Interdiffusion of solvent into glassy polymer films: a molecular dynamics study

Large scale molecular dynamics and grand canonical Monte Carlo simulation techniques are used to study the behavior of the interdiffusion of a solvent into an entangled polymer matrix as the state of the polymer changes from a melt to a glass. The weight gain by the polymer increases with time t as t(1/2) in agreement with Fickian diffusion for all cases studied, although the diffusivity is found to be strongly concentration dependent especially as one approaches the glass transition temperature of the polymer. The diffusivity as a function of solvent concentration determined using the one-dimensional Fick's model of the diffusion equation is compared to the diffusivity calculated using the Darken equation from simulations of equilibrated solvent-polymer solutions. The diffusivity calculated using these two different approaches are in good agreement. The behavior of the diffusivity strongly depends on the state of the polymer and is related to the shape of the solvent concentration profile.     

Modeling diffusion in miscible polymer blend films

Recent experiments designed to probe polymer transport in the bulk and in the vicinity of surfaces have examined the interdiffusion of multilayer sandwiches of isotopically labeled polymers. The measured time dependent concentration profiles normal to the surface are typically fit to Fick's law, with a single fitting parameter, the mutual binary diffusion coefficient (MBDC). The resulting MBDCs are found to vary over a broad range of film thicknesses and time, with the time dependence being viewed as a unique signature of the reptation mechanism of long chain motion, and the thickness dependence being attributed to the slowing down of chain dynamics near surfaces. Since the experiments are conducted at finite concentration, the MBDC, which is a product of the bare mobility and the concentration derivative of the chemical potential, could be dominated by the time and thickness dependence of this second term (which is ignored in Fick's law). To quantify this conjecture we consider the more rigorous Cahn formulation of the diffusion problem in terms of chemical potential gradients. We use square gradient theory to evaluate chemical potentials, and fit the resulting time dependent concentration profiles to the analytical solution of Fick's law. By thus mimicking the experimental analysis we find that the apparent MBDCs vary with time as t(-1/2) at short times, in good agreement with existing experiments. We show that this time dependence reflects the system's desire to minimize concentration gradients, a fact ignored in Fick's law. Since these arguments make no reference to the mechanism of chain motion, we argue that the time dependence of MBDC derived from interdiffusion experiments does not provide unequivocal support for the reptation mechanism of long chain transport. The MBDC values, which also vary with the degree of confinement, are predicted to increase with decreasing thickness for model parameters corresponding to experimental systems. In contrast, since the experimental fits yield an opposite trend, we suggest that the bare mobility of the chains decreases strongly with decreasing thickness. These findings strongly support the idea that the chains are "pinned" irreversibly to the surfaces, in good agreement with other, independent experiments.     

Process of drying a polymeric paint by diffusion-evaporation and shrinkage. Determination of the concentration-dependent diffusivity

The process of drying a paint made of a dispersion of a polymer in a solvent is experimentally and theoretically studied at various constant temperatures. The diffusion of the solvent through the paint and evaporation from the surface is considered, as well as the subsequent shrinkage. From measurements made at the beginning and at the end of the drying operation on the kinetics of drying, the diffusivity is found to largely depend on the solvent concentration. An exponential relationship of the diffusivity versus the concentration is thus found and successfully tested for the whole process of drying, the diffusivity increasing with the solvent concentration.     

Simulation of Non-Linear Flow Density in Transition State from the Mathematical Model of the Diffusion of Metal Anions Through a Flat Sheet Supported Liquid Membrane

A mathematical model is developed for the carrier facilitated transport of metal ions through a flat sheet support liquid membrane (FSSLM) in transition state from Fick's second law. From this model, and from Fick's first law, the flow density is derived as a non-linear concentration gradient. Both expressions, concentration and flow density, depend on the thickness of the membrane and on time. Since the rate constant plays an important role in the model, it is considered as the parameter that controls the system and an equation for it is obtained. This equation explains the velocity of the co-transport process. The proposed model takes into account the species co-transported together with the metal ions. An equation for the number of moles of this species is obtained as a function of the metal species. The concentration gradient of this species explains the behaviour of pH in the feed phase during the process. The model is tested against experimental data corresponding to the transport of metal anions in acidic solution and it is shown that the co-transport process is reproduced with high accuracy.     

Internal concentration gradients of guest molecules in nanoporous host materials: measurement and microscopic analysis

Evolution of internal concentration profiles of methanol in 2-D pore structure of ferrierite crystal was measured in the pressure range of 0 to 80 mbar with the help of the recently developed interference microscopy technique. The measured profiles showed that both a surface barrier and internal diffusion controlled the kinetics of adsorption/desorption. Furthermore, they indicated that in the main part of the crystal, the z-directional 10-ring channels were not accessible to methanol and that the transport of methanol mainly occurred via 8-ring y-directional channels. The roof-like part of the crystal was almost instantaneously filled/emptied during adsorption/desorption, indicating accessible 10-ring channels in this section. The measured profiles were analyzed microscopically with the direct application of Fick's second law, and the transport diffusivity of methanol in ferrierite was determined as a function of adsorbed phase concentration. The transport diffusivity varied by more than 2 orders of magnitude over the investigated pressure range. Transport diffusivities, calculated from measured profiles from small and large pressure step changes, were all found to be consistent. Simulated concentration profiles obtained from the solution of Fick's second law with the calculated functional dependence of diffusivities on concentration compared very well with the measured concentration profiles, indicating validity and consistency of the measured data and the calculated diffusivities. The results indicate the importance of measuring the evolution of concentration profiles as this information is vital in determining (1) the direction of internal transport, (2) the presence of internal structural defects, and (3) surface/internal transport barriers. Such detailed information is available neither from common macroscopic methods since, they measure changes in macroscopic properties and use model assumptions to predict the concentration profiles inside, nor from microscopic methods, since they only provide information on average displacement of diffusing molecules.     

Simulation of polymer--polymer interdiffusion using the dynamic lattice liquid model

In this paper, we present computer simulation results concerning interdiffusion of fully compatible components in symmetric binary (AB) polymer mixtures in solutions. The simulation is performed in two dimensions using the algorithm based on the dynamic lattice liquid model. The solvent molecules are taken into account explicitly. The evolution of the concentration profiles in time at an interface is studied for chain lengths N=2,4,8,16 for three polymer concentrations phi=0.1,0.5,0.9. The tracer diffusion coefficients for polymer chains and for the solvent are obtained by monitoring the mean square displacements of their center of mass. The relationships between coefficients of interdiffusion and self-diffusion are tested.     

Solvent-polymer interaction: I. Molecular transport of some selected organic liquids in polymer membrane

Methods and microtechniques for determining solubility, diffusivity, thermodynamic properties, and kinetic parameters of 12 selected organic liquid solvents in polyurethane membrane by thermogravimetry (TG) are described. TG provides a simple, sensitive, rapid, and accurate microtechnique for measuring a minute change in weight (or mass) of a substance as a function of time at isothermal condition or as a function of temperature at dynamic manner. Thus from a single isothermal TG-desorption experiment, solubility and diffusivity of solvent molecules in polyurethane membrane were obtained simultaneously. Furthermore, by a dynamic TG-desorption run, kinetic parameter such as activation energy of desorption of solvent molecules from polyurethane membrane was determined. In addition, much other useful information such as equilibrium sorption constant, the changes in standard enthalpy and standard entropy of sorption, permeability, the activation energy of diffusion and so forth for solvents in polyurethane membrane are also evaluated and discussed. Finally, the correlation between the microscopic molecular structure and macroscopic properties of solvent molecules in polymer membrane is interpreted in terms of linear free energy relationships.     

Interdiffusion in Polymer Film by Ellipsometry

The long-time approximation for the time-dependence concentration of solvent into film is derived. The interdiffusion coefficients for different temperatures and the different polystyrene molecular weights were calculated by using asymptotic formulae and time-dependence concentrations of solvent into polymer films. The apparent activation energies of interdiffusion were estimated. Copyright 1999 Academic Press.     

Termination rate constant in free-radical polymerization

The specific rate constant for the termination reaction between two flexible polymer molecules with active chain ends has been considered in relation to the segmental diffusion of chain ends in solution. The probability of reaction between two chain ends per unit time when the centers of gravity of two polymer molecules are at a distance of separation has been calculated by using the Smoluchowski equation and a Gaussian distribution of chain ends. The time during which two polymer molecules are in contact has also been calculated by using the diffusion equation and the potential energy function for intermolecular interaction. The rate constant may then be completely expressed as a complex function of the intramolecular linear expansion factor, molecular weight, and the frictional properties of the reacting polymers' segment. This expression predicts that the rate constant is inversely proportional to solvent viscosity, decreasing with increasing molecular weight to some extent, and is affected by the excluded volume effect and chain flexibility. The complete expression for the rate constant has been simplified and the result compared with experimental data. Close agreement is found between the calculated rate constants and those experimentally obtained.     

Gas Diffusion in Anisotropic Nonhomogeneous Polymers

In an effort to address the situation of gas diffusion in polymers that undergo swelling during the absorption process, two mass flux constitutive relations, independent of Fick's law, are evaluated. Both mass flux relations utilize the mass diffusivity as a tensor quantity. The diffusivity tensor considered can be a function of both time and spatial coordinates. For the case of one‐dimensional diffusion, approximations are found relating the diffusivity, when it is considered to be time varying and a function of position in the polymer, to the constant diffusivity assumed by Fick's law and its predicted concentration profile. Analyzing a case of oxygen absorption by glassy poly(ethylene terephthalate) shows a time‐dependent diffusivity with negligible spatial variation.

     

聚合物熔体界面分子链扩散的流变学研究

以2种完全相容的多分散的聚合物树脂为研究对象,利用动态流变学手段实现了界面分子链扩散过程的实时检测.研究结果发现,完全相容的聚合物树脂之间的扩散动力学参数α具有相对分子质量差异的依赖性,两者之间相对分子质量差异越大,α值越大.通过流变学参数还可获得另一扩散动力学参数—特征时间tc,从而得以计算出组分之间扩散系数,研究结果表明所获得的扩散系数具有频率依赖性.     

Evaluation of permselectivity of a liquid anion-exchange membrane by diffusion dialysis

The permselectivity of liquid anion-exchange membrane containing high molecular weight amines to some organic and inorganic anions was evaluated by diffusion dialysis. The relative ion exchange constant of the membrane solution was determined by a solvent extraction procedure. The apparent diffusivity of several ion pairs through a supported liquid membrane was determined by the time lag method. The permselectivity to the anions was approximately consistent with the Hofmeister anion series and it was correlated with the diffusivity of ion pairs in the membrane and the relative ion exchange constant at the membrane-solution interface. The relative ion exchange constant was found to be the dominant factor in permselectivity. The concentration profile of ion pairs in a stack of liquid membranes was linear with the distance at steady state.     

Simulation of an ultrafiltration process of bovine serum albumin in hollow-fiber membranes

This work presents a numerical simulation of an ultrafiltration process of bovine serum albumin in solution, using hollow-fiber membranes. Such membranes are constituted of tiny polymer cylinders disposed in a tube-and-shell arrangement. The concentrate flows through the interior of the fibers and the pure solvent is recovered in the shell, assuming perfect solute rejection. In modeling the process, the flow of concentrate inside the fibers was considered to be laminar, with constant density, viscosity and solute diffusivity. Axial diffusion and angular effects were ignored. The model combines the effect of concentration polarization and adsorption, which are the two main limiting phenomena in ultrafiltration processes. The pressure on the shell side was considered constant and inside the fibers a linear pressure profile, dependent on the axial position, was adopted. The solution of the problem was achieved with the method of orthogonal collocation, with adequate choice of the weight function in the radial direction. In the axial direction, a finite-difference method was used. The numerical results were compared with experimental data available in the literature.     

Diffusivity of solvent in a polymer solution—expansive free volume effect

The concentration dependency of the diffusivity of a solvent in a polymer solution is derived on the basis of a free volume theory. Applying a molecular kinetics approach, the Fujita-Doolittle equation is modified. The result of numerical simulation reveals that the diffusivity of solvent in a polymer solution depends largely on both the polymer chain structure and its concentration. The applicability of the analytical expression derived is justified by fitting the experimental data for n-alkyl acetate-(poly-methyl acrylate) polymer solutions in the literature.     

Evidence for the existence of an effective interfacial tension between miscible fluids. 2. Dodecyl acrylate-poly(dodecyl acrylate) in a spinning drop tensiometer

We studied drops of dodecyl acrylate in poly(dodecyl acrylate) (molecular weight of 25,000) in a spinning drop tensiometer to determine whether an effective interfacial tension (EIT) existed between these two miscible fluids. We found convincing evidence. We estimated the mechanical relaxation time from an immiscible analogue (1-propanol and poly(dodecyl acrylate)) and showed that the dodecyl acrylate drops maintained quasi-steady diameters long after this relaxation period. Drops continuously grew in length and became more diffuse, but the width of the transition zone did not grow with t(1/2) as expected from Fick's law although this system had been shown to follow Fick's law in a static configuration (Antrim, D.; Bunton, P.; Lewis, L. L.; Zoltowski, B. D.; Pojman, J. A. J. Phys. Chem. B 2005, 109, 11842-11849). The EIT was determined from Vonnegut's equation, EIT = (Deltarho)omega(2)r(3)/4; both the inner and outer diameters were measured, yielding values of 0.002 and 0.02 mN m(-1), respectively. The EIT was found to be independent of the rotation rate above 6000 rpm and independent of the initial drop volume. The EIT was found to decrease with temperature and increase with the difference in concentration between the monomer drop and polymer-monomer fluid. The square gradient parameter, k, was determined from EIT = k(Deltac(2)/delta), where Deltac is the difference in mole fraction and delta is the width of the transition zone. The square gradient parameter was on the order of 10(-9) N. The square gradient parameter was found to decrease with temperature, to be independent of concentration, and to increase with the molecular weight of the polymer.     

The permeation of organophosphorus compounds in silicone rubber membranes

The permeabilities, solubilities, and diffusivities of eight organophosphorus chemicals in silicone rubber were measured at saturation concentration using two different experimental methods: permeation experiments and absorption experiments. All tests were carried out at 25°C (±3°C). The eight organophosphorus chemicals investigated are dimethyl methylphosphonate, diethyl methylphosphonate, dimethyl hydrogenphosphonate, diethyl hydrogenphosphonate, trimethylphosphate, triethylphosphate, trimethylphosphite, and triethylphosphite. These eight chemicals were selected based on their similarities to organophosphorus chemicals used as pesticides and chemical warfare agents. The experimental data were analyzed using solutions of Fick's second law of diffusion and boundary conditions representative of the experimental settings. An unsteady-state diffusion model using boundary conditions that represent uniform initial concentration in the polymer and constant but different surface concentrations was used to interpret the permeation experimental data. In this model, the effective diffusivity calculated from the steady-state permeability and equilibrium solubility of each chemical was used and was assumed to be constant.     

Measuring the mutual diffusion coefficient for dodecyl acrylate in low molecular weight poly(dodecyl acrylate) with laser line deflection (Wiener's Method) and the fluorescence of pyrene

Diffusion of small molecules into glassy polymers is quite complicated and almost always non-Fickian. Little work has been done with the diffusion of low molecular weight polymers that are liquids at room temperature (such as poly(dodecyl acrylate)) into their miscible monomers. We have studied three molecular weights under 20 000 to determine if poly(dodecyl acrylate) diffusion into dodecyl acrylate could be treated with Fick's law and if so to determine the values of the diffusion coefficients. We compare two methods for measuring the diffusion of dodecyl acrylate into poly(dodecyl acrylate): We used laser line deflection (Wiener's method) and improved upon the method from published reports. We also used the dependence of pyrene's fluorescence on the viscosity to measure the concentration distribution, and thus to extract the diffusion coefficient. After an initial relaxation period, diffusion in all cases followed Fick's law with a single concentration-independent diffusion coefficient. Comparison of the diffusion coefficients obtained by both methods yielded the same order of magnitude for the diffusion coefficients (10(-7) cm2/s) and showed the same trend in the dependence on the average molecular weight of the polymer (a decrease in the diffusion coefficient with an increase in the molecular weight).     

Studies on potentiometric stripping analysis

Theoretical and experimental investigations have been made on the method of potentiometric stripping analysis. By solving Fick's second law of diffusion for the metal in the amalgam, it was found that τ = C⁰_Rl/k[Ox], where τ is the elapsed time, l is the thickness of the mercury film, C⁰_R is the concentration of the metal in the amalgam, [Ox] is the concentration of the oxidant in the solution and k is a constant. Since C⁰_R is proportional to the concentration of that particular ion in the solution under given pre-electrolysis conditions, the relations given by the equation above can all be verified experimentally. The equation of the potential—time curve and the effect of complex formation on the elapsed time were also investigated and discussed. In the absence of complexation reactions in the solution and with proper control of the concentration of the oxidant, the lowest limit of detection for Pb was found to be 10⁻¹²M for a 4-min pre-electrolysis, with dissolved oxygen as oxidant.     

Diffusive solvent dynamics in a polymer gel electrolyte studied by quasielastic neutron scattering

A quasielastic neutron scattering study has been performed on a polymer gel electrolyte consisting of lithium perchlorate dissolved in ethylene carbonate/propylene carbonate and stabilized with poly(methyl methacrylate). The dynamics of the solvent, which is crucial for the ion conduction in this system, was probed using the hydrogen/deuterium contrast variation method with nondeuterated solvent and a deuterated polymer matrix. Two relaxation processes of the solvent were studied in the 10-400 microeV range at different temperatures. From analysis of the momentum transfer dependence of the processes we conclude that the faster process ( approximately 100 microeV) is related to rotational diffusion of the solvent and the slower process ( approximately 10 microeV) to translational diffusion of the solvent. The translational diffusion is found to be similar to the diffusion in the corresponding liquid electrolyte at short distances, but geometrically constrained by the polymer matrix at distances beyond approximately 5 A. The study indicates that the hindered diffusion of the solvent on a length scale of the polymer network interchain distance ( approximately 5-20 A) is sufficient to explain the reduced macroscopic diffusivity and ion conductivity of the gel electrolyte compared to the liquid electrolyte.     

Solubility,diffusivity, and mobility of n-pentane and ethanol in poly(1-trimethylsilyl-1-propyne)

The diffusion coefficient of ethanol and of n-pentane in PTMSP, at 27°C, was measured as a function of concentration up to a penetrant content of about 12% by weight, for polymer samples obtained through different processes; differential sorptions and desorptions with vapor phases were considered. In the case of ethanol a nonmonotonous behavior was observed for the diffusivity, while in the case of n-pentane the same property was found to monotonously decrease with increasing the penetrant content. The sorption isotherms were also reported, indicating that n-pentane exhibits a typical dual mode behavior, while ethanol follows an unusual s-shape curve. The chemical potential of the dissolved penetrants, calculated directly from the isotherms, shows the very different importance of the energetic interactions of the two penetrants with the polymer units. In spite of the remarkably different concentration dependencies observed for both solubility and diffusivity of the two penetrants, the mobility factors are in both cases monotonously decreasing with the penetrant concentration, and follow very similar trends. The significant differences observed for the concentration dependence of the diffusion coefficients are, thus, associated to the thermodynamic contributions, which are very different for n-pentane and ethanol. Different polymeric films, obtained through different solvent evaporation processes, show quite different solubility, diffusivity and mobility for both ethanol and n-pentane. On the other hand, the ratio between the mobility of the two penetrants as well as the slope of mobility as function of the concentration remains the same for all the different samples inspected. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 2245–2258, 1997