Mutual diffusion and thermodynamics in the blends of polystyrene and tetramethylbisphenol-A polycarbonate

A study was made of miscible polymer blends of deuterated polystyrene (d-PS) and tetramethylbisphenol-A polycarbonate (TMPC). The Flory interaction parameter χ was obtained from the relation between mutual and tracer diffusion coefficients, D? and D^*, which were measured by forward recoil spectrometry. The temperature dependence of diffusion at PS weight fractions ω of 0.25 and 0.5, and the composition dependence at temperatures 45°C above the glass transition temperature, T_g, were investigated. A stronger dependence of χ on both temperature (at ω = 0.5) and composition was observed in comparison with other miscible binary polymer blends involving PS. Analysis using the generalized lattice-fluid model of Sanchez and Balazs¹ showed that the incorporation of a significant specific interaction is needed to explain the temperature dependence of χ. The diffusion coefficients obtained in the one-phase region were extrapolated to the two-phase region, and these were compared with the effective diffusion coefficient extracted from phase separation dynamics measured by light scattering.² A significant discrepancy between the extrapolated and effective diffusion coefficients was observed. © 1995 John Wiley & Sons, Inc.     

Diffusion of camphorquinone in uniaxially drawn polycarbonate films

Employing the laser-induced holographic grating relaxation technique, we have measured tracer diffusion coefficients of a phtochromous dye, camphorquinone, in uniaxially drawn polycarbonate films as a function of stretch ratio. Anisotropy in the tracer diffusion coefficient has been observed with D_∥ greater than D_⟂ by at least a factor of 4 for the film stretched to the stretch ratio δ = 2.3. The diffusion coefficient along the direction of stretch D_∥ increases significantly with increasing δ, whereas D_⟂ decreases slightly with increasing δ. The stretch ratio dependence of D_∥ and D_⟂ is interpreted according to a modified free volume theory. The strain rate and stretch temperature dependence of the anisotropic tracer diffusion coefficient has also been investigated. © 1992 John Wiley & Sons, Inc.     

Free volumes in simple and polymeric liquids

Values of ε₀ν₀ the vaporization energy and volume in the hypothetical liquid state at 0°K., are derived for some simple polar and nonpolar molecules used as models for vinyl polymers. The following empirical relationship between the free volume fraction, f = (v ? v₀)/v, and the liquid compressibility coefficient β is demonstrated: ?f² ∝? This is applied to several vinyl polymer liquids near their glass transition temperatures, T_g, giving. f_g ? 0.17, if the “hard-core” volume v^* is considered to be independent of pressure and temperature, (i.e., v^* = v₀); or, f_g ?0.12, if the P,T dependence of v^* is considered to be the same as that of the glass. These agree with f_g values derived by Simha and Boyer from thermal expansion coefficients for the two analogous cases. An empirical viscosity-free volume equation of the Doolittle form: η = ATⁿe^b/f is applied to the glass transition, on assuming that this is an isoviscosity state and with the use of reported values for the expansion and compressibility coefficients and dT_g/dP for three polymers: polystyrene, poly(methyl methacrylate), and poly(vinyl acetate). Reasonable values of b/n are thus obtained. This viscosity equation is critically examined in the light of molecular theories of liquid viscosity.     

Diffusion Coefficients for Binary, Ternary, and Polydisperse Solutions from Peak-Width Analysis of Taylor Dispersion Profiles

Binary mutual diffusion coefficients D can be estimated from the width at half height W _1/2 of Taylor dispersion profiles using D=(ln 2)r ² t _R/(3W ² h) and values of the retention time t _R and dispersion tube radius r. The generalized expression D _h=−(ln h)r ² t _R/(3W ² _h ) is derived to evaluate diffusion coefficients from peak widths W _h measured at other fractional heights (e.g., (h = 0.1, 0.2,…,0.9). Tests show that averaging the D _h values from binary profiles gives mutual diffusion coefficients that are as accurate and precise as those obtained by more elaborate nonlinear least-squares analysis. Dispersion profiles for ternary solutions usually consist of two superimposed pseudo-binary profiles. Consequently, D _h values for ternary profiles generally vary with the fractional peak height h. Ternary profiles with constant D _h values can however be constructed by taking appropriate linear combinations of profiles generated using different initial concentration differences. The invariant D _h values and corresponding initial concentration differences give the eigenvalues and eigenvectors for the evaluation of the ternary diffusion coefficient matrix. Dispersion profiles for polymer samples of N i-mers consist of N superimposed pseudo-binary profiles. The edges of these profiles are enriched in the heavier polymers owing to the decrease in polymer diffusion coefficients with increasing polymer molecular weight. The resulting drop in D _h with decreasing fractional peak height provides a signature of the polymer molecular weight distribution. These features are illustrated by measuring the dispersion of mixed polyethylene glycols.     

Self‐diffusion of styrene, 2,2′‐azobisisobutyronitrile,and polystyrene in bulk polymerization by pulsed‐gradient spin‐echo nuclear magnetic resonance spectroscopy

The self‐diffusion of styrene, polystyrene, and 2,2′‐azobisisobutyronitrile has been determined in the bulk polymerization of styrene with pulsed‐gradient spin‐echo nuclear magnetic resonance at 25 °C. Data on small molecules are discussed with respect to recent diffusion models. They can fit self‐diffusion coefficient data of small molecules in dilute or semidilute polymer solutions; in concentrated solutions, however, there is a breakdown. A semiempirical model based on scaling laws is used to describe the self‐diffusion of styrene and 2,2′‐azobisisobutyronitrile over the whole range of concentrations studied. The dependence of the polystyrene self‐diffusion coefficient on the polymer concentration is described with a stretched exponential function, D = D₀ exp(?αc^ν), where α depends on the molecular weight of the polymer and ν depends on the kind of solvent. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1605–1614, 2003     

Dispersive aspects of the high-field hopping mobility of molecularly doped solids with dipolar disorder

The time-of-flight mobility of photoinjected charges in molecularly doped polymers obeys a Poole–Frenkel law, μ ∝ exp($\gamma \sqrt E $), which is commonly viewed as arising from hopping transport among sites with a large degree of energetic disorder. Recent theoretical investigations have focused on long-range correlations that characterize site energies when the dominant mechanism for energetic fluctuations is the interaction of charge carriers with randomly-oriented permanent dipoles of the dopant and host polymer. An exact calculation of the steady-state drift velocity v_d for a one-dimensional system with correlated dipolar disorder predicts a Poole–Frenkel law similar to that observed. In order to investigate another feature commonly observed in the high-field measurements, namely, the anomalous dispersion of the current–time transients, we have performed an exact calculation of the field-dependent diffusion constant D for the same dipolar disorder model. In the bulk limit we obtain an expression D = (KT/e) ∂v_d/∂E that generalizes the normal Einstein relation and predicts a strongly field-dependent diffusion constant. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 2803–2809, 1997     

Diffusion-controlled reaction. V. Effect of concentration-dependent diffusion coefficient on reaction rate in graft polymerization

The effect of diffusion on radiation-initiated graft polymerization has been studied with emphasis on the single- and two-penetrant cases. When the physical properties of the penetrants are similar, the two-penetrant problem can be reduced to the single-penetrant problem by redefining the characteristic parameters of the system. The diffusion-free graft polymerization rate is assumed to be proportional to the v power of the monomer concentration C, in which the proportionality constant a = k_pR/k, where k_p and k_t are the propagation and termination rate constants, respectively, and R_i is the initiation rate. The values of v, w, and z depend on the particular reaction system. The results of our earlier work were generalized by allowing a non-Fickian diffusion rate, obtained from an extension of the Fujita free-volume theory, which predicts an essentially exponential dependence on the monomer concentration of the diffusion coefficient, D = D₀ [exp(δC/M)], where M is the saturation concentration. It was shown that a reaction system is characterized by the three dimensionless parameters v, δ, and A = (L/2)[aM^(v?1)/D₀]^1/2, where L is the polymer film thickness. Graft polymerization tends to become diffusion controlled as A increases. Larger values of δ and v cause a reaction system to behave closer to the diffusion-free regime. The transition from diffusion-free to diffusion-controlled reaction involves changes in the dependence of the reaction rate on film thickness, initiation rate, and monomer concentration. Although the diffusion-free rate is w order in initiation rate, v order in monomer, and independent of film thickness, the diffusion-controlled rate is w/2 order in initiator rate and inverse first-order in film thickness. The dependence of the diffusion-controlled rate on monomer is dependent in a complex manner on the diffusional characteristics of the reaction system.     

Broken orbital symmetry study of low-lying excited and N15 ionized states of pyrazine

Ab initio ΔE_SCF calculations of the low-lying n-π^* and π-π^* transitions in pyrazines are reported with D_2h and C_2v symmetry adapted molecular orbitals. The use of broken orbital symmetry is essential for interpreting the emission properties of pyrazine at the computational level used. The energy of the N_is orbital is calculated using these two symmetry constraints with C_2v orbitals leading to results in better agreement with experiment.     

Comparison of the Diffusion of Aqueous Glycine Hydrochloride and Aqueous Glycine

A Taylor dispersion tube has been used to measure mutual diffusion in aqueous solutions of glycine hydrochloride at 25°C and concentrations from 0.0005 to 0.5 M. Analysis of the dispersion profiles shows that the diffusion of glycine hydrochloride (GlyHCl) produces a subtantial additional flow of hydrochloric acid that is liberated by the dissociation: GlyH⁺ + Cl^- Gly + H⁺ + Cl^-. Diffusion in this system is, therefore, a ternary process described by the equations J ₁(GlyHCl) = – D ₁₁C ₁ – D ₁₂C ₂ and J ₂(HCl) = –D ₂₁C ₁ – D ₂₂C ₂ for the coupled fluxes of total glycine hydrochloride (1) and hydrochloric acid (2) components. The ratio D ₂₁/D ₁₁ of measured diffusion coefficients indicates that up to two moles of HCl are cotransported per mole of GlyHCl. Although protonated glycine diffuses with relatively mobile Cl^– counterions, the main diffusion coefficient of glycine hydrochloride, D ₁₁, is lower than or nearly identical to the diffusion coefficient of aqueous glycine. A model for the diffusion of protonated solutes is developed to interpret this result and the large coupled flows of HCl. Diffusion coefficients are also reported for the aqueous hydrochlorides of 3- and 4-aminobenzoic acids.     

Kinetics of Methyl Iodide Tribosorption on the Solid-Phase Salt KI

It was found that the tribosorption of methyl iodide from the gas phase of a closed reactor onto a matrix of KI is described by the rate equation for a reversible first-order reaction v _g – v _{, g} = mv _ts ^* exp(–_td D _sp), where v _g and v _{, g} are, respectively, the current and equilibrium amounts of methyl iodide in the gas phase; v _ts ^* is the equilibrium amount of methyl iodide tribosorbed per gram of salt; m is the mass of potassium iodide; _td is a constant, which characterizes the efficiency of tribodesorption (_td = 0.011 ± 0.005 g/J); and D _sp is the specific dose of mechanical energy absorbed by the KI powder. The value of v _ts ^* monotonically increased with increasing equilibrium partial pressure of methyl iodide and reached a maximum value of 25 mol/g. The lower limit of the constant _td, which characterizes the efficiency of tribosorption, was estimated at 0.1 g/J.     

Small-Molecule Diffusion in Semicrystalline Polymers as Revealed by Experimental and Simulation Studies

Summary: Diffusion of n-hexane in poly(ethylene-co-1-hexene)s with 15–75 wt.% crystallinity was studied by desorption experiments analyzing data using the Fickian equations with a concentration dependent diffusivity. The effect of the impenetrable crystalline phase on the penetrant diffusivity (D) is described by D = D_a/(τβ), where D_a is the diffusivity of the amorphous polymer, τ is the geometrical impedance factor and β is a factor describing the constraining effect of the crystals on the non-crystalline phase. For a polymer with 75 wt.% crystallinity, τβ varied markedly with penetrant concentration (v_1a) in the penetrable phase: 1000 (v_1a = 0) and 10 (v_1a = 0.15). This penetrant-uptake had no effect on the gross crystal morphology, i.e. β must be strongly dependent on v_1a. Samples saturated in n-hexane exhibited a penetrant-induced loosening of the interfacial structure, as revealed by an increase in crystal density that require an increased mobility in the interfacial component and by a decrease in the intensity of the asymmetric X-ray scattering associated with the interfacial component. The geometrical impedance factor has been modelled by mimicking spherulite growth and τ was obtained as the ratio of the diffusivities of the fully amorphous and semicrystalline systems. The maximum τ obtained from these simulations is ca. ten, which suggests that β in the systems with v_1a = 0.15 takes values close to unity. The simulations showed that the geometrical impedance factor is insensitive to the ratio of the crystal width and the crystal thickness. A free path length scaling parameter characteristic of the amorphous phase correlated with τ.     

Soret Coefficients for Aqueous Polyethylene Glycol Solutions and Some Tests of the Segmental Model of Polymer Thermal Diffusion

A thermogravitational cell is used to measure Soret coefficients (s) for dilute binary aqueous solutions of ethylene glycol, diethylene glycol, triethylene glycol, tetraethylene glycol, and polyethylene glycol (PEG) fractions with average molecular weights from 200 to 20,000 g-mol^–1. The cell design allows the top and bottom halves of the solution column to be withdrawn and injected into a high-precision HPLC differential refractometer detector for analysis. Previously reported mutual diffusion coefficients D and the measured Soret coefficients are used to calculate thermal diffusion coefficients D _T. s and D vary with the PEG molecular weight M as M ^+0.53 and M ^–0.52, respectively; hence, D _T = sD is essentially independent of M. The segmental model of polymer thermal diffusion predicts D _T = D_seg U _S/RT ², where D _seg is the segment diffusion coefficient, U _S the solvent activation energy for viscous flow, R the gas constant, and T the temperature. The predicted D _T values, although independent of M, are too large by a factor of five. Additional tests of the segmental model are provided using literature data for polystyrene + toluene, n-alkane + CCl₄, and n-alkane + CHCl₃ solutions. Agreement with experiment is not obtained. In particular, the measured D _T values for the alkane solutions are negative.     

Chain dynamics in entangled polymers: Diffusion versus rheology and their comparison

This review article scrutinizes and reanalyzes the extensively available literature data on the tracer and self chain diffusion coefficients D_tr and D_s along with the corresponding zero‐shear viscosity η₀ to show that D_s ∝ M^?ν starts with ν > 2.0 and converges to the asymptotic scaling exhibited by D_tr ∝ M^?2.0 as the molecular weight M increases beyond M/M_e = 10–20, in contrast to the onset of the asymptotic scaling M³ for η₀ taking place typically for M/M_e ? 10–20. A coherent analysis of these observations leads to the suggestion that the observed crossover in D_s is due to the constraint release effect, which diminishes around M/M_e = 10–20 and is negligible in measurements of D_tr when the matrix molecular weight P is much greater than M. The contour length fluctuation (CLF) effect, which is believed to cause the molecular weight scaling of η₀ to deviate significantly from its limiting behavior of M³, has little direct influence on the chain diffusion. The absence of the CLF effect on D_s leads to a much stronger than linear dependence of the product η₀D_s on M, which has been observed previously. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1589–1604, 2003     

Linear polymers in nonhomogeneous flow fields. I. Translational diffusion coefficient

The translational diffusion coefficient D, needed in treatments of flow-induced polymer migration phenomena, is studied. Explicit results are reported for bead–spring model macromolecules with two, three, and four beads, respectively. If the number of beads is large, an integral formulation is used. The results obtained show that D depends upon the hydrodynamic interaction parameters h^* and h^*N^1/2. Nondraining behavior of D is predicted irrespective of h if h^* is close to 0.2357. Approximations for D are obtained and the error involved is estimated.     

Fluorescence recovery after photobleaching measurements of polymers in a surface forces apparatus

The surface forces apparatus has been combined with fluorescence recovery after photobleaching to measure translational diffusion of polymer confined between mica sheets. This article presents findings using polydimethylsiloxane with number‐average molecular weight M_n = 2200 g mol^?1, the chains end‐labeled with soluble fluorescent dye. Melts with thickness 10 nm display a translational diffusion coefficient (D) with a bulk component and a slower component assigned to surface diffusion. Reduction of thickness to 1.8 nm causes mobility to split into two populations—an immobile fraction (immobile on the time scale of 30–60 min) and a mobile fraction who's D slow only weakly with diminishing film thickness. However, when load causes the confining mica sheets to flatten, D of the mobile fraction drops by up to an additional order of magnitude, depending on the local pressure that squeezes on the polymer. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2010     

Ternary Solution Mutual Diffusion Coefficients and Densities of Aqueous Mixtures of Sucrose with NaCl and Sucrose with KCl at 25°C

Ternary solution isothermal mutual diffusion coefficients (interdiffusion coefficients) have been measured for aqueous mixtures of 0.250 mol-dm^–3 sucrose (component 1) with 0.5 and 1.0 mol-dm^–3 NaCl or with 0.5 and 1.0 mol-dm^–3 KCl (salt = component 2) at 25.00°C using Rayleigh interferometry with computerized data acquisition. Densities were also measured. The volume-fixed diffusion coefficients (D _ij)_V show the following characteristics. At all compositions (D ₂₁)_V is much larger than (D ₁₂)_V and (D ₂₁)_V is a fairly significant fraction (33 to 68%) of (D ₁₁)_V. In addition, (D ₁₂)_V is slightly larger for mixtures containing NaCl than for those containing KCl at the same concentration, whereas (D ₂₁)_V is significantly larger for mixtures containing KCl. Values of (D ₁₁)_V are slightly larger for solutions containing KCl than for solutions containing NaCl. The observed trends imply that (D ₂₁)_V will probably exceed (D ₁₁)_V in both mixtures if concentrations of NaCl or of KCl are increased much further while maintaining the sucrose concentration at 0.250 mol-dm^–3. Finally, the solvent-fixed cross-term diffusion coefficients (D ₁₂)₀ and (D ₂₁)₀ are significantly larger than their corresponding (D ₁₂)_V and D ₂₁)_V.     

Toluene diffusion in butyl rubber

The sorption isotherm and the polymer mass-fixed diffusion coefficients, D, for toluene in butyl rubber have been measured by the incremental sorption method to concentrations of 130%, corresponding to a solvent volume fraction of 0.578. The increase in D with concentration is strongly exponential to a concentration of 30% and then begins to level out. Since the nature of the dimensional change occurring in vapor sorption was not known, the values of D were converted to solvent self-diffusion coefficients, D₁, assuming both swelling in the thickness direction (1D) and isotropically (3D). The free volume (FV) theory of Fujita was fitted to the resulting D₁ with the zero concentration diffusion coefficient and the self-diffusion coefficient of toluene as limiting values leaving only a single arbitrary parameter. In this form the FV theory was able to describe the trend of the experimental D₁ for the 1D and 3D cases equally well. Values of D were back-calculated from the FV relations for the 1D and 3D cases for comparison with the experimental results and with the diffusion coefficient determined by immersion in toluene. These comparisons favor the assumption that swelling is isotropic. It appears that the simple free volume relation is capable of providing a satisfactory representation of the experimental data with only a single fitting parameter, although there are moderate quantitative discrepancies. © 1994 John Wiley & Sons, Inc.     

Crosslinking,filler, or transition constraint of polymer networks. II

The theory of Part I is developed by application to filler reinforcement of NR and SBR. For unswollen but prestretched networks it quantifies entire stress–strain curves and applies new concepts of extensibility and strain hardening. Constraint of swelling is expressed by a constant φ, termed linkage reinforcement, and by an effective hard fraction C_h per cubic centimeter of compound. For rubber–filler swelling v_c the modified Flory functions F(v_c) in part 1 need 3% correction. Then, relative to gum fix points 1/F₀(v_r): where C_h ≤ 1.15C for filler concentrations C per cubic centimeter of compound. The effective C_h comprises the volume fraction C* of bonded particles and 5–10 Å of surface–bound rubber that has been stretched hard by swelling. When needed, the actual crosslink density and intrinsic linkage reinforcement φ₀ can be obtained by dividing by (1 + 2.5C_φ) where C_φ = (C ln φ)/(1 + 2.5C). The case C_h ≤ 1.15C with Graphon or inert fillers is identified and assessed by equations: Results C_h > 1.15C are invalid, but then C_h ≈ 1.15C* ≈ 1.15C, e.g., for carbon blacks. Even Graphon is distinguished from inert fillers at low concentrations C by substantial constraint reinforcement F₀(v_r)/F(v_c) > 1. For prestressed dry rubber a modulus G, network extensibility α_b ? 1, and upturn coefficient μ_h express the whole curve; G and μ_h show identical constraint strength distributions. Network extensibility α_b ? 1 is the microbreaking strain (prestretch); for pure elastomer it is elongation at break. The relation of stress F to extension ratio α is: where C₂* = 0.7 and j = 0.4 from NR/MPC data. Strain-hardening coefficients h are obtained from μ_h by the theory given in Part I. Hard modulus components G_h = 0.7 ln (h/h₀) vanish as h → h₀ (gum) = 110. After high prestresses the residual ln-(h^*/h₀) due to strong carbon-rubber linkages implies G_h* = 0.42 kg/cm², i.e., ca. 10% of the normal cure crosslinks.     

Effect of the solvent upon the diffusion coefficient of polydisperse polystyrene and styrene-acrylonitrile copolymer in dilute solution

A laser homodyne spectrometer was used to obtain translational diffusion coefficients for dilute polystyrene and styrene-acrylonitrile copolymer solutions at room temperature. Data were obtained in the concentration range from 0.01 to 2.0 g polymer per 100 cm³ solution for polystyrene in benzene and in decalin; and for copolymer in dimethyl formamide, in methyl ethyl ketone, and in benzene. The samples were polydisperse polystyrenes of weight average molecular weights between 80,000 and 350,000 and polydisperse copolymers of weight average molecular weights between 200,000 and 800,000. The SAN copolymers were random copolymer samples containing 24% by weight acrylonitrile. For each of the systems investigated the concentration dependence of the diffusion coefficient was linear over the concentration range studied, and was expressed as D(c) = D₀(1+k_Dc). Values of D₀ could be explained with a modified Kirkwood-Riseman expression. Values of the parameter k_D obtained from the slopes could be interpreted using the two-parameter theory approach as suggested by Vrentas and Duda. The value of k_D is positive for high-molecular-weight polymers and negative for low-molecular-weight polymers. For a particular polymer, the molecular weight at which k_D changes sign is greater for poor solvents than for good solvents. Observed values of D₀ were 1 × 10^?7 to 7 × 10^?7 cm²/sec.     

Physical aging and dye diffusion in polysulfone below the glass transition temperature

The effect of physical aging on the tracer diffusion coefficient D of camphorquinone in polysulfone is investigated. It is shown that if the sample is sufficiently annealed and physical aging is nearly complete, the temperature dependence of D will reflect the primary α-relaxation process of the host polymer. In the temperature range between T_g (=185°C) and 165°C, D is found to be a function of time, and the time dependence of D is given by D = At^?μ, with μ approximately equal to unity. © 1994 John Wiley & Sons, Inc.