Statistical branching of heterochains

A general theory for the statistical branching of heterochains is proposed on the basis of the random sampling technique. Consider a polymer mixture that consists of N types of chains whose weight fractions are w_i (i = 1, 2, …, N), and number- and weight-average chain lengths are P̄_np,i and P̄_wp,i, respectively. Suppose the transition probability that a branch point on a chain of type i is connected to a chain end of a type j chain is given by p_ij. When the branching density of chains of type i is ρ_i, the weight-average chain length is given by $\bar P_w = W\sum\nolimits_{m = 0}^\infty T ^m \sum\nolimits_{n = 0}^\infty {SU} ^n 1$, where S is the diagonal matrix whose elements are $S_{ii} = \bar P_{wp,i}$, 1 is the column vector whose elements are all unity, U is the transition matrix whose elements are given by $u_{ij} = \rho _i p_{ij} P_{np,j} ,T$ is another transition matrix whose elements are given by t_ij = (w_j/w_i)U_ji, and W is the row vector whose elements are w_i. Simpler expressions of P̄_w are presented for binary systems. In addition to the multicomponent systems, the present equation could also be used such as for free-radical polymerization with long-chain branching, by considering primary chains formed at different times as different types of polymer chains. For the prediction of the full molecular weight distribution, a Monte Carlo simulation method is used to illustrate the resulting distribution profiles.     

Markovian approach to nonlinear polymer formation: Free-radical polymerization with chain transfer to polymer

A Markovian model is proposed for nonrandom branching reactions, by using free-radical polymerization that involves chain transfer to polymer as an example. Free-radical polymerizations are kinetically controlled; therefore, each primary polymer molecule experiences different history of branched structure formation. By assuming that the primary chains with the identical birth time conform to the same chain connection probabilities, the nonlinear structural development can be viewed as a system in which the primary chains formed at different birth times are combined into nonlinear polymers in accordance with the first-order Markov chain statistics. An explicit formula for the weight-average chain length is derived in a matrix form. The onset of gelation is simply stated as a point at which the largest eigenvalue of the transition matrix X reaches unity, i.e., det(X − I) = 0. This criterion for the onset of gelation can be considered as an extension of the Flory/Stockmayer theory to a nonequilibrium reaction system. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36 : 357–371, 1998     

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     

Heterochain model for simultaneous long‐chain branching and crosslinking. II. Application to free‐radical polymerization

The matrix formula developed in the context of hetrochain theory, M?_w = M?_wp + WF ( I ? M )^?1 S , was applied to describe the molecular weight development during free‐radical homopolymerization. All of the required probabilistic parameters are expressed in terms of the kinetic‐rate constants and various concentrations. In free‐radical polymerization, the primary chains are formed consecutively, and the number of heterochain types, N, is extrapolated to infinity. Practically, such extrapolation can be conducted on the basis of the calculated values for only three different N values with sufficient accuracy. This matrix formula is valid regardless of the chemical and reactor systems used, as long as the primary chain‐connection statistics is considered Markovian. The gel point can be determined simply by solving an equation det( I ? M ) = 0. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2791–2800, 2004     

Blends of poly(n-vinyl-2-pyrrolidone)/poly(monoitaconates) II. The coefficient of linear expansion

Blends of poly(monoitaconates)^b) containing different side chain structures with poly(N-vinyl-2-pyrrolidone) (PVP) of three different weight-average molecular weights (M̄_w) were studied by thermomechanical analysis. Blends containing PVP of M̄_w =10000 shows a monotonous variation of the coefficient of linear expansion α against composition but PVP samples of higher molecular weights present a minimum which is attributed to polymer-polymer complex due to strong specific interactions.     

Modelling free-radical copolymerization kinetics—evaluation of the pseudo-kinetic rate constant method, 1. Molecular weight calculations for linear copolymers

The moment equations for binary copolymerization in the context of the terminal model have been solved numerically for a batch reactor operating over a wide range of conditions. Calculated number- and weight-average molecular weights were compared with those found using pseudo-kinetic rate constants with the method of moments and with the instantaneous property method for homopolymerization. With the pseudo-kinetic rate constant method under polymerization conditions where number-average molecular weights (M̄_n) are below about 10³ the error in calculating M̄_n exceeds 5%. The error increases rapidly with decrease in molecular weight for M̄_n < 10³. M̄_n measured experimentally for polymer chains (homo- and copolymers) have error limits of greater than ±5% at the 95% confidence level. Therefore, for all practical purposes, the pseudo-kinetic rate constant method is valid for M̄_n greater than 10³. Errors in calculating weight-average molecular weights (M̄_w) or higher averages are always smaller than those for M̄_n when applying the pseudo-kinetic rate constant method. The assumptions involved in molecular weight modelling using the pseudo-kinetic rate constant approach are thus proven to be valid, and therefore it is recommended that the pseudo-kinetic rate constant method be employed with the instantaneous property method to calculate the full molecular weight distribution and averages for linear chains synthesized by multicomponent chain growth polymerization.     

Modelling free-radical copolymerization kinetics—evaluation of the pseudo-kinetic rate constant method, 2. Molecular weight calculations for copolymers with long chain branching

The full moment equations and equations using pseudo-kinetic rate constants for binary copolymerization with chain transfer to polymer in the context of the terminal model have been developed and solved numerically for a batch reactor operating over a wide range of conditions. Calculated number- and weight-average molecular weights (M̄_n and M̄_w) were compared with those found using the pseudo-kinetic rate constant method (PKRCM). The results show that the weight-average molecular weights calculated using PKRCM are in agreement with those found using the method of full moments for binary copolymerization when polymeric radical fractions φ₁^˙ and φ₂^˙ of type 1 and 2 (radical centers are on monomer types 1 and 2 for a binary copolymerization) are calculated accounting for chain transfer to small molecules and polymer reactions in addition to propagation reactions. Errors in calculating M̄_w using PKRCM are not always negligible when polymer radical fractions are calculated neglecting chain transfer to small molecules and polymer. In this case, the relative error in M̄_w by PKRCM increases with increase in monomer conversion, extent of copolymer compositional drift and chain transfer to polymer rates. The errors in calculating M̄_w, however, vanish over the entire monomer conversion range for all polymerization conditions when chain transfer reactions are properly taken into account. It is theoretically proven that the pseudo-kinetic rate constant for chain transfer to polymer is valid for copolymerizations. One can therefore conclude that the pseudo-kinetic rate constant method is a valid method for molecular weight modelling for binary and multicomponent polymerizations.     

Ethylene polymerizations with novel tantalum(V) aminopyridinato complex/MAO systems

The use of two kinds of tantalum(V) aminopyridinato complexes, bis(2-benzylaminopyridinato)trichlorotantalum(V) and trichlorobis[2,6-di(phenylamino)pyridinato-N,N′]-tantalum(V), activated by methylaluminoxane was studied in polymerization of ethylene. The activities of these homogeneous catalyst systems are comparable to those of metallocenes. The weight-average molecular weights (M̄_w) of the produced polyethylenes are between 60 000 and 200 000 and M̄_w/M̄_n ≈ 2.     

Statistical thermodynamics in the framework of the lattice fluid model, 2. Binary mixture of polydisperse polymers of special distribution

In this paper, the Gibbs free energy, the equation of state and the chemical potentials of polydisperse multicomponent polymer mixtures are derived. For general binary mixtures of polydisperse polymers, we also give the Gibbs free energy, the equation of state and the chemical potentials and derive the stability criteria and spinodal. Furthermore, binary polydisperse polymer mixtures of special distribution, i.e., Flory distribution, uniform distribution and Schulz distribution, are discussed and the influence of polydispersity on the interaction energy parameter is considered. For these special-distribution systems, the spinodal curves are simulated and the influence of chain length and polydispersity on the spinodal curves is discussed. The results suggest that the spinodal temperature of the mixture with a given volume fraction of one component decreases with increasing polydispersity and the extent of the shift decreases with increasing degree of polymerization when η = M̄_w/M̄_n is given. In addition, the variations of the spinodal curves with polydispersity and chain length are shown and they are qualitatively compared with the experimental results.     

Synthesis and characterization of multiarm star-branched polyisobutylenes: Effect of arm molecular weight

A series of star-branched polyisobutylenes with varying arm molecular weights was synthesized using the 2-chloro-2,4,4-trimethylpentane/TiCl₄/pyridine initiating system and divinylbenzene (DVB) as a core-forming comonomer (linking agent). The resulting star-branched polymers were characterized with regard to the weight-average number of arms per star molecule (N̄_w) and dilute solution viscosity behavior. As the molecular weight of the arm (M̄_{w, arm}) was increased, dramatically longer star-forming reaction times were needed to produce fully developed star polymers. It was calculated that N̄_w varied from 50 to 5 as the M̄_{w, arm} was increased from 13,000 to 54,000 g/mol. The radius of gyration, R_g, of the star polymers was observed to increase as M̄_{w, arm} was increased. The solution properties of the star polymers were evaluated in heptane using dilute solution viscometry. It was determined that the stars had a much higher [η] compared to the respective linear PIB arms, but a much lower [η] compared to a hypothetical linear analog of an equivalent molecular weight. The dependence of [η] on temperature for the stars and linear arms was very small over the temperature range 25 to 75°C, with only a very slight decrease with increasing temperature. [η]_star was also determined to increase with increasing M̄_{w, arm}, but decrease with increasing M̄_{w, star}. The branching coefficient, g′, calculated for the stars at 25°C, increased as N̄_w decreased and agre ed well with literature values for other star polymer systems. © 1997 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 35: 3767–3778, 1997     

Liquid crystalline,highly luminescent poly(2,5-dialkoxy-p-phenyleneethynylene)

A high-molecular-weight poly(2,5-dialkoxy-p-phenyleneethynylene) derivative has been prepared by the Heck reaction of 1,4-bis(2-ethylhexyloxy)-2,5-diiodobenzene and 1,4-diethynyl-2,5-dioctyloxybenzene. The highly luminescent polymer exhibits excellent solubility and can readily be processed into high-optical-quality films. The weight-average molecular weight M̄_w was 240000 g · mol⁻¹, with a polydispersity index of 2.9. Thermal analysis revealed a glass transition around 90°C, and an onset of chemical crosslinking at 130°C. The high M̄_w and the remarkable solubility enabled the preparation of liquid crystalline solutions of the new PPE.     

Mechanism of the anionic copolymerization of anhydride-cured epoxies – analyzed by matrix-assisted laser desorption ionization time-of-flight mass spectrometry (MALDI-TOF-MS)

The mechanism of the strictly alternating anionic copolymerization of phenyl glycidyl ether (PGE) and phthalic anhydride (PA) was initiated by various imidazoles. Because of the strictly alternating copolymerization polyesters with a repeating unit of PGE-PA were obtained. The mechanism of the reaction was analyzed by means of matrix-assisted laser desorption ionization time-of-flight mass spectrometry (MALDI-TOF-MS). With this technique the molar masses of the oligomers, the molar mass of the repeating unit, the weight-average molar mass M̄_w and the number-average molar mass M̄_n, their ratio M̄_w/M̄_n and the residual molar mass could be calculated. The strictly alternating copolymerization was easy to prove because the molar masses of PGE and PA are slightly different. The question whether the initiator remains chemically bound during the whole reaction could be solved. To this end polyesters obtained by initiation with various imidazoles with different molar masses were synthesized. The calculated residual molar masses correspond exactly to the molar masses of the imidazoles.     

Polymerization of ethene initiated with a mixture of siloxane-bridged mono- and dinuclear zirconocenes

In the polymerization of ethene cocatalyzed with modified methylaluminoxane, the catalyst activities of the siloxane-bridged dinuclear zirconocenes, tetramethyldisiloxanediylbis(cyclopentadienylindenylzirconium dichloride) ( 3 ) and hexamethyltrisiloxanediylbis(cyclopentadienylindenylzirconium dichloride) ( 4 ) were lower than that obtained with the siloxane-bridged mononuclear zirconocene, tetramethyldisiloxanediyldicyclopentadienyldimethylzirconium ( 1 ). On the other hand, weight-average molecular weight M̄_w and ratio of weight- to number-average molecular weights M̄_w/M̄_n of polyethene (PE) obtained with 3 or 4 were higher than those of PE obtained with 1. For a binary mixture of 1/3 or 1/4 , it was found that the obtained PE exhibits a bimodal molecular weight distribution for an appropriate composition of the mixed zirconocenes. M̄_w/M̄_n of PE could be adjusted by changing the relative concentrations of the two zirconocenes.     

Influence of polymolecularity on the second virial coefficient of polymers

Scaling theory is applied to derive expressions describing the influence of polymolecularity on the second virial coefficient, A₂, as obtained from osmotic pressure and light scattering measurements. Numerical values of polymolecularity correction factors are calculated for Schulz-Zimm and logarithmic normal distributions of the molecular weight, different qualities of the solvent and several ratios of the weight-average and the number-average molecular weights M̄_w/M̄_n. It is found that in the equation $ A_2 = K_{A_2 } \cdot M_{{\rm av}}^{a_{A_2 } } $ the weight-average molecular weight is a good approximation for M_av if A₂ is measured via light scattering, while the number-average molecular weight can be inserted for M_av if A₂ stems from osmotic pressure measurements.     

Salt-catalyzed addition reaction of monoepoxides with philosamia cynthia ricini and bombyx mori silk fibroins

The heterogeneous addition reaction of various monoepoxides with silk fibroins of Philosamia cynthia ricini and Bombyx mori was investigated at 45–75°C by use of aqueous solutions of various salts as padding catalysts. The effects of salt on the epoxide–silk fibroin reactions were attributed mainly to the nucleophilicity of the anions and also to the acidity or the electronegativity of the cations. The effect of the substituent of the epoxide on the add-ons was elucidated by the modified Taft equation, (log W ? log W₀)/σ* = ρ_p + ρ_sE_s/σ*, where W₀ and W are the add-ons for the reaction of a given compound and of its substituted derivatives, σ* and E_s are the polar and the steric substituent constants, ρ_p and ρ_s are the polar and the steric reaction constants, respectively. Histidine, lysine, arginine, tyrosine, serine, and acidic amino acids were found to react. The reactivity difference between Philosamia cynthia ricini and Bombyx mori fibroins towards the epoxide was discussed in the light of the observed phenomena.     

Molecular Weight Development in Free‐Radical Polymerization with Polyfunctional Chain‐Transfer Agents, 1. Equal Reactivity Model

The analytic expression for the weight‐average molecular weight development in free‐radical polymerization that involves a polyfunctional chain‐transfer agent is proposed. Free‐radical polymerization is kinetically controlled; therefore, the probability of chain connection with a polyfunctional chain‐transfer agent as well as the primary chain‐length distribution changes during the course of polymerization. We consider the primary chains formed at different times as different types of chains, and the heterochain branching model is used to obtain the weight‐average chain length at a given conversion level in a matrix formula, described as P_w = W { D _w + ( I + T ) SP ( I – TSP )^–1 D_f }. Because the primary chains are formed consecutively, the number of chain types N is extrapolated to infinity, but such extrapolation can be conducted with the calculated values for only three different N values. The criterion for the onset of gelation is simply described as a point at which the largest eigenvalue of the product of matrixes, TSP reaches unity, i. e., det  ( I – TSP ) = 0. The present model can readily be extended for the star‐shaped polyfunctional initiators, and the relationships between the model parameters and kinetic rate expression for such reaction systems are also shown.     

Postgel properties in the statistical crosslinking of heterochains. I. Systems with N types of chains

The heterochain crosslinking model describes nonrandom crosslinking of polymer chains and is an extension of the classical Flory/Stockmayer gelation theory. We consider the postgelation relationship for the system consisting of N types of polymer chains, in which the probability that a crosslink point on an i‐type chain is connected to a j‐type chain is explicitly given by p_ij. The analytical solutions for the weight fraction of the sol, the number‐average and weight‐average molecular weights within the sol fraction, and the crosslinking density within the sol and gel fractions are derived for the systems, with each type of chain conforming to the Schulz–Zimm distribution. Illustrative calculations are shown for the systems consisting of two and three types of chains, and the obtained results agree with those from the Monte Carlo method. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2333–2341, 2000     

Narrow-polydispersity polystyrene/poly[styrene-co-(butyl methacrylate)] block copolymers by an N-oxyl mediated free radical polymerization process

Polystyrene/poly[styrene-co-(butyl methacrylate)] block copolymers with controlled molecular weights and with polydispersities generally below M̄_w/M̄_n = 1,45 and partially as low as M̄_w/M̄_n = 1,19 were synthesized by a free radical bulk copolymerization using a 2,2,6,6-tetramethylpiperidine-N-oxyl (TEMPO)-capped polystyrene macroinitiator. The influence of the macroinitiator concentration on the block copolymerization was studied. The polymerization rates are independent of the macroinitiator concentration and are close to that of thermally self-initiated styrene/butyl methacrylate copolymerizations showing the important role of self-initiation for N-oxyl mediated free radical polymerizations.     

Synthesis and mesomorphic behavior of poly(dipropylsilylenemethylene)

Based on the reaction of trichloro(chloromethyl)silane ( 1 ) with 2 equivalent amounts of the respective Grignard-reagent and subsequent cyclization, 1,1,3,3-tetrapropyldisilacyclobutane ( 3 ) has been prepared. Catalytic polymerization with H₂PtCl₆ was employed to prepare the corresponding poly(dipropylsilylenemethylene) (PDPSM, 4 ) with strictly alternating SiR₂/CH₂ backbone structure. A high-molecular-weight fraction of the material (weight-average molecular weight M̄_w = 166 500 and number-average molecular weight M̄_n = 115 200) obtained by fractionating precipitation was investigated with respect to glass transition and formation of conformationally disordered mesomorphic phases. The glass transition temperature T_g = 232 K of PDPSM evidenced lower backbone flexibility than observed for the analogous poly(dipropylsiloxane), (PDPS). PDPSM exhibited mesomorphic behavior. In contrast to poly(dipropylsiloxane), PDPSM showed a surprisingly narrow mesomorphic regime between 355 K and 365 K. Based on polarizing microscopy and ²⁹Si-MAS (magic angle spinning) solid-state NMR the mesophase is described as a conformationally disordered state, which is most probably columnar in analogy to PDPS.     

Polymerization of Methyl Methacrylate Initiated with Organomanganate Reagents

Organomanganate reagents [R₃Mn]^–Li⁺ (R = Bu, Me) were found to polymerize methyl methacrylate in the presence of potassium tert‐butylate. A conversion of the tacticity of the resulting poly(methyl methacrylate)s from heterotactic (mr = 54%) to isotactic (mm = 58%) was observed upon changing the R group of the initiator from Bu to Me. The addition of triisobutylaluminium was found to efficiently control M̄_w and M̄_w/M̄_n of the resulting polymers.