Relationship between chain termination rate constant and conversion in radical polymerization

At low and high conversions, the chain termination rate constant for bimolecular termination between polymeric radicals given by k_t = A_tD_s, where A_t is a constant and D_s is the diffusion constant of radical chain end, is completely correct. This termination rate constant does not depend on solution viscosity, but conversion.     

A study of dimethylferrocene diffusion and electrode kinetics using microelectrode voltammetry in poly(ethylene glycol) solvent

The heterogeneous electron transfer rate constant (k _s) of dimethylferrocene (DMFc) was estimated using cyclic voltammetric peak potential separations taken typically in a mixed diffusion geometry regime in a polyelectrolyte, and the diffusion coefficient (D) of DMFc was obtained using a steady-state voltammogram. The heterogeneous electron transfer rate constant and diffusion coefficient are both smaller by about 100-fold in the polymeric solvent than in the monomeric solvent. The results are in agreement with the difference of longitudinal dielectric relaxation time (τ_L) in the two kinds of solvents, poly(ethylene glycol) (PEG) and CH₃CN, indicating that k _s varies inversely with τ_L; k _s is proportional to D of DMFc. Both D and k _s of DMFc in PEG containing different supporting electrolytes and at different temperatures have been estimated. These results show that D and k _s of DMFc increase with increasing temperature in the polyelectrolyte, whereas they vary only slightly with changing the supporting electrolyte. Received: 5 February 1998 / Accepted: 23 July 1998     

Effect of viscosity on termination rate constant in radical polymerization at low conversion

By using the expression, k_t = A₁D_s for the chain termination rate constant (where A₁ is a constant and D_s is the diffusion constant of radical chain end), a familiar chain termination rate constant, k_t = A₂/η_s (where A₂ is a constant and η_s is solvent viscosity) was examined with variation of conversion x. It was found that the proportionality of chain termination rate constant and solution viscosity is a valid relation at conversion 0 but is approximate at conversion x_c ≥ x > 0. Here x_c denotes a critical conversion under the average distance around spherical polymers formed in polymerization solution is zero. At conversions above x_c, the inverse relation between chain termination rate constant and solution viscosity is not correct.     

Dynamic and electrophoretic light scattering of poly(dimethyldiallylammonium chloride) in salt-free solutions

Dynamic and electrophoretic light scattering were used to study the diffusion and electrophoretic mobility of poly(dimethyldiallylammonium chloride) as a function of polymer molecular weight in salt-free solutions. Two relaxation modes characterized as fast diffusion (D_f) and slow diffusion (D_s) were obtained from dynamic light scattering. Although the slow diffusion coefficient D_s strongly depends on molecular weight (M_w), the fast diffusion coefficient D_f was found to be independent of M_w over the range in the study. The fast diffusion was considered as the diffusion of a part of the polymer chain; the slow diffusion was interpreted by multichain diffusion. Electrophoretic light scattering results in the salt-free solution show that the electrophoretic mobility of the polymer is independent of M_w. © 1996 John Wiley & Sons, Inc.     

Termination rate constants in radical polymerization

A rate constant is generally derived by using Fick's equation corresponding to the spherical interdiffusion of particles. By using this rate constant, chain and primary radical termination rate constants can be approximated to rate constants for the bimolecular reactions between two radical chain ends, and primary radical and radical chain end, respectively. The former is given by k_s = 8πN_LD_sL_s exp { ? L_s/R_s} × 10^?3 1./mole-sec. The latter is given by k_si = 4πN_L(D_s + D_i)L_si exp { ? L_si/R_si} × 10^?3 1./mole-sec. Here, N_L is Avogadro's number; D_s and D_i are the diffusion constants of radical chain end and primary radical, respectively; L_s and L_si are, respectively, the distances between two radical chain ends and between a primary radical and a radical chain end at a thermal energy equal to the coulombic energy of interaction of the net charges; and R_s and R_si are, respectively, the average distances between two radical chain ends and primary radical on a collision.     

Edge profiles at lamellar phase/melt interfaces

We study in the framework of the continuum theory of dislocations the structure of the interface between an AB diblock copolymer lamellar film deposited on a solid substrate and an A-homopolymer melt. The dislocation inside the lamellar phase induces steps at the interface. The shape of the profile at the edge of a step (edge profile) depends on the distance of the dislocation from the interface. The profile and the equilibrium location of the dislocations are both studied as a function of the film thickness, D. For large D, the dislocation is stabilized at a finite distance, h_eq, from the interface, due to the small surface tension and large surface bending elastic constant, K_s. For zero surface tension, h_eq ≈ K_s/(2K), where K is the bulk bending elastic constant. For small D, h_eq is mainly determined by the proximity of the solid substrate. The edge profile along the interface is a monotonic function of the distance along the interface for large D of the film and becomes nonmonotonic for small D. Also the dislocation energy strongly depends on D for small D. The theory is discussed in connection to recent experimental studies of diblock copolymer films deposited on a solid substrate.     

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     

Anisotropic and enhanced self-diffusion of a macromolecular chain under simple shear flow as revealed by Monte Carlo simulation on lattices

Two-dimensional simple shear flow of a self-avoiding macromolecular chain is simulated by a lattice Monte Carlo (MC) method with a pseudo-potential describing the flow field. The simulated velocity profile satisfies the requirements of simple shear flow unless the shear rate is unreasonably high. Some diffusion problems for a free-draining bead-spring chain with excluded volume interaction are then investigated at low and relatively high shear rates. Three diffusion coefficients are defined and examined in this paper: the conventional self-diffusivity in zero field, D_self, the apparent self-diffusivity in flow field, D_app, and the flow diffusivity in simulation, D_flow reflecting actually the transport coefficient. It is found that these three diffusivities for a flexible chain are different from each other. What is more important is that self-diffusion exhibits a high anisotropy in the flow field. The apparent self-diffusion along the flow direction is enhanced to a large extent. It is increased monotonically with the increase of shear time or shear strain, whereas the chain configuration can achieve a stationary anisotropic distribution following an interesting overshoot of the coil shape and size. Besides a single self-avoiding chain, an isolated Brownian bead and a group of self-avoiding beads with a quasi-Gaussian spatial distribution are also simulated. According to the comparison, the effects of the connectivity of the chain on the diffusion behavior are revealed. Some scaling relations of D_app versus t are consistent with the theoretical analyses in the pertinent literature.     

Can the Theoretical Fitting of the Proton‐Nuclear‐Magnetic‐Relaxation‐Dispersion (Proton NMRD) Curves of Paramagnetic Complexes Be Improved by Independent Measurement of Their Self‐Diffusion Coefficients?

The self‐diffusion (D_c) coefficients of various lanthanum(III) diamagnetic analogues of open‐chain and macrocyclic complexes of gadolinium used as MRI contrast agents were determined in dilute aqueous solutions (3–31 mM ) by pulsed‐field‐gradient (PFG) high‐resolution ¹H‐NMR spectroscopy. The self‐diffusion coefficient of H₂O (D_w) was obtained for the same samples to derive the relative diffusion constant, a parameter involved in the outersphere paramagnetic‐relaxation mechanism. The results agree with an averaged relative diffusion constant of 2.5 (±0.1)×10^?9 and of 3.3 (±0.1)×10^?9 m² s^?1 at 25 and 37°, respectively, for 'small' contrast agents (M_r 500–750 g/mol), and with the value of bulk H₂O (2.2×10^?9 and 2.9×10^?9 m² s^?1 at 25° and at 37°, respectively) for larger complexes. The use of the measured values of D_c for the theoretical fitting of proton NMRD curves of gadolinium complexes shows that the rotational correlation times (τ_R) are very close to those already reported. However, differences in the electronic relaxation time (τ_SO) at very low field and in the correlation time (τ_V) related to electronic relaxation were found.     

Diffusion behavior of pharmaceutical O/W microemulsions studied by dynamic light scattering

Dynamic light scattering experiments have been performed at various concentrations, of pharmaceutical oil-in-water microemulsions consisting of Eutanol G as oil, a blend of a high (Tagat O2) and a low (Poloxamer 331) hydrophilic–lipophilic balance surfactant, and a hydrophilic phase (propylene glycol/water). We probe the dynamics of these microemulsions by dynamic light scattering. In the measured concentration range, two modes of relaxation were observed. The faster decaying mode is ascribed classically to the collective diffusion D _c (total droplet number density fluctuation). We show that the slow mode is also diffusive and suggest that its possible origin is the relaxation of polydispersity fluctuations. The diffusion coefficient associated with this mode is then the self-diffusion D _s of the droplets. It was found that D _c and D _s had opposite volume fractions of oil plus surfactants (ϕ) dependence and a common limiting value D ₀ for ϕ=0. Average hydrodynamic radius (R _h=10.5 nm) of droplets was calculated from D ₀. R _h is supposed to compose the inner core, a surfactant film including possible solvent molecules, which migrate with the droplet. The concentration dependence of diffusion coefficients reflects the effect of hard sphere and the supplementary repulsive interactions which arises due to loss of entropy, when absorbed chains of surfactant intermingle on the close approach of the two droplets. This mechanism could also explain the observed stability of our systems. The estimated extent of polydispersity is 0.22 from the amplitude of slower decaying mode. The polydispersity in microemulsion systems is dynamic in origin. Results indicate that the time scale for local polydispersity fluctuations is at least three orders of magnitude longer than the estimated time between droplet collisions.     

Small-angle neutron scattering study on the miscibility and concentration fluctuation of hydrogen-bonded polymer blends

Thermal concentration fluctuation in the blends of deuterated poly(ethylene oxide) (dPEO) and poly(vinyl acetate-co-vinyl alcohol) [P(VAc-VOH)] with various VOH contents f_OH were examined by small angle neutron scattering techniques at a fixed blend composition, dPEO/P(VAc-VOH) = 20/80 which is close to the critical composition. Blends at the highest f_OH (=0.35) showed a non-Lorentzian scattering profile: specifically the scattering intensities at the low q (angle) region were suppressed compared to those expected from the random phase approximation (RPA) theory. However, for the blends at lower f_OH (≤0.28), the profiles could be represented by the RPA. Using the RPA we determined effective values of the Flory-Huggins interaction parameter χ_eff as a function of f_OH (=0–0.28). The χ_eff showed the minimum around f_OH = 0.1–0.18 meaning the highest miscibility of the blend at these f_OH. On the basis of the random copolymer theory, we evaluated the three interaction parameters χ_Ac–EO, χ_EO–OH, and χ_Ac–OH separately from the χ_eff(f_OH) and found the order of magnitude; χ_Ac–EO < 0 < χ_EO–OH < χ_Ac–OH. The largest positive χ_Ac–OH showing intrachain interaction in the P(VAc-VOH) copolymer was concluded to be the origin of the enhanced miscibility at around f_OH = 0.1–0.18. On the basis of the Coleman and Painter's theory, the effects of hydrogen bonding on these three χ_A-B were discussed. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 2556–2565, 2008     

Cross-linking kinetics study and high temperature mechanical properties of ethylene–octene copolymer (EOC)/dicumylperoxide(DCP) system

Ethylene–octene copolymer (EOC) was cross-linked by dicumyl peroxide (DCP) at various temperatures (150–200 °C). Six concentrations of DCP in range 0.2–0.7 wt.% were investigated. Cross-linking was studied by rubber process analyzer (RPA). From RPA data analysis real part modulus s′, tan(delta) and reaction rate constant K were investigated as a function of peroxide content and temperature. The highest s′_max and the lowest tan(delta) were found for 0.7% of DCP at 150 °C. The quantitative analysis confirmed that the DCP–EOC cross-linking was occurring as first order reaction. The highest cross-linking kinetics constant K was found for 0.6% of peroxide at 200 °C. The activation energy of cross-linking E_A obtained by Arrhenius plot had maximum at 0.5–0.6% of peroxide. While at 190–200 °C temperature range there was no detectable degradation for 0.2% of peroxide, for 0.4–0.7% of peroxide there was increasing level of degradation with increasing peroxide content. Generally, at low temperatures (150–180 °C) the increasing peroxide content caused increase in cross-linking kinetics. However at higher temperatures (190–200 °C) increase in kinetics (for 0.2–0.5% of peroxide) was followed by decrease. Especially in 0.6–0.7% peroxide level range the cross-linking is in competition with degradation which lowers the overall cross-linking kinetics. Gel content of the cross-linked EOC samples was found to be increasing with increase in peroxide content, which is caused by the increased cross-link network. Cross-linked samples were subjected to creep studies at elevated temperature (150 °C) and the result was found in agreement with the gel content and RPA results. Storage modulus and tan(delta) values obtained by Dynamic Mechanical Analysis (DMA) also support the RPA results.     

Surface diffusion and relaxation of partially adsorbed polymers

We studied by lattice simulation the surface diffusion and relaxation of isolated, self‐avoiding polymers partially adsorbed onto a flat surface. The key parameters describing the system are the number of segments in the chain, N, the adsorption energy of a segment, expressed as a dimensionless surface temperature T_s, and the segmental friction factor on the surface relative to that in the bulk, ζ_s/ζ_b. The simulation data indicate Rouse scaling of the surface diffusion coefficient, D_∥, and in‐plane relaxation time, τ, versus N for all values of T_s and ζ_s/ζ_b studied. A simple application of the Rouse model to a partially adsorbed chain, which ignores fluctuations in adsorbed trains, yields a formula for D_∥ with the correct N‐scaling. It can account for the effects of T_s when ζ_s/ζ_b is finite (≲10), but it fails when ζ_s/ζ_b diverges, predicting no surface diffusion at all, whereas simulations indicate finite surface mobilities facilitated by a caterpillar‐like motion. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1146–1154, 2000     

Behavior of polyelectrolyte solutions in a wide range of solvent dielectric constant

We exploit the strong temperature dependence of dielectric constant of N-methylformamide to control the strength of electrostatic interactions in polyelectrolyte solutions of sodium polystyrene sulfonate, and investigate the dynamic properties of salt-free solutions over a broad temperature range, from 54 to −58 °C. The traditional fast and slow diffusion processes are observed. The ratio of diffusion coefficients, D_s/D_f, increases and the ratio of their amplitudes, A_s/A_f, decreases, both by a factor of about two in this temperature range confirming the expected temperature variation with the Bjerrum length.     

Miscibility and ferroelectric behavior in blends of poly(vinylidene fluoride/trifluoroethylene) and poly(vinyl acetate)

The blend system containing a poly(vinylidene fluoride/trifluoroethylene) [P(VDF/TrFE)] copolymer (68/32 mol %) and poly(vinyl acetate) (PVAc) was miscible from the results of differential scanning calorimetry (DSC) studies that exhibit the presence of a single, composition‐dependent glass transition temperature (T_g) and a strong melting point depression for the semicrystalline P(VDF/TrFE) component. However, differences between the DSC and dielectric measurements, which showed a separate P(VDF/TrFE) T_g peak, suggests that the P(VDF/TrFE)/PVAc blends are actually partially miscible. Because of the lower dielectric constant of PVAc and the reduced sample crystallinity caused by the addition of PVAc, both the dielectric constant and the remanent polarization of the copolymer blends decrease with increasing PVAc content. The presence of a small amount of PVAc stabilized the anomalous ferroelectric behavior of ice–water‐quenched P(VDF/TrFE), and the blend portrayed normal polarization reversal behavior after adding only 1 wt % PVAc. The piezoelectric response suggests small changes with an increasing number of poling cycles. It is believed that PVAc affects the D‐E hysteresis behavior at the interface between crystalline and amorphous phases, although much work remains to be done to confirm this hypothesis. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 927–935, 2003     

The effect of plasticization on the transport of gases in and through glassy polymers

The effects of plasticization on the transport of gases and vapors in and through glassy polymers are examined from the viewpoint of the “dual-mode” sorption model with partial immobilization. The analysis assumes the existence of two penetrant populations with different mobilities in the Henry's law and Langmuir domains of the glassy polymers. These mobilities are characterized by their mutual diffusion coefficients D_D and D_H. The plasticization of the polymer by penetrant gases is reflected in the concentration dependence of D_D and D_H. Expressions for the effective (apparent) diffusion and permeability coefficients are derived assuming that D_D and D_H are exponential functions of the penetrant concentration in the polymers. The results of this study are compared with a similar analysis which assumed the existence of a single mobile penetrant population. The present analysis provides information on the effects of plasticization on the penetrant transport in the Henry's law and Langmuir domains separately. The effects of antiplasticization or clustering of penetrant molecules on the effective diffusion and permeability coefficients are also examined.     

Estimation of molecular weight distribution based on chain length dependence of termination rate

When a chain length dependence of polymer-polymer termination is given by k_t,ns = const. (n^?2a + s^?2a) where n and s are the chain lengths for the polymer radicals and a is parameter, an instantaneous weight fraction of the non-reacting polymers is derived as: where h and k? are the kinetic parameters, p is a parameter depending on a, and p_n is instantaneous number-average chain length. Such a weight fraction corresponds to the experimental one over a wide range of conversion in the polymerization of styrene. On the scope of this correspondence, the polymer-polymer termination rate is estimated as: k?_t = 8πR₀D₁/100 ( = 4πR_sD_s) where R₀ is reaction radius between monomer radicals and D₁ is the diffusion coefficient of the monomer; R_s is reaction radius between segment radicals with n ? 100 and D_s is the diffusion coefficient of the segment. The Fujita-Doolittle theory applies to such a rate. Further, the rate also yields 1.5 × 10⁷1./mole-sec, which is the observable extent at conversions less than 0.2.     

Static and dynamic light-scattering of colloidal gases,liquids and crystals

Static and dynamic light-scattering measurements are made for colloidal-crystals,-liquids and-gases of silica spheres, 103 nm in diameter, in the exhaustively deionized suspension and in the presence of sodium chloride. Sharp peaks in the scattering curve are observed, for the first time, for the colloidal crystals in very diluted aqueous suspension. The product of the effective diffusion coefficient and the scattered light intensity is found constant over the whole range of the scattering angle measured for the colloidal crystals and liquids. Three and two dynamic processes have been extracted separately from time profiles of autocorrelation function of colloidal crystals and liquids, respectively from Marquadt histogram analysis. Decay curves of colloidal gases are characterized by a single translational diffusion coefficient,D ₀.D ₀ of the gases is always lower than the calculation from the Stokes-Einstein equation with the true diameter of spheres, and increases as ionic concentration increases. These experimental results emphasize the important role of the expanded electrical double layers on the diffusive properties in the colloidal crystals, liquids and gases.     

Effects of type of adsorption isotherms on parallel diffusion of sulfonated dyes into porous cellulose membrane

Transport phenomenon of three sulfonated azo dyes, C.I. Acid Red 88, C.I. Direct Yellow 12, and C.I. Direct Blue 15 into water-swollen cellulose membranes has been analyzed on the basis of parallel transport theory by surface and pore diffusion. Langmuir equation was applied into the mass balance equation to estimate dye concentration in the pores. The results were compared with the results obtained by applying Freundlich equation in our previous papers. The surface diffusivity (D _s) and the pore diffusivity (D _p) for the parallel diffusion model obtained by applying Langmuir equation agreed with those obtained by applying Freudlich equation. The theoretical concentration profiles for parallel diffusion calculated usingD _s andD _p coincided accurately with the experimental data when we applied either Langmuir or Freundlich equations.     

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.