The nature of the chain-length dependence of the propagation rate coefficient and its effect on the kinetics of free-radical polymerization. 1. Small-molecule studies

In this paper we summarize and analyze the currently available small-molecule data, both experimental and theoretical, that is relevant to chain-length-dependent propagation in free-radical polymerization (FRP). We do this in order to appreciate the nature of chain-length-dependent propagation, because workers are becoming increasingly cognizant of its necessity in reaching a complete understanding of FRP kinetics. We show that studies of addition in small-molecule (model) systems support a chain-length dependence (at short chain lengths i) which is described by the following functional form, which therefore can be said to be physically realistic: , where the values of C₁ and i_1/2 are of the order of 10 and 1, respectively. These results are supported by transition state theory, which predicts a very similar behavior for the Arrhenius frequency factor. We illustrate that in systems with low number-average degree of polymerization (DP_n), this chain-length dependence can dramatically affect the observed (chain-length-averaged) propagation rate coefficient 〈k_p〉, which can be significantly higher than the long chain value, k_p. However, this effect is only observed if the activation energy for the first radical addition is similar to that for propagation. In the case that the former is significantly higher (e.g., when choosing a less than optimal initiator or in the case of retardative chain transfer), the chain-length-dependent propagation predicted by our model will not be observed, and in fact a significant lowering of 〈k_p〉 can in cases be expected up to relatively high DP_n.     

Chain-length-dependent termination rate constant: Effect on molecular weight distributions formed by pulsed laser radical polymerization

For polymerization initiated by an arbitrary sequence of laser pulses a numerical technique for calculating molecular weight distributions (MWDs) is developed, which takes into consideration the chain length dependence of the termination rate constant k_t. The MWDs for methyl methacrylate and styrene are calculated by use of α and k₀ values (for the law k = k₀(i)^−α of termination of radicals with chain length i) and averages $ \overline {(i,{\rm }j)} $ (for rate constants k = k₀$ \overline {(i,{\rm }j)} $ of termination of radicals with different degrees of polymerization) taken from the literature. The dependences of the overall termination constant 〈k_t〉 on initiation parameters (pulse repetition rate (v) and pulse intensity for initiation by periodic laser pulses) are presented. Two methods are proposed for α and k₀ determination: (a) by experiments on polymerization with periodic laser pulses where monomer-to-polymer conversions per pulse are determined for different v; (b) by experiments on polymerization with packets of pulses where the constant k_p (the rate constant of propagation), α and k₀ can be determined simultaneously from MWD. For both methods simple analytical equations are derived for evaluation of the constants. The limits of application of the methods are determined by use of the numerical technique for MWD calculation.     

Chain-length dependent termination in pulsed-laser polymerization, 6. The evaluation of the rate coefficient of bimolecular termination kt for the reference system methyl methacrylate in bulk at 25°C

The values for the rate coefficient of chain termination k_t in the bulk polymerization of methyl methacrylate at 25°C were formally calculated (i) from the second moment of the chain-length distribution and (ii) from the rate equation for laser-initiated pseudostationary polymerization (both expressions were originally derived for chain-length independent termination) by inserting the appropriate experimental data including the rate constant of chain propagation k_p. These values were treated as average values, k and k , respectively. They exhibited good mutual agreement, even the predicted gradation (k < k by about 20%) was recovered. The log-log plot of k_t vs. the average degree of polymerization of the chains at the moment of their termination v′ yielded exponents b of 0.16–0.17 in the power-law k _t = A · v′^−b, A ranging from 1.1 × 10⁸ to 1.3 × 10⁸ (L · mol⁻¹ · s⁻¹). A 70% contribution of disproportionation to overall termination has been assumed in the calculations.     

Analysis of branched polymers by high resolution multidetector size exclusion chromatography: Separation of the effects of branching and molecular weight distribution

This article presents the SEC analysis of branched polyisobutylene PIB and polystyrene PS with high molecular weight and broad multimodal molecular weight distribution. Both polymers were synthesized using an inimer technique, which results in long‐chain branched polymers with statistical branching and broad multimodal distributions. Using high resolution multidetector Size Exclusion Chromatography SEC the polymers were analyzed based on three branching factors: g = (R_z,br/R_z,lin)_Mw; h = (〈R_h〉_z,br/〈R_h〉_z,lin)_{Mw ;} and ρ = (R ^1/2/〈R_h〉_z). It is generally accepted that for monodisperse branched polymers g and h < 1. In the case of our polydisperse PIB and PS, it was seen that g and h > 1, and ρ increases with molar mass and the number of chain ends as predicted earlier. The multidetector SEC system allowed for the separation of branching and polydispersity, reported here for the first time experimentally. The g parameter as a function of DP_i was compared to the theory developed by Zimm and Stockmayer. The plots followed a similar trend, but were shifted by a factor related to the average chain length between branching points. The ρ parameter decreased with increasing DP_i, as predicted theoretically by Kajiwara. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Determination of Propagation Rate Coefficient of Acrylates by Pulsed‐Laser Polymerization in the Presence of Intramolecular Chain Transfer to Polymer

Unusual difficulties are faced in the determination of propagation rate coefficients (k_p) of alkyl acrylates by pulsed‐laser polymerization (PLP). When the backbiting is the predominant chain transfer event, the apparent k_p of acrylates determined in PLP experiments for different frequencies should range between k_p (propagation rate coefficient of the secondary radicals) at high frequency and k at low frequency. The k value could be expressed from kinetic parameters: , where k_fp is the backbiting rate coefficient, k_p2 is the propagation rate coefficient of mid‐chain radicals, and [M] is the monomer concentration.

Apparent propagation rate coefficients determined for different frequencies by simulating the PLP of n‐butyl acrylate at 20 °C. Horizontal full lines show the values of k_p and k.     

Dilute solution properties of a polybenzimidazole

Light scattering and viscometric studies have been carried out on dilute solutions of a polybenzimidazole in N,N-dimethylacetamide. The data, which span the molecular weight range 2.9 ≦ 10^?4M_w ≦ 23.3, and the temperature range 290 ≦ T/K ≦343, yield the dependence of the mean-square radius of gyration 〈s²〉_LS, the second virial coefficient A₂, and the intrinsic viscosity [η] on molecular weight M_w and temperature. The unperturbed mean-square radius 〈s〉_LS was calculated using experimental values of 〈s²〉_LS and A₂. It was found that excluded volume effects on 〈s²〉_LS are very small. The unperturbed hydrodynamic chain dimension 〈s〉_η was estimated by considering draining effects. A small value of the draining parameter was obtained. Analysis of the temperature dependence of A₂ and [eta;] leads to the conclusion that this system approaches a lower theta temperature with increasing temperature. The steric factor σ = 〈s〉/〈s〉f, based on the value of 〈s〉_f calculated for the polymer chain with free rotation, is nearly unity. Most of these properties can be interpreted in terms of long rotational units within the main chain.     

Chain-length dependent termination in pulsed-laser polymerization, 5. The evaluation of the rate coefficient of bimolecular termination kt for the reference system styrene in bulk at 25°C

Following earlier suggestions the values for the rate coefficient of chain termination k_t in the bulk polymerization of styrene at 25°C were formally calculated (a) from the second moment of the chainlength distribution (CLD) and (b) from the rate equation for laser-initiated pseudostationary polymerization (both expressions originally derived for chain-length independent termination) by inserting the appropriate experimental data including the rate constant of chain propagation k_p. These values were treated as average values, k and k , respectively. They exhibited good mutual agreement, even the predicted gradation (k < k by about 20%) was recovered. The log-log plot of k_t vs. the number-average degree of polymerization of the chains at the moment of their termination yielded exponents b of 0.16–0.18 in the power-law k_t = A · P_n ^−b, A ranging from 2.3 × 10⁸ to 2.7 × 10⁸ L · mol⁻¹ · s⁻¹. These data are only slightly affected if termination is not assumed to occur by recombination only and a small contribution of disproportionation is allowed for.     

Influence of short-range interactions on the shape of an unperturbed polymer chain: polyethylene

The average instantaneous shape of an unperturbed polyethylene chain is studied with a Monte Carlo technique. Different short-range interactions in the polyethylene chain are considered. The shape is evaluated as the ratio 〈L〉:〈L〉:〈L〉, where L₁≤L₂≤L₃ are the orthogonal components in the system of principal axes of gyration. Differences are found for different interactions in short- and medium-length chains, while for long chains all ratios converge to a common limit, which is about 1:2.7:12.0 for polyethylene chains.     

Magnesium chloride supported high-mileage catalysts for olefin polymerization. XVII. Effect of lewis base on propylene polymerization

A systematic study has been made on the functions of external Lewis base (B_e, methyl-p-toluate, MPT) and internal Lewis base (B_i, ethyl benzoate, EB) for the CW-catalyst system MgCI₂/EB/PC/AlEt₃/TiCl₄–AlEt₃/MPT (PC, p-cresol). B_i is a nonstereoselective modifier. It increases the active site concentrations and rate constants of propagation, k_p, of both the isospecific and nonspecific sites, and thus the productivities of the stereoregular and irregular polypropylenes by five- to tenfold. It seems that B_i complexes with the MgCl₂ support to lower the electronegativity of the surface Mg atoms. It also acts to lower the rate constant of chain transfer to aluminum alkyl, k, by two- to fourfold. The action of B_e is highly stereospecific. The isotacticity index of polypropylene is ? 95% in the presence of B_e but ? 68% without it. Addition of B_e decreases nonspecific [Ti^*]_a by about (11 ± 2)-fold; there is only about a twofold reduction of the isospecific [Ti^*]_i. It decreases k_p,a about three times but has no effect on k_p,i, so that the latter is (21 ± 4) times the former. B_e decreases k for transfer with aluminum alkyl much more than it does to k; but it does not affect the rates of chain transfer with monomer and by β-hydride elimination or the rate of catalyst deactivation. The number of active sites without B_e is [Ti^*]_i = 15% and [Ti^*]_a = 55% for a total of 70%. In the presence of B_e they are both about 6%. Optimum performance in propylene polymerizations requires both B_i and B_e in the case of the CW-catalyst.     

SP–PLP–EPR Investigations into the Chain‐Length‐Dependent Termination of Methyl Methacrylate Bulk Polymerization

Termination kinetics of methyl methacrylate (MMA) bulk polymerization has been studied via the single pulsed laser polymerization–electron paramagnetic resonance method. MMA‐d₈ has been investigated to enhance the signal‐to‐noise quality of microsecond time‐resolved measurement of radical concentration. Chain‐length‐dependent termination rate coefficients of radicals of identical size, k, are reported for 5–70 °C and up to i = 100. k decreases according to the power‐law expression . At 5 °C, k_t for two MMA radicals of chain‐length unity is k = (5.8 ± 1.3) · 10⁸ L · mol⁻¹ · s⁻¹. The associated activation energy and power‐law exponent are: E_A(k) ≈ 9 ± 2 kJ · mol⁻¹ and α ≈ 0.63 ± 0.15, respectively.

     

Scope for Accessing the Chain Length Dependence of the Termination Rate Coefficient for Disparate Length Radicals in Acrylate Free Radical Polymerization

A method that utilizes reversible addition fragmentation chain transfer (RAFT) chemistry is evaluated on a theoretical basis to deduce the termination rate coefficient for disparate length radicals k in acrylate free radical polymerization, where s and l represent the arbitrary yet disparate chain lengths from either a “short” or “long” RAFT distribution. The method is based on a previously developed method for elucidation of k for the model monomer system styrene. The method was expanded to account for intramolecular chain transfer (i.e., the formation of mid-chain radicals via backbiting) and the free radical polymerization kinetic parameters of methyl acrylate. Simulations show that the method's predictive capability is sensitive to the polymerization rate's dependence on monomer concentration, i.e., the virtual monomer reaction order, which varies with the termination rate coefficient's value and chain length dependence. However, attaining the virtual monomer reaction order is a facile process and once known the method developed here that accounts for mid-chain radicals and virtual monomer reaction orders other than one seems robust enough to elucidate the chain length dependence of k for the more complex acrylate free radical polymerization.     

Kinetics of crosslinking reactions. 2. Reaction rates of intermolecular and intramolecular crosslinkings by using a soluble microgel

Soluble microgels with several pendant vinyl groups were synthesised by radical copolymerization of methyl methacrylate (MMA) with p-divinyl benzene (p-DVB). The competitive reactions of intermolecular and intramolecular crosslinkings of these microgels were carried out at 40°C in the presence of 1-buten-3-ol as a degradative chain transfer agent. The rate constant of intermolecular crosslinking (k) was estimated by GPC (gel permeation chromatography) analysis on the polymer produced from intermolecular propagation between bimolecules. The k depended strongly on the internal structure of microgels. Network formation was discussd inclusive of informations for the rate constant of intramolecular crosslinking (k).     

Solvay type VCl3 isospecific catalysts

Solvay type S –VCl₃ catalyst has 7% of catalytically active vanadium sites ([C^*]) with k_p (rate constant of propagation) = 31 (M s)^?1 for ethylene polymerization. Addition of a comonomer, propylene of 4-methylpentene-1 (4-MP) significantly raised the ethylene polymerization activity. S –VCI₃ catalyst has very small amounts of catalytically active vanadium for propylene polymerizations: [C] = 0.19% with k_p,i = 857 (M s)^?1 and [C] = 0.45% with k_p,a = 23 (M s)^?1 for isospecific and nonspecific sites, respectively. Addition of a conomer, ethylene or 4-MP. lowered the propylene polymerization activity. S –VCI₃ is more easily reduced to the divalent ion by AIR₃ than S –TiCl₃. Methyl-p-toluate moderates the reducting power of AIR₃; it increase the productivity and stereoselectivity of the S –YiCl₃ catalyst, VCI₃ supported on MgCl₂ (CW–V catalyst) has enhanced rate constant of propylene polymerization but has the opposite effects on the S –TiCl₃ Catalyst. VCI₃ supported on MgCl₂ (CW–V catalyst) has enhances rate constant of propylene polymerization but only a minute fraction of the supported vanadiums are catalytically active: [C] = 0.019% and k_p,i = 1580 (Ms)^?1, [C] = 0.057% and k_p,i = 58 (M s)^?1. This is compared with far greater number of catalytically active titanium sites in the TiCl₃ supported on MgCl₂ catalyst: [C] = 6% and k_p,i = 200 (M s)^?1, [C] = 6% and k_p,a = 16(M s)^?1. Therefore, both the S –VCI₃ and CW–V catalysts are highly stereoselective but low in efficiency with respect to the utilization of the vanadium ion in the catalysis.     

Free‐radical copolymerization of styrene with butyl acrylate. II. Elemental kinetic copolymerization step predictions from homopolymerization data

The free‐radical copolymerization of styrene and butyl acrylate has been carried out in benzene at 50 °C. The lumped k _p/k parameter (where k _p and k _t are the average copolymerization propagation and termination rate constants, respectively) has been determined. Applying the implicit penultimate unit model for the overall copolymerization propagation rate coefficient and the terminal unit effect for the overall copolymerization termination rate coefficient and using the homopolymerization kinetic coefficients, we have found good qualitative agreement between the experimental and theoretical k _p/k values. The variation of the copolymerization rate in solution with respect to the values previously found in bulk has been ascribed to a chain length effect on the copolymerization termination rate coefficient. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 130–136, 2004     

A molecular approach to chain entanglement in linear polymers

The phenomenon of chain entanglement in undiluted linear amorphous polymers is treated by calculating the probability of forming closed intramolecular loops. Adoption of the rotational isomeric state model of polymer chains permits an appropriate accounting of the detailed molecular structure to be made through the configurational characteristics of the polymer. The second (〈r〉₀) and fourth (〈r〉₀) moments of the vector r_hk connecting groups h and k in the isolated polymer chain and averaged over all chain conformations are calculated and used to evaluate the probability W_x(0) that r_hk is 0, or that an intramolecular loop of x = k ? h bonds is formed. Several linear polymers with widely differing molecular structures are treated. An attempt is made to correlate the degree of chain entanglement they manifest in the bulk with their ability to fold back upon themselves to form closed intramolecular loops.     

Effects of CH3OH‐H2O and CH3OH solvents on rate of reaction of phthalimide with piperidine

Pseudo‐first‐order rate constants (k_obs) for the cleavage of phthalimide in the presence of piperidine (Pip) vary linearly with the total concentration of Pip ([Pip]_T) at a constant content of methanol in mixed aqueous solvents containing 2% v/v acetonitrile. Such linear variation of k_obs against [Pip]_T exists within the methanol content range 10%–∼80% v/v. The change in k_obs with the change in [Pip]_T at 98% v/v CH₃OH in mixed methanol‐acetonitrile solvent shows the relationship: k_obs = k[Pip]_T + k[Pip], where respective k and k represent apparent second‐order and third‐order rate constants for nucleophilic and general base‐catalyzed piperidinolysis of phthalimide. The values of k_obs, obtained within [Pip]_T range 0.02–0.40 M at 0.03 M NaOH and 20 as well as 50% v/v CH₃OH reveal the relationship: k_obs = k₀/(1 + {k_n[Pip]/k_OX[⁻OX]_T}), where k₀ is the pseudo‐first‐order rate constant for hydrolysis of phthalimide, k_n and k_OX represent nucleophilic second‐order rate constants for the reaction of Pip with phthalimide and for the XO⁻‐catalyzed cyclization of N‐piperidinylphthalamide to phthalimide, respectively, and [⁻OX]_T = [NaOH] + [⁻OX_re], where [⁻OX_re] = [⁻OH_re] + [CH₃O_re⁻]. The reversible reactions of Pip with H₂O and CH₃OH produce ⁻OH_re and CH₃O_re⁻ ions. The effects of mixed methanol‐water solvents on the rates of piperidinolysis of PTH reveal a nonlinear decrease in k with the increase in the content of methanol. © 2000 John Wiley & Sons, Inc. Int J Chem Kinet 33: 29–40, 2001     

Absorption line shape of two coupled oscillators decaying in two different mediums

The linear response theory is used to study the line shape of two coupled oscillators, decaying in two different baths, when one of the oscillators is coupled to the radiation field. The baths are considered to be thermally excited. The analysis is based on the computation of the correlation functions 〈a (t)a_i (0)〉 and 〈a_i (t)a(0)〉 of each oscillator, taking the coefficients associated with the operators a and a_i in the Heisenbemg representation as the starting point of the calculation. The line shape in different cases is also discussed.     

Melt‐state polymer chain dimensions as a function of temperature

The unperturbed chain dimensions (〈R²〉_o/M) of cis/trans‐1,4‐polyisoprene, a near‐atactic poly(methyl methacrylate), and atactic polyolefins were measured as a function of temperature in the melt state via small‐angle neutron scattering (SANS). The polyolefinic materials were derived from polydienes or polystyrene via hydrogenation or deuteration and represent structures not encountered commercially. The parent polymers were prepared via lithium‐based anionic polymerizations in cyclohexane with, in some cases, a polymer microstructure modifier present. The polyolefins retained the near‐monodisperse molecular weight distributions exhibited by the precursor materials. The melt SANS‐based chain dimension data allowed the evaluation of the temperature coefficients [dln 〈R²〉_o/dT(κ)] for these polymers. The evaluated polymers obeyed the packing length (p)‐based expressions of the plateau modulus, G = kT/np³ (MPa), and the entanglement molecular weight, M_e = ρN_anp³ (g mol^?1), where n_t denotes the number (～21) of entanglement strands in a cube with the dimensions of the reptation tube diameter (d_t) and ρ is the chain density. The product np³ is the displaced volume (V_e) of an entanglement that is also expressible as pd or kT/G. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1768–1776, 2002     

Structural analysis of aggregates formed by a thermoresponsive homopolymer in dilute aqueous solutions

The aqueous solution of a thermoresponsive polymer, poly[2‐(2‐ethoxy) ethoxyethyl vinyl ether] poly(EOEOVE), contains a tiny amount of large polymer aggregates at low polymer concentrations far below the lower critical solution temperature (～40 °C). The molar mass M_w,slow, radius of gyration 〈S²〉, and hydrodynamic radius R_H,slow of the aggregating component of poly(EOEOVE) were obtained by simultaneous static and dynamic light scattering as functions of the polymer concentration and temperature, while the weight fraction w_slow of the component was estimated by size‐exclusion chromatography. The M_w,slow dependencies of 〈S²〉 and R_H,slow, as well as the ratio 〈S²〉/R_H,slow, indicated that the poly(EOEOVE) aggregate takes a sparsely branched polymer‐like conformation. We have analyzed the structure of the aggregate, using the branched polymer model of random type. The M_w,slow dependence of 〈S²〉 obtained was favorably compared with this model with reasonable structural parameters. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1179–1187, 2006     

Free radical dispersion polymerization of styrene in a mixture of 2-propanol and tetrahydrofuran

Free radical dispersion polymerization of styrene in a mixture of 2-propanol and tetrahydrofuran was carried out at 70°C up to high conversions. The influence of the change of the critical chain length on the evolution of the insoluble polymer component was examined. Monomer conversion and the formation of the insoluble polymer component were measured in order to test a mathematical model presented in our previous article. The critical polymer chain length i₀, the initiation rate constant k_d, and the ratio k_p/k, where k_p and k_t are propagation and termination rate constants, respectively, have been obtained and compared with those reported in the literature. © 1998 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 36: 2757–2761, 1998