Molecular dynamics simulation of solvent-polymer interdiffusion: Fickian diffusion

The interdiffusion of a solvent into a polymer melt has been studied using large scale molecular dynamics and Monte Carlo simulation techniques. The solvent concentration profile and weight gain by the polymer have been measured as a function of time. The weight gain is found to scale as t(1/2), which is expected for Fickian diffusion. The concentration profiles are fit very well assuming Fick's second law with a constant diffusivity. The diffusivity found from fitting Fick's second law is found to be independent of time and equal to the self-diffusion constant in the dilute solvent limit. We separately calculated the diffusivity as a function of concentration using the Darken equation and found that the diffusivity is essentially constant for the concentration range relevant for interdiffusion.     

Monte Carlo simulation of polymer welding

Monte Carlo Modelling of random polymer chains, course grained onto a cubic F lattice, provides the ability to monitor the long range relaxation processes and the dynamic parameters of chains up to 400 units long. The model, described and verified by Haire et al. (Haire KR, Carver TJ, Windle AH. A Monte Carlo model for dense polymer systems and its interlocking with molecular dynamics simulation. Computational and Theoretical Polymer Science 2000; in press), is here applied to the study of molecular parameters in the vicinity of different types of surface and also to the process of polymer welding, whereby adhesion between two adjacent surfaces is achieved by the interpenetration of chains which are across the surface.The model demonstrates that a surface distorts the conformation of chains adjacent to it to give an oblate molecular envelope, that the concentration of vacant sites and chain ends increases near to the surface and that the density of points representing the centres of mass of the chains increases in the sub-surface regions. These results confirm earlier predictions and provide additional confidence in the model.Modelling of the welding process leads to the parameter intrinsic weld time, t_w, which is the time from initial perfect contact of the surfaces to the achievement of a weld within which the chain conformation is indistinguishable from the bulk. After the initial period in which the mating surfaces roughen, the welding proceeds according to the t^1/4 law predicted by reptation theory. The time to a given level of interdiffusion across the boundary is proportional to the chain length l, a comparatively weak dependence, while t_w is proportional to l³, a strong dependence. This is the same dependence on length as for the relaxation time of the chain end-to-end vectors. In fact, the agreement between the relaxation time, measured on the model of the bulk, and t_w is surprisingly close, at least for the monodisperse polymers investigated here.     

Interdiffusion of polymers across interfaces

Neutron Reflection (NR) and Dynamic Secondary Ion Mass Spectroscopy (DSIMS) experiments were conducted on symmetrically deuterated polystyrene triblock bilayers (HDH/DHD) which directly probed the interdiffusion dynamics of the chains during welding. The HDH chains had their centers deuterated 50%, the DHD chains had their ends deuterated (25% at each end) such that each chain contained approximately 50% D. During welding, anisotropic motion of the chains produces a time-dependent oscillation (ripple) in the H and D concentration at the interface, which bears the characteristic signature of the polymer dynamics. These oscillations were compared with those predicted by Rouse, polymer mode coupling (PMC), and reptation dynamics. The following conclusions can be made from this study. (a) During the interdiffusion of high molecular weight HDH/DHD pairs, higher mobility of the chain ends caused a concentration oscillation which increased to a maximum amplitude, and eventually vanished at times, t > τ_D. The amplitude, or excess enrichment found, was appreciably more than that predicted by Rouse and PMC simulations, and was only slightly less than that predicted from reptation simulations. (b) The oscillations were completely missing in the 30 and 50K HDH/DHD polymers, which are only weakly entangled. The lack of oscillations for the 30 and 50K pairs may be due to a combination of surface roughness and fluctuations of order 30 Å. (c) It was found that the position of the maximum in this ripple stayed at the interface during its growth. This is also consistent with reptation and has not been explained by other theories. (d) All dynamics models for linear polymers produce ripples, many of which are qualitatively similar to that predicted for reptation. However, each ripple bears the fingerprint of the dynamics in terms of its time-dependent shape, position, and magnitude, and the models are clearly distinguishable. Our results, in summary, support reptation as a candidate mechanism of interdiffusion at polymer(SINGLEBOND) polymer interfaces and its uniqueness is being further pursued. © 1996 John Wiley & Sons, Inc.     

An ultrasonic method to study the interdiffusion phenomenon between two polydimethylsiloxanes

We present an original method that allows the monitoring of the interdiffusion process between two miscible polymers. This acoustical method is nondestructive, and does not need any chain marking. It is based on the variations of the angular reflection coefficient R(θ) measured at fixed frequency, caused by the ultrasonic parameters modifications in the interphase during the diffusion process. We applied it to the interdiffusion between two polydimethylsiloxanes A and B of very different molecular weights. The evolution of R(θ) was also computed using successively the slow and the fast theories of diffusion, after calculation of the concentration profiles. The adjustment of the theoretical reflection spectra given by the slow theory on the experimental spectra allowed us to find the value the interdiffusion coefficient D₀, which is characteristic of the system. This value is very near the value determined from the molecular dynamics laws (Rouse, reptation, Stokes-Einstein). But in the studied time domain, the fast theory accounts for the experimental results as well as the slow theory. Finally, we discuss the precision of the method and the means to improve it.     

Measuring flux distributions for diffusion in the small-numbers limit

For the classical diffusion of independent particles, Fick's law gives a well-known relationship between the average flux and the average concentration gradient. What has not yet been explored experimentally, however, is the dynamical distribution of diffusion rates in the limit of small particle numbers. Here, we measure the distribution of diffusional fluxes using a microfluidics device filled with a colloidal suspension of a small number of microspheres. Our experiments show that (1) the flux distribution is accurately described by a Gaussian function; (2) Fick's law, that the average flux is proportional to the particle gradient, holds even for particle gradients down to a single particle difference; (3) the variance in the flux is proportional to the sum of the particle numbers; and (4) there are backward flows, where particles flow up a concentration gradient, rather than down it. In addition, in recent years, two key theorems about nonequilibrium systems have been introduced: Evans' fluctuation theorem for the distribution of entropies and Jarzynski's work theorem. Here, we introduce a new fluctuation theorem, for the fluxes, and we find that it is confirmed quantitatively by our experiments.     

Molecular weight dependence of healing and brittle fracture in amorphous polymers above the entanglement molecular weight

A model based on chain entanglements is developed to predict the molecular weight dependence of the fracture stress for brittle, amorphous polymers. It is then used in conjunction with the reptation model of polymer diffusion to predict the time and molecular weight dependence of the fracture stress in the closely related healing problem. Theoretical predictions are found to be in close agreement with experimental data.     

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.     

The concentration dependence of the diffusion and permeability in a homogeneous membrane

For the sorption and diffusion coefficient dependence on the concentration of the penetrant the transport properties of a homogeneous medium are calculated. The diffusion current is assumed to be proportional to the negative gradient of the chemical potential. This is in contrast with the first Fick's law that assumes this current to be proportional to the negative gradient of the concentration of the penetrant. The difference between the two cases depends on the concentration dependence of the sorption coefficient. In a homogeneous membrane the chemical potential formulation leads to an equation which is very similar to the Fickian expression. The apparent diffusion coefficient, however, depends not onlly on the transport resistance but also on the deviation of the sorption coefficient from constancy.     

Self-diffusion of polystyrene in solution 2. Discussion of experimental results on the basis of the reptation mechanism and entanglements

The expressions for polymer self-diffusion in semidilute solutions, theoretically derived from the reptation mechanism, the blob concept and scaling considerations, are discussed and compared against experimental data from the authors' investigations and the literature. In the nonentangled (from viscoelastic data) semidilute solution, the experimentally observed concentration and molar mass exponents are in fair agreement with those derived theoretically. However, a quantitative estimation shows that the experiments cannot be explained by reptation. Experiments with polymer mixtures also give strong evidence against reptation. It is concluded, that in the nonentangled semidilute solution, the polymer self-diffusion is more complicated than simple reptation. This is also supported by recently observed long-range density fluctuations or cluster formation in this concentration region detected by scattering techniques and NMR-PFGT. In the entangled semidilute solution, the self-diffusion data are in accordance with the reptation mechanism; reptation being within a tube having approximately 20 blobs between entanglements.     

Full‐chain dynamics of entangled linear and star polymers

The full‐chain dynamics and the linear viscoelastic properties of monodisperse, entangled linear and star polymers are simulated consistently via an equilibrium stochastic algorithm, based on a recently proposed full‐chain reptation theory 1 that is able to treat self‐consistently mechanisms of chain reptation, chain‐length fluctuations, and constraint release. In particular, it is the first time that the full‐chain simulation for star polymers is performed without subjecting to the great simplifications usually made. To facilitate the study on linear viscoelasticity, we employ a constraint release mechanism that resembles the idea of tube dilation, in contrast to the one used earlier in simulating flows, where constraint release was performed in a fashion similar to double reptation. Predictions of the simulation are compared qualitatively and quantitatively with experiments, and excellent agreement is found for all investigated properties, which include the scaling laws for the zero‐shear‐rate viscosity and the steady‐state compliance as well as the stress relaxation and dynamic moduli, for both polymer systems. The simulation for linear polymers indicates that the full‐chain reptation theory considered is able to predict very well the rheology of monodisperse linear polymers under both linear viscoelastic and flow conditions. The simulation for star polymers, on the other hand, strongly implies that double reptation alone is insufficient, and other unexplored mechanisms that may further enhance stress relaxation of the tube segments near the star center seem crucial, in explaining the linear viscoelasticity of star polymers. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 248–261, 2000     

Michaelis-Menten relations for complex enzymatic networks

Most biological processes are controlled by complex systems of enzymatic chemical reactions. Although the majority of enzymatic networks have very elaborate structures, there are many experimental observations indicating that some turnover rates still follow a simple Michaelis-Menten relation with a hyperbolic dependence on a substrate concentration. The original Michaelis-Menten mechanism has been derived as a steady-state approximation for a single-pathway enzymatic chain. The validity of this mechanism for many complex enzymatic systems is surprising. To determine general conditions when this relation might be observed in experiments, enzymatic networks consisting of coupled parallel pathways are investigated theoretically. It is found that the Michaelis-Menten equation is satisfied for specific relations between chemical rates, and it also corresponds to a situation with no fluxes between parallel pathways. Our results are illustrated for a simple model. The importance of the Michaelis-Menten relationship and derived criteria for single-molecule experimental studies of enzymatic processes are discussed.     

Viscoelastic Properties of Aqueous Anionic Surfactant (Potassium Oleate) Solutions

The dependence of the viscosity of potassium oleate salt-containing solutions on the surfactant concentration is studied. The potassium oleate concentration corresponding to the formation of semidilute solution of cylindrical micelles is determined. It is shown that, in the semidilute regime, one can distinguish two regions characterized by different exponential dependences of the viscosity on the surfactant concentration. The first region corresponds to unbreakable micelle chains that have no time to be destroyed during the characteristic time of reptation, whereas the second region corresponds to “living chains” whose lifetime is shorter than the characteristic time of reptation. In the second region, rheological properties of semidilute potassium oleate solutions are adequately described by the Maxwell model with a single relaxation time.     

Surface anchoring of nematic liquid crystal 8OCB on a DMOAP-silanated glass surface

Dynamic light scattering spectroscopy has been used to determine the temperature dependence of the anchoring strength of the nematic liquid crystal 8OCB on DMOAP-silanated glass surfaces inducing homeotropic alignment. Wedge-type glass cells with known thickness profile starting from 150 nm to several microns have been used in the experiments. The relaxation rates of the nematic fluctuations with the wave vector perpendicular to the confining surfaces have been measured as a function of the cell thickness. Fitting of the thickness dependence of the relaxation rate allows for straightforward determination of the surface extrapolation length and therefore also the strength of the surface anchoring, which is 1×10^-4 J m^-2. The overall experimental accuracy of the experiments is discussed.     

Cooperative polymer dynamics under nanoscopic pore confinements probed by field-cycling NMR relaxometry

Reptational dynamics of bulk polymer chains on a time scale between the Rouse mode relaxation time and the so-called disengagement time is not compatible with the basic thermodynamic law of fluctuations of the number of segments in a given volume. On the other hand, experimental field-cycling NMR relaxometry data of perfluoropolyether melts confined in Vycor, a porous silica glass of nominal pore dimension of 4 nm, closely display the predicted signatures for the molecular weight and frequency dependences of the spin-lattice relaxation time in this particular limit, namely T1 proportional M-1/2nu1/2. It is shown that this contradiction is an apparent one. In this paper a formalism is developed suggesting cooperative chain dynamics under nanoscopic pore confinements. The result is a cooperative reptational displacement phenomenon reducing the root-mean-squared displacement rate correspondingly but showing the same characteristic dependences as the ordinary reptation model. The tube diameter effective for cooperative reptation is estimated on this basis for the sample system under consideration and is found to be of the same order of magnitude as the nominal pore diameter of Vycor.     

Disentanglement and reptation during dissolution of rubbery polymers

The dissolution mechanism of rubbery polymers was analyzed by dividing the penetrant concentration field into three regimes that delineate three distinctly different transport processes. The solvent penetration into the rubbery polymer was assumed to be Fickian. The mode of mobility of the polymer chains was shown to undergo a change at a critical penetrant concentration expressed as a change in the diffusion coefficient of the polymer. It was assumed that beyond the critical penetrant concentration, reptation was the dominant mode of diffusion. Molecular arguments were invoked to derive expressions for the radius of gyration, the plateau modulus, and the reptation time, thus leading to an expression for the reptation diffusivity. The disentanglement rate was defined as the ratio between the radius of gyration of the polymer and the reptation time. Transport in the second penetrant concentration regime was modeled to occur in a diffusion boundary layer adjacent to the polymer-solvent interface, where a Smoluchowski type diffusion equation was obtained. The model equations were numerically solved using a fully implicit finite difference technique. The results of the simulation were analyzed to ascertain the effect of the polymer molecular weight and its diffusivity on the dissolution process. The results show that the dissolution can be either disentanglement or diffusion controlled depending on the polymer molecular weight and the thickness of the diffusion boundary layer. © 1996 John Wiley & Sons, Inc.     

Surface anchoring of nematic liquid crystal 8OCB on a DMOAP‐silanated glass surface

Dynamic light scattering spectroscopy has been used to determine the temperature dependence of the anchoring strength of the nematic liquid crystal 8OCB on DMOAP‐silanated glass surfaces inducing homeotropic alignment. Wedge‐type glass cells with known thickness profile starting from 150 nm to several microns have been used in the experiments. The relaxation rates of the nematic fluctuations with the wave vector perpendicular to the confining surfaces have been measured as a function of the cell thickness. Fitting of the thickness dependence of the relaxation rate allows for straightforward determination of the surface extrapolation length and therefore also the strength of the surface anchoring, which is 1×10^?4 J m^?2. The overall experimental accuracy of the experiments is discussed.     

Relationship between radial diffusion of copper ions released from a metal disk and cytotoxic effects. Comparison with results obtained using extracts

The extended use of metallic biomaterials yields to increasing sources of metal ions within the human body and may result in inflammation of the surrounding tissues, cell damage, and cancer. The aim of this study was to investigate the relationship between the radial diffusion of metal ions released from a metal disk by the corrosion process and the toxic effect on a cell line that grew around it. Results obtained with the metal disks (direct contact) were compared with assays made with extracts obtained from the dissolution of a metallic sample ex situ and then added to the cell culture to elucidate the cause of apparent inconsistencies in previous reports. The change of copper concentration due to corrosion and transient diffusion of copper ions from the copper disks into the cell line was evaluated according to Fick's 2nd law. Surviving cells distribution was interpreted considering the radial and time-dependence of copper concentration. We concluded that the toxic effect on those cells close to metallic biomaterials may be underestimated when only the extract methodology is employed for cytotoxic tests or when during the experiments with disks the presence of concentration gradients and the non-homogeneous distribution of dead cells are disregarded.     

Molecular basis of healing and fracture at polymer interfaces

The interdiffusion of polymer chains across a polymer–polymer interface, and subsequent fracture to re-create the interface is reviewed. In particular, films formed via latex coalescence provide a very large surface area. Of course, latex film formation is a very important practical problem. Healing of the interface by interdiffusion is treated using the de Gennes reptation theory and the Wool minor chain reptation model. The self-diffusion coefficients of polystyrene and the polymethacrylates obtained by small-angle neutron scattering, SANS, direct non-radiative energy transfer, DET, and other techniques are compared. Reduced to 150,000 g/mol and 135°C, both polystyrene and poly(methyl methacrylate) have diffusion coefficients of the order of 10^?16?10^?17 cm²/sec. Variations in the diffusion coefficient values are attributed to the experimental approaches, theoretical treatments and molecular weight distribution differences. An activation energy of 55 kcal/mol was calculated from an Arrhenius plot of all polystyrene data reduced to a number-average molecular weight of 150,000 g/mol, using an inverse square molecular weight conversion method. Interestingly, this is in between the activation energies for the α and β relaxation processes in polystyrene, 84 and 35 kcal/mol, respectively. Fracture of polystyrene was considered in terms of chain scission and chain pull-out. A dental burr apparatus was used to fracture the films. For low molecular weights, chain pull-out dominates, but for high molecular weights, chain scission dominates. At 150,000 g/mol, the energy to fracture is divided approximately equally between the two mechanisms. Above a certain number average molecular weight (about 400,000 g/mol), the number of chain scissions remains constant at about 10²⁴ scissions/m³. Energy balance calculations for film formation and film fracture processes indicate that the two processes are partly reversible, but have important components of irreversibility. From the interdiffusion SANS data, the diffusion rate is calculated to be about 1 Å/min, which is nine orders of magnitude slower than the dental burr pull-out velocity of about 0.8 cm/sec.     

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).