Mixing by shear, dilation, swap, and diffusion

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Abstract

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 non-equilibrium 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.

Introduction

Diffusion and flow in liquids and solids require molecules to change neighbors. Diffusion refers to the migration of a particle relative to its surrounding molecules, and the diffusivity D characterizes the rate of diffusion. Flow refers to the molecular motion to relax an applied stress, and the viscosity η gives the proportionality ratio between the stress and strain rate. Einstein (1905) assumed that a particle diffuses in a liquid by the Stokes creep of the liquid, and related the diffusivity D of the particle to the viscosity η of the liquid:DηkT=1Cawhere k is Boltzmann constant, T is the temperature, a is a length of the order of the size of the particle, and C is a dimensionless number. Einstein derived this relation for a diffusing particle much larger than the molecules of the surrounding liquid, but the relation holds remarkably well for particles of sizes down to that of the molecules of the surrounding liquid, including self-diffusion (Edward, 1970). For a particle diffusing in a viscous liquid, the Stokes-Einstein relation holds because both the diffusion of the particle and the flow of the liquid are rate-limited by a single molecular process: molecules of the liquid change neighbors.

The Stokes-Einstein relation, however, fails in most materials. The failure simply indicates that more than a single molecular process mediates diffusion and flow. For example, in a physically crosslinked gel, flow requires the breaking and reforming of the physical crosslinked network, whereas diffusion of the solvent can happen through the mesh of the polymer without rearranging the network (Phillips et al., 1989). As a second example, at a low temperature, flow of a crystalline solid is often mediated by dislocations, and is unrelated to self-diffusion. At a high temperature, flow of a crystalline solid can be mediated by self-diffusion, but the viscosity also depends on the grain size, as in the Herring creep (Herring, 1950) and the Coble creep (Coble, 1963). As a third example, in a supercooled liquid, molecules form a dynamically heterogeneous structure. Molecules move fast in some regions and slow in others (Berthier, 2011, Ediger, 2000, Karmakar, Dasgupta, Sastry, 2014). Flow requires the rearranging of the slow moving regions, but diffusion can be facilitated by molecules hopping in the fast moving regions (Li, Liu, Brassart, Suo, 2014, Liu, Huang, Suo, 2015).

This paper formulates a continuum theory of coupled flow and diffusion in a solution of two species of molecules. We view the solution as a body of many small pieces. Each piece consists of many molecules and evolves through a sequence of homogeneous states. Different pieces communicate through the compatibility of deformation, conservation of molecules, and balance of forces. For a homogeneous solution of fixed numbers of both species of molecules, the volume of the solution is taken to be independent of stress. We call such an idealized solution a molecularly incompressible solution. In our previous work (Brassart et al., 2016), we have identified that the homogeneous state of a molecularly incompressible solution of two molecular species can evolve in three modes: shear, dilation, and swap.

  • Shear changes the shape of a piece, but preserves the number of each species of molecules in the piece.

  • Dilation changes the volume of a piece by inserting molecules into, or removing molecules from, the piece, but preserves the ratio of the two species of molecules and the shape of the piece.

  • Swap changes the ratio of the two species of molecules, but preserves both the shape and volume of the piece.

The shape, volume, and composition evolve when a piece of homogeneous binary solution is subject to a combination of mechanical and chemical loads. All three modes break and form intermolecular bonds. We place the three modes on equal footing, as distinct, concurrent, non-equilibrium processes. Our theory thus removes the bias that assumes local chemical equilibrium but allows the non-equilibrium process of shear. Specifically, we do not assume chemical equilibrium at the scale of the individual piece. While diffusion is assumed as infinitely fast at the scale of a single piece, change in composition by insertion and removal of species (leading to dilation or swap) may take time. The apparent paradox is resolved by recalling the presence of fast diffusion paths that allow the fast redistribution of species across the piece.

The relaxation of the local chemical equilibrium is a fundamental departure from most the classical thermodynamic theory of irreversible processes (de Groot and Mazur, 1984). The idea was first proposed by Brassart and Suo (2012, 2013) in the context of chemically reactive host-guest systems. In our previous work (Brassart et al., 2016), we presented a thermodynamic theory of mixing in a body small enough so that diffusion is taken to be infinitely fast. We now generalize our theory into a theory that couples flow and diffusion. We also discuss the implementation of the theory using the finite element method.

To focus on the essential idea of coupled diffusion and flow without assuming local chemical equilibrium, here we neglect elasticity. Elasticity is obviously important in solids, but including elasticity here will obscure the coupling between diffusion and the three local non-equilibrium processes, shear, dilation and swap. The coupling between elasticity, flow and diffusion will be addressed elsewhere. Also see a coupled theory of viscoelasticity and poroelasticity (Hu and Suo, 2012).

Continuum mechanics describes deformation of a body by the movements of material particles. Such a description assumes that each material particle is a piece of the material with a fixed identity and number of molecules - that is, each piece is a closed thermodynamic system. However, the notion of material particles is unsuitable when all molecules can diffuse, because small pieces of material no longer preserve the identity and number of molecules in time - that is, each small piece is an open thermodynamic system. Following Darken (1948), we imagine markers dispersed in the body, such as those used to visualize the flow in hydrodynamic and metallurgical experiments. The markers are inert particles, and do not react with the molecules of the body. An individual marker is small enough to be carried with the flow without perturbing it, but is large enough to diffuse negligibly. We track the flow in the body by the movements of the markers, and identify the diffusion flux of each species of molecules by the flux relative to the markers. In effect, we use the markers to label small pieces of materials. Thus, the notion of markers replaces the conventional notion of material particles. As we will demonstrate below, with some care in interpretation, most equations in conventional continuum mechanics are applicable for markers.

Our theory based on markers differs from the classical Navier-Stokes equations for solutions, which is based on the barycentric velocity and does not allow for independent diffusion fluxes of the two species (de Groot, Mazur, 1984, Landau, Lifshitz, 1987). Our theory is also distinct from that due to Darken (1948) and Stephenson (1988), who assumed local chemical equilibrium. Darken (1948) only considered a special case that flow is so fast that the motion of the markers is limited by the diffusion of molecules. In the case of single species, our theory is consistent with the theory for coupled creep and self-diffusion (Li, Liu, Brassart, Suo, 2014, Pharr, Zhao, Suo, Ouyang, Liu, 2011, Suo, 2004). Our theory based on markers also differs from the theory coupling diffusion and creep in a crystal, which uses a crystalline lattice to label material particles (Fischer, Svoboda, 2014, Larché, Cahn, 1985, Mishin, Warren, Sekerka, Boettinger, 2013, Villani, Busso, Forest, 2015). Our theory also differs from the theory of coupled diffusion and deformation in a gel, which uses a covalent polymer network to label material particles (Bouklas, Landis, Huang, 2015, Chester, Anand, 2010, Drozdov, Christiansen, 2013, Hong, Zhao, Zhou, Suo, 2008). Analogous to Biot’s theory of poroelasticity (Biot, 1941), the theory developed in this paper may be called poroviscosity.

Like other theories of coupled diffusion and creep, our theory of poroviscosity specifies a length, which we call the poroviscous length:Λ=DηVkT,where V is a volume characteristic of the molecular volume. The poroviscous length represents the length scale where diffusional relaxation and viscous relaxation happen at comparable rate. When the size of a composition heterogeneity is small relative to the poroviscous length, homogenization is limited by viscous flow, rather than by diffusion. In our theory, this is also true in the absence of any applied stress. When the size of a composition heterogeneity is large relative to the poroviscous length, homogenization is limited by diffusion. This is the case when the Stokes–Einstein relation (1) holds, and the poroviscous length is of the order of the molecular size. Our theory then recovers the theory due to Stephenson (1988). Section 2 formulates the theory of poroviscosity for binary solutions within a thermodynamically-consistent and Lagrangian setting. Specific forms of the free energy function and kinetic models are given in Section 3. Boundary-value problems are discussed in Section 4, and a numerical implementation based on the finite element method is outlined in Section 5. In Section 6 we illustrate the theory in two examples: interdiffusion in a binary couple and swelling of a particle. Appendix A formulates the theory in an Eulerian setting and compares the theory to the Navier–Stokes equations. An analytical solution to the linearized 1D interdiffusion problem is presented in Appendix B.

Section snippets

Theory of poroviscosity

We develop poroviscosity of binary solutions; extension to solutions of more than two molecular species should not cause conceptual difficulty. Two species of molecules, A and B, are taken to be miscible in any proportion to form a solution. Subject to a history of mechanical and chemical loads, a body of the solution flows, and its composition varies in space and time (Fig. 1). Imagine markers dispersed in the body. Associated with a marker is a small piece of the solution. Each individual

Free energy function

We specify the partial free energy function FA(ξ) and FB(ξ) by adopting the model for ideal mixing of molecules of unequal size proposed in Brassart et al. (2016). In contrast with classical thermodynamic models of solutions (e.g. Flory, 1942), this model does not rely on a lattice model, and can therefore handle nonzero volume of mixing. The model is based on the very strong assumption that molecules do not interact with each other and are therefore free to explore the entire configuration

Summary of the theory of poroviscosity and boundary value problems

Sections 2 and 3 form a complete theory of poroviscosity. Given continuous fields of chemical potential μA and μB and body force B, Eqs. (4) and (5) (kinematics), (6) (force balance), (9) (conservation of species), (12) (incompressibility), (30) (flow), (33) and (34) (insertion), and (36) (diffusion) provide 34 relations to evolve 34 fields: χ, F, d, σ, RA, RB, CA, CB, JA, JB. The composition is calculated from the nominal concentrations using Eq. (8). Boundary conditions are written in terms

Weak formulation and finite element solution procedure

In this section we turn to the numerical implementation of the theory of poroviscosity with the finite element method. As is standard, the weak form of the mechanical problem is obtained by multiplying the balance Eq. (6) by a vector test function u^ that vanishes on Su, integrating over the reference volume of the body, using the divergence theorem and the boundary conditions (41) and (42) to finally obtain:P:u^dV=u^·BdV+STu^·T¯dS.Similarly, the weak form of the chemical problem is

Interdiffusion in an 1D unbounded, periodic system

We consider a periodic system consisting of alternating layers of A-rich and B-rich solutions (Fig. 3). The in-plane dimensions of the layers are assumed much larger than the out-of-plane dimension, so that diffusion takes place only in the out-of-plane, x-direction. We write the stretch in the x-direction: λx=x/X, and the swelling ratio is given by Ω=λx. The system is subject to no applied traction in the x-direction, so that mechanical equilibrium reduces to σxx=0. We assume that the system

Concluding remarks

We have proposed a theory of poroviscosity that couples viscous flow and diffusion in binary solutions. The theory relaxes the classical assumption of local chemical equilibrium. This generalization is motivated by the decoupling between molecular processes for viscous flow and diffusion that may arise in complex liquids and amorphous solids. We have also proposed a Lagrangian, finite element-based approach to solve boundary-value problems. The implications of the theory are discussed in two

Acknowledgment

The work at Harvard was supported by MRSEC (DMR-14-20570).

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