Glass rheology: From mode-coupling theory to a dynamical yield criterion

Abstract

The mode coupling theory (MCT) of glasses, while offering an incomplete description of glass transition physics, represents the only established route to first-principles prediction of rheological behavior in nonergodic materials such as colloidal glasses. However, the constitutive equations derivable from MCT are somewhat intractable, hindering their practical use and also their interpretation. Here, we present a schematic (single-mode) MCT model which incorporates the tensorial structure of the full theory. Using it, we calculate the dynamic yield surface for a large class of flows.arrest | solidification | plasticity T he 20th Century saw formidable advances in the subject known as theoretical rheology-whose aim is to predict or explain the nonlinear flow behavior of materials. Ideally, for each class of material, one wishes to gain a "constitutive equation" that predicts the stress tensor at time t as a functional of the strain tensor at all earlier times (or vice versa). There are two broad approaches to this task. The more traditional one focuses on symmetry, conservation, and invariance principles (often of some subtlety) and then proposes empirical equations that respect these principles (1). In the second approach, the goal is to start from a first-principles analysis of molecular motion and, then, by judicious (though possibly uncontrolled) approximation, arrive at a continuum-level constitutive model. This is clearly far more ambitious, and success has so far been restricted to relatively few classes of material. Perhaps the most striking success has been the Doi-Edwards theory for solutions and melts of entangled linear polymers (2, 3) [extended later to branched (4) or breakable (5) chains]. In their resting state, such polymers are ergodic and therefore attain the Boltzmann distribution: Moreover their local structure is weakly perturbed from this, even under flow.Glasses at rest, in contrast, are nonergodic on experimental time scales. This poses major obstacles to the rheological theory of glasses and is responsible for aging and other phenomena that have been partially addressed by using mesoscopic models (6). The onset of arrest at the glass transition is, familiarly, accompanied by the onset of an elastic modulus. Window glass is a brittle solid: it deforms elastically for low stresses but shatters under large ones. However, some other glasses-most notably in colloidal suspensions (whose glass transition, for hard spheres, is found experimentally at ≈58% volume fraction) are not brittle solids but show continuous yielding behavior. Although experiments suggest a more complex picture (7,8), the simplest explanation is that, above some yield stress, the glass melts. If a steady stress above the yield level is maintained, the resulting fluid can be expected to attain an ergodic (though non-Boltzmann) steady state.This restoration of ergodicity under steady flow offers one motivation for an approach to glass rheology based on mode-coupling theory (MCT). In particular, it mitigates a well-known sho...

From equilibrium to steady-state dynamics after switch-on of shear

“…Setting h 1 ≡ 1 further reduces our model to the F (γ) 12 model [17], originally proposed for the analysis of steady shear flows. This model provides excellent fits to the flow curves from large scale simulations [24], and the extension setting h 1 ≡ h 2 does not qualitatively change these flow curves, while keeping a closer connection to the more general time-dependent flows [22]. The F 12 model has glass transitions along a line of coupling parameters (v c 1 , v c 2 ), where the long time limit f = lim τ →∞ φ e (τ ) jumps discontinuously from zero to its critical value f c .…”

“…Recently, a schematic model allowing to treat arbitrarily time-dependent flow has been proposed [22], whose equations of motion in the case of steady simple shear reduce to…”

“…Its precise form is not crucial, and we follow the choice of Ref. [22] by choosing a simple decaying function that is even in the strain. There, γ c = 0.1 was introduced to model the typical cage-breaking length scale: strains of about 10% mark the point where noticeable strain reduction of memory effects sets in.…”

Simple Model for the Deformation-Induced Relaxation of Glassy Polymers

“…For uniaxial elongation at strain rateε, (4) reduces to λ = µ √ 3|ε|. Note that in this simple fluidity model, the rejuvenation of τ is essentially strain-induced [9] but, in contrast to the factorable model of [26], can be rapidly reversed by subsequent aging. Our model is completed by the standard equations of mass and force balance for an incompressible fluid of negligible inertia: ∇.v = 0, and ∇.…”

“…Second, with flow present, τ would, in the absence of such aging, itself undergo deformation-induced relaxation towards τ 0 which is a 'fully rejuvenated' value. This relaxation occurs at a rate λ, proportional to a scalar measure of flow rate (with µ another dimensionless coefficient [2,9]). In steady shear (λ = µγ), τ then varies inversely with strain rateγ in accord with microscopic theory [8].…”