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.     

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.     

Free‐Radical Termination Kinetics Studied Using a Novel SP‐PLP‐ESR Technique

Summary: A novel method for measuring termination rate coefficients, k_t, in free‐radical polymerization is presented. A single laser pulse is used to instantaneously produce photoinitiator‐derived radicals. During subsequent polymerization, radical concentration is monitored by time‐resolved electron spin resonance (ESR) spectroscopy. The size of the free radicals, which exhibits a narrow distribution increases linearly with time t, which allows the chain‐length dependence of k_t to be deduced. The method will be illustrated using dodecyl methacrylate polymerization as an example.

Two straight lines provide a very satisfactory representation of the chain‐length dependence of k_t over the entire chain‐length region (c_R = radical concentration).     

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.     

Free‐Radical Propagation Rate Coefficients via n‐Pulse Periodic Polymerization

Aspects of applying n‐pulse periodic initiation in pulsed laser polymerization/size‐exclusion chromatography (PLP/SEC) experiments are studied via simulation of molecular weight distributions (MWDs). In n‐pulse periodic PLP/SEC, sequences of n laser pulses at successive time intervals Δt₁ up to Δt_n are periodically applied. With the dark time intervals being suitably chosen, n‐modal MWDs with n well separated peaks occur. The n‐pulse periodic PLP/SEC method has the potential for providing accurate propagation rate coefficients, k_p. Among several measures for k_p, the differences in molecular weights at the MWD peak positions yield the best estimate of k_p under conditions of medium and high pulse laser‐induced free‐radical concentration. Deducing k_p from n dark time intervals (corresponding to n regions of free‐radical chain length) within one experiment at otherwise identical PLP/SEC conditions allows addressing in more detail a potential chain‐length dependence of k_p. Simulations are compared with experimental data for 2‐pulse periodic polymerization of methyl methacrylate.

Measured MWD (solid line) and associated first derivative curve (dotted line) for a 2‐pulse periodic bulk polymerization experiment of MMA at 20 °C.     

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 Importance of Chain-Length Dependent Kinetics in Free-Radical Polymerization: A Preliminary Guide

The effect of chain-length dependent propagation at short chain lengths on the observed kinetics in low-conversion free-radical polymerization (frp) is investigated. It is shown that although the values of individual propagation rate coefficients quickly converge to the high chain length value (at chain lengths, i, of about 10), its effect on the average propagation rate coefficients, 〈k_p〉, in conventional frp may be noticeable in systems with an average degree of polymerization (DP_n) of up to 100. Furthermore it is shown that, unless the system is significantly retarded, the chain-length dependence of the average termination rate coefficient, 〈k_t〉, is not affected by the presence of chain-length dependent propagation and that there exists a simple (fairly general) scaling law between 〈k_t〉 and DP_n. This latter scaling law is a good reflection of the dependence of the termination rate coefficient between two i-meric radicals, k, on i. Although simple expressions seem to exist to describe the dependence of 〈k_p〉 on DP_n, the limited data available to date does not allow the generalization of these expressions.     

A Method of Improving the Theoretical Basis of kp Determination from PLP‐SEC Measurements

Making use of hitherto ignored features (such as the peak width) contained in the chain‐length distributions of polymers prepared by pulsed‐laser polymerization (PLP), corrections are calculated from simulated chain‐length distributions for improving the accuracy of the “characteristic chain length” L₀ data on which the evaluation of the propagation rate constant k_p is based. These corrections refer to a wide range of chain lengths and primary radical production, slightly chain‐length‐dependent termination by disproportionation or combination, and a reasonable extent of axial dispersion introduced by the chromatographic device used in the evaluation of the chain‐length distribution. They can be applied to the point of inflection on the low‐molecular‐weight side of the extra peaks as well as to the peak maximum. The remaining mean error which, of course, concerns the evaluation of L₀ only, is shown to be of the order of 1.0–1.5%, if the mode of termination is unknown, and comes down to about half that value if information on the mode of termination is available. Although all the other errors inherent in the size exclusion chromatography (SEC) method are still present, this method constitutes substantial progress with respect to the accuracy of determining k_p data from PLP experiments followed by chromatographic analysis.

Hyper mass distributions calculated for L₀ = 200, C = 5 and b = 0.16 for termination by disproportionation considering Poissonian and Gaussian broadening.     

Laser Single Pulse Initiated RAFT Polymerization for Assessing Chain‐Length Dependent Radical Termination Kinetics

Summary: A novel method combining RAFT polymerization with pulsed‐laser initiation for determining chain‐length dependent termination rate coefficients, k_t, is presented. Degenerative chain‐transfer in RAFT enables single‐pulse pulsed‐laser polymerization (SP‐PLP) traces to be measured on systems with a narrow radical distribution that remains essentially unchanged during the experiment. SP‐PLP‐RAFT experiments at different polymerization times allow for determining k_t as a function of chain length via classical kinetics assuming chain‐length independent k_t.

Single‐pulse pulsed‐laser polymerization trace for BMPT‐mediated RAFT polymerization of butyl acrylate.     

Termination in Dilute‐Solution Free‐Radical Polymerization: A Composite Model

Literature data are summarized for the chain‐length‐dependence of the termination rate coefficient in dilute solution free‐radical polymerizations. In essence such experiments have yielded two parameter values: the rate coefficient for termination between monomeric free radicals, kequation/tex2gif-stack-1.gif, and a power‐law exponent e quantifying how k_t values decrease with increasing chain length. All indications are that the value e ≈ 0.16 in good solvent is accurate, however the values of kequation/tex2gif-stack-2.gif which have been deduced are considerably lower than well‐established values for small molecule radicals. This seeming impasse is resolved by putting forward a ‘composite’ model of termination: it is proposed that the value e ≈ 0.16 holds only for long chains, with e being higher for small chains – the value 0.5 is used in this paper, although it is not held to dogmatically. It is then investigated whether this model is consistent with experimental data. This is a non‐trivial task, because although the experiments themselves and the ways in which they are analyzed are elegant and not too complicated, the underlying theory is sophisticated, as is outlined. Simulations of steady‐state polymerization experiments are first of all carried out, and it is shown that the composite model of termination both recovers the e values which have been found and beautifully explains why these experiments considerably underestimate the true value of kequation/tex2gif-stack-3.gif. Simulations of pulsed‐laser polymerizations find the same, although not quite so strikingly. It is therefore concluded that our new termination model, which retains the virtue of simplicity and in which all parameter values are physically reasonable, is consistent with experimental data. Taking a wider view, it seems likely that the situation of the exponent e varying with chain length will not just be the case in dilute solution, but will be the norm for all conditions, which would give our model and our work a general relevance.

Normalized chain length distributions from PLP simulations.     

On the (im)possibility of evaluating correct individual rate constants of chain propagation kp and chain termination kt by combining kp2/kt and kp/kt data for chain‐length dependent termination

On the basis of simulated data two ways of evaluating individual rate constants by combining k_p²/k_t and k_p /k_t (k_p , k_t = rate constants of chain propagation and termination, respectively) were checked considering the chain‐length dependence of k_t. The first way tried to make use of the fact that pseudostationary polymerization yields data for k_p²/k_t as well as for k_p /k_t referring to the very same experiment, in the second way k_p²/k_t (from steady state experiments) and k_p/k_t data referring to the same mean length of the terminating radical chains were compared. In the first case no meaningful data at all could be obtained because different averages of k_t are operative in the expressions for k_p /k_t and k_p²/k_t. In spite of the comparatively small difference between these two averages (≈15% only) this makes the method collapse. The second way, which can be regarded as an intelligent modification of the “classical” method of determining individual rate constants, at least succeeded in reproducing the correct order of magnitude of the individual rate constants. However, although stationary and pseudostationary experiments independently could be shown to return the same k_t for the same average chain‐length of terminating radicals within extremely narrow limits no reasonable chain‐length dependence of k_t could be derived in this way. The reason is an extreme sensitivity of the pair of equations for k_p/k_t and k_p²/k_t towards small errors and inconsistencies which renders the method unsuccessful even for the high quality simulation data and most probably makes it even collapse for real data. This casts a characteristic light on the unsatisfactory situation with respect to individual rate constants determined in the classical way, regardless of a chain‐length dependence of termination. As a consequence, all efforts of establishing the chain‐length dependence of k_t are recommended to avoid this way and should rather resort to methods based on inserting a directly determined k_p into the equations characteristic of k_p²/k_t or k_p/k_t, properly considering the chain‐length dependent character of k_t.     

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.     

Depolymerization kinetics of di(4‐tert‐butyl cyclohexyl) itaconate and Mark‐Houwink‐Kuhn‐Sakurada parameters of di(4‐tert‐butyl cyclohexyl) itaconate and di‐n‐butyl itaconate

Multipulse pulsed laser polymerization coupled with size exclusion chromatography (MP‐PLP‐SEC) has been employed to study the depropagation kinetics of the sterically demanding 1,1‐disubstituted monomer di(4‐tert‐butylcyclohexyl) itaconate (DBCHI). The effective rate coefficient of propagation, k, was determined for a solution of monomer in anisole at concentrations, c, 0.72 and 0.88 mol L^?1 in the temperature range 0 ≤ T ≤ 70 °C. The resulting Arrhenius plot (i.e., ln k vs. 1/RT) displayed a subtle curvature in the higher temperature regime and was analyzed in the linear part to yield the activation parameters of the forward reaction. In the temperature region where no depropagation was observed (0 ≤ T ≤ 50 °C), the following Arrhenius parameters for k_p were obtained (DBCHI, E_p = 35.5 ± 1.2 kJ mol^?1, ln A_p = 14.8 ± 0.5 L mol^?1 s^?1). In addition, the k data was analyzed in the depropagatation regime for DBCHI, resulting in estimates for the associated entropy (?ΔS = 150 J mol^?1 K^?1) of polymerization. With decreasing monomer concentration and increasing temperature, it is increasingly more difficult to obtain well structured molecular weight distributions. The Mark Houwink Kuhn Sakurada (MHKS) parameters for di‐n‐butyl itaconate (DBI) and DBCHI were determined using a triple detection GPC system incorporating online viscometry and multi‐angle laser light scattering in THF at 40 °C. The MHKS for poly‐DBI and poly‐DBCHI in the molecular weight range 35–256 kDa and 36.5–250 kDa, respectively, were determined to be K_DBI = 24.9 (10³ mL g^?1), α_DBI = 0.58, K_DBCHI = 12.8 (10³ mL g^?1), and α_DBCHI = 0.63. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 1931–1943, 2007     

Chain‐length dependent termination in rotating sector polymerization, 1. The evaluation of the rate constant of chain propagation kp

The chain‐length distributions (CLDs) of polymers prepared by rotating‐sector (RS) techniques under pseudostationary conditions were simulated for the case of chain‐length dependent termination and analysed for their suitability of determining the rate constant of chain propagation k_p from the positions of their points of inflection. The tendency to underestimate k_p is a little more pronounced than in pulsed‐laser polymerization (PLP) but, interestingly, the situation improves in the presence of chain‐length dependent termination. The estimates also were found to be more precise a) for smaller rates of initiation, b) for higher order points of inflection, c) if termination is by combination, d) if the role played by the shorter one of the two chains becomes less dominant. Taken in all, the determination of k_p from the points of inflection in the CLD of RS‐prepared polymers may well compete with the more famous PLP method, especially if some care is taken with respect to the choice of experimental conditions.     

Pulsed‐laser polymerization‐gel permeation chromatographic determination of the propagation‐rate coefficient for the methyl acrylate dimer: A sterically hindered monomer

The methyl acrylate dimer (MAD) is a sterically hindered macromonomer, and the propagating radical can fragment to an unsaturated end group. The propagation‐rate coefficient (k_p) for MAD was obtained by pulsed‐laser polymerization (PLP). The Mark–Houwink–Sakaruda parameters required for the analysis of the molecular weight distributions (MWDs) were obtained by multiple‐detector gel permeation chromatography (GPC) with on‐line viscometry. The small radical created by the fragmentation results in a short‐chain polymer that means the MWD may no longer be given by that expected for “ideal” PLP conditions; simulations suggest that the degree of polymerization required for “ideal” PLP conditions can be obtained from the primary point of inflection provided the GPC traces also show a clear secondary inflection point (radicals terminated by the second, rather than the first, pulse subsequent to initiation). Over the temperature range of 40–75 °C, the data can be best fitted by k_p/dm³ mol^?1 s^?1 = 10^6.1 exp(?29.5 kJ mol^?1), with a moderately large joint confidence interval for the Arrhenius parameters. The data are consistent with an increased activation energy and reduced frequency factor as compared with acrylate or methacrylate; both of these changes can be ascribed to hindrance. © 2001 John Wiley & Sons, Inc. J Polym Sci Part A: Polym Chem 39: 3902–3915, 2001     

One‐range Addition Theorems for Complete Sets of Modified Exponential Type Orbitals and Noninteger n Slater Functions in Standard Convention

Using the L ‐generalized Laguerre polynomials L ‐GLPs) and φ ‐generalized exponential type orbitals φ ‐GETOs) introduced by the author in standard convention, the one‐ and two‐center onerange addition theorems are established for the complete sets of Ψ^(α*) ‐modified exponential type orbitals (Ψ^(α*) ‐METOs) and noninteger n χ‐Slater type orbitals (χ‐NISTOs), where p_l* = 2l + 2 ‐ α* and α* is the integer (α* = α, ?∞ < α ≤2) or noninteger (α* ≠ α, ?∞ < α* < 3) self‐frictional quantum number. It should be noted that the origin of the L ‐GLPs, φ ‐GETOs and Ψ^(α*) ‐METOs, therefore, of the one‐range addition theorems presented in this work is the Lorentz damping or self‐frictional field produced by the particle itself.     

Easy Access to Chain‐Length‐Dependent Termination Rate Coefficients Using RAFT Polymerization

Chain‐length‐dependent termination rate coefficients of the bulk free‐radical polymerization of styrene at 80 °C are determined by combining online polymerization rate measurements (DSC) with living RAFT polymerizations. Full k_t versus chain‐length plots were obtained indicating a high k_t value for short chains (2 × 10⁹ L · mol⁻¹ · s⁻¹) and a weak chain‐length dependence between 10 and 100 monomer units, quantified by an exponent of −0.14 in the corresponding power law 〈k_t^i,i〉 = k_t⁰ · P^−b.

Double logarithmic plots of 〈k_t^i,i〉 versus P, evaluated from experimental time‐resolved R_p data according to the procedure described in the text, for different CPDA and AIBN concentrations. The best linear fit for (10 < P < 100) is indicated as full line.     

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.

     

New RuII(arene) Complexes with Halogen‐Substituted Bis‐ and Tris(pyrazol‐1‐yl)borate Ligands

[RuCl(arene)(μ‐Cl)]₂ dimers were treated in a 1:2 molar ratio with sodium or thallium salts of bis‐ and tris(pyrazolyl)borate ligands [Na(Bp)], [Tl(Tp)], and [Tl(Tp^{iPr, 4Br})]. Mononuclear neutral complexes [RuCl(arene)(κ²‐Bp)] ( 1 : arene=p‐cymene (cym); 2 : arene=hexamethylbenzene (hmb); 3 : arene=benzene (bz)), [RuCl(arene)(κ²‐Tp)] ( 4 : arene=cym; 6 : arene=bz), and [RuCl(arene)(κ²‐Tp^{iPr, 4Br})] ( 7 : arene=cym, 8 : arene=hmb, 9 : arene=bz) have been always obtained with the exception of the ionic [Ru₂(hmb)₂(μ‐Cl)₃][Tp] ( 5′ ), which formed independently of the ratio of reactants and reaction conditions employed. The ionic [Ru(CH₃OH)(cym)(κ²‐Bp)][X] ( 10 : X=PF₆, 12 : X=O₃SCF₃) and the neutral [Ru(O₂CCF₃)(cym)(κ²‐Bp)] ( 11 ) have been obtained by a metathesis reaction with corresponding silver salts. All complexes 1 – 12 have been characterized by analytical and spectroscopic data (IR, ESI‐MS, ¹H and ¹³C NMR spectroscopy). The structures of the thallium and calcium derivatives of ligand Tp, [Tl(Tp)] and [Ca(dmso)₆][Tp]₂ ? 2 DMSO, of the complexes 1 , 4 , 5′ , 6 , 11 , and of the decomposition product [RuCl(cym)(Hpz^{iPr, 4Br})₂][Cl] ( 7′ ) have been confirmed by using single‐crystal X‐ray diffraction. Electrochemical studies showed that 1 – 9 and 11 undergo a single‐electron Ru^II→Ru^III oxidation at a potential, measured by cyclic voltammetry, which allows comparison of the electron‐donor characters of the bis‐ and tris(pyrazol‐1‐yl)borate and arene ligands, and to estimate, for the first time, the values of the Lever E_L ligand parameter for Bp, Tp, and Tp^{iPr, 4Br}. Theoretical calculations at the DFT level indicated that both oxidation and reduction of the Ru complexes under study are mostly metal‐centered with some involvement of the chloride ligand in the former case, and also demonstrated that the experimental isolation of the μ³‐binuclear complex 5′ (instead of the mononuclear 5 ) is accounted for by the low thermodynamic stability of the latter species due to steric reasons.