Mass transport with relaxation in polymers (original) (raw)
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Non-Fickian mass transport in polymers
Journal of Rheology, 2002
The model of isothermal diffusion in a polymeric medium derived by El Afil et al. ͓A. El Afif, M. Grmela, and G. Lebon, J. Non-Newtonian Fluid Mech. 86, 253 ͑1999͔͒ is investigated in the absence of an overall flow and in mechanical equilibrium. First, we derive its more macroscopic reduced versions and compare them with the models introduced previously in the literature. Next, we investigate the wave propagation of disturbances in the solvent concentration. Subsequently, we specify the free energy and kinetic coefficients that appear in the general governing equations and solve ͑using both qualitative and numerical methods͒ the governing equations expressed in the material coordinates. In this way we obtain the time evolution of the solvent concentration, the diffusion flux, the swelling, the internal deformations and stresses, and the internal viscosity associated with the solvent penetration and the swelling. The governing equations involve three parameters that express the individual nature of the mixture: the relaxation time of the polymeric structure, the relaxation time of the diffusion flux, and one parameter that expresses coupling of the polymeric structure and the solvent concentration in free energy. As an illustration, we show that with these three characteristic parameters we can reproduce results of observations that we have selected from the literature ͓N. L. Thomas and A. H. Windle, Polymer 19, 255 ͑1978͔͒. In particular, we reproduce the case II type diffusion observed in the absence of a glass-rubber transition.
Rate type equations for the diffusion in polymers: Thermodynamic constraints
AIChE Journal, 1993
ABSTRACT Conditions imposed by the second law of thermodynamics on viscoelastic rate type constitutive equations for the diffusive mass flux are considered. The analysis of three different rate type models proposed in the literature points out that presently physically unrealistic predictions are possible in desorption processes. The thermodynamic analysis of such models, based on the entropy inequality and on the stability requirement of the equilibrium states, leads to precise relationships among relaxation times, diffusion coefficients, and entropy equations of state. In particular, the analysis shows that relaxation times and diffusion coefficients cannot be simply constant numbers. When the thermodynamic constraints imposed on the constitutive equations are introduced, the models do not show physically unrealistic behaviors any more; Fickian diffusion close to the pure penetrant or pure polymer regions is also recovered. Finally, it is shown that the stability requirement for the equilibrium states may introduce very rigid requirements for the model feasibility, well beyond what appears explicitly from the kinetic equations alone.
Non-Fickian behaviors in mass transport are very common in polymeric materials. The classical Fickian law, dictating the instantaneous response of the mass flux to changes in the driving force (namely the concentration gradient), seems to be too simplistic for mass transport in viscoelastic materials, generally linked through characteristic times to stress relaxation or structural changes. A constitutive equation in which the evolution of the mass flux is governed by a characteristic relaxation time thus appeared to be the most natural choice in order to catch anomalous behaviors. This has led since the '80s to the hyperbolic formulation of the transport equations, known as Cattaneo-Maxwell equations in the one-dimensional case. However, hyperbolic equations use for mass transport applications were limited by the well known but never understood the unpredictable appearance of inconsistencies (read unphysical concentration overshoots and negative concentration values). Recently, inconsistencies are explained by letting the hyperbolic formulation of transport equations in polymeric systems be derived from the theory of stochastic processes possessing finite propagation velocity, namely the Poisson-Kac processes. In this paper, the hyperbolic transport model based on partial probability waves is introduced and discussed. The model was applied to two relevant literature cases, showing to be suitable for the description of anomalous diffusional phenomena.
Modeling of Mass Transport into Immiscible Polymeric Blends
Macromolecules, 2003
A nonlinear 3D model for mass transport into immiscible polymeric blends is developed by explicitly incorporating the interface dynamics into the transport equations. The interface is characterized, on a mesoscopic level of description, by a scalar Q(r,t) and a second-order tensor q(r,t) respectively describing the local size and anisotropy densities of the interfacial area. The newly obtained constitutive equation for the diffusion mass flux density extends Fick's first law by involving two additional terms accounting for the local changes of the interface morphology. The model provides an expression for the distribution of both isotropic (Laplace) and anisotropic stresses created by mass transport within the immiscible polymeric blend. The governing equations are parametrized by the free energy density that includes a mixing part and an excess energy term attributed to the presence of the interface. We investigate in more detail a one-dimensional sorption process of a solvent into a thin immiscible blend consisting of a matrix and a dispersed phase. Three dimensionless groups of physical parameters arise in the 1D dimensionless formulation; two are coupling constants that explicitly relate diffusion to the interface dynamic changes, and one is the diffusion Deborah number. Numerical results show that diffusion becomes non-Fickian for values of Deborah number approaching unity. The time evolution of the calculated mass uptake, swelling, stresses, and total size and anisotropy densities provides a good indication of the effects of diffusion-interface interaction on both mass transport and the morphology of the interface.
Molecular Transport in Viscoelastic Materials: Mechanistic Properties and Chemical Affinities
SIAM Journal on Applied Mathematics, 2014
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.
Flow and mass transport in blends of immiscible viscoelastic polymers
Rheologica Acta, 2009
The purpose of this paper is to derive a nonlinear 3D model that investigates, on two levels of description, the flow–diffusion–structure interaction occurring in mixtures consisting of a Newtonian fluid and a blend of immiscible viscoelastic polymers embedding an interface. The internal structure is characterized on the kinetic level of description by two distribution functions and on the mesoscopic level by a scalar and two symmetric second-order tensors. The morphology of the interface and the conformation of the macromolecules are found to be strongly dependent on the flow and diffusion. In return, the behavior of the flow and mass transport are shown to be influenced by the deformation of the internal structure. In the absence of flow, the behavior of diffusion is examined in more detail, and the occurrence of Non-Fickian mass transport is discussed. The dimensionless form of the governing equations includes three Deborah numbers and three coupling constants. The nature of propagation of both nonlinear hyperbolic and linear dispersive waves is examined and explicit formulas for the characteristic speed, phase velocity, and attenuation are provided. The stability conditions determine the range of validity of the kinetic coefficients involved in the model equations.
Packaging Technology and Science, 2019
The unsteady, conjugate, mass transfer of the packaging constituents to a food product has been analysed. The system was considered motionless (hydrostatic conditions). The diffusion coefficient inside the packaging material was considered concentration variable while the diffusion coefficient inside the food product was considered constant. The mass balance equations were solved numerically. The influence of the concentration-variable diffusivity on the mass transfer mechanism and rate was analysed for different values of the partition coefficient and packaging thickness.
A mathematical model for a dissolving polymer
AIChE Journal, 1995
In certain polymer-penetrant ,systems, nonlinear viscoelastic effects dominate those of Fickian diffusion. This behavior is often embodied in a memory integral incorporating nonlocal time effects into the dynamics; this integral can be derived from an augmented chemical potential. The mathematical framework presented is a moving boundary-value problem. The boundary separates the polymer into two distinct states: glassy and rubbery, where different physical processes dominate. The moving boundary condition that results is not solvable by similarity solutions, but can be solved by perturbation and integral equation techniques. Asymptotic solutions are obtained where sharp fronts move with constant speed. The resultant profiles are quite similar to experimental results in a dissolving polymer. It is then demonstrated that such a model has a limit on the allowable front speed and a self-regulating mass uptake.
Chemical Engineering Science, 2007
The detailed transports of both small and large molecules in heterogeneous media including either random disorder or periodic obstacles are known to decrease the value of macroscopic diffusion coefficients. This work proposes to analyze the successive displacements of medium-sized molecules in polymer materials according to the dispersion and topology of sorption sites from a modified application of the transition state theory. In absence of available information on the dispersion of rate constants between sorption macrosites for such molecules, their transport mechanisms at molecular scale is related to their sorption properties, which are more likely to be available. Simulations by kinetic Monte Carlo (KMC) techniques are presented for different distributions of occupancy values randomly allocated in space or distributed in self-similar clusters. Network structures are generated from the equilibrium occupancy on the basis of transition-state theory formulation on 2D lattice approximations. Different reconstruction strategies on 2D hexagonal lattices are examined regarding maximum likelihood principles including maximization of obstruction effects and minimization of either local or global variance of conductances between sorption sites. Effects of short time scales are assessed by comparing results obtained with networks verifying reversible and non-reversible random walks.