Mechanics of Supercooled Liquids

Liu

Suo

2018

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This paper presents a theory of poroviscosity for binary solutions. Subject to mechanical forces and connected to reservoirs of molecules, a binary solution evolves by concurrent flow and diffusion. Our theory generalizes the classical theory of interdiffusion by decoupling the molecular processes for flow and diffusion. We further remove the assumption of local chemical equilibrium, so that the insertion of molecular into a material element, accompanied by a change in volume, is treated as nonequilibrium process and is put on the same footing as the process of shear deformation by viscous flow. The theory of poroviscosity has an intrinsic length scale, called the poroviscous length, below which the homogenization of a composition heterogeneity is limited by viscous flow, rather than by diffusion. The theory has implications for the analysis of interdiffusion in systems that display a decoupling between flow and diffusion, such as supercooled liquids, glasses, and physical gels. We illustrate the theory with numerical examples of a layered structure and a spherical particle. We discuss the results for feature sizes below and above the poroviscous length.

Section: Introductionsupporting

confidence: 81%

“…Molecules move fast in some regions and slow in others (Ediger (2000); Berthier (2011); Karmakar et al (2014)). Flow requires the rearranging of the slow moving regions, but diffusion can be facilitated by molecules hopping in the fast moving regions (Li et al (2014); Liu et al (2015)).…”

Section: Introductionmentioning

confidence: 99%

Mixing by shear, dilation, swap, and diffusion

Liu

Suo

2018

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“…When L ă Λ, the behaviour is non-Fickian, and the relaxation time tends to a constant value that coincides with the time for diffusion in an inclusion of radius a. The two-scale model predictions are similar to the predictions of phenomenological theories for coupled interdiffusion and viscous flow , as well as a theory for coupled self-diffusion and viscous flow (Li et al, 2014).…”

Section: Limiting Regimessupporting

confidence: 65%

“…These two limiting regimes were previously predicted by phenomenological theories for coupled diffusion and viscous flow (Li et al, 2014;. These theories rest on the postulate that long-range transport of species and local concentration changes are mediated by distinct molecular processes with different kinetics.…”

Section: Introductionmentioning

confidence: 54%

“…The latter could represent for example the kinetics of breaking and reforming chemical bonds or creep relaxation associated with volume change due to species insertion (Brassart et al, 2016). In the context of supercooled liquids (Li et al, 2014), the theory assumes fast diffusion through regions of high mobility and creep-limited species insertion dominated by regions of low mobility. In the present case, the local kinetics of species insertion will be associated to diffusion in inclusions with low diffusivity.…”

Section: Introductionmentioning

confidence: 99%

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Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities

Stainier

2019

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This paper presents a homogenisation-based constitutive model to describe the effective transient diffusion behaviour in heterogeneous media in which there is a large contrast between the phase diffusivities. In this case mobile species can diffuse over long distances through the fast phase in the time scale of diffusion in the slow phase. At macroscopic scale, contrasted phase diffusivities lead to a memory effect that cannot be properly described by classical Fick's second law. Here we obtain effective governing equations through a two-scale approach for composite materials consisting of a fast matrix and slow inclusions. The micro-macro transition is similar to first-order computational homogenisation, and involves the solution of a transient diffusion boundary-value problem in a Representative Volume Element of the microstructure. Different from computational homogenisation, we propose a semi-analytical mean-field estimate of the composite response based on the exact solution for a single inclusion developed in our previous work [Brassart, L., Stainier, L., 2018. Effective transient behaviour of inclusions in diffusion problems. Z. Angew Math. Mech. 98, 981-998]. A key outcome of the model is that the macroscopic concentration is not one-to-one related to the macroscopic chemical potential, but obeys a local kinetic equation associated with diffusion in the slow phase. The history-dependent macroscopic response admits a representation based on internal variables, enabling efficient time integration. We show that the local chemical kinetics can result in non-Fickian behaviour in macroscale boundary-value problems.

Shear, dilation, and swap: Mixing in the limit of fast diffusion

Liu

Suo

2016

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Molecules of different species mix by local rearrangement and long-range migration. Under certain conditions, the molecules are partially jammed: they rearrange slowly, but migrate fast. Here we formulate a theory of mixing when the long-range migration of molecules is fast, and the local rearrangement of molecules sets the time needed for mixing. In this limit, the time needed for mixing is independent of the length scale of inhomogeneity. We identify three modes of local rearrangement: shear, dilation, and swap. All three modes break and form intermolecular bonds. We place the three modes on equal footing, as distinct, concurrent, nonequilibrium processes. Our theory thus removes the bias that assumes local chemical equilibrium but allows the nonequilibrium process of shear. We propose a kinetic model of four independent viscosity-like coefficients, and a thermodynamic model of ideal mixing of molecules of unequal sizes and nonzero volume of mixing. We illustrate the theory with several examples, including the development of growth stress, the homogenization of a bilayer, and the disappearance of an inclusion in a matrix.