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Onsager Consistency Checks for Multicomponent Diffusion Models
Abstract
The Onsager reciprocity relations are applied to several recently proposed multicomponent diffusion models in an attempt to gauge their validity and ascertain their applicability. Each of these friction-based diffusion models stems from the more general Bearman formalism through various assumptions regarding the individual friction coefficients. By assessing the compliance of the Bearman model with respect to the Onsager relations, we ascertain the validity of the simplifications introduced to each diffusion model and suggest which postulates lead to results consistent with the Onsager relations. Although some models are not consistent with the Onsager relations, each model predicts the multicomponent drying of polymer films reasonably well. The necessity for consistency with the Onsager development is, therefore, revisited. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 1496–1504, 2001
... Originellement, cette hypothèse a été formulée par Bitter [21], qui a alors réécrit l'expression Maxwell-Stefan dans la description molaire. Zielinski et Hanley [165] ont fait de même pour une description massique : ils formulent alors leur hypothèse par la relation (ik/ Mk = (ij / Mj ('Vk, j) [163]. Bearman, par l'hypothèse des solutions régulières [18], puis Alsoy et Duda [5] ont implicitement écrit la forme volumique par l'hypothèse : ...
... (ik/Vk = (ij/10 (V'k, j) 110 [163]. On aboutit donc aux relations reportées sur la troisième ligne du tableau 3.3. ...
... L'effet de co-solubilité n'est pas négligeable systématiquement. 163 Ceci est réalisé en imposant des valeurs nulles aux dérivées croisées de l'activité : Aij,if.J = 0 Le degré d'interdépendance diffusionnelle peut être apprécié lors du calcul de la diffusivité traceur de i en négligeant la fraction volumique des autres pénétrants. Pour un système à deux pénétrants, l'expression de la diffusivité du composé 2 devient indépendante de la fraction volumique <I> 1 (et inversement) : ...
La séparation hydrocarbures / hydrogène par membrane à sélectivité inverse est étudiée. Le premier objectif est d'élaborer des matériaux dont la sélectivité est accrue. Des composites silicone - charges minérales sont réalisées, mais l'amélioration attendue n'est pas observée. Le second objectif est de modéliser le transport de plusieurs pénétrants dans les élastomères. En effet, le modèle de perméabilité ne permet pas de décrire correctement la perméation lorsque des vapeurs sont présentes. Dès lors, plusieurs phénomènes sont considérés: l'interdépendance diffusionnelle, la synergie de sorption, le gonflement et l'entraînement convectif. La dépendance des diffusivités avec la composition est représentée par la théorie du volume libre. La résolution numérique du modèle permet de prédire correctement les flux en mélange à partir des mesures sur les corps purs. Pour les systèmes étudiés, l'interdépendance diffusionnelle est prépondérante et la synergie de sorption ne peut être négligée.
... By setting different values to ˛i, the theories can be recovered. [16] checked the consistency of multicomponent diffusion models using Onsager reciprocal relations (ORR). They showed that Zielinski and Hanley [8] model satisfies the Onsager relation for low molecular weight species but fails for higher molecular weight species. ...
... gives the exactly same results as that of Zielinski and Alsoy [16] Onsager analysis. Zielinski and Alsoy [16] Onsager analysis get the results which is same as the assumptions made in model development. ...
... gives the exactly same results as that of Zielinski and Alsoy [16] Onsager analysis. Zielinski and Alsoy [16] Onsager analysis get the results which is same as the assumptions made in model development. Therefore, all the models satisfy the Onsager reciprocal relations. ...
... The collective diffusion coefficients (also known as main or principal diffusion coefficients 30 ) are the diagonal entries of the diffusion matrix D in (23b). For instance, the collective diffusion of the blue species is ...
... where d is the problem dimension, D s is given in (30), and c ≡ φ for the system in equilibrium. This relation, known as the Einstein relation, is more commonly expressed as ...
... For instance, many valid multicomponent diffusion models have been found to be inconsistent with these relations. 30 We proceed to investigate in which cases, if any, our cross-diffusion model (23) is consistent with the Onsager relations. It is appropriate to consider the free-energy gradient with respect to the densities to be the force X driving the system to its minimum free-energy equilibrium state (Ref. ...
Stochastic models of diffusion with excluded-volume effects are used to model many biological and physical systems at a discrete level. The average properties of the population may be described by a continuum model based on partial differential equations. In this paper we consider multiple interacting subpopulations/species and study how the inter-species competition emerges at the population level. Each individual is described as a finite-size hard core interacting particle undergoing Brownian motion. The link between the discrete stochastic equations of motion and the continuum model is considered systematically using the method of matched asymptotic expansions. The system for two species leads to a nonlinear cross-diffusion system for each subpopulation, which captures the enhancement of the effective diffusion rate due to excluded-volume interactions between particles of the same species, and the diminishment due to particles of the other species. This model can explain two alternative notions of the diffusion coefficient that are often confounded, namely collective diffusion and self-diffusion. Simulations of the discrete system show good agreement with the analytic results.
... For example, in Rational Thermodynamics [13], Truesdell remarks that "Onsager's and Casimir's claim that their assertions follow from the principle of microscopic reversibility which has been accepted with little question . . . the reversibility theorem and Poincare's recurrence theorem make irreversible behavior impossible for dynamical systems in a classical sense. Something must be added to the dynamics of conservative systems, something which is not consistent with it, in order to get irreversibility at all" [16]. ...
In this work, we investigate the validity of axioms such as Onsager Reciprocal Relations (ORR) for heat transfer in irreversible thermodynamics close to equilibrium. We show that the ORR for this case could be directly derived by introducing the widely accepted concept of heat transfer coefficients into the entropy production rate and by assuming that the thermal conductivity coefficients are uniquely defined. It is believed that this work can not only be used for pedagogical purposes but may also be generalized to other processes beyond heat transfer, thus leading to a generalized framework for transport phenomena and irreversible thermodynamics.
... Modeling multicomponent transport in polymer systems, however, is a delicate task, not all possible expressions are feasible for the diffusion coefficients, and a more thorough analysis of the mutual interaction of the diffusion coefficients of the different species is required. Thermodynamic limitations, indeed, have to be carefully considered in order to ensure the consistency and the robustness of the modeling approach, which has clearly to fulfill all thermodynamic constraints (in terms of Gibbs-Duhem relationship) and possibly symmetry requirements (Onsager criterion in linear relationships) [25,26]. For these reasons, dedicated transport procedures are available for modeling multicomponent transport, appropriately accounting for the effect of mutual interactions. ...
The modeling description of basic transport phenomena of either liquid, gas or vapor molecules in dense polymeric membranes is of tremendous impact for the separation industry, which relies on solid models for the design of optimal process conditions, for the selection of the most suitable membrane materials as well as for the development of novel ones. Such models need to deal with several physical aspects and phenomena, spanning over broad time and length scales, thus requiring multiple approaches. The solid frameworks now available mainly rely on the solution–diffusion theory, in which equation of state models and free volume theories are applied for the description of thermodynamic and kinetic properties, to be coupled in appropriate transport schemes.
... According to this model, multicomponent diffusivities are only a function of the self-diffusion coefficients and the thermodynamic factor of the corresponding component and they can be easily determined from readily accessible thermodynamic and self-diffusion data. The validity of this theory has previously been confirmed by: (a) predicting experimental values of the principal and cross coefficients for a ternary organic mixture [30]; (b) applying the diffusion model to predict the drying behaviour of different ternary systems [29,31]. Equations in the SS model involve friction coefficients that no experimental measurements are available on these coefficients or how they change as a function of composition [17]. ...
Asymmetric membranes were prepared by dry-cast phase inversion technique from a cellulose acetate, acetone, water solution in order to assess the validity of the mathematical model recently developed by us. Based on the model predictions, general structural characteristics of the membranes were determined by plotting the composition paths on the ternary phase diagram and polymer concentration profile at the first moment of precipitation. Composition paths on the ternary phase diagram enable the assessment of whether a phase separation occurs and allow prediction of inception time and duration of the phase separation. The polymer distribution at the moment of precipitation provides a rough thickness of the high polymer concentration region near the interface and a pore distribution of the sublayer structure. The effects of polymer/nonsolvent ratio in the casting solution, the initial film thickness, evaporation temperature, relative humidity and velocity of air were investigated. Model predictions were compared with the morphological analysis conducted using scanning electron microscopy. Results show that diffusion formulation plays an important role in capturing the accurate structure of the membrane from the model predictions.
High throughput screening of new generation nanoporous materials, e.g. metal organic frameworks (MOFs) and zeolitic imidazolate frameworks (ZIFs), for gas mixture separations have produced large amounts of data. Reported gas permeabilities have been mainly calculated using a simplified (approximate) approach. Permeability predictions of an alternative method (new method), proposed previously by our group, have shown to significantly improve the predictions of the approximate method for noble gas mixtures. Permeabilities calculated using Onsager coefficients, detailed method, are accepted as correct answers, however constructing Onsager coefficient matrix is computationally cumbersome and not feasible especially for the high throughput screening purposes. In this work, we question in what accuracy the approximate and new methods can predict gas permeabilities and permeation selectivities of gases with respect to the detailed method. We perform Grand Canonical Monte Carlo and Molecular Dynamics simulations for six ZIFs, namely ZIF-6, ZIF-10, ZIF-60, ZIF-69, ZIF-79, and ZIF-81. Permeabilities of H2/CH4 and H2/CO2 mixtures are selected as cases, since the dominating interactions of the gas species with the pores of the membranes are different. For the H2/CH4 mixture, approximate and new methods predict H2 permeability results of the detailed method sufficiently good. CH4 permeabilities of the approximate method reveal deviations from the correct answers, yet the new method improves the predictions of the approximate approach. In the case of H2/CO2 mixture, H2 permeabilities calculated by the approximate approach are in agreement with the detailed method, except for ZIF-60 and ZIF-79. For the CO2 permeability, approximate and new methods give significant deviations from the results calculated using the detailed method.
Work on interdiffusion has been mainly carried out in binary systems in the past, and this work has focused on polymer–solvent (S) systems and polymer blends. To understand and predict the interdiffusion of two solids in the presence of one S, we present a new mathematical model based on the Onsager approach. Within our model, interdiffusion kinetics are described with a modification of the reptation model for long polymer chains, and the chemical potential gradient is used as the driving force behind mass transfer. The chemical potential is calculated with a Flory–Huggins approach. The model was validated with 29 Raman spectroscopy experiments in poly(vinyl acetate)–poly(methyl methacrylate)–toluene systems at 20 °C. Monomer mobilities (L i,0s) were determined for both polymers to show the independence of L i,0 from the chain length. The L i,0s were found to be strongly dependent on the S content. With the knowledge of phase equilibria and L i,0s, interdiffusion in the ternary polymer–polymer–S system could be predicted by the introduced model. © 2018 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2019, 136, 47092.
Multicomponent diffusion in polymer solutions is a complex and not yet completely understood phenomenon. Available studies are so far almost exclusively limited to ternary systems containing a polymer and two solvents. In this work measurement data in form of isothermal drying curves in quaternary systems, consisting of the polymer poly(vinyl acetate) (PVAC) and the solvents dichloromethane, methanol and toluene obtained by inverse micro-Raman spectroscopy (IMRS) is presented. The measurements show that the diffusive behaviour is a function of the overall composition of the mixture, as additional solvents accelerate diffusion. Since no numerical model for such systems has been validated in literature, a framework for the simulation of mass transfer processes in quaternary systems has been developed. The shrinking of the film due to evaporation is taken into account by a transformation of the frame of reference. The concentration-dependent diffusion coefficients are calculated by an expression only requiring binary data, previously only tested in ternary systems, enabling predictions in uncharacterized multicomponent systems. Cross-term diffusion is neglected. The numerical model, validated by the measurement data, is able to predict the experimental results reliably for varying initial conditions.
Multicomponent diffusion of solvents in polymeric systems is not completely understood, despite many scientific contributions to the topic. Literature scarcely offers measurement data on diffusion for model validation in such systems. In this work, the ternary systems consisting of poly(vinyl acetate) and the solvents toluene and methanol was investigated experimentally and numerically. By means of inverse micro Raman spectroscopy (IMRS) concentration gradients in drying thin films have been measured. Initial composition of the samples has been varied systematically in order to detect mutual influence of the solvents' diffusive behavior. It was shown that the mobility of the different species is increased in the presence of other solvents as predicted by theory. This experimental data is provided for model validation. A new expression to calculate the diffusion coefficients in ternary mixtures is proposed which only requires binary data. This expression is tested by means of a model-based simulation to predict the drying of ternary polymer solutions in terms of concentration profiles and residual solvent content. The results are in very good agreement with the experiments. Cross terms diffusion coefficients and thermodynamic factors were not found to be necessary for a satisfying prediction. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016, 133, 43899.
Governing equations for evolution of concentration and temperature in three-component systems were derived in the framework of classical irreversible thermodynamics using Onsager variational principle and were presented for solvent/solvent/polymer and solvent/polymer/polymer systems. The derivation was developed from the Gibbs equation of equilibrium thermodynamics using the local equilibrium hypothesis, Onsager reciprocal relations and Prigogine theorem for systems in mechanical equilibrium. It was shown that the details of mass and heat diffusion phenomena in a ternary system are completely expressed by a 3x3 matrix whose entries are mass diffusion coefficients (4 entries), thermal diffusion coefficients (2 entries) and three entries that describe the evolution of heat in the system. The entries of the diffusion matrix are related to the elements of Onsager matrix that are bounded by some constraints to satisfy the positive definiteness of entropy production in the system. All the elements of diffusion matrix were expressed in terms of derivatives of exchange chemical potentials of the components with respect to concentration and temperature. The spinodal curves of ternary polymer solutions were derived from the governing equations and their correctness was checked by the Hessian of free energy density. Moreover, it was proved that setting cross-diffusion coefficients to zero results in a contradiction, and, the governing equations without cross-diffusion coefficients do not express the actual phase behavior of the system.
Diffusion coefficients of methanol and toluene in ternary methanol-toluene-poly (vinyl) acetate systems are presented at temperatures from 40°C and 60°C. An apparatus using a magnetic suspension balance [16, 17] was constructed for the measurements. The diffusion coefficients of methanol in ternary systems were correlated with free-volume theory [1,18,19] and compared with some available literature data (Surana et al. [20]). The introduction of a second solvent in polymer increases the free volume of the system and accordingly increases the diffusion coefficient of the first solvent. It has been shown that the diffusion coefficient of toluene can increase with three orders of magnitude when the methanol content in polymer is 0.1.
Sorption equilibrium and diffusion coefficients of solvents in a polymer were determined by use of a magnetic suspension balance (MSB). The binary systems poly(vinyl acetate)–methanol and poly(vinyl acetate)–toluene at 20, 40 and 60 ◦C were investigated. The solvent activity was calculated with the Flory–Huggins and UNIFAC-FV model and the diffusion coefficient with the free-volume theory. The measured values are compared with literature data and Flory–Huggins interaction parameters and free-volume parameters were adapted to the measured data.
The transport of gases and solvents in polymeric membranes is an important phenomenon with many technical applications, such as the mixture separation using polymeric membranes, the polymeric membrane preparation, the polymer devolatilization and the selection of appropriate polymeric membranes for a certain process. In most membrane separation operations and manufacturing, equilibrium and diffusion data are required.
Useful information on equilibrium and diffusion can be obtained from the solvent sorption study. The phase equilibrium for polymer solvent systems can be successfully described using the correlative model or the predictive model based on the UNIFAC group contribution such as the UNIFAC-FV. The diffusion coefficients in polymers depend strongly on temperature and their composition. First researches concerning the diffusion in polymers were reported by different studies. The free-volume theory is the most widely used theory for correlating and predicting the diffusion in polymer-solvent systems. It has been extended to predict the diffusion in ternary polymer-solvent-solvent systems using the pure component viscosity and density data and the binary polymer-solvent diffusion data.
The aim of this work is to present some literature models used to calculate the solvent-polymer equilibrium and the diffusion behaviors, including Fickian and non-Fickian diffusion. These models will be used to correlate equilibrium and diffusion data in solvent polymer systems, such as: methanol - polyvinyl acetate, toluene - polyvinyl acetate, toluene - polystyrene, dichlormethane - cellulose triacetate, water-polyvinyl alcohol at different temperatures.
Concentrations of the solvents were measured using confocal laser Raman Spectroscopy for two ternary systems, poly(styrene)—tetrahydrofuran—p‐xylene and poly(methyl methacrylate)—ethylbenzene—tetrahydrofuran, during drying at room temperature. The concentrations were compared with predictions of drying models, which utilize several existing theories for mutual diffusion coefficients for polymer solvent systems. Of the nine free volume parameters required to predict diffusion coefficients of binary systems, four for each of the four pairs studied here were estimated as suggested by the literature. Estimation was done by minimizing the difference between predictions of the model and experimental weight loss data for each binary pair. It is found that the predictions of the models which include cross term diffusion coefficients are in better agreement with measured concentrations than those which ignore the cross terms. © 2012 Wiley Periodicals, Inc. J. Appl. Polym. Sci., 2013
In the manufacturing process of adhesive tapes usually there is a step where the solvents present in the solution are evaporated in order to allow the formation of a semi-solid layer of material over a substrate. As the final product requires high coating weight and as the solvent concentration on adhesive solution is high as well, the drying becomes, most of the time, the constraint of the overall process. Productivity increase on machines that can produce hundreds of thousands of square meters of adhesive tapes per month can represent additional sale volumes or reduction on operating costs. The analysis of drying process of polymeric solutions coated over impermeable substrates through the development of a mathematical model to represent the heat and mass transfer is presented. The power of the computer-aided simulation is showed by a case study in the manufacturing of adhesive tapes at 3M Brazil.
An extended version of the Maxwell−Stefan (EMS) equation is proposed to describe transport in multicomponent solutions of molecules that are starkly different in size and whose segments are all accessible for mutual frictional interactions. The proposed modification corrects the friction factor between the colliding molecules by replacing the molar species concentration by the molar segment concentration, or equivalently, by the volume fraction of the component. The new expression for the friction factor is consistent with the kinetic theory of Curtiss and Bird for polymer solutions in the limit of linear isotropic systems. When the classical generalized Maxwell−Stefan equation is applied to diffusion in a membrane-solvent system, the (unknown) molecular weight of the membrane must be specified. In EMS, however, such a specification is not required. Further, in contrast to a previous equation for polymer solutions proposed by Heintz and Stephan (J. Membr. Sci. 1994, 89, 153), EMS is consistent with restrictions given by the Gibbs−Duhem equation and by Onsager's reciprocity relations. The multicomponent diffusion equation proposed here is appropriate for modeling mass transfer in a variety of technologies, including membrane-separation operations, polymer drying and coating, and multicomponent transport in biological membranes, cells, and drug-delivery systems.
The aim of this paper is to evaluate free-volume theory when applied to diffusion in liquids. A classical strategy for computing diffusive mass transfer consists of using the Vignes equation, but this empirical interpolation is restricted to binary systems. Thus, particular emphasis has been placed on the development of a multicomponent model based on Vrentas and Duda approach, which is widely accepted for polymeric systems. For weakly interacting species, it is shown that robust mass-transfer computations can be achieved if local tracer diffusivities, which can be theoretically computed by free-volume theory, are known. A case study based on the benzene−cyclohexane system has been performed and shows that the simplest free-volume formalism does not achieve a good prediction of experimental tracer diffusion data. Nevertheless, accurate interpolations are obtained when the excess volume is taken into account. Finally, the empirical Vignes relationship is shown to be consistent with basic free-volume arguments.
Three models, two of them relying on free volume—the Cohen–Turnbull–Fujita (CTF) model and the Vrentas–Duda (VD) model, and the third being empirical using an exponential concentration dependence of the diffusivity, were applied to desorption data for a series of alkane penetrants (2,2-dimethylbutane, cyclohexane, n-hexane, n-decane, and n-tetradecane) in low-density polyethylene. The CTF model described the desorption data very well and better than the exponential diffusion law. The VD model with the attractive feature of being based on independently determined parameters was unsuccessful in describing the desorption data. Diffusivity data indicated that the three components outside the crystal core were less accessible to n-tetradecane than to the other penetrants. This indication was further substantiated by solubility data. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 723–734, 2007
In the present work a comprehensive methodology is proposed for calculating the Fickian diffusion coefficients in a multi-component solvents–polymer system as a function of the solvents self-diffusion coefficients, the constituent chemical potentials and the process conditions. This methodology is based on the application of well-established principles such as the Gibbs-Duhem theorem for diffusing systems along with the friction coefficient concepts. The case of constant friction coefficient ratios is re-examined leading to a screening of the existing theories for multi-component polymer systems. Finally, the described methodology is applied to the formamide–acetone–cellulose acetate (CA) system which is used in the asymmetric membrane manufacture. The acetone evaporation process from this system is studied as a one-dimensional numerical experiment. For this purpose, the evaporation process is modeled as a coupled heat and mass transfer problem with a moving boundary. The Galerkin finite element method is used to simultaneously solve the non-linear governing equations. All the model parameters were estimated from literature measurements leading to a fully predictive model. The model predictions are in excellent agreement with experimental data for polymer solution weight and surface temperature vs. time thus validating the applied methodology for the calculation of friction coefficients.
The effect of glassy skin formation on the drying of semicrystalline polymers was investigated with a comprehensive mathematical model developed for multicomponent systems. Polymers with high glass-transition temperatures can become rubbery at room temperature under the influence of solvents. As the solvents are removed from the polymer, a glassy skin can form and continue to develop. The model takes into account the effects of diffusion-induced polymer crystallization as well as glassy–rubbery transitions on the overall solvent content and polymer crystallinity. A Vrentas–Duda free-volume-based diffusion scheme and crystallization kinetics were used in our model. The polymer–solvent system chosen was a poly(vinyl alcohol) (PVA)–water–methanol system. The drying kinetics of PVA films were obtained by gravimetric methods with swollen films with known water/methanol concentrations. The overall drying behaviors of the polymer system determined by our model and experimental methods were compared and found to match well. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 3191–3204, 2005
Recent publications by Zielinski and Hanley and by Alsoy and Duda have proposed relationships between self- and mutual-diffusion coefficients in multicomponent systems that can be derived from Bearman's friction-based theory. Subsequently, Zielinski and Alsoy published an article calling into question the ability of these new theories to satisfy the Onsager reciprocal relations, and Nauman and Savoca also reported that the Zielinski and Hanley theory can, in some cases, fail to satisfy material balance constraints. Here, the existing friction-based theories are reviewed and then previous approaches to relating self- and mutual-diffusion coefficients are generalized. The result is an expression that both unifies previous results and is itself a previous result. This generalization prompts us to address the issues raised regarding the existing theories. Some example calculations are also presented, and some directions for future areas of research are suggested.
Although most industrial drying and separation technologies are based on transport of multiple components, a convenient and practical means of describing multicomponent diffusion over wide temperature and concentration ranges has not been developed for polymer solutions or even for simple liquid mixtures. We have been able to demonstrate that all of the diffusion coefficients required to describe molecular transport in multicomponent systems can be easily predicted from self-diffusion coefficients and thermodynamic information when the mass flux with respect to the mass-average velocity is expressed in terms of the frictional force experienced by a molecule as it undergoes Brownian motion. This approach enables the flux equations for any combination of materials to be written in terms of experimentally measurable quantities. The model was tested by: (1) predicting literature data for principal diffusion coefficients in a ternary system; (2) applying the diffusion model to predict the drying behavior of three ternary systems; (3) predicting experimental values of the principal and cross coefficients for a ternary organic solvent mixture from self-diffusion data measured by PFG-NMR and available thermodynamic data; and (4) examining whether the limiting cases in this model are physically realistic. The applicability of the model to gas transport and polymer interdiffusion is also briefly examined.
So as to describe the interactions of bodies composed of identifiable constituents, we set up the mathematical theory ofmixtures. The theory aims to represent the phenomena of diffusion, dissociation, combination, and chemical reaction in the broadest sense. It finds long antecedence in the special theories of diffusions used by chemists and physicists and in the theories of homogeneous chemical reactions, not to mention special static and kinetic theories of heterogeneous media, but a general framework for all such theories seems first to have been laid down in a note of mine1, published in 1957. It has been extended by KELLY 2 to include electromagnetic effects, couple stresses, and singular surfaces, but today I shall need only its simplest mechanical and thermodynamic aspects.
Several expressions previously proposed for the mutual diffusion coefficient of a binary liquid system are shown to follow from the Bearman theory when a geometric-mean relationship between certain friction coefficients is assumed. The Rice-Allnatt extension of the Rice-Kirkwood theory, which can be tested without the introduction of this assumption, is shown to give virtually quantitative agreement with experiment for both viscosity and diffusion when the binary system is close to ideality. Under these conditions, however, their equations likewise approximate closely to the assumption of a geometric-mean relationship between the friction constants.
Two equations have recently been proposed to relate the mutual diffusion coefficient of a binary polymer-polymer mixture to the two tracer (self) diffusion coefficients. We show how these two equations can be obtained from a generalized statistical-mechanical approach in the hope of clarifying some of the current problems that arise in polymer-polymer diffusion. We discuss the different sets of assumptions needed in the statistical-mechanical approach to obtain the equations.
Examples of coupled irreversible processes like the thermoelectric phenomena, the transference phenomena in electrolytes and heat conduction in an anisotropic medium are considered. For certain cases of such interaction reciprocal relations have been deduced by earlier writers, e.g., Thomson's theory of thermoelectric phenomena and Helmholtz' theory for the e.m.f. of electrolytic cells with liquid junction. These earlier derivations may be classed as quasi-thermodynamic; in fact, Thomson himself pointed out that his argument was incomplete, and that his relation ought to be established on an experimental basis. A general class of such relations will be derived by a new theoretical treatment from the principle of microscopic reversibility. (§§1-2.) The analogy with a chemical monomolecular triangle reaction is discussed; in this case a a simple kinetic consideration assuming microscopic reversibility yields a reciprocal relation that is not necessary for fulfilling the requirements of thermodynamics (§3). Reciprocal relations for heat conduction in an anisotropic medium are derived from the assumption of microscopic reversibility, applied to fluctuations. (§4.) The reciprocal relations can be expressed in terms of a potential, the dissipation-function. Lord Rayleigh's "principle of the least dissipation of energy" is generalized to include the case of anisotropic heat conduction. A further generalization is announced. (§5.) The conditions for stationary flow are formulated; the connection with earlier quasi-thermodynamic theories is discussed. (§6.) The principle of dynamical reversibility does not apply when (external) magnetic fields or Coriolis forces are present, and the reciprocal relations break down. (§7.)
The Bearman statistical mechanical theory, which couples the mutual-diffusion and self-diffusion coefficients via friction factors, has been applied to polystyrene/toluene solutions with polystyrene molecular weights of 18 kDa and 900 kDa. Toluene and polystyrene self-diffusion coefficients, obtained from the literature and measured here, along with polystyrene/toluene binary mutual-diffusion coefficients and thermodynamic data, were employed to independently calculate the three friction coefficients (ζ12, ζ11, and ζ22) required to describe transport within binary solutions. Results reveal that the frequently used geometric mean approximation (GMA) for relating the friction coefficients consistently underestimates ζ22 and worsens with increasing polymer molecular weight. The GMA is found to be appreciably more accurate than the arithmetic mean approximation. A formalism recently proposed by Vrentas et al. has also been evaluated for this system and is found to describe the concentration dependence of ζ11 very well. The Vrentas model, therefore, can accurately predict mutual-diffusion coefficients from solvent self-diffusion coefficients for polystyrene/toluene solutions. © 1996 John Wiley & Sons, Inc.
Expressions for the interdiffusion and the tracer-diffusion coefficients in terms of the time-integrals of force-correlation functions are obtained using a microscopic description of the inter-diffusion and tracer-diffusion processes in a two-component liquid. The present approach provides a systematic and direct derivation, as well as an extension, of the results of Bearman et al. in their pioneering work in the late 1950s which have been used recently by Cohen and Foley to discuss the two possible relationships between the interdiffusion and tracer diffusion coefficients, referred to as the “fast-mode expression” of Hartley and Crank and the “slow-mode expression” based on the random phase approximation. In our analysis we find an additional three-particle friction term in the exact expression for the interdiffusion coefficient, which is not present in Bearman's work. It is concluded that not much can be said in favor of the fast-mode or slow-mode expressions, unless this additional three-particle friction is neglected. We also show that the center-of-volume reference frame, in which the hydrodynamic fluxes are measured, naturally follows from the microscopic description in the isothermal-isobaric limit.
Theories of the constitution of bodies suppose them either to be continuous and homogeneous, or to be composed of a finite number of distinct particles or molecules. In certain applications of mathematics to physical questions, it is convenient to suppose bodies homogeneous in order to make the quantity of matter in each differential element a function of the coordinates, but I am not aware that any theory of this kind has been proposed to account for the different properties of bodies. Indeed the properties of a body supposed to be a uniform plenum may be affirmed dogmatically, but cannot be explained mathematically.
Diffusion coefficients, partial molal volumes and refractive index derivatives are reported for six compositions of this ternary system; some new data are presented for diffusion coefficients and refractive index derivatives of the systems H2O-glycine and H2O-KCl. The Onsager reciprocal relation connecting the four diffusion coefficients at each composition of the ternary system is tested by using data from the diffusion experiments reported in this paper and existing activity coefficient data for the systems H2O-glycine, H2O-KCl and H2O-glycine-KCl; the relation is satisfied within the estimated experimental error. Equations connecting Onsager phenomenological coefficients for the solvent- and volume-fixed frames of reference are derived and discussed.
Equations are derived relating experimentally measured ternary diffusion coefficients Dij with the appropriate phenomenological coefficients Lij of the thermodynamic theory of irreversible processes. The necessary and sufficient condition that the Onsager reciprocity relations be satisfied is also given. These equations, together with the measured values of Dij and approximations for the required - but unavailable - activity data, permit a preliminary test of the Onsager Reciprocity Relations. It is found that these relations are satisfied within the estimated error for eight of the nine cases for which Dij have been experimentally determined. The activity coefficient approximations and their probable errors are discussed.
Experimental self-diffusion results are reported for the binary systems benzene-cyclo-hexane, acetone-chloroform, acetone-benzene, and acetone-water. Data were recorded as a function of concentration at 25° using the nmr spin-echo method. The data are discussed in connection with mutual diffusion results, previously published, with particular emphasis on the equations D = (∂ In a1/∂ In x1)(x1D2+x2D1) (Hartley-Crank and Darken) and D = D1(∂ In a1/∂ In c1) (Bearman and Eyring). These equations are qualitatively but not quantitatively descriptive of the experimental data. A theoretical analysis is presented in which the mutual and self-diffusion coefficients are expressed in terms of integrals of molecular velocity correlation functions. This analysis approaches a molecular view of the three diffusion coefficients and gives some insight into the nature of the Hartley-Crank relationship.
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A derivation is presented of a ternary diffusion model to describe the mass transfer processes associated with the quench bath period of the phase inversion process for membrane formation. The complete governing equations, initial conditions, and boundary conditions in the casting film and coagulation bath are presented. Equations for ternary chemical potentials and diffusion coefficients are consistently based on constant specific volume formulations. The model is applied to the analysis of mass transfer paths and their effects on membrane structure formation. Precipitation times are determined for given sets of conditions by superposing calculated mass transfer paths on the ternary phase diagram and observing when the miscibility gap is crossed. Comparisons are made with an earlier reported study on the membrane-forming system: water-acetone-cellulose acetate (CA). Agreement between predicted and measured precipitation times is found to be excellent. The polymer film composition profile at the moment of precipitation is shown to be a useful indicator of both skin and sublayer structures, allowing distinctions to be made between conditions leading to spongelike and fingerlike morphologies. The influence of model parameters on the mass transfer paths and associated polymer profiles is also discussed.
The drying of volatile solvents from a coated film is a complicated process since it involves simultaneous heat and mass transfer controlled by complex transport and thermodynamic behavior of polymer solutions. In this work, a model is developed to describe the drying behavior of multicomponent polymer solutions deposited on the impermeable substrate. A key component of the model is incorporation of multicomponent diffusion coefficients that consist of thermodynamic factors and self-diffusivities. Vrentas-Duda free-volume theory was used for predicting concentration and temperature dependency of self-diffusion coefficients. Drying processes in ovens with different zones in which temperature, velocity, and/or partial pressure of each solvent vary from zone to zone are considered. The predictions from the models provide detailed information about the relative importance of internal and external resistances to drying, effect of operating conditions, effect of the multiple-zone approach, as well as sensitivity of all physical properties on the drying rate. The validity of the model was confirmed by comparing predictions with the drying data available in the literature and the data collected in our laboratories. The experimental data and the simulation results are in good agreement.
Over a century ago, researchers discovered that membranes could separate gases. However, this technology has been commercialized only recently, mainly because the early membranes had low permeance and permselectivity, making them uneconomical. With the discovery of thin barrier membranes, either in dense, asymmetric, or thin-film composite form, gas separation by membranes has become commercially viable. Furthermore, the discovery of hollow fibre technology has permitted low-cost membrane fabrication and high-efficiency module design. Homogeneous dense hollow fibre membranes are fabricated from polymer melts using a special die. Asymmetric membranes with integral dense skins, however, are made by controlled coagulation of polymer solutions. Finally, thin-film composite membranes, where a thin dense barrier layer is made separately on a microporous substrate, are made either by simple dip coating or by complex interfacial processes. The latter concept is versatile because it permits formation of thin barrier layers by a variety of methods. The apparatus used to fabricate various membrane types and the critical factors influencing the membrane properties are discussed.
In order to validate the dry-cast model developed by us, a multifaceted experimental approach was undertaken whereby three process variables can be followed independently in real time. Since it is extremely difficult to determine experimentally the concentration and temperature profiles within the cast polymer solution, experiments were designed so as to provide information on the coupled mass- and heat-transfer processes by measuring other process variables. The experimental data-acquisition technique combined gravimetric, inframetric, and light-reflection analyses which provided information on the overall mass change, surface temperature, and the onset and duration of phase separation, respectively. Structural studies were conducted using scanning electron microscopy. These studies revealed that macrovoids or “fingers” can be fomred in dry-cast membranes. A hypothesis for the formation of fingers based on the model predictions and experimental observations is proposed.
The dry-cast membrane-formation process is a major phase-inversion technique by which asymmetric membranes are manufactured. In this paper a fully predictive model which incorporates coupled heat and mass transfer is developed to describe the evaporation of both solvent and nonsolvent from an initially homogeneous polymer/solvent/nonsolvent system. This unsteady-state, one -dimensional, coupled heat- and mass-transport model allows from local film shrinkage owing to excess volume of mixing effects as well as evaporative solvent and nonsolvent loss. The model can predict composition paths into the ternary phase diagram which determine the onset of phase transition. The ternary phase diagram is predicted using the Flory-Huggins theory allowing for composition-dependent interaction parameters. The model is applied to the well-characterized cellulose acetate/acetone/water-system for which sufficient experimental data are available to permit determination of the friction coefficients in the ternary mass-transport model.The model is solved numerically using a software package based on the method of lines which is capable of handling moving boundary problems. The modeling studies indicate that for a given polymer/solvent/nonsolvent/support system, the most influential parameters are the gas-phase mass transport, initial cast film thickness, and initial composition. Of particular importance, the model can predict the general morphological characteristics associated with the formation of dense polymer films and symmetric as well as asymmetric membranes.
A model is developed to explain the behavior of composite gas separation membranes which consist of a porous asymmetric substrate of one polymer, and a coating of a second polymer. An analogy between gas permeation and electrical flow is made, and the various portions of the composite membrane are described in terms of their resistance to gas permeation. It is shown that substrate porosity can vary significantly without altering the separating properties of such a composite, and that substrate and coating properties can be matched to optimize the flux and separation factor. Several major problems previously associated with the development of useful hollow fibers for gas separation are discussed, and it is shown that the use of Resistance Model composites can help to resolve these problems.
Thesis (Ph. D.)--University of Pennsylvania, 1962. Includes bibliography. Photocopy.
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