Molecular Transport in Viscoelastic Materials: Mechanistic Properties and Chemical Affinities

Abstract

The effects of mechanical properties and chemical affinities of materials on the transport of solutes were studied. Simulations show that the kinetics of permeant fluids in matrices depends on the rheological and chemical properties of the polymer. Fick's law failed to describe transport through viscoelastic materials because of the force exerted on the incoming fluid causing a delay. Reversible binding to immobilizing sites also retarded permeation of molecules. An integro-differential equation was applied to model transport in the presence of solute membrane interactions. While the differential part of the equation was represented by an elliptic operator, the integral part, composed of two integrals, described the contributions of stress and reversible binding. Using Laplace transforms, the steady-state flux and effective time constant were calculated. The latter parameter represents a statistical interpretation of the waiting time to achieve equilibrium in the system. The lag time, that defines the first moment when a detectable concentration is measured in a receiver cell, was also studied using multiple integration. Subsequent analyses revealed the dependence of the steadystate flux, the effective time constant, the lag time on the Young modulus, the viscosity and the binding/unbinding rates. The results presented in this paper make it possible to tune the mechanical and chemical properties to achieve a desired transport profile.

On the exponential decay of waves with memory

“…where τ is a parameter. If we assume that the kernel function Ker, when τ → 0, is such that the integral term in equation (6) formally reduces to −D 2 ∆u(t), then equation (6) is replaced by the wave equation…”

“…Indeed, the wave equation with memory is reduced to the classical wave equation. Figure 1 illustrates the behaviour of the solution of the IBVP defined by (6) with Ω = (−1, 1) 2 , with homogeneous Dirichlet boundary conditions, a gaussian profile u(x, y, 0) = e −10(x 2 +y 2 ) , (x, y) ∈ Ω, as initial data and Ker (s; τ ) = τ −1 e − s τ , ρ = c = 1 for different values of τ at t = 4. When the memory parameter τ decreases, we observe that the corresponding solution approximates the case with no memory and wave coefficient (…”

“…Let us consider now in equation (6) the damping factor c = 1 and ρ → 0. Formally, we obtain the following diffusion equation with memory Figure 2 illustrates the behaviour of the solution of the IBVP defined by (6) with Ω = (−1, 1) 2 , with homogeneous Dirichlet boundary conditions, the same gaussian profile used for obtaining We present now two different estimates for the energy…”

“…of the solution of the IBVP defined by (12) with homogeneous Dirichlet boundary conditions. The first one is obtained using the energy method and the second one is similar to the one established for instance in [6].…”

Generalized Diffusion–Relaxation Model for Solvent Sorption in Polymers

“…However, they neglected any relation between the flux and the gradient in chemical potential. This approach was further developed by Ferreira et al ,− with a series of marvelous works on the mathematical modeling for various application areas where anomalous sorption curves were observed like drying or drug release. ,, Different coefficients were required for describing the two fluxes, as the concentration gradient and stress gradient were superimposed to the total flux. The Fickian diffusion coefficient was used to relate the concentration gradient and flux, while a different proportionality constant was introduced for the stress gradient.…”