rheophysical classification of concentrated suspensions and granular pastes. Phys Rev E 1999;59:4445᎐4457. w48x Edwards SF, Grinev DV. The statistical-mechanical theory of stress transmission in granular materials. Physica A 1999;263:545᎐553. w49x Cates ME, Wittmer JP, Bouchaud J-P, Claudin P. Jamming and static stress transmission in granular materials. Chaos 1999;9:511᎐522.