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.     

Collision between polymeric radicals for termination rate constants

The motion of each polymeric radical during a collision between the polymeric radicals with the same radius is treated as completely random motion. The result obtained is: k_t = 0.250k_s (where k_t is the chain-termination rate constant and k_s is the reaction rate constant between radical chain ends). On taking the motion of the primary radical during a collision between a primary radical and a large polymeric radical to be completely random, the result obtained is: k_ti = 0.250k_si (where k_ti is the primary radical termination rate constant and k_si is the reaction rate constant between primary radical and radical chain end). On substituting k_s for k_si in the second equation, the rate constant obtained becomes the chain termination rate constant between the very small polymeric radical and the very large polymeric radical, and identical to the former equation. This identity indicates that the effect of the difference of the size of the polymeric radicals on the collision process relating to the chain termination rate constant should not be large.     

Primary radical termination rate constant at high conversion in radical polymerization

A primary radical termination rate constant given by: k_ti = A_1iD_i, where A_1i is a constant and D_i is the diffusion constant of the primary radical, was examined on the basis of the variation of conversion. It was proved that this rate constant is correct at high conversion. A relationship between primary radical termination rate constant and conversion was derived. The effect of variation of conversion on the gel effect is discussed.     

三代碳硅烷光致变色液晶树状物的光化学研究——端基含108个4-丁氧基偶氮苯介晶基元

报道了新化合物含108个丁氧基偶氮基元端基的三代(D3)碳硅烷光致变色液晶树状物在各溶液中的反-顺光异构化(光致变色)反应速率常数k_p, 光化学回复异构化正/逆反应速率常数k_t和k_c, 热回复异构化反应速率常数k_H, 光化学回复异构化反应平衡常数k_t/k_c, 活化能E, 异构化转换率及热回复异构化反应中的反-顺异构体组分比. D3的光致变色反应速率常数为10^－1 s^－1, 而含偶氮基元的光致变色液晶聚硅氧烷的光致变色反应速率常数为10^－8 s^－1, 因此, D3的光响应速度比后者快10⁷倍.     

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.     

Calculations of Monomer Conversion and Radical Concentration in Reversible Addition‐Fragmentation Chain Transfer Radical Polymerization

Simple expressions are derived for the development of monomer conversion, as well as propagating radical, adduct radical, dormant chain, and dead chain concentrations in reverse addition‐fragmentation transfer polymerization (RAFT). The relations for the profiles of propagating radical concentration and conversion versus time are derived and depend on group parameters of rate constants and chemical recipe. The analytical equations are verified against numerical solutions of the mass‐balance differential equations. This derivation involves the steady‐state hypothesis for radical and RAFT agent concentrations. The errors introduced by these assumptions are negligible when the fragmentation rate constant, k_f, is higher than 10 s⁻¹ or when the cross‐termination rate constant, k_ct, is higher than 10⁵ L · mol⁻¹ s⁻¹.

Calculated concentration profiles (points: numerical, lines: analytical) of propagating radical R, adduct radical A, dormant T, and dead D (= P + P′) chains.     

Gamma radiation-initiated polymerization of styrene at high pressure. II. Chain termination

The pressure dependence of the termination rate constant k_t for the free radical polymerization of monomers such as styrene is a function of polymer chain length, chain stiffness, and monomer viscosity, all of which influence the rate of segmental diffusion of an active radical chain end out of the coiled polymer chain to a position in which it can react with a proximate radical. Although k_t is not sensitive to changes in chain length, the large increase in molecular weight is responsible for a significant reduction in k_t at high pressures. For most of the common vinyl polymers, which exhibit some degree of chain stiffness, k_t is inversely proportional to a fractional power of the monomer viscosity because it depends in part on the resistance of chain segments to movement and in part on the influence of viscosity in controlling diffusion of the chain ends. The fractional exponent appears to increase with pressure and this is interpreted as evidence that the polymer chains become more flexible in a more viscous solvent. Because the fractional exponent is higher for more flexible chains, the value of the activation volume for chain termination is an indication of the degree of flexibility of the polymer chains, provided that the monomer is a good solvent for the polymer and that chain transfer is negligible.     

Kinetics of high-intensity electron-beam polymerization of a divinyl urethane

The kinetics of high-intensity electron beam-induced polymerization of di(2′-methacryloxyethyl)-4-m-phenylenediurethane during the network formation has been studied up to complete gelation and up to 56% conversion of unsaturation. From experimentally determined gel fractions, rate of disappearance of unsaturation, kinetic chain length, and intensity dependence, it is proposed that the polymerization takes place in a swollen network where the growing chains undergo unimolecular termination, and where gel-gel reaction is prohibited. The rate expression derived is: In [α(1 ? g)^0.545] = In α0 ? 2.51 k_ik_pt/k_t where α is the total unsaturation and g is the gel fraction. The value of k_p/k_t is found to be 2.1 and that of G_R, the free radical yield per 100 eV absorbed, to be 16; these high values are ascribed to the high viscosity of the polymerizing system.     

Modeling the Effects of Primary Radicals in Free‐Radical Polymerization

The effects of non‐ideal initiator decomposition, i.e., decomposition into two primary radicals of different reactivity toward the monomer, and of primary radical termination, on the kinetics of steady‐state free‐radical polymerization are considered. Analytical expressions for the exponent n in the power‐law dependence of polymerization rate on initiation rate are derived for these two situations. Theory predicts that n should be below the classical value of 1/2. In the case of non‐ideal initiator decomposition, n decreases with the size of the dimensionless parameter α ≡ (k_tz /k_dz) √r_ink_t, where k_tz is the termination rate coefficient for the reaction of a non‐propagating primary radical with a macroradical, k_dz is the first‐order decomposition rate coefficient of non‐propagating (passive) radicals, r_in is initiation rate, and k_t is the termination rate coefficient of two active radicals. In the case of primary radical termination, n decreases with the size of the dimensionless parameter β ≡ k_t,s r_in^1/2/k_p,s M r_t,l^1/2, where k_t,s is the termination rate coefficients for the reaction of a primary (“short”) radical with a macroradical, k_t,l is the termination rate coefficients of two large radicals, k_p,s is the propagation rate coefficient of primary radicals and M is monomer concentration. As k_t is deduced from coupled parameters such as k_t /k_p, the dependence of k_p on chain length is also briefly discussed. This dependence is particularly pronounced at small chain lengths. Moreover, effects of chain transfer to monomer on n are discussed.     

Modeling Termination Kinetics of Non‐Stationary Free‐Radical Polymerizations

Pulsed‐laser induced polymerization is modeled via an approach presented in a previous paper.^[1] An equation for the time dependence of free‐radical concentration is derived. It is shown that the termination rate coefficient may vary significantly as a function of time after applying the laser pulse despite of the fact that the change in monomer concentration during one experiment is negligible. For the limiting case of t ≫ c^–1 (k_pM)^–1, where c is a dimensionless chain‐transfer constant, k_p the propagation rate coefficient and M the monomer concentration, an analytical expression for k_t is derived. It is also shown that time‐resolved single pulse‐laser polymerization (SP–PLP) experiments can yield the parameters that allow the modeling of k_t in quasi‐stationary polymerization. The influence of inhibitors is also considered. The conditions are analyzed under which M (t) curves recorded at different extents of laser‐induced photo‐initiator decomposition intersect. It is shown that such type of behavior is associated with a chain‐length dependence of k_t.     

Modeling termination kinetics in free‐radical polymerization using a reduced number of parameters

An approach for modeling chain‐length dependent termination rate coefficients is presented. The method is based on the assumption that free‐radical chain length may be considered as a continuous variable. As compared to discrete numerical methods, in continuous modeling the number of independent dimensionless parameters can be significantly reduced. As a consequence, for a wide variety of monomers the conversion dependence of k_t can be predicted without extensive numerical calculations. The method may also be used to determine polymerization conditions under which simpler models of k_t (which neglect effects arising from the dependence of k_t on chain length) may be applied. Calculations for methyl methacrylate, styrene, and butyl acrylate bulk polymerizations up to high degrees of monomer conversion show that the impact of chain length on termination varies with conversion and strongly depends on the type of monomer.     

THE OVERALL KINETICS OF FREE RADICAL (CO)POLYMERIZATION WITH HIGHLY DIFFUSION CONTROLLED TERMINATION

The overall reaction rate kinetics of polymerization of diethyleneglycol dimethacrylate and copolymerization of it with styrene in bulk and in the presence of inert diluents were investigated. Theresults indicated that these reactions can be treated as free radical polymerization with highly diffu-sion controlled termination reaction in which the termination rate constant is an empirically derivedfunction of monomer conversion: K_t=K_(to)(1-c ln[M]/ [M_0])~(-1) in which K_(to) is the initial terminationrate constant and c is a factor related to the magnitude of diffusion co?re The following equationof monomer conversion as a function of time could then be derived: U=1-exp {1/c [1-(1+ckt/2)~2]}in which k=K_P(R_i/2K_(to))~(1/2) and t is the time of reaction. Excellent agreement between the theoreticaland experimental overall reaction kinetic curves was obtained. The equation is valid for crosslinkingand noncrosslinking free radical polymerizations in which the self-acceleration effcct is effective fromthe very beginning of the reaction. The equation can be expressed in a more generally applicableform: U=1--exp{1/e[1--(1+?t/n)~n] in which n≥0.     

Termination rate coefficients derived from PLP-SEC data

A novel procedure is outlined by which the termination rate coefficient, k_t, may be deduced from molecular weight and monomer conversion data of pulsed laser polymerization (PLP) – size exclusion chromatography (SEC) experiments. For this k_t analysis only the central part of the molecular weight distribution (MWD) between the first point of inflection (POI), that is also used for k_p analysis, and the third such POI is taken into account. Within this region a characteristic ratio of areas under the MWD is fitted either by using PREDICI or by applying a lumping scheme method. The success of the lumping scheme procedure is demonstrated for the bulk polymerization of butyl methacrylate. The k_t values derived by this method refer to small initial degrees of monomer conversion as are typical of PLP-SEC investigations. The relatively fast and efficient lumping scheme technique is restricted to situations where k_t may be considered independent of chain length and where chain transfer processes are not important.     

Relationship between the autoacceleration of polymerization rate and conversion in radical polymerization

By using a simple treatment for the kinetics of radical polymerization with primary radical termination, the ratio k_ty/k_tx of chain termination rate constant k_ty at conversion y to that k_tx at conversion x and the ratio k_tiy/k_tix of the primary radical termination rate constant k_tiy at conversion y to k_tix at conversion x were calculated for the polymerizations of methyl methacrylate and ethyl acrylate in the conversion range 0 to 0.4. k_ty/k_tx and k_tiy/k_tix were treated by using the following equations based on the variation of conversion: where g(T,y) is the average fractional free volume of radical chain end at conversion y and absolute temperature and β(T) is a function depending on T, and where g_i(T,y) is the average fractional free volume of primary radical at conversion y and T and β_i(T) is a function depending on T. The autoacceleration for the above monomers was successfully interpreted by the above treatment.     

Polymerization of neat 2‐ethylhexyl acrylate induced by a pulsed electron beam

The bulk polymerization of 2‐ethylhexyl acrylate (2‐EHA), induced by a pulsed electron beam, was investigated with pulse radiolysis, gravimetry, and Fourier transform infrared spectroscopy. The roles of the dose rate, pulse frequency, and added acrylic acid (AA) in the polymerization of 2‐EHA were examined at ambient temperature. In the range of 12.6–71.2 Gy/pulse, the polymerization of 2‐EHA was dose‐rate‐dependent: at the same total dose, a lower dose rate yielded a higher conversion. Also, a lower pulse rate gave a higher conversion at the same total dose. The addition of up to 10 wt % AA showed no increase in the conversion of 2‐EHA at a low conversion (8 kGy), but at a higher conversion (16 kGy), a 20 wt % increase in the conversion of 2‐EHA was observed. The estimated values (1.6 ± 0.3) × 10^?3 (dm³ s)^3/2 mol^?1 s^?1/2 for k_p(G/2k_t)^1/2 and 2.6 ± 0.8 dm³ s J^?1 for 2k_tG (where k_p is the rate constant of propagation, k_t is the rate constant of bimolecular termination, and G is the yield of free radicals) were obtained at relatively low conversions. The reaction rate constant of the addition of 2‐EHA^· free radicals to the monomer was measured by pulse radiolysis and found to be 2.8 × 10² mol^?1 dm³ s^?1. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 196–203, 2003     

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.     

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.     

The Influence of Chain Length‐Dependent Propagation on the Evaluation of the Chain Length Dependence of the Rate Coefficient of Bimolecular Termination, 1. PLP–SEC Methods

Chain length distributions have been calculated for polymers prepared by pulsed laser polymerization (PLP) under the condition that not only chain termination but also chain propagation is subject to chain length dependence. The interplay between these two features is analyzed with the chain length dependence of the rate coefficient of termination k_t introduced in the form of a power law and that of propagation k_p modeled by a Langmuir‐type decrease from an initial value for zero chain length to a constant value for infinite chain lengths. The rather complex situation is governed by two important factors: the first is the extent of the decay of radical concentration [R] during one period under pseudostationary conditions, while the second is that termination events are governed by [R]² while the propagation goes directly with [R]. As a consequence there is no general recommendation possible as to which experimental value of k_p is best taken as a substitute for the correct average of k_p characterizing a specific experiment. The second point, however, is apparently responsible for the pleasant effect that the methods used so far for the determination of k_t and its chain length dependence (i.e., plotting some average of k_t versus the mean chain‐length of terminating radicals on a double‐logarithmic scale) are only subtly wrong with regard to a realistic chain length dependence. This is especially so for the quantity k_t* (the average rate coefficient of termination derived from the rate of polymerization in a PLP system) and its chain length dependence.     

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.     

Initiator efficiency of 2,2′‐azobis(isobutyronitrile) in bulk dodecyl acrylate free‐radical polymerizations over a wide conversion and molecular weight range

An Erratum has been published for this article in J. Polym. Sci. Part A: Polym. Chem. (2004) 42(21) 5559 . The initiator efficiency, f, of 2,2′‐azobis(isobutyronitrile) (AIBN) in dodecyl acrylate (DA) bulk free‐radical polymerizations has been determined over a wide range of monomer conversion in high‐molecular‐weight regimes (M_n ? 10⁶ g mol^?1 [? 4160 units of DA)] with time‐dependent conversion data obtained via online Fourier transform near infrared spectroscopy (FTNIR) at 60 °C. In addition, the required initiator decomposition rate coefficient, k_d, was determined via online UV spectrometry and was found to be 8.4 · 10^?6 s^?1 (±0.5 · 10^?6 s^?1) in dodecane, n‐butyl acetate, and n‐dodecyl acetate at 60 °C. The initiator efficiency at low monomer conversions is relatively low (f = 0.13) and decreases with increasing monomer to polymer conversions. The evolution of f with monomer conversion (in high‐molecular‐weight regimes), x, at 60 °C can be summarized by the following functionality: f^{60 °C} (x) = 0.13–0.22 · x + 0.25 · x² (for x ≤ 0.45). The reported efficiency data are believed to have an error of >50%. The ratio of the initiator efficiency and the average termination rate coefficient, 〈k_t±, (f/〈k_t〉) has been determined at various molecular weights for the generated polydodecyl acrylate (M_n = 1900 g mol^?1 (? 8 units of DA) up to M_n = 36,500 g mol^?1 (? 152 units of DA). The (f/〈k_t〉) data may be indicative of a chain length‐dependent termination rate coefficient decreasing with (average) chain length. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 5170–5179, 2004