PoS(SMPRI2005)045 nited http://pos.sissa.it/ rocess of ‘granulation’, whereby a pensions under shear can be viewed resses but which seizes up above a mmed material can be brittle in the ne of two states, a solid granule and a corporating ideas from mode coupling stresses. This scenario can be used to , admits in certain parameter ranges an ticity and Related Instabilities e Commons Attribution-NonCommercial-ShareAlike Licence. ∗ [email protected] Speaker. sense that it will fracture, not shear-melt, at still higher certain threshold of order the Brownian stress scale. Theexplain ja some bizarre experimental findingssmall concerning volume the of colloidal p suspension can bistably exist in o I review recent work on the ideaas that jamming a of stress-induced colloidal glass sus transition. Atheory schematic but model, entirely in neglecting hydrodynamic interactions extreme shear-thickening material that is flowable at low st flowable droplet. ∗ Copyright owned by the author(s) under the terms of the Creativ c

International conference on Statistical Mechanics of Plas August 29-September 2, 2005 Indian Institute of Science, Bangalore, India Kingdom E-mail: Michael Cates Stress-induced solidification in colloidal suspensions SUPA, School of Physics, JCMB Kings Buildings, Edinburgh EH9 3JZ, U PoS(SMPRI2005)045 verges of shear ) ˙ γ suspensions ( Michael Cates Σ for dense colloidal lass of behaviors. ˙ γ r range of behaviors e; in the discontinu- icted by [2], but also phenomena are well- s at a certain volume g and shear-thinning in olloidal glass transition. tive slope. This region is euristic adaptations of the that is a plot up and then down. vs strain rate ˙ γ Σ ies on ramping 2 the colloidal fluid phase is replaced by an amorphous solid g φ ≥ φ of colloid. As this is approached from below, the structural relaxation time di g φ = Various schematic flow curves of steady state shear stress φ Ref. [2] in fact provides a rather detailed theoretical account of yieldin Continuous shear thickening involves upward curvature on the flow curv In this brief review I discuss recent speculative models of jamming in colloidal colloidal glasses using an extension of the MCTestablished methodology. experimentally Although [3], these they form inEach fact type only of two behavior members is of characterized a by wider a c steady-state ‘flow curve’, based on the idea of amode stress-induced coupling glass theory transition. (MCT) These [1] models that are hasThis h been theory widely used predicts to address correctly thefraction many c features of that transition which occur phase of finite yield stress [2]. 2. Flow curves for dense suspensions stress against strain rate in simpleseen in shearing. colloids, Figure including 1 not shows onlycontinuous some shear-thinning and of and discontinuous the the shear wide yielding thickening. solid pred suspensions. Arrows denote possible hysteretic trajector 1. Introduction in a specific fashion, and for Figure 1: Stress-induced solidification ous case, the curve seemingly develops an S-shape with a region of nega PoS(SMPRI2005)045 . ˙ γ Σ 0). > α 0. Michael Cates rate, within a ology. This is → ˙ γ nd static jamming ictly requires only ‘static jam’ in Fig. tion as a candidate s. It is conceivable (of which there are bands. The upward posed and explored tion. The measured r subject here.) g and yielding-solid straining destroys the e of shear-thickening, g is often found close xis. What this means ue on a rheometer, an shear thickening (e.g., volving shear-banding. egions of high and low orted in Ref. [9], albeit in Ref. [10]; we discuss er, indirect evidence for steresis); moreover, un- hear-thinning [8]. (The g from lubrication forces g a sphere trapped initially istently at high stresses nomenological parameters; ish when the rate of strain . Setting the latter to zero gives a , combined with ever-decreasing gaps α ∞ → t 0 as 3 → ′ dt ) ′ t ( ˙ γ t 0 R 1 − in the model of Refs. [11, 12] gives a new coupling, effectively t α and a ‘jammability’ parameter 0, but v > ) t ( , whereby the presence of a shear stress promotes arrest (we assume ˙ γ | Σ | then controls how close one is to the static glass transition. α v + v → v The observation of static jamming has important consequences for colloid rhe An extreme version of discontinuous shear thickening is the curve denoted A heuristic extension of the MCT calculations to address shear-thickening a The reader should however be warned that a ‘static jamming’ flow curve str The additional parameter 1. Here the S-shapeis curve that has a extended system allcertain that the window is way of fluid finite back at stress toin values. low the a stresses This colloidal vertical kind is system a with of maintained rather behaviormuch in poorly-characterized the has interactions. a same been picture Howev state rep in of well-characterized,this hard-sphere zero further colloids shear below. is given within a glass-transition context isseveral presented technical in variants, [11, which 12]. we Thea do resulting ‘glassiness’ not model discuss parameter here) has two phe that the strain rate tends tothat zero creep, in in steady which state within the given window of stres because shear-thickening is often attributed toin enhanced dissipation the arisin narrow gaps between colloids [3]. However, all such forces van setting 3. A simple shear-thickening model is zero. Accordingly, if one acceptsone static must jamming seek as a an special alternative (extreme) mechanisticto cas explanation. the colloidal Since glass shear transition, thickenin itmechanism. is natural to look to the physics of glass forma Stress-induced solidification mechanically unstable [4], and likely to beThat bypassed is, by if a hysteresis one cycle slowly in apparently increases vertical section the of applied the stress, flowstress as curve coexist is measured in encountered, by layers, within the which withtorque torq r layer then comes normals from along a theand volume-weighted vorticity average downward (neutral) of section direc the of stresses the insteady, possibly the discontinuous two chaotic, curve flow, need is not[5]). often coincide found Simple (hy to models accompany[6, discontinuous showing 7], such chaotic as behavior havelatter analagous have can recently behaviors include colloids in been with systems pro strong showing attractive interactions, discontinuous but that s is not ou between the colloids, could conspire to maintain a static stress in the formal limit This is plausible, since shear stress requires distortion of cages. Imaginin The parameter good schematic representation ofcalculations the of complicated [2]. The MCT-based main shear-thinnin physicscaging of effect those that calculations is causes that the continuous glass transition, restoring fluidity selfcons PoS(SMPRI2005)045 4), > v in which Michael Cates eneous yielding ro stress ( al new regimes that t between the origin glass), and moreover no yield stress beyond picts a nonequilibrium zero in finite strain. sed by the tendency of ting an elliptical shape) use something to happen: within the model of [11, 12]. ˙ γ 1) there are flow curves in Fig. 2 where vs strain rate ≥ Σ lso schematic (e.g. they are shown straight for α 4 parameter space, and the flow curves in each regime. ) 0, effectively) in Ref. [2]. α , v = ( α Schematic flow curves of steady state shear stress Hence there is an antagonistic tendency, captured in a simple way by the model, As well as the curves shown in Fig. 1, the model of [11, 12] shows sever the restoration of fluidanisotropic behavior cage distortions at (that large, is,‘phase cage-breaking stress) diagram’ of strains to the promote is regimes arrest. in oppo the Figure 2 de were not expected. In particular, at high jammability ( 4. Brittle jamming and brittle glasses the resulting flow curve is degenerate: itand becomes infinity. simply This a describes vertical a line material segmen that that becomes is ever solid more at low arrestedwhich stresses as (a ordinary stress colloidal fluidity is is applied. restored.but Accordingly Of the there course, is natural a candidate sufficientpredicted stress is for will colloidal now ca glasses brittle (with failure, not the shear-melting or homog reduces the free volume available to the sphere, and in fact reduces this to clarity though the real ones are somewhat curved). within a spherical cage, one notes that affinely shearing the cage (crea The ‘phase boundaries’ delineating different regimes are a the jammed state persists up to indefinite stress. For a system that is a glass at ze Figure 2: Stress-induced solidification PoS(SMPRI2005)045 u- ow ility fail by jam the will give φ n a granule stresses; and topic and W. sses above a Michael Cates k into a granule, n in colloids, or ss, such physics process, a shear- rittle jam will be The latter arise as ittle jamming sce- . 3) that combines n called granulation creasing s-transition physics, ctor (brittle glass) or ular state is achieved, at (i) a granule reverts face as a result of their xt to an unmelted gran- in amount of the solvent, nting into small granules, y EPSRC/GRS10377. ditions, so that there is no le [10]. e, once the interior is jammed, . g φ 5 of colloids nor on the interaction potential between φ 1 one finds (alongside another more complex regime not discussed > below the static glass transition α φ 4 but < v between the initial flowable bulk state and a final granule. The resulting bistab φ These peculiar phenomena can be explained by a scenario (depicted in Fig Due to the extremely schematic nature of the model, it is not possible to say exactly h The possibility of brittle jamming, suggested by this admittedly a very simple model, is intrig Similarly at I thank M. Fuchs, M. Haw, C. Holmes, P. Sollich for their collaboration on this Although the observation of granulation thus lends strong support to the br should depend on the volume fraction α , v the particles. However, it seemsa safe trajectory to across assume Figure that, 2 forthe most from bottom interactions, bottom right in left sector towardsencountered (yielding either in glass). the a range top of In right either se case it is possible that the b of material properties is made evidentto by a data fluid (reviewed droplet in on [10]) vibration;ule, showing (ii) a th will fluid fuse droplet with created it, this causingat way, it placed least to ne temporarily, melt; by (iii) poking a it fluid with droplet a can spatula. be transformed bac here) a material which is fluid at low stresses, but is statically jammed for all stre certain minimum. This is ansystem extreme more case and of more, static so jamming, thatfracture, in the not which jammed by higher state homogeneous stresses is yielding. just again brittle (‘brittle jamming’) and will 5. Discussion Stress-induced solidification brittle jamming in the droplet interiorsoon with as strong colloidal capillary particles forces force at their the waydilatant perifery. partially tendency through under the shearing; fluid-air moreover, inter suchcapable capillary of forces sustaining ar a shear stress throughout the interior of thenario, granu it remains tosomething be more seen mechanical in whether origin. this Thethere is author suspects is truly that, once an a the aspect strong gran which of mechanical has the contact glass at network transitio its betweenmight core particles be their essential so Brownian in that explaining motion,this glas how ceases is such a to a core material be can elementand relevant. fluidize a in fluid completely the Nonethele droplet at mechanical with low bistability precisely the that same allows contents. transformation betwee 6. Acknowledgements ing from a physical viewpoint. In[10], particular, that it is may help widely explain usedthickening a in dense phenomeno suspension industry is but sheared littleeach relentlessly understood containing and by hundreds ends physicists. or up thousands fragme surrounded In of by this colloidal air. particles and This adifference can in certa be done under controlled evaporation con Poon, M. Greenall, T. Voigtmann for discussions. Work funded in part b PoS(SMPRI2005)045 J. ). The a dius Michael Cates , 267 (2003); is the interfacial σ 123 , 104 (2002). , 296 , where ξ / σ Science Faraday Discuss. lity. The flow curve for a brittle jamming , 120 (2002); M. E. Cates, D. A. Head and . Between this onset stress and the fracture 57 nsistently maintained by capillary forces. At 3 ure into small granules by ingress of air. For a , a l stages of granulation require the fracture stress / , 084502 (2004). 0]. equired is of order T 95 (1999). 92 B 4 , k , , 241 (1992); W. M. van Megen, S. M. Underwood 472 (1996). , 397 (2005). , 87 (1994). 6 1 , 248304 (2002); 55 70 , 54 , , 89 , , Europhys. Lett. , 1586 (1991); K. N. Pham, et al., Phys. Rev. Lett. 67 is a structural length-scale (no smaller than the colloid ra , ξ Europhys. Lett. Rep. Prog. Phys. Phys. Rev. Lett. , 025202 (2002). 66 , , S1681 (2005). 17 , Phys. Rev. Lett. Curr. Opin. Colloid Interface Sci. J. Non-Newtonian Fluid Mech. Curr. Opin. Colloid Interface Sci. Phys. Rev. E Schematic explanation of granulation and granular bistabi and P. N. Pusey, Phys. Cond. Mat. A. Ajdari, [1] W. Goetze and L. Sjoegren, [2] M. Fuchs and M. E. Cates, [3] J. F. Brady, [4] P. D. Olmsted, [6] D. A. Head, A. Ajdari and M. E. Cates, [7] A. Aradian and M. E. Cates, [5] H. M. Laun, [8] S. M. Fielding and P. D. Olmsted, stress, one has amuch granule lower in stresses, which a flowable an droplet arrested can arise. solid The is initia selfco to be exceeded, causing the homogeneous suspension to fract onset of jamming is set by the Brownian stress scale more complete description of this simplified picture, see [1 References tension between solvent and air, and Figure 3: Stress-induced solidification material is supplemented with a fracture curve; the stress r PoS(SMPRI2005)045 Michael Cates , 237 (2005). , S2517 (2005). 49 17 , , , 240 (2003). 63 , , 60401(R) (2002). J. Rheology 66 , 7 J. Phys. Cond. Matt. Europhys. Lett. Phys. Rev. E [9] E. Bertrand, J. Bibette and V. Schmitt, [12] C. B. Holmes, M. Fuchs, M. E. Cates and P. Sollich, Stress-induced solidification [10] M. E. Cates, M. D. Haw and C. B. Holmes, [11] C. B. Holmes, M. Fuchs and M. E. Cates,