Gel formation with multiple interunit junctions. I—Molecules carrying different functional groups

Number‐ and weight‐average molecular weight of condensation polymers formed by primary molecules carrying different species of functional groups {A_i} (i = 1, 2, …, s) are derived by cascade theory. These functional groups are allowed to form multiple junctions of variable multiplicity k. The gel point condition is found to be given by ∑ w_i/|μ_w,i + 1/∑ f_i ? 1 = 0, where f_i is the number of A_i groups specified by the index i on a primary molecule, w_i ≡ f_i/∑ f_i the number fraction of the species i it carries, and |μ_w,i the weight average multiplicity of the junctions formed by the groups A_i. The explicit form of the molecular weight distribution function is found for the simplest case of two components. Possible application to thermoreversible gelation is suggested. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2405–2412, 2003     

Synthesis of polycarbosilanes by A2 + Bn (n = 2, 3, and 4) type hydrosilylation reaction and evaluation of their refractive index properties

Polycarbosilanes were synthesized by hydrosilylation reaction of A₂ monomer containing bis Si? H moieties and B_n (n = 2, 3, and 4) monomers containing di‐, tri‐, and tetra‐vinyl groups in the presence of Karstedt's catalyst. The corresponding linear polycarbosilanes (LPC) and hyperbranched polycarbosilanes (HBPC) having M_n 2200–51,500 were obtained in 34–94% yield, without any gel product. The values of refractive index (n_D) of the synthesized LPC and HBPC were in the range from 1.460 to 1.711, and were consistent with the structures of the synthesized products. In the case of HBPC, the values of n_D increased with increase of number‐average molecular weight (M_n), molecular weight distribution (M_w/M_n), and glass transition temperature (T_g), apparently because of increased density due to the presence of microgels, that is, high refractive index hyperbranched carbosilanes could be synthesized by A₂ + B_n (n = 3 and 4) method. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2010     

Bulk hydrosilylation reaction of poly(dimethylsiloxane) chains catalyzed by a platinum salt: Effect of the initial concentration of reactive groups on the final extent of reaction

Model silicone networks obtained by curing linear poly(dimethylsiloxane) (PDMS) chains with end‐vinyl groups, (B₂), with a polyfunctional silane‐terminated crosslinker of functionality f, (A_f), through a hydrosilylation reaction have been widely used. In these networks, the principal characteristics of their ultimate molecular structure are strongly affected by the final extent of reaction reached during the crosslinking reaction. This work analyzes the effect of the initial concentration of the reactive end groups on the maximum attainable extent of reaction under normal bulk crosslinking conditions. This was accomplished by examining the reaction between linear B₂ PDMS chains with difunctional and trifunctional silanes. The experimental results were fitted by an exponential equation to have an empirical equation able to predict the maximum extent of reaction to be obtained as a function of the initial concentration of reactive groups. Molecular parameters relevant to this study, such as the degree of polymerization, the weight‐average molecular weight for the A₂ + B₂ system, or the weight fraction of solubles for the A₃ + B₂ system, were calculated with a mean field theory (recursive approach). © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 1099–1106, 2003     

Reaction of oligoalcohols with diepoxides: An easy,one‐pot way to star‐shaped,multibranched polymers. II. Poly(ethylene oxide) stars—synthesis and analysis by size exclusion chromatography triple‐detection method

An Erratum has been published for this article in Journal of Polymer Science Part A: Polymer Chemistry (2004) 42(10) 2575‐2576 Starlike, highly branched (A_xB_yA_z) macromolecules having from a few to 100 arms and molar masses up to 10⁵ were prepared in three stages with the one‐pot, arms‐core‐arms method (B_y stands for y molcules of former diepoxides introduced into the core). Oligoalcohols, at least partially converted into their alcoholate counterpart states, reacted with diepoxy compounds giving star‐shaped, highly branched macromolecules. With the properly chosen conditions, complete conversion of both starting components was achieved. In this article homostars built with the first and second generation of poly(ethylene oxide) arms (A_x and A_z, respectively) are described. The number of arms (f) was determined either by direct measurements of the number‐average molcular weight (M_n) of the first and second stars (M_n of arms A_x and A_z is known) or by calculating f from branching indices g and g′ determined from the radius of gyration and the limited viscosity number measured with size exclusion chromatography (SEC) triple detection with TriSEC software. For a few samples, M_n was measured with high‐speed membrane osmometry. The progress of the stars' formation was monitored by ¹H NMR, SEC, and matrix‐assisted laser desorption/ionization time‐of‐flight methods. Functionalization of the ? OH end groups in the second generation of arms was observed by ¹H and/or ³¹P NMR. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 1576–1598, 2004     

Characterization of novel optically active conjugated polyarylenes and poly(aryleneethnylene)s by a combination of off-line static and dynamic light scattering with gel permeation chromatography

By combining the offline static and dynamic laser light scattering (LLS) and gel permeation chromatography (GPC) results of a broadly distributed polymer sample, we were able to characterize a series of chiral binaphthyl-based polyarylenes and poly(aryleneethnylene)s in THF at 25°C. For each of the samples, we obtained not only the weight-average molar mass M_w, the second virial coefficient A₂ and the z-average translational diffusion coefficient 〈D〉, but also two calibrations: V = A + Blog(M) and D = k_D M^−αD, where V, D, and M are the elution volume, the translational diffusion coefficient and the molar mass for monodisperse polymer chains, respectively, and A, B, k_D, and α_D are four calibration constants. Using these calibrations, we estimated the molar mass distributions of these novel polymers. We showed that using polystyrene to calibrate the GPC columns could lead to a smaller M_w. Our results indicate that all the polymers studied have a rigid chain conformation in THF at 25°C and the introduction of the —NO₂ groups into the monomer can greatly promote the polymer solubility in THF.© 1998 John Wiley & Sons, Inc. J. Polym. Sci. B Polym. Phys. 36: 2615–2622, 1998     

Accessing reaction rate constants in on-column reaction chromatography: an extended unified equation for reaction educts and products with different response factors

An extension of the unified equation of chromatography to directly access reaction rate constants k ₁ of first-order reaction in on-column chromatography is presented. This extended equation reflects different response factors in the detection of the reaction educt and product which arise from structural changes by elimination or addition, e.g., under pseudo-first-order reaction conditions. The reaction rate constants k ₁ and Gibbs activation energies DG ¹ \Delta G^{ \ne } of first-order reactions taking place in a chromatographic system can be directly calculated from the chromatographic parameters, i.e., retention times of the educt E and product P ( t_\textR^\textA t_{\text{R}}^{\text{A}} and t_\textR^\textB t_{\text{R}}^{\text{B}} ), peak widths at half height (w _A and w _B), the relative plateau height (h _p) of the conversion profile, and the individual response factors f _A and f _B. The evaluation of on-column reaction gas chromatographic experiments is exemplified by the evaluation of elution profiles obtained by ring-closing metathesis reaction of N,N-diallytrifluoroacetamide in presence of Grubbs second-generation catalyst, dissolved in polydimethylsiloxane (GE SE 30).      

Preparation of monodisperse nanoparticles containing poly(propylene imine)(NH2)32‐polystyrene

Polypropylenimine dendrimer (DAB‐Am‐32, generation 4.0) was converted into a macroinitiator DAB‐Am‐32‐Cl via reaction with 2‐chloropropionyl chloride. Monodisperse nanoparticles containing poly(propylene imine)(NH₂)₃₂‐polystyrene were prepared by emulsion atom transfer radical polymerization (ATRP) of styrene (St), using the DAB‐Am‐32‐Cl/CuCl/bpy as initiating system. The structure of macroinitiator was characterized by FTIR spectrum, ¹H NMR, and ¹³C NMR. The structure of poly(propylene imine)(NH₂)₃₂‐polystyrene was characterized by FT‐IR spectrum and ¹H NMR; the molecular weight and molecular weight distribution of poly(propylene imine)(NH₂)₃₂‐polystyrene were characterized by gel permeation chromatograph (GPC). The morphology, size and size distribution of the nanoparticles were characterized by photon correlation spectroscopy (PCS), transmission electron microscopy (TEM), and atomic force microscopy (AFM). The effects of monomer/macroinitiator ratio and surfactant concentration on the size and size distribution of the nanoparticles were investigated. It was found that the diameters of the nanoparticles were smaller than 100 nm (30–80 nm) and monodisperse; moreover, the particle size could be controlled by monomer/macroinitiator ratios and surfactant concentration. With the increasing of the ratio of St/DAB‐Am‐32‐Cl, the number‐average diameter (D_n), weight‐average diameter (D_w) were both increased gradually. With enhancing the surfactant concentration, the measured D_h of the nanoparticles decreased, while the polydispersity increased. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 2892–2904, 2009     

Influence of Molar Mass Distribution on the Compatibility of Polymers

Abstract

Phase equilibria were calculated by means of a new method (direct minimization of the Gibbs energy of mixing) for polymer blends consisting of monodisperse polymer A and polydisperse polymer B. The results obtained for a Schulz-Flory distribution of B (molecular nonuniformity U = (M _w/M _n) ?1 = 1 and 100 components of model B) agree quantitatively with that of computations on the basis of continuous thermodynamics. The influence of U _B on the miscibility of A and B in 1:1 mixtures was studied for constant M _w of B, quantifying the incompatibility of the polymers by the length of the tie lines. The outcome of these calculations demonstrates that the typical effect of an augmentation of U _B (keeping M _w and the overall composition constant) consists in an enlargement of the mutual solubility of A and B. However, for an almost compatible pair of polymers (i.e., interaction parameters g are only slightly larger than the critical values for U _B = 0), this statement remains true only in the case of sufficiently small U _B. In order to gain some understanding of these findings, calculations were also performed for ternary systems (A and two species B). They demonstrate that it is the distance of the overall composition in the Gibbs phase triangle to the critical line (connecting the critical points for different U _B) which governs the changes in compatibility. Normally the critical point comes closer to the overall composition as U _B is raised, except for low g values where the critical point — after an initial approach — drifts apart as U _B becomes larger.     

Fe(CO)4 flexible as a two‐level system avoided conical intersection

This article reports new square‐planar Fe(CO)₄ D_4h structures that are optimized, using the Hartree–Fock (HF) approach, and multiconfiguration self‐consistent field (MCSCF) theory in active space [2b_2g2e_ga_1ga_2u]⁸, and which energy increased in sequence: ³B_2g TS < ¹A_1g TS < ¹A_1g GS. A triple ζ valence basis set supplemented with 4f for Fe and 3d for C and O polarization shells [TZV (DF)] was used. At the HF/TZV (DF) level, ¹A_1g TS and ³B_2g TS (³B_2g TS energetically more favorable), there are transition states of tetrahedral inversion (defining stereochemical flexibility of Fe(CO)₄) between known equivalent ¹A₁ and ³B₂ Jahn–Teller distorted tetrahedron C_2v structures with activation energy at ～0.96 kcal/mol according to the experimental data. ¹A_1g TS differs from ¹A_1g GS in electronic configuration by occupation of a_1g and a_2u MOs. At the MCSCF/ TZV (DF) level, ¹A_1g TS and ¹A_1g GS are optimized as near‐pure states in different potential energy surfaces (PES) avoided conical intersection with near‐equal interatomic distances, and define electronic flexibility of Fe(CO)₄. Estimation of the energy separation in a two‐level system that avoids a conical intersection from vibrational spectrum is based on the effective Hamiltonian of the perturbation theory. The energy gap between two square‐planar Fe(CO)₄ D_4h ¹A_1g TS < ¹A_1g GS is 0.27 kcal/mol. The energy gap between ¹A_1g GS and ¹A₁ is 1.28 kcal/mol. It is possible to observe ³B₂, ¹A₁ and ¹A_1g GS separately in the course of the experiment. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Bio‐based aromatic copoly(ether ester)s with enhanced toughness and degradability: Influence of insertion of phenoxy‐ether linkage and eugenol‐derived composition on properties

Two series of renewable nipagin and eugenol‐based copoly(ether ester)s, PDN1_1?xE1_x and PDN1_1?xE2_x (x = 0%, 10%, 20%, 30%, 40%, 50%), were prepared in the melt with 1,10‐decanediol as a comonomer. The synthesized poly(ether ester)s have weight‐average molecular weights (M_w) in the range of 20,400–37,200 g mol^?1, and dispersity (D) values between 1.7 and 1.9. Thermal gravimetric analysis (TGA) reveal that all the poly(ether ester)s exhibit a two‐step degradation mechanism with an initial degradation temperature above 350 °C. Results from differential scanning calorimetric (DSC) and wide‐angle X‐ray diffraction (WXRD) analyses demonstrate that the poly(ether ester)s are all semicrystalline materials with glass transition temperature (T_g) values ranging between ?21.3 and ?8.3 °C. The insertions of phenoxy‐ether linkage and eugenol‐derived composition have significant influence on the T_g, crystallinity, Young's modulus, and tensile strength, as well as the toughening effect and degradability. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2016 , 54, 2171–2183     

Conformationally asymmetric block copolymers

The standard parameters controlling AB diblock copolymer phase behavior are χN and f_A, where χ is an A-B segment interaction parameter, N is the overall degree of polymerization, and f_A is the volume fraction of the A block. Recently, it has been recognized that the ratio of the A and B statistical segment lengths α_A/α_B also represents another important parameter. Here, we theoretically examine the effects of this latter parameter on the phase behavior using the standard Gaussian chain model. Calculations are performed using both self-consistent field theory (SCFT) and strong segregation theory (SST). The ratio α_A/α_B is shown to have strong effects on order-order phase boundaries. Furthermore, it significantly affects the relative stability of the complex phases. In particular, it enhances the metastability of the perforated lamellar phase and may actually cause it to become an equilibrium structure. We also illustrate that varying α_A/α_B produces large changes in the relative domain spacings at order-order phase boundaries, which could strongly affect the kinetics of these transitions. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 945–952, 1997     

Reaction parameters in the RAFT mediated dispersion polymerization of styrene

Monodisperse polystyrene (PS) particles were prepared by a living radical dispersion polymerization with a reversible addition‐fragmentation chain transfer (RAFT) agent in an ethanol medium. In the presence of RAFT agent, the effects of various reaction parameters on the characteristics of PS particles were systematically investigated. When no RAFT agent was involved, the number‐average molecular weight (M_n) of the PS particles increased from 17,800 to 30,000 g/mol, but the weight‐average diameter (D_w) decreased from 2.54 to 2.06 μm with the increase of poly(N‐vinylpyrrolidone) content from 4.0 to 16.0 wt %. No correlation between the M_n and the coefficient of variation (CV) was observed. However, when the RAFT concentration varied from 0 to 2.0 wt %, all of the conversion, M_n, D_w, CV, and polydispersity index (M_w/M_n) decreased. This indicates that the RAFT agent alters the inverse behavior between the molecular weight (MW) and particle size shown in the conventional dispersion polymerization. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 872–885, 2008     

Heterochain model for simultaneous long‐chain branching and crosslinking. III. Multicomponent polymerization

The matrix formula developed in the context of heterochain theory, M?_w = M?_wp + WF ( I ? M )^?1 S , was applied to describe the molecular weight development during free‐radical multicomponent polymerization. All of the required probabilistic parameters are expressed in terms of the kinetic‐rate constants and the various concentrations associated with them. In free‐radical polymerization, the number of heterochain types, N, needs to be extrapolated to infinity, and such extrapolation is conducted with only three different N values. This matrix formula can be used as a benchmark test if other approximate approaches can give reasonable estimates of the weight‐average molecular weights. The moment equations with the average pseudo‐kinetic‐rate constants for branching and crosslinking reactions may provide poor estimates when the copolymer composition drift during polymerization is very significant. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2801–2812, 2004     

Amphiphilic silica/fluoropolymer nanoparticles: Synthesis,tem‐responsive and surface properties as protein‐resistance coatings

A series of amphiphilic silica/fluoropolymer nanoparticles of SiO₂‐g‐P(PEGMA)‐b‐P(12FMA) were prepared by silica surface‐initiating atom transfer radical polymerization (SI‐ATRP) of poly(ethylene glycol) methyl ether methacrylate (PEGMA) and poly dodecafluoroheptyl methacrylate (P12FMA). Their amphiphilic behavior, lower critical solution temperature (LCST), and surface properties as protein‐resistance coatings were characterized. The introduction of hydrophobic P(12FMA) block leads SiO₂‐g‐P(PEGMA)‐b‐P(12FMA) to form individual spherical nanoparticles (～150 nm in water and ～170 nm in THF solution) as P(PEGMA)‐b‐P(12FMA) shell grafted on SiO₂ core (～130 nm), to gain obvious lower LCST at 36–52 °C and higher thermostability at 290–320 °C than SiO₂‐g‐P(PEGMA) (LCST = 78–90 °C, T_d = 220 °C). The water‐casted SiO₂‐g‐P(PEGMA)‐b‐P(12FMA) films obtain much rougher surface (125.3–178.4 nm) than THF‐casted films (11.5–16.9 nm) and all SiO₂‐g‐P(PEGMA) films (26.8–31.3 nm). Therefore, the water‐casted surfaces exhibit obvious higher water adsorption amount (Δf = ?494 ～ ?426 Hz) and harder adsorbed layer (viscoelasticity of ΔD/Δf = ?0.28 ～ ?0.36 × 10^?6/Hz) than SiO₂‐g‐P(PEGMA) films, but present loser adsorbed layer than THF‐casted films (ΔD/Δf = ?0.29 ～ ?0.63 × 10^?6/Hz). While, the introduction of P(12FMA) segments does not show obviously reduce in the protein‐repelling adsorption of SiO₂‐g‐P(PEGMA)‐b‐P(12FMA) films (△f = ?15.7 ～ ?22.3 Hz) compared with SiO₂‐g‐P(PEGMA) films (△f = ?8.3 ～ ?11.3 Hz) and no obvious influence on water adsorption of ancient stone. Therefore, SiO₂‐g‐P(PEGMA)‐b‐P(12FMA) is suggested to be used as protein‐resistance coatings. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2016 , 54, 381–393     

Conformational structures and thermodynamic properties of compact polymer chains confined between two parallel plane boundaries

In this paper, the authors investigated the adsorption phenomenon of compact chains confined between two parallel plane boundaries using a pruned‐enriched Rosenbluth method. The authors considered three cases with different adsorption energies of ε = 0, ?1, and ?3 (in units of k_BT) for the confined compact chains of different chain lengths N, respectively. Several parameters were employed to describe the size and shape of compact chain, and some special behaviors in the conformational structures were investigated for the first time. For example, the size and shape of confined compact chains undergo distinct changes in the adsorption cases of ε = ?1 and ?3, and pass through the maximum values at the characteristic distances D_c. The authors found that this characteristic distance D_c could be scaled as D_c～ (N + 1)^ν (ν = 0.56 ± 0.01) in the case of ε = ?3. In addition, the microstructures of chains were investigated, and several significant results were obtained by analyzing the segment density distribution and the mean fractions of segment in tails, trains, bridges, and loops structures. On the other hand, the thermodynamic properties were also investigated for the confined compact chains, such as average energy per bond, Helmholtz free energy per bond, and elastic force per bond. Results show that elastic forces f have different behaviors in three cases, indicating that it is not necessary to exert an external force on the boundaries in the nonadsorption case. At the same time, the average contact energy of compact chain obviously changes when the distance between the two parallel boundaries D increases, which is similar to those of the size and shape parameters. The authors also conclude that these thermodynamic properties of compact chains depend strongly on not only the adsorption energies but also the chain lengths and the confined condition. In addition, several results of the conformational and thermodynamic parameters, such as the segment density distribution and free energy, were compared with the results from the self‐consistent field theory. These investigations may help us to deepen the knowledge about the adsorption phenomenon of confined compact chains. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2888–2901, 2006     

Viscometric,light scattering,and size‐exclusion chromatography studies on the structural changes of aqueous poly(vinyl alcohol) induced by γ‐ray irradiation

The crosslinking processes of aqueous poly(vinyl alcohol) (PVA) by γ‐ray irradiation were studied by viscometry, dynamic and static light scattering (DLS and SLS), as well as size exclusion chromatography (SEC). Increases in the intrinsic viscosity ([η]), molecular weight (M_w), hydrodynamic radius (R_h), and radius of gyration (R_g), and a decrease in second virial coefficient (A₂) were observed after γ‐ray irradiation. However, both the values of [η] and A₂ for irradiated PVA fell below the data of unirradiated PVA solutions, suggesting a conformational change of PVA chains after γ‐ray irradiation. This structural change of PVA as a result of γ‐ray irradiation was also indicated by the decreases in R_g/R_h from 1.5 to 1.39 by SLS and DLS, and in Mark–Houwink exponent α_η from 0.54 to 0.26 by SEC‐Viscometry. The broadening of the M_w distribution (MWD) as indicated by the polydispersity index increased from 2.2 to 6.5 because of γ‐ray irradiation. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 214–221, 2000     

Microwave‐improved polymerization of ϵ‐caprolactone initiated by carboxylic acids

The ring‐opening polymerization of ε‐caprolactone (ε‐CL), initiated by carboxylic acids such as benzoic acid and chlorinated acetic acids under microwave irradiation, was investigated; with this method, no metal catalyst was necessary. The product was characterized as poly(ε‐caprolactone) (PCL) by ¹H NMR spectroscopy, Fourier transform infrared spectroscopy, ultraviolet spectroscopy, and gel permeation chromatography. The polymerization was significantly improved under microwave irradiation. The weight‐average molecular weight (M_w) of PCL reached 44,800 g/mol, with a polydispersity index [weight‐average molecular weight/number‐average molecular weight (M_w/M_n)] of 1.6, when a mixture of ε‐CL and benzoic acid (25/1 molar ratio) was irradiated at 680 W for 240 min, whereas PCL with M_w = 12,100 and M_w/M_n = 4.2 was obtained from the same mixture by a conventional heating method at 210 °C for 240 min. A degradation of the resultant PCL was observed during microwave polymerization with chlorinated acetic acids as initiators, and this induced a decrease in M_w of PCL. However, the degradation was hindered by benzoic acid at low concentrations. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 13–21, 2003     

Manipulation of the molecular weight and branching structure of hyperbranched poly(arylene ether phosphine oxide)s prepared via an A2 + B3 approach

A series of hyperbranched poly(arylene ether phosphine oxide)s (HB PAEPOs) were prepared via an A₂ + B₃ polymerization scheme with tris(4‐fluorophenyl)phosphine oxide as B₃, and a variety of bisphenols as A₂. The effects of the reactivity of the A₂ monomer, the A:B ratio, the addition mode, the solvent, and the concentration on the final molecular weight, polydispersity index (PDI), and degree of branching (DB) were studied. Soluble HB PAEPOs with weight‐average molecular weights of up to 299,000 Da were achieved. Reactions in which the A₂ component was added slowly resulted in lower DBs (0.2–0.5), whereas the slow addition of the B₃ component provided samples with DBs of approximately 0.75. Reactions performed under high‐dilution conditions afforded completely soluble materials with weight‐average molecular weights of 9000–12,100 Da and PDI values as low as 2.20. The molecular weights achieved under high‐dilution conditions were independent of the mode of monomer addition. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 3871–3881, 2003     

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.     

General matrix formula for the weight-average molecular weights of crosslinked polymer systems

On the basis of the first-order Markovian statistics, we propose a general matrix formula for the weight-average molecular weight of crosslinked polymer systems, explicitly given by M̄_w = M̄_w,0 + WX₀ (I − X)⁻¹ S_f . This equation is valid for both step and chain-growth polymerizations, including those in a nonequilibrium state irrespective of the reactor types used. In the context of the present theory, the onset of gelation is simply stated as a point at which the largest eigenvalue of the matrix X reaches unity (i.e., det( I − X ) = 0). The present theory provides a unified point of view for various types of gelling systems. © 1998 John Wiley & Sons, Inc. J. Polym. Sci. B Polym. Phys. 36: 2423–2433, 1998