Molecular Weight Distribution Generated on Initiation of Free RadicalPolymerization by Sequence of Laser Pulses

An investigation of molecular weight distributions (MWDs) resulting from initiation of free radical polymerization by an arbitrary sequence of short laser pulses has been undertaken. The analytical expressions have been derived to calculate MWDs for a polymerization scheme that contains reactions of chain initiation, propagation and termination by recombination or disproportionation. The MWDs produced by pulse-periodic irradiation have been calculated for a wide range of initiating pulses repetition rates (f = 0.05–100 Hz). The MWD dynamics in the act of polymerization and the influence of the duration of polymerization pseudostationary regime establishment upon the MWD have also been studied. The suitability of the derived expressions for describing the MWD generated by CW radiation before and after the establishment of polymerization quasi-stationary regime has been considered.


INTRODUCTION
Polymerization of vinyl monomers and oligomers photoinduced by pulsed laser radiation is an intensively studied process. [1][2][3][4][5][6][7][8][9][10][11] The interest in this way of polymerization initiation is due both to the possibility of exerting easily control over energy input to the polymerization medium and, consequently, to some extent over the polymerization process in itself, and to a principal opportunity of extracting some information on this process kinetic parameters from the experiments on pulsed polymerization. The set of reaction constants that describe the polymerization process eventually governs molecular weight distribution (MWD). [12][13][14] The estimated MWD enables to qualitatively describe the significant characteristics of polymer systems, i.e. static moments, polydispersity factor, etc. 13 '14 Alternatively, the MWD features under pulsed irradiation allow in a number of cases to immediately evaluate the rate constants of polymerization (kp) and a polymer chain termination (kt) in the same experiment 4-6'15-18 which is not feasible under polymerization by CW radiation. Though, the calculation of MWD and, especially, the analysis of MWD dynamics during polymerization involve severe difficulties and require numerical techniques. 84 A.V. EVSEEV AND A.N. NIKITIN At the present time, several papers 2'15'19'2 are availalle that consider the features of MWDs resulting from medium irradiation by pulsed laser. A polymerization scheme was studied that comprised the reactions of polymer chains initiation, propagation and termination by recombination of growing polymer chains or their disproportionation. The MWDs under pseudostationary polymerization 15 with small degree of monomer-to-polymer conversion were calculated on the assumption that the Flory principle aa is satisfied and growing radicals are instantaneously generated. It was shown that under pseudostationary regime of polymerization the equidistant peaks exist in the MWD spectrum, the heights of them falling rapidly with chain length increase. For the same polymerization scheme the MWD was obtained under single-pulse irradiation by method of generating functions, and the MWD dynamics in the polymerization process was studied. 9 The same problem, but with allowance for inhibition action of radicals resulting from photoinduced decomposition of non-symmetrical initiator by laser pulse, was solved too. 2 The present paper studies theoretically the MWDs resulting from polymerization initiation by an arbitrary sequence of short laser pulses.

SCHEME AND KINETIC MODEL OF POLYMERIZATION
Consideration was given to the pulsed-laser-initiated free radical polymerization of monomers (M) having a single reactive double bond. Polymerization of such compounds produces linear macromolecules differing in the number of elementary links (i). It was supposed that with radiation absorption the initiator (I) instantaneously dissociates into two radicals similar in reactivity: hv Though, as a probability of M involvement in the chain propagation reaction (3) is slight at the small degree of monomer-to-polymer conversion, M were regarded as dead polymer. It was supposed that the Flory principle la was satisfied, whereby the rate constants of propagation kp and those of chain termination kt b and kt are not functions of chain length i. An option to simplification of the model is equality of rate constants of initiation (2) and chain propagation (3)   Let the initiating radicals with W2 concentration be generated in the medium at T interval after the first pulse. Shift now the time origin to the moment of these radicals appearance. Then, for the initial concentration of radicals A 2) we can write: A 2= W260 + W1(1 + VI T )-(UT) exp(-UT) i=0 was used as a generating one. With Z 1 eq. (12) gives a summary concentration of radicals in the polymerization medium after the action of two laser pulses. As the kinetic equations (6)-(8) are valid for A 2) as well, then, differentiating (12), we obtain dFT) ZUF 7) UF 7) k F 7) F? (13) dt When integrating (13) at Z 1, we find that --1 + k F(2)(0)t 1 + V2' and, allowing for (11) and (12), obtain The general solution of (13) has the form where FT)(0) =W 2 +W(1 + VT) -1 E zj(UT')exp(-UT) j=o J! (18) Relying on the expansion exp(-UZt) (19) and regarding (18) F can be represented as Distribution of dead macromolecules concentrations is found by integrating (8). With allowance for (14) and (21), we obtain: Performing the same mathematics, as used in calculating A] 2) and B] 2) distributions after the N-pulse that produces initiating radicals with W N concentration in the 88 A.V. EVSEEV AND A.N. NIKITIN medium, we derive: Thus, on medium irradiation, with an arbitrary sequence of N pulses at any moment after the N pulse, distribution of growing radicals comprises a sum of N particle groups having Poisson distribution. Coefficients S present the remaining by the moment of k-pulse concentration of radicals produced by (k-j + 1) pulse. Q is the time interval between the (k-+ 1) and k pulses. There exists a simple relation between V k parameters and summary concentration of radicals F k) (0) immediately after the k pulse: The degree of monomer-to-polymer conversion can be determined by integrating With small degree of monomer-to-polymer conversion after the N-pulse, concentra- This expression was used to verify whether the degree of monomer-to-polymer conversion is small.

CALCULATION RESULTS AND DISCUSSION
The derived above expressions for AI N) and BI N) were used in the study of distributions produced by pulse-periodic irradiation and by irradiation with a periodic sequence of pulses packets. Before the calculation results are cited, we shall consider the question of the transition period duration, i.e. the time of polymerization pseudostationary regime establishment. This regime features a periodic variation in the summary concentration of radicals (R(t)) with frequency equal to pulse repetition rate. 15 By this, the values of maximum and minimum radicals concentration does not vary with time.
For lesser values of k satisfying the ktWT < 1 condition, this regime is attained only following a number of pulses N ps. If k 10 6 1/(mol.s), the amplitude of radicals concentration vibrations accounts for W and is much lower than average value of R. The reduction of ktW T causes the decrease in vibration amplitude, and polymerization regime approaches the quasi-stationary state. With quasi-stationary polymerization, the initiation velocity vi, is equal to that of chain termination vt. So, for where a is the e-dependent coefficient. For example, at e =0.1 a 1.5, at e 0.01 a, 2.7 and at e 0.001 a 3.8. The A spectra present a aggregate of radical groups, each having Poisson distribution. A radical group with the shortest chain lengths (the first one in the spectrum) corresponds to the last pulse of N-sequence. With time, the number of radicals in the group reduces, the group maximum shifts to large i-values (~t) and the group width increases (~t/2). So, at large N (for N 25 and 50 in Figure 3a) only groups produced by the last pulses show up in A spectra. The radical groups generated by earlier pulses shift to the region of large and broaden, thus forming the continuous spectrum. Calculated for kt, I-M] 300s -t, k 107 1/(mols) and W 10 -s mol/1 and T 0.2s.
The reduction of radicals number in the group is due both to recombination of them with each other and, more substantially, to their interaction with the radicals of other groups, the summary concentration of which increases within the period of pseudostationary state attainment. As follows from (35) and (38), the effective lifetime of radicals approximates N T and the pattern of A distribution is governed mainly by interaction of radical groups produced by the last N pulses. The radicals spectrum in the pseudostationary state is a periodic function of (A (t) A(t + T)).
The distribution of dead macromolecules along the chain length varies significantly as well at the period of pseudostationary state attainment and slightly changes after this regime has been established (Figure 3b). The abrupt termination of Bi spectrum at N 1 depends upon the fact that it has been calculated for t 0.1 s, and by this moment only macromolecules with chain length i< 2 Ut have managed to RADICAL POLYMERIZATION 93 appear. At the period of pseudostationary regime establishment, the general maximum of Bi distribution is shifted to large i-values with N growth, and the width of distribution increases. The abrupt increase in dead macromolecules concentrations with i kp[M-ITn (n <N-1) is a consequence of recombination of a radical group produced by the last pulse with those generated by earlier pulses. As by the moment of the N-pulse onset the concentration of radicals produced by the (N-1) pulse is the largest, then the contribution of these radical groups recombination to a dead polymer formation is the greatest. In the pseudostationary state, N growth is responsible for the linear increase in the summary number of dead macromolecules (by W/2 after each pulse). In this case, location of B maximum and the distribution pattern do not practically change (Bi(N)(t)/N Bi iN+ 1)(t + T)/(N + 1)).
At the period of pseudostationary state attainment, the major contribution to the summary distribution A d-B is made by growing radicals (N 1,5,10). By the moment of the regime establishment, the concentration of dead macromolecules turns essential (N 25); with N further increase this contribution will grow and the portion of growing radicals will reduce. Let us consider now in greater detail the effect of parameters of initiating pulses (T and W) and polymerizing medium (kp and kt) upon MWD. Thus, the analysis of distributions produced by pulse-periodical irradiation with high pulse repetition rate is evidence that the formulae to calculate the distributions under pulsed irradiation, reported in this work, can be used to study the distributions produced by continuous irradiation; for example, in calculating of MWD in the case that the condition of quasi-stationarity is not met. The relation UT kp[M]T 10 approximately determines the boundary separating the region where pulse-periodical irradiation has practically no effect upon the MWD pattern (UT < 10) from that with the strong effect of this kind (UT > 10). This relation was found from the condition of absence of local extremums in B distribution.

Influence of Irradiation Period upon MWD Pattern
In the case of large T values favoring the fulfillment of the ktWT >> 1 condition, the spectru B features a comparitively small number of peaks, the amplitudes of which decrease with growth ( Figure 4 for T 10s). Here, the pseudostationary regime of polymerization is virtually attained after the second pulse. The distance between maxima, as well as between minima in Bi spectrum is   (41) can not be used because no separate peaks will be visible in Bi spectrum.
With ktWT < 1 and UT > 10, a structure induced by pulse-periodical nature of polymerization initiation appears in Bi spectrum on the background of distribution approximating that under quasi-stationary regime (Bi distributions for T >0.1 s in Figure 4). The comparison between the distributions for T 0.3, 0.8 and 2.0 s alludes to the fact that the technique of kp evaluation benefits much from polymerization with ktWT < 1. In this case both the number of maxima and the amplitudes of some peaks in B spectrum exceed essentially those existing with T 10s.

Effect of Polymerizing Medium and Radiation Parameters upon MWD
Constant kp defines peak widths in distributions of growing radicals and dead macromolecules, as it enters into the parameter of Poisson distribution (see (9), (10), (25), (26)) and the distance between peaks (39). The number of peaks distinguishable in A and B distributions grows with k, as the distance between the peaks extends faster than their width. By this, both ima for separate peaks and width of distributions are increased.
The pattern of growing radicals and dead molecules distribution does not change with k and W, if V ktWT--const. Reduction of V extends the period of polymerization pseudostationary regime attainment (Figure 2). The absolute and relative amplitudes of peaks vary with V; therewith, location and width of peaks in spectra Ai and B are retained. With increase in V, the relative amplitude of each subsequent peak in A and B spectra is lowered with growth, and the general maximum B is shifted to lesser i-values.
Chain Termination by Disproportionation Figure 5 shows B distributions for the case of chain termination by disproportionation, calculated for various pulse repetition rates. In calculations, values parameters were used, for which B distributions presented in Figure 4 were found. All the distributions in Figure 5  termination by recombination. Though it should be noted that in the absence of pseudostationary regime under chain termination by disproportionation, distributions can be produced approximating in pattern those presented in Figure 4 for ktWT < 1.

M WD on Irradiation with Sequence of Pulses Packets
At now let's consider the features of distributions generated by polymerization induced with the same number of pulses broken variously down into packets. Here, a case is considered when pulse repetition rate in a packet exceeds much that of the packets on their own. Figure 6 illustrates the dead macromolecules distributions on polymerization initiated with pulse packets, the spacing between then 20 s, for the cases of chain A monotonically decreasing distribution is transformed to that typical for chain termination by radicals recombination. The growing radicals with chain length 0 < < npUT interact for a time npT i/U with the summary concentration of radicals gradually increasing from Wi/TU to Wnp (if variation in concentration because of disproportionation reaction is neglected), and for a time i/U they interact with concentration Wnp. This gives rise to maximum origination in distribution B i.

CONCLUSIONS
In the present paper the expressions have been derived that enable to study the distributions of polymer molecules along the chain length on initiation of linear polymerization by an arbitrary sequence of short pulses. These expressions are valid if the constants of chain propagation and termination do not depend on the chain length; the constant of initiation is equal to or much greater than kp; the degree of monomer-to-polymer conversion is not large ([M] const) the processes of chain inhibition and transfer are lacking. On these assumptions, a computer investigation of MWD produced on pulse-periodical initiation of polymerization has been undertaken for a broad range of initiating pulses repetition rates and durations. The computing results have brought out that the derived expressions can be applied to study MWD on initiation of polymerization both by arbitrary-duration pulses and by CW beam.
The research into the temporal dynamics of MWD has disclosed that the pattern of distribution of growing radicals and dead macromolecules greatly varies in the process of pseudostationary state attainment, and under pseudostationary state Ai(t) Ai(t + T) and BN)(t)/N B N/ 1)(t + T)/(N + 1). Consideration of MWD with ktWT < 1 regardless of the period of pseudostationary regime establishment gives a false idea of the pattern of polymer molecules concentrations distribution under the pseudostationary state. Only with the constraint ktWT >> 1, when pseudostationary state is virtually achieved following the second pulse, the period of its attainment need not be accounted for. Initiating of polymerization by pulsed radiation allows to substantially govern the pattern of MWD formed in the process of polymerization and to gain fractions of polymer molecules with certain average weight by appropriate selection of beam parameters. Thus, for example, when conducting polymerization by pulse packets, it is possible to obtain distributions typical for the period of establishment of polymerization pseudo-or quasistationary regime. And finally, results of calculations (see Figure 4 and Figure 5) directly indicate the range of repetition rate of initiating pulses in which the more contrast MWD spectrum may be obtained.