Gravimetric determination of the diffusion characteristics of polymers using small specimens

An investigation has been undertaken of the factors controlling the rate of ingress of water into relatively small epoxy specimens. The application of a three-dimensional diffusion model, fully acknowledging that the specimen geometry controls the rate of ingress, has successfully resolved the problems of determining the diffusion characteristics—the diffusion coefficient D and the uptake M_∞ corresponding to saturation—and making theoretical predictions of the uptake M_t by small specimens. On the theoretical front, simulations have been carried out to demonstrate that (1) theoretical predictions require two equations to be used to obtain correct relative uptake M_t/M_∞ values for short and long exposure times; (2) a cube of the side h is saturated faster than an infinite slab of the thickness h, and (3) determination of the D value by applying the model derived for an infinite slab to specimens of finite dimensions results in an apparent value that may grossly exceed the true diffusion coefficient. On the experimental front, suggestions are made regarding (4) elimination of a systematic error that might be introduced by single weighing of a specimen prior exposure, (5) determination of the product M_∞ D^1/2 from the gravimetry data for short exposure times, and (6) splitting this product into absolute values D and M_∞ by means of a minimization procedure. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42:2122–2128, 2004     

Frequency spectra of disordered metals: especially liquid Rb

The phonon frequency spectrum g(ω) of a crystal, such as body centred cubic (bcc) Rb, is known to be characterized by the Van Hove singularities at ω?≠?0. However, for a liquid metal like Rb, g(ω) has a single, hydrodynamic-like singularity, namely a cusp ∝ ω ^(1/2), at ω?=?0. Here, we note first that computer simulation on liquid Rb near freezing has revealed a rather well-defined Debye frequency ω_D. Therefore, we propose here a zeroth-order model g ₀ (ω ) of g(ω) for Rb, which combines the Debye model with the ‘hydrodynamic’ ω ^(1/2) cusp. The corresponding velocity autocorrelation function 〈 v (t)·v (0)〉 has correctly a long-time tail ∝ t ^-(3/2). The terms from g ₀ (ω ) involving ω_D are then damped by weak exponential factors exp (-α _i t), and the resulting first-order approximation, g ₁ (ω ) say, to the frequency spectrum is found to have features in common with the molecular dynamics (MD) simulation form. Thus ω_D is fixed, as well as transport coefficients for the known thermodynamic state. The article concludes with a more qualitative discussion on supercooled liquids, and on metallic glasses such as Fe, for which MD simulations exist.     

Identification of unknown diffusion coefficient in pure diffusive linear model of chronoamperometry. I. The theory

Coefficient identification problem for diffusion equation u _t (x, t) = (D(x)u _x (x, t)) _x arising in chronoamperometry is studied. The adjoint problem approach is developed for the case when the output measured data is given in the form of left/right flux. Analytical formulas for determination of the values D(0), D(L) at the endpoints x = 0; L, of the unknown coefficient D(x), via the solution v(x, t) of the constant coefficient equation v _t (x, t) = D v _xx (x, t) is obtained. The integral identity relating solutions of the forward and corresponding adjoint problems is derived. This integral identity permits one to prove the monotonicity and invertibility of input-output map, as well as formulate the gradient of the cost functional via the solutions of the direct and adjoint problems.     

Role of diffusion in gel permeation chromatography

A theory is developed that describes the diffusion of solute into the gel particles during a gel permeation chromatographic experiment. The particles are treated as homogeneous spheres of radius a, into which diffusion takes place with diffusion coefficient D_s. The concentration in the mobile phase at any level at any time is supposed to be uniform throughout the cross-section of the column. It is shown that in the usual columns the effect of diffusion in the mobile phase is unimportant. A determinative quantity in the process is the parameter a²/D_st, where t is the time. For large values of a²/D_st an explicit expression for concentration versus time in the mobile phase at the end of the column is derived [eq. (26) and Fig. 1]. It shows a relatively long tail at large efflux volumes V, where the concentration varies at V^?3/2. For arbitrary values of a²/D_st the first three moments of the concentration versus time curve are calculated [eqs. (33)–(37)]. Pronounced skewness of the curve is found unless a²/D_st is small.     

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.     

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 of protons in silica hydrogel

The effect of preparation pH of silica hydrogel on the effective diffusion coefficient of protons in silica hydrogel (D _e , m²/s), on surface area of silica gel (S, m²/s) and on particle size of silica gel (D _p , mm) was studied. Silica hydrosols were obtained by adding water glass to sulfuric acid. The effective diffusion coefficient of proton in silica hydrogel was determined by the method of diffusion from silica hydrogel plane sheet to a stirred solution of a limited volume. A numerical solution was obtained for the diffusion equation using the Regula Falsi method. Regression analyses of experimental data were conducted.Diffusion of protons in silica hydrogel is a complicated process due to a decelerating effect of the porous structure of silica hydrogel and to the accelerating effects of slow ions such as Na⁺ and surface diffusion. The effective diffusion coefficient increased with surface area of silica gel, indicating the diffusion of protons on the surface of the silica particles.     

Complexation of poly(vinyl alcohol)–congo red aqueous solutions. 3. Dynamic light scattering study

Mesoscopic structures of poly(vinyl alcohol)(SINGLEBOND)Congo red (PVA(SINGLEBOND) CR) complexes in aqueous solutions were investigated in terms of dynamic light scattering (DLS) technique. The intensity-intensity time correlation function, g⁽²⁾(t), was analyzed with an equation including a single and a stretched exponential function. Two diffusion coefficients, D_f (fast) and D_s (slow) were evaluated. D_f was converted to the apparent correlation length, ξ_app, via the mode-mode coupling hypothesis. The estimated ξ_app was insensitive to the sol(SINGLEBOND)to(SINGLEBOND)gel transition but decreased with CR concentration. This change may be related to the electrostatic screening effect. On the other hand, D_s oscillates with increasing CR concentration at a specific PVA concentration range. This explains well the reentrant sol(SINGLEBOND)gel(SINGLEBOND)sol(SINGLEBOND)gel transition behavior observed in the PVA(SINGLEBOND)CR systems. D_s seems to represent the mobility of the PVA(SINGLEBOND)CR complexes, which annihilates at the gel point. © 1996 John Wiley & Sons, Inc.     

Autoxidation of poly-4-methylpentene-1. I. The role of diffusion in autoxidation kinetics

An apparatus is described for the measurement of oxygen uptake into a polymer sample at constant oxygen pressures in the range 20–1000 mm Hg. Measurements of the rate of oxygen uptake into poly-4-methylpentene-1 show that the rate is accurately first-order in oxygen pressure over the range 50–800 mm pressure for temperatures ranging from 122 to 154°C and film thickness in the range 0.001–0.025 cm. A theoretical treatment of the kinetics of a reaction in which oxygen diffuses into both faces of a thin film, in which it is consumed by a first-order reaction shows that the oxidation rate ρ per unit area of film surface is given by ρ = ρ_∞ tanh ßL/2 where ρ_∞ is the limiting oxidation rate for a thick film, L is the film thickness, and ß = (k/D)^1/2, k being the oxidation rate constant and D the diffusion constant. Values of D and the activation energy for diffusion calculated from autoxidation data are in good agreement with values determined directly.     

Symmetry Adapted Analysis of Linear Molecules

Symmetry groups of the linear molecules belong to the C_∞v and D_∞h infinite groups. The symmetry adapted analysis of such types of molecule, is usually not systematically performed in the text book or paper. Since the standard formulas of symmetry adapted analysis are usually applicable for the finite groups only, one has to analyze the different subgroups of the linear molecules indirectly and correlates them with the irreducible representation of D_∞h and C_∞v. In this work, a systematic symmetry adapted analysis are introduced for the C_∞v and D_∞h molecules. It is a uniquely convenient way for molecular orbital calculations and vibrational normal mode analysis of the linear molecules.     

Emission of occluded volatiles during deformation of polycarbonate due to strain-enhanced diffusion

Measurements of the emission of purposely entrained volatiles (Ar and D₂O) during the loading and unloading of a bisphenol-A polycarbonate in vacuum are made by quadrupole mass spectrometry. Transient loading events are accompanied by dramatic increases in emission, reflecting a similar rise in the diffusion constant of the measured species. We attribute this change to an increase in size of molecular voids in the polymer network, which accompany the increase in sample volume under load. The results are interpreted in terms of the Dolittle relation in which the diffusion constant depends exponentially upon v^*/v_f0, the ratio between an activation volume for diffusion and the average size of the relevant voids in the polymer network. Our data suggests that v^*/v_f0 is unusually low in the D₂O-polycarbonate system, which we attribute to a relatively large value of v_f0; this would be consistent with the relatively long distance between flexible links in the polycarbonate structure. © 1994 John Wiley & Sons, Inc.     

Infrared spectroscopic studies of CO2 sorbed in glassy and rubbery polymeric membranes

Infrared spectra of CO₂ sorbed in rubbery and glassy polymers were measured to examine the relationships between the spectroscopic data and physical properties of the polymeric membranes. The “V-shape” tendency in the plot of W₁ [i.e., half-width of CO₂ peak sorbed in the membranes] vs glass-transition temperature (T_g) is observed, and has exactly the same tendency that is widely known from the plot of log D (diffusion coefficient) vs T_g. It is suggested that the membranes having a wider W₁ give a faster diffusion coefficient, since W₁ is inversely related to the moment of inertia of CO₂ in the membranes. Two distinct peaks of CO₂ were not observed in the infrared spectra of CO₂ sorbed in the glassy polymers. This suggests that the states of CO₂ in the Henry mode and Langmuir mode in the glassy polymers are similar in the spectroscopic measurements. © 1994 John Wiley & Sons, Inc.     

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.     

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.     

Twinkling fractal theory of the glass transition

In this paper we propose a solution to an unsolved problem in solid state physics, namely, the nature and structure of the glass transition in amorphous materials. The development of dynamic percolating fractal structures near T_g is the main element of the Twinkling Fractal Theory (TFT) presented herein and the percolating fractal twinkles with a frequency spectrum F(ω) ∼ ω^df–1 exp −|ΔE|/kT as solid and liquid clusters interchange with frequency ω. The Orbach vibrational density of states for a fractal is g(ω) ∼ ω^df–1, where d_f = 4/3 and the temperature dependent activation energy behaves as ΔE ∼ (T² − T). The key concept of the TFT derives from the Boltzmann population of excited states in the anharmonic intermolecular potential between atoms, coupled with percolating solid fractal structures near T_g. The twinkling fractal spectrum F(ω) at T_g predicts the correct dynamic heterogeneity behavior via the spatio-temporal thermal fluctuation autocorrelation relaxation function C(t). This function behaves as C(t) ∼ t^−1/3 (short times), C(t) ∼ t^−4/3 (long times) and C(t) ∼ t⁻² (ω < ω_c), which were found to be in excellent agreement with published nanoscale AFM dielectric force fluctuation experiments on a glassy polymer near T_g. Using the Morse potential, the TFT predicts that T_g = 2D_o/9k, where D_o is the interatomic bonding energy ∼ 2–5 kcal/mol and is comparable to the heat of fusion ΔH_f. Because anharmonicity controls both the thermal expansion coefficient α_L and T_g, the TFT uniquely predicts that α_L×T_g ≈ 0.03, which is found to be universal for a broad range of glassy materials from Pyrex to polymers to glycerol. Below T_g, the glassy structure attains a frustrated nonequilibrium state by getting constrained on the fractal structure and the thermal expansion in the glass is reduced by the percolation threshold p_c as α_g ≈ p_cα_L. The change in heat capacity ΔC_p = C_pL–C_pg at T_g was found to be related to the change in dimensionality from D_f to 3 in the Debye approximation as the ratio C_pL/C_pg = 3/D_f, where D_f is the fractal dimension of the glass. For polymers, the TFT describes the molecular weight dependence of T_g, the role of crosslinks on T_g, the Flory-Fox rule of mixtures and the WLF relation for the time-temperature shift factor a_T, which are traditionally viewed in terms of Free-Volume theory. The TFT offers new insight into the behavior of nano-confined glassy materials and the dynamics of physical aging. It also predicts the relation between the melting point T_m and T_g as T_m/T_g = 1/[1−p_c] ≈ 2. The TFT is universal to all glass forming liquids. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 2765–2778, 2008     

Glass transition kinetics of some Se-Te-Ag chalcogenide glasses

Summary The present paper reports the Differential Scanning Calorimetric (DSC) study of some Ag doped Se-Te chalcogenide glasses. DSC runs were taken at different heating rates. Well-defined endothermic and exothermic peaks were obtained at glass transition and crystallization temperatures. The variation of glass transition temperature T_gwith Ag concentration has been studied. It has been found that T_gdecreases with increase in Ag concentration. The heating rate dependence of T_gis used to evaluate the activation energy of glass transition (DE_t). The value of<span style='font-size:10.0pt; font-family:"SymbolProp BT";mso-bidi-font-family:"SymbolProp BT"'>DE_thas been found to increase with increase in Ag concentration followed by nearly constant value at higher concentrations of Ag.     

Transport through a slab membrane governed by a concentration-dependent diffusion coefficient: Part IV. D(C)/D0=1+(αC)−β(αC)

Using the specific functional form D(C)/D₀=1+(αC)−β(αC)² an investigation has been made of (isothermal) transport through a slab membrane under ‘simple’ boundary conditions and governed by a diffusion coefficient, D(C), which, with increasing concentration, at first increases, passes through a maximum value and finally decreases. The flux, integral diffusion coefficient and concentration profile characteristic of steady-state permeation have been evaluated; special attention has been paid to the positions of such profiles in relation to the corresponding linear distribution associated with a constant diffusion coefficient.The corresponding transient-state transport has been studied within a framework of the time-lag ‘early-time’ and ‘ ’ procedures. Expressions for the ‘adsorption’ and ‘desorption’ time-lags are given. The concentration-dependence of these time-lags, of the (four) integral diffusion coefficients derived from them and of the arithmetic-mean time-lag ratios have been considered in some detail. The ‘early-time’ and ‘ ’ finite-difference procedures have likewise been employed to derive four further integral diffusion coefficients, so enabling a comparison to be made of the nine integral coefficients pertaining to established experimental techniques.Particular interest attaches to the situation for which n≡β(αC₀)=1 (where C₀ is the ingoing or upstream concentration of diffusant) resulting in D(C₀) being symmetrical about C₀/2. Some consideration has been given, in general, to features of transient-state transport when governed by a symmetrical D(C).     

Development of high ductility and tensile properties by a two-stage draw of poly(acrylonitrile): Effect of molecular weight

Ultradrawing of atactic poly(acrylonitrile) (PAN) was investigated for a M_v series, ranging 8.0 × 10⁴–2.3 × 10⁶. Samples for the draw were prepared from 0.5–30 wt % solutions of PAN in N,N′-dimethylformamide. The solutions were converted to a gel by quenching from 100 to 0°C. The dried gel films were initially drawn uniaxially by solid-state coextrusion (first-stage draw) to an extrusion draw ratio (EDR) of 16, followed by further tensile draw at 100–250°C (second-stage draw). The maximum total draw ratio (DR_t,max) and tensile properties achieved by two-stage draw increased remarkably with sample M_v. Other factors affecting ductility were the solution concentration from which gel was made and the second-stage draw temperature. The effects of these variables became more prominent with increasing M_v. The temperature for optimum second-stage draw increased with sample M_v. Both the initial gel and the drawn products showed no small-angle X-ray long period scattering maximum, suggesting the absence of a chain-folded lamellae structure, which had been found in our previous study on the drawing of nascent PAN powder. The chain orientation function (f_c) and sample density (ρ_s) increased rapidly with DR_t in the lower range (DR_t < 30) and approached constant values of f_c = 0.980–0.996 and ρ_s = 1.177–1.181 g/cm³, respectively, at higher DR_t > 30–100. The tensile modulus also showed a similar increase with DR_t. The tensile strength increased linearly with DR_t, reaching a maximum, and decreased slightly at yet higher DR_t. The highest modulus of 28.5 GPa and strength of 1.6 GPa were achieved with the highest M_v of 2.3 × 10⁶. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 629–640, 1998     

Feasibility study for the rapid screening of target molecules using translational diffusion coefficients: diffusion-ordered NMR spectroscopy of biological toxins

A panel of 15 biological toxins ranging between ~60–28,000 g/mol was used to evaluate the feasibility of screening aqueous samples for toxin analytes based on their translational diffusion coefficients, D _t. Toxin D _t values were measured by pulsed-field gradient ¹H NMR spectroscopy using a bipolar pulse pair, longitudinal eddy current delay pulse sequence incorporating water suppression to achieve the maximum dynamic range for toxin signals. To collect data for an effective screening protocol, reference D _t values were determined from five independent measurements at both 25 and 37 °C for all toxins in the panel. In the protocol, D _t values are measured at both temperatures for a suspected toxin target in a sample, and for assignment as a potential toxin analyte, the measurements are required to fall within ±0.25 × 10⁻⁶ cm²/s of both reference D _t values for at least one toxin in the panel. Only solution viscosity was found to influence sample D _t measurements appreciably; however, the measurements are easily corrected for viscosity effects by calculating the D _t value of the suspected toxin at infinite dilution. In conclusion, the protocol provides a rapid and effective means for screening aqueous samples for all toxins in the panel, narrowing toxin identification to ≤2 possibilities in virtually all cases.     

Zymomonas mobilis as catalyst for the biotechnological production of sorbitol and gluconic acid

The conversion of glucose and fructose into gluconic acid (GA) and sorbitol (SOR) was conducted in a batch reactor with free (CTAB-treated or not) or immobilized cells of Zymomonas mobilis. High yields (more than 90%) of gluconic acid and sorbitol were attained at initial substrate concentration of 600 g/L (glucose plus fructose at 1:1 ratio), using cells with glucose-fructose-oxidoreductase activity of 75 U/L. The concentration of the products varied hyperbolically with time according to the equations (GA)=t(GA)_max/(W_GA +t), (SOR)=t (SOR)_max/(W_Sor+t), v_GA=[W_GA (GA)_max]/(W_GA+t)² and V_SOR=[W_SOR (SOR)_max]/(W_SOR+t)². Taking the test carried out with free CTAB-treated cells as an example, the constant parameters were (GA)_max= 541 g/L, (SOR)_max=552 g/L, W_GA=4.8h, W_SOR=4.9h, υ_GA=112.7 g/L· and υ_SOR=112.7 g/L·.