Flow-Induced Phase Transitions: Papers

Polymer Solutions:

Apparent Fracture in Polymer Fluids under Step Shear
OS Agimelen and PD Olmsted, Physical Review Letters 110 (2013) 204503

Recent step strain experiments in well-entangled polymeric liquids demonstrated a bulk fracturelike phenomenon. We study this instability by using a modern version of the Doi-Edwards theory for entangled polymers, and we find close quantitative agreement with the experiments. The phenomenon occurs because the viscoelastic liquid is sheared into a rubbery state that possesses an elastic constitutive instability [G. Marrucci and N. Grizzuti, J. Rheol. 27, 433 (1983)]. The fracture is a transient manifestation of this instability, which relies on the amplification of spatially inhomogeneous fluctuations. This mechanism differs from the fracture in glassy materials and dense suspensions.

A meniscus unstability in shear banding fluids driven by second normal stress differences
S Skorski and PD Olmsted, Journal of Rheology 55 (2011) 1219-1246 .

dge fracture occurs frequently in non-Newtonian fluids. A similar instability has often been reported at the free surface of fluids undergoing shear banding and leads to expulsion of the sample. In this paper, the distortion of the free surface of such a shear banding fluid is calculated by balancing the surface tension against the second normal stresses induced in the two shear bands, and simultaneously requiring a continuous and smooth meniscus. We show that wormlike micelles typically retain meniscus integrity when shear banding, but in some cases can lose integrity for a range of average applied shear rates during which one expects shear banding. This meniscus fracture would lead to ejection of the sample as the shear banding region is swept through. We further show that entangled polymer solutions are expected to display a propensity for fracture because of their much larger second normal stresses. These calculations are consistent with available data in the literature. We also estimate the meniscus distortion of a three-band configuration, as has been observed in some wormlike micellar solutions in a cone and plate geometry.

Transient shear banding in entangled polymers: a study using the Rolie-Poly model
JM Adams, SM Fielding, and PD Olmsted, Journal of Rheology 55 (2011) 1007-1032..

Spatially inhomogeneous shear flow occurs in entangled polymer solutions, both as steady state shear banding and transiently after a large step strain or during start up to a steady uniform shear rate. Steady state shear banding is a hallmark of models with a non-monotonic constitutive relation between total shear stress and applied shear rate, but transient banding is sometimes seen in fluids that do \textit{not} shear band at steady state. We model this behavior using the diffusive Rolie-Poly model in a Newtonian solvent, whose constitutive behavior can be monotonic or non-monotonic depending on the degree of convected constraint release (CCR). We study monotonic constitutive behaviour. Linear stability analysis of start up to a sufficiently high shear rate shows that spatial fluctuations are unstable at early times. There is a strong correlation between this instability and the negative slope of the (time dependent) constitutive curve. If the time integral of the most unstable eigenvalue is sufficiently large then the system exhibits transient shear bands that later vanish in steady state. We show how perturbations, due to fluctuations or the inhomogeneous stresses, can trigger this instability. This transient behavior is similar to recent observations in entangled polymer solutions.

Reply to Comment of SQ Wang
JM Adams and PD Olmsted, Physical Review Letters 103 (2009) 219802.

Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions
JM Adams and PD Olmsted Physical Review Letters 27 (2009) 067801.

Recent experiments on entangled polymer solutions may indicate a constitutive instability, and have led some to question the validity of existing constitutive models. We use a modern constitutive model, the Rolie-Poly model plus a solvent viscosity, and show that (1) this simple class of models captures instability; (2) shear banding phenomena is observable for weakly stable fluids in flow geometries with sufficiently inhomogeneous total stress; (3) transient phenomena exhibit inhomogeneities resembling shear banding, even for weakly stable fluids.

Layered Phases:

Vorticity Banding During the Lamellar-to-Onion Transition in a Lyotropic Surfactant Solution in Shear Flow
GMH Wilkins and PD Olmsted, European Physical Journal E 21 (2006) 133.

Equilibrium Onions?
L Ramos, D Roux, PD Olmsted, and ME Cates Europhysics Letters 66 (2004) 888-894.

Dynamical coarse-graining of highly fluctuating membranes under shear flow
SW Marlow and PD Olmsted, Physical Review E66 (2002) 061706.

The effect of strong shear flow on highly fluctuating lamellar systems stabilized by intermembrane collisions via the Helfrich interaction is studied. Advection enters the microscopic equation of motion for a single membrane via a non-linear coupling. Upon coarse-graining the theory for a single bilayer up to the length scale of the collision length, at which a hydrodynamic description applies, an additional dynamical coupling is generated which is of the form of a wavevector-dependent tension that is non-linear in the applied shear rate. This new term has consequences for the effects of strong flow on the stability and dynamics of lamellar surfactant phases.

The Effect of Shear Flow on the Helfrich Interaction in Lyotropic Lamellar Systems
SW Marlow and PD Olmsted, European Physical Journal E 8 (2002) 485-497, (preprint) .

We study the effect of shear flow on the entropic Helfrich interaction in lyotropic surfactant smectic fluids. Arguing that flow induces an effective anisotropic surface tension in bilayers due to a combination of intermonolayer friction, bilayer collisions and convection, we calculate the reduction in fluctuations and hence the renormalised change in effective compression modulus and steady-state layer spacing. We demonstrate that non-permeable or slowly permeating membranes can be susceptible to an undulatory instability of the Helfrich-Hurault type, and speculate that such an instability could be one source of a transition to multilamellar vesicles.

Experimental Collaboration:

Influence of boundary conditions and confinement on nonlocal effects in flows of wormlike micellar systems
C Masselon, A Colin, and PD Olmsted, Physical Review E 81 (2010) 021502.

Birefringence Banding in a Micellar Solution or the Complexity of Heterogeneous Flow
S. Lerouge, JP Decruppe, and P Olmsted, Langmuir 20 (2004) 11355-11365.

Time scales in shear banding of wormlike micelles
O Radulescu, PD Olmsted, JP Decruppe, S Lerouge, J-F Berret, and G Porte, Europhysics Letters, 62 (2003) 230-236.

We show the existence of three well defined time scales in the dynamics of wormlike micelles after a step between two shear rates on the stress plateau. These time scales are compatible with the presence of a structured interface between bands of different viscosities and correspond to the isotropic band destabilization during the stress overshoot, reconstruction of the interface after the overshoot and travel of a fully formed interface. The last stage can be used to estimate a stress diffusion coefficient.

Phenomenology:

A meniscus unstability in shear banding fluids driven by second normal stress differences
S Skorski and PD Olmsted, Submitted to Journal of Rheology (2010).

Edge fracture occurs frequently in non-Newtonian fluids. A similar instability has often been reported at the free surface of fluids undergoing shear banding, and leads to expulsion of the sample. In this paper the stability of the free surface of such a shear banding fluid is calculated by balancing the surface tension against the second normal stresses induced in the two shear bands, and simultaneously requiring a continuous and smooth meniscus. We show that wormlike micelles are typically stable when shear banding, but in some cases can become unstable for a range of average applied shear rates during which one expects shear banding. This instability would lead to ejection of the sample as the shear banding region is swept through. We further show that entangled polymer solutions are expected to be much more unstable, because of their much larger second normal stresses. These calculations are consistent with available data in the literature. We also estimate the stability of a three band configuration, as has been observed in some wormlike micellar solutions in a cone and plate geometry.

Two-dimensional perturbations in a scalar model for shear-banding
JLA Dubbeldam and PD Olmsted European Physical Journal E 29 (2009) 363-378.

The interplay between boundary conditions and flow geometries in shear banding: Hysteresis, band configurations, and surface transitions
JM Adams, SM Fielding, and PD Olmsted, J Non-Newtonian Fluid Mechanics 151 (2008) 101-118.

Nonlinear Dynamics of an Interface between Shear Bands
SM Fielding and PD Olmsted Physical Review Letters 96 (2006) 104502

Spatio-temporal oscillations and rheochaos in a simple model of shear banding
Suzanne Fielding, Peter Olmsted, Physical Review Letters 92 (2004) 084502 .

We study a simple model of shear banding in which the flow-induced phase is destabilised by coupling between flow and microstructure (wormlike micellar length). By varying the strength of the instability and the applied shear rate, we find a rich variety of oscillatory and rheochaotic shear banded flows. At low shear and weak instability, the induced phase pulsates in width next to one wall of the flow cell. For stronger instability, single or multiple high shear pulses ricochet across the cell. At high shear rates we observe oscillating bands on either side of a defect. In some cases, multiple such defects exist and propagate across the cell to interact with each other. We discuss our results in the context of recent observations of oscillating and fluctuating shear banded flows.

Flow phase diagrams for concentration-coupled shear banding instability SM Fielding and PD Olmsted, European Physical Journal E 11 (2003) 65-83.

After surveying the experimental evidence for concentration coupling in the shear banding of wormlike micellar surfactant systems, we present flow phase diagrams spanned by shear stress (or strain rate ) and concentration, calculated within the two-fluid, non-local Johnson-Segalman (d-JS- ) model. We also give results for the macroscopic flow curves for a range of (average) concentrations . For any concentration that is high enough to give shear banding, the flow curve shows the usual non-analytic kink at the onset of banding, followed by a coexistence "plateau" that slopes upwards, . As the concentration is reduced, the width of the coexistence regime diminishes and eventually terminates at a non-equilibrium critical point . We outline the way in which the flow phase diagram can be reconstructed from a family of such flow curves, , measured for several different values of . This reconstruction could be used to check new measurements of concentration differences between the coexisting bands. Our d-JS- model contains two different spatial gradient terms that describe the interface between the shear bands. The first is in the viscoelastic constitutive equation, with a characteristic (mesh) length l. The second is in the (generalised) Cahn-Hilliard equation, with the characteristic length for equilibrium concentration-fluctuations. We show that the phase diagrams (and so also the flow curves) depend on the ratio , with loss of unique state selection at r=0. We also give results for the full shear-banded profiles, and study the divergence of the interfacial width (relative to l and ) at the critical point.

Kinetics of the shear banding instability in startup flows
SM Fielding and PD Olmsted (2002), Physical Review E68 (2003) 036313.

Motivated by experiments on wormlike micelles, we study the early stages of the shear banding instability using a two-fluid Johnson-Segalman model. We perform a linear stability analysis for coupled fluctuations in shear rate, micellar strain and concentration about an initially homogeneous state. First we calculate the ``spinodal'' onset of instability in sweeps along the intrinsic constitutive curve. For startup ``quenches'' into the unstable region, the instability usually occurs before the intrinsic constitutive curve can be attained so we analyse the fluctuations with respect to the homogeneous startup flow to find the selected length and time scales at which inhomogeneity first emerges. In the uncoupled limit, fluctuations in shear rate and micellar strain are independent of those in concentration, and are unstable when the intrinsic constitutive curve has negative slope; but no length scale is selected. When coupled to concentration, this instability is enhanced at short length scales; a length scale is selected, as seen experimentally. The unstable region is then broadened. Far from an underlying (zero-shear) demixing instability, the broadening is slight and the instability is still dominated by shear rate and micellar strain. Close to demixing, instability sets in at very low shear rate, where it is demixing triggered by flow.

Early stage kinetics in a unified model of shear-induced demixing and mechanical shear banding instabilities
SM Fielding and PD Olmsted, Physical Review Letters 90 (2003) 224501.

We study the early stages of the shear banding instability in semidilute wormlike micelles using the non-local Johnson-Segalman model with a two- fluid coupling of the concentration (phi) to the shear rate (gamma_dot) and micellar strain (tensor{W}). We calculate the ``spinodal'' limit of stability for sweeps along the homogeneous intrinsic flow curve. For startup ``quenches'' into the unstable region, the instability in general occurs before the homogeneous startup flow can attain the intrinsic flow curve. We predict the selected time and length scales at which inhomogeneity first emerges. In the ``infinite drag'' limit, fluctuations in the mechanical variables (gamma_dot and \tensor{W}) are independent of those in phi, and are unstable when the slope of the intrinsic flow curve is negative; but no length scale is selected. For finite drag, the mechanical instability is enhanced by coupling to phi and a length scale is selected, in qualitative agreement with recent experiments. For systems far from an underlying zero-shear demixing instability this enhancement is slight, while close to demixing the instability sets in at low shear rates and is essentially demixing triggered by flow.

A Minimal Model for Vorticity and Gradient Banding in Complex Fluids
JL Goveas and PD Olmsted, European Physical Journal E 6 (2001) 79-89, (preprint)

A general phenomenological reaction-diffusion model for flow-induced phase transitions in complex fluids is presented. The model consists of an equation of motion for a nonconserved composition variable, coupled to a Newtonian stress relations for the reactant and product species. Multivalued reaction terms allow for different homogeneous phases to coexist with each other, resulting in banded composition and shear rate profiles. The one-dimensional equation of motion is evolved from a random initial state to its final steady-state. We find that the system chooses banded states over homogeneous states, depending on the shape of the stress constitutive curve and the magnitude of the diffusion coefficient. Banding in the flow gradient direction under shear rate control is observed for shear-thinning transitions, while banding in the vorticity direction under stress control is observed for shear-thickening transitions.

Matched Asymptotic Solutions for the Steady Banded Flow of the Johnson-Segalman Model in Various Geometries
O Radulescu and PD Olmsted, Journal of Non-Newtonian Fluid Mechanics 91 (2000) 141-162 (preprint).

 We present analytic solutions for steady flow of the Johnson-Segalman (JS) model with a diffusion term in various geometries and under controlled strain rate conditions, using matched asymptotic expansions.  The diffusion term represents a singular perturbation that lifts the continuous degeneracy of stable, banded, steady states present in the absence of diffusion. We show that the stable steady flow solutions in Poiseuille and cylindrical Couette geometries always have two bands.  For Couette flow and small curvature, two different banded solutions are possible, differing by the spatial sequence of the two bands.

The Johnson-Segalman model with a diffusion term: a mechanism for stress selection
PD Olmsted, O Radulescu, and CYD Lu, Journal of Rheology 44 (2000) 257-275 (preprint).

We study the Johnson-Segalman (JS) model as a paradigm for some complex fluids which are observed to phase separate, or  ``shear-band'' in flow. We analyze the behavior of this model in cylindrical Couette flow and demonstrate the history dependence inherent in the local JS model.  We add a simple gradient term to the stress dynamics and demonstrate how this term breaks the degeneracy of the local model and prescribes a much smaller (discrete, rather than continuous) set of banded steady state solutions. We investigate some of the effects of the curvature of Couette flow on the observable steady state behavior and kinetics, and discuss some of the implications for metastability.

Effect of Non-local Stress on the Determination of Shear Banding Flow
C-YD Lu, PD Olmsted, and RC Ball, Physical Review Letters 84 (2000) 642-645 , (preprint). .

We analyze the steady planar shear flow of the modified Johnson-Segalman model, which has an added non-local term. We find that the new term allows for unambiguous selection of the stress at which two ``phases'' coexist, in contrast to the original model. For general differential constitutive models we show the singular nature of stress selection in terms of a saddle connection between fixed points in the equivalent dynamical system.  The result means that stress selection is unique under most conditions for space non-local models.  Finally, illustrated by simple models, we show that stress selection generally depends on the form of the non-local terms (weak universality).

Two-state shear diagrams for complex fluids in shear flow
PD Olmsted,  Europhysics Letters 48 (1999) 339-345. (preprint)

The possible ``phase diagrams'' for shear-induced phase transitions between two phases are collected. We consider shear-thickening and shear-thinning fluids, under conditions of both common strain rate and common stress in the two phases, and present the four fundamental shear stress vs. strain-rate curves and discuss their concentration dependence. We outline how to construct more complicated phase diagrams, discuss in which class various experimental systems fall, and sketch how to reconstruct the phase diagrams from rheological measurements.

Shear-banding in reaction-diffusion models
O Radulescu and PD Olmsted, Rheologica Acta 38 (1999) 606-613.

Shear banding occurs in the flow of complex fluids: various types of shear thinning and shear thickening micelles solutions and liquid crystals. In order to cope with the strongly inhomogeneous interface between the bands, constitutive models used in standard rheology must be supplemented by non-local terms.  This leads rather generally to non-linear partial differential equations of the reaction-diffusion type. We use this formalism in order to explain some observed experimental features and as a guide for future research in this field.

Phase Coexistence of Complex Fluids in Shear Flow
PD Olmsted, Faraday Discussions 112 (1999) 183-194. (preprint in ps or pdf formats.)

We present some results of recent calculations of rigid rod-like particles in shear flow, based on the Doi model. This is an ideal model system for exhibiting the generic behavior of shear-thinning fluids (polymer solutions, wormlike micelles, surfactant solutions, liquid crystals) in shear flow. We present calculations of phase coexistence under shear among weakly-aligned (paranematic) and strongly-aligned phases, including alignment in the shear plane and in the vorticity direction (log-rolling). Phase coexistence is possible, in principle, under conditions of both common shear stress and common strain rate, corresponding to different orientations of the interface between phases. We discuss arguments for resolving this degeneracy. Calculation of phase coexistence relies on the presence of inhomogeneous terms in the dynamical equations of motion, which select the appropriate pair of coexisting states. We cast this condition in terms of an equivalent dynamical system, and explore some aspects of how this differs from equilibrium phase coexistence.

Liquid Crystals:

Transient and stationary flow behaviour of side chain liquid crystalline polymers: Evidence of a shear-induced isotropic to nematic phase transition
C. Pujolle-Robic, P.~D. Olmsted, and L. Noirez, Europhysics Letters 59 (2002) 364-369.

This letter describes the non-linear rheology of the isotropic phase of a thermotropic side chain liquid-crystal polymer (SCLCP), from which we infer a flow- induced iso- tropic-to-nematic (IN) phase transition above a critical shear stress and construct non-equilib- rium phase diagrams. In contrast to the well- studied wormlike-micellar solutions and predictions for simple liquid-crystalline systems, the critical stress does not vanish as the equilibrium transition temperature is approached from the above. We postulate that this is due to: i) the coupling between mesogens and the polymer backbone, whose equilibrium oblate nematic backbone conformation contrasts with the prolate non-equilibrium conformation; and ii) the peculiar topological constraints in SCLCP melts, which have been previously postulated as leading to long-lived clusters.

Phase Separation of Rigid-Rod Suspensions in Shear Flow
PD Olmsted and C-YD Lu, Physical Review E 60 (1999) 4397-4415. (preprint)

We analyze the behavior of a suspension of rigid rod-like particles in shear flow using a modified version of the Doi model, and construct diagrams for phase coexistence under conditions of constant imposed stress and constant imposed strain rate, among paranematic, flow-aligning nematic, and log-rolling nematic states. We calculate the effective constitutive relations that would be measured through the regime of phase separation into shear bands. We calculate phase coexistence by examining the stability of interfacial steady states and find a wide range of possible ``phase'' behaviors.

Coexistence and Phase Separation in Sheared Complex Fluids P. D. Olmsted and C.-Y. D. Lu, Physical Review E 56 (1997) R56-59.

We demonstrate how to construct dynamic phase diagrams for complex fluids that undergo transitions under flow, in which the conserved composition variable and the broken-symmetry order parameter (nematic, smectic, crystalline, etc.) are coupled to shear rate. Our construction relies on a selection criterion, the existence of a steady interface connecting two stable homogeneous states. We use the (generalized) Doi model of lyotropic nematic liquid crystals as a model system, but the method can be easily applied to other systems, provided non-local effects are included.

Nematogenic Fluids Under Shear Flow: State Selection, Coexistence, Phase Transitions, and Critical Behavior P. D. Olmsted and P. Goldbart, Physical Review A 46 (1992) 4966-4993.

Macroscopic fluid motion can have dramatic consequences near the isotropic-nematic transition in fluids of nematogens. We explore some of these consequences using both deterministic and stochastic descriptions involving coupled hydrodynamic equations of motion for the nematic order parameter and fluid velocity fields. By analyzing the deterministic equations of motion we identify the locally stable states of homogeneous nematic order and strain rate, thus determining the homogeneous nonequilibrium steady states which the fluid may adopt. By examining inhomogeneous steady states we construct the analogue of a first-order phase boundary i.e. a line in the nonequilibrium phase diagram spanned by temperature and applied stress, at which nonequilibrium states may coexist, and which terminates in a nonequilibrium analogue of a critical point. From an analysis of the nematic order parameter discontinuity across the coexistence line, along with properties of the interface between homogeneous states, we extract the analogue of classical equilibrium critical behavior near the nonequilibrium critical point. We develop a theory of fluctuations about biaxial nonequilibrium steady states by augmenting the deterministic description with noise terms, to simulate the effect of thermal fluctuations. We use this description to discuss the scattering of polarized light by order parameter fluctuations near the nonequilibrium critical point and also in weak shear flow near the equilibrium phase transition. We find that fluids of nematogens near an appropriate temperature and strain rate exhibit the analogue of critical opalescence, the intensity of which is sensitive to the polarizations of the incident and scattered light, and to the precise form of the critical mode.

Theory of the Non-Equilibrium Phase Transition for Nematic Liquid Crystals Under Shear Flow P. D. Olmsted and P. Goldbart, Physical Review A 41 (1990) 4578-4581.

We consider the impact of shear flow on the isotropic-nematic transition in crystalline liquids by generalizing Leslie-Ericksen nematodynamics to include amplitude and biaxial degrees of freedom. Neglecting fluctuations, we find steady state solutions to the equations of motion for the nematic order parameter and fluid velocity and interpret them in terms of non-equilibrium steady states. We predict a transition temperature increasing with shear rate up to a non-equilibrium critical point, and discuss the singular behavior of the order parameter and external stress near this point.

Nematogenic Fluids in Shear Flow: A Laboratory for Nonequilibrium Physics , P. M. Goldbart and P. D. Olmsted, Proc. of Complex Fluids: XII Sitges Conf. (Sitges-Barcelona, Spain; June 1-5, 1992) ed. L. Garrido (Springer-Verlag: Berlin) (Lecture Notes in Physics 415) (1993).

Light Scattering Near the Shear-Induced Critical Point in Nematic Liquid Crystals , P. D. Olmsted and P. Goldbart. Complex Fluids, Proceedings of 1991 MRS Fall Meeting (Boston, MA), MRS Proceedings Vol. 248 , pp. 179-184, Eds. E. Sirota, D. Weitz, T. Witten, J. Israelachvili.

We study linear fluctuations about the steady states of thermotropic liquid crystals under planar shear flow. Near the non-equilibrium critical point we find a single critical mode whose singular behavior at the critical point may be selectively probed by polarized light scattering. This critical mode is a combination of both amplitude and orientation fluctuations of the order parameter, which distinguishes it from conventional Ising-like critical modes, which are purely amplitude fluctuations. There are at least two distinct signatures of the critical point, depending on whether the critical strain rate is in the (material-dependent) strong or weak shear regimes.

Non-Equilibrium Phase Transitions in Nematic Liquid Crystals Under Shear Flow , P. D. Olmsted and P. Goldbart, Mol. Cryst. Liq. Cryst. 198 (1991) 265-271 (presented at the 13th International Liquid Crystal Conference (Vancouver BC, Canada) 1990).

The effect of shear flow on the isotropic-nematic transition in liquid crystals P. D. Olmsted, Ph.D. Thesis , Department of Physics, University of Illinois @ Urbana-Champaign (Advisor, Professor Paul Goldbart). August 1991. (353Kb .dvi file)

Reviews:

Perspectives on Shear Banding in Complex Fluids
PD Olmsted Rheologica Acta 47 (2008) 283-300.

Dynamics and flow-induced phase separation in polymeric fluids, PD Olmsted, Current Opinion in Colloid and Interface Science, 4 (1999) 95-100. Preprint available in PostScript or; Adobe PDF formats.

The past few years have seen many advances in our understanding of the dynamics of polymeric fluids. These include improvements on the successful reptation theory; an emerging molecular theory of semiflexible chain dynamics; and an understanding of how to calculate and classify `phase diagrams' for flow-induced transitions. Experimentalists have begun mapping out the phase behavior of wormlike micelles, a `living' polymeric system, in flow: these systems undergo transitions into shear-thinning or shear-thickening phases, whose variety is remarkably rich and poorly understood. Polymeric ideas must be extended to include the delicate charge and composition effects which conspire to stabilize the micelles and are strongly influenced by flow.

Wormlike micelles stir up a storm, PD Olmsted, Physics World June 1998, 22-23.

Review of the PRL by Sarah Keller, where they used freeze-fracture techniques to try and visualize the weird gel-like phase that occurs in their wormlike micelles. We still don't know what it is!