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Calculation of the Conformational Entropy of Polysilane T Polymer Journal, Vol. 28, No.3, pp 246---248 (1996) Calculation of the Conformational Entropy of Polysilane t Mengbo Luo and Jianmin Xu* Department of Physics, Hangzhou University, Hangzhou. 310028. People's Republic of China (Received July 4, 1995) ABSTRACT: Exact enumeration method is used to calculate the conformational entropy of polysilane chain, submitted to short-range interaction and long-range interaction, with an off-lattice RIS model. The conformational entropy is in proportion to chain length at any temperature (from 50 K to 400 K). It varies with temperature, can be expressed as the function of temperature. And we find the second-order transition temperature (T2 ) of polysilane is about 65 K. KEY WORDS Exact Enumeration Method I Conformational Entropy I Polysilane I Conformations of polymer chains depend largely on silicon, the bond length of the Si-Si is fixed as 2.34 A, conformational entropy which is associated with the the bond angle of the Si-Si-Si 1s I 09.4 a. The primary order of the system. Calculation of the conformational statistical weight matrix is 11 entropy is an important and hard work. The entropy (J of a polymer chain at 8-state in a dilute solution (the (I) chain submitted only to short-range interaction) can be calculated with the method of Flory's matrix multi­ CJW 1 plication. But this method will be ineffective when where long-range interaction prevails among the chain seg­ ments.2 CJ = exp(- E,)kT) There are several methods for calculating the con­ 1/J = exp(- E"'jkT) formational entropy of polymer chain perturbed by w=exp( -E,)kT) long-range interaction. Exact enumeration2- 4 of all the possible conformations can calculate the entropy of and Eu = -0.278 kcal mol- 1, E"' = -0.240 kcal mol- 1, such system, but it is only able to deal with short chain Ew = 0.387 kcal mol- 1, k is the Boltzmann constant. (10--20 segments) due to the tremendous number of The long-range interaction between non-neighbor conformations for long chain. Meirovitch's scanning atoms i and j is represented by a hard sphere potential 2 7 method can treat relatively long chain. 5 •6 More recently, which represents an excluded volume · Collet and Premilat 7 proposed a method for calculating 00, rij < rc { the conformational entropy of much longer polymer E;i= 0, (2) chains based on the mean probabilities obtained from Monte Carlo simulation and applied it successfully on rii is the distance between two atoms, and I i-j I must polyethylene chain. greater than 4. Simply, we only consider the skeleten Among all these methods which are used to calculate atoms Si, chain side atoms Hare neglected, 12 rc is chosen the entropy of perturbed chain, including those not as the Vander Waals radii of Si-Si in this paper, so rc mentioned here, the exact enumeration can provide most is 4.10 A.U Since the statistical properties of an isolated precise results, others can only provide approximate chain (chain in a dilute solution) are independent of its results. So, it is widely used in polymer science,2- 4 •8 - 10 external coordinates, they only depend on the internal which results are often treated as an important check of rotation of bonds of the chain, so conformations of a other methods. 5 In the present work, we propose to chain with N bonds are determined by the internal calculate the conformational entropy of the polysilane rotation of N- 2 bonds, thus the conformational entropy chain with the exact enumeration method. The calcula­ of the chain depends on the rotation of N- 2 bonds.1 tion is performed on an off-lattice three rotational Therefore, the first two bonds are fixed in our calculation, isomeric state (RIS) model of polysilane. The polymer the next N- 2 bonds are generated with an off-lattice chain was submitted not only to short-range interaction model, each has three possible orientation (t, g+, g-). (RIS), but also to long-range interaction (excluded The successive two rotational angles are determined by11 volume in the work) in a manner similar to that proposed by Jianmin et al. 2 tt tg gt METHOD OF CALCULATION The skeleton atoms of the polysilane chain are all the ¢,,¢,+1 (±120°, +120°) t This work was supported by the Natural Science Foundation of Zhejiang Province, China. All the possible conformations of the polysilane chain ** To whom all correspondence should be addressed. of N bonds are generated. Then the partition function 246 Conformational Entropy of Polysilane can be calculated as2 •13 Table I. The number of all possible configurations of the polysilane chain submitted to short-range interaction Z = L exp(- EdkT) (3) and long-range interaction• N 7 8 9 10 II 12 where E; is the energy of conformation i and the summation is over all the possible conformations. T is Q 143 363 937 2405 6137 15701 the temperature, k is the Boltzmann constant as in eq I. The mean energy of the polymer chain is: N 13 14 15 16 17 Q 40115 102401 261297 666125 1697653 (E) =Z- 1 · 'I E;·exp( -EJkT) (4) ------·--- ------ i -- -- ·-·--- ------- -- • N is the chain length (defined as the number of bonds of the chain), Then the conformational entropy per mol of the chain Q is the number of all the possible configurations. is: R S=R·lnZ+(E)/T (5) 14 400K here R=N0 k, N 0 is Avogadro constant, R= 1.987 1 1 300K calK - mol- . 12 .. 250K In this paper, the conformation entropies Sat different temperature are calculated, the temperatures range from 200K 50 K to 400 K. Since it takes a great deal of CPU time IO of a computer, the chain length is limitted to 17 bonds. 8 150K RESULTS AND DISCUSSION Vl The conformational entropies of short polysilane chain 6 submitted to short-range interaction (RIS) and long­ range interaction (hard sphere) at temperature from 50 K 4 to 400 K are calculated, while the chain length N is from ---------lOOK 7 up to 17. The number of all possible conformations of such chain is listed in Table I. 2 The conformational entropy per mol S versus N-2 is shown in Figure I for the temperature from 50 K to -------------------------50K-------------- 0 400 K. Results show that the conformational entropy is 5 6 7 8 9 10 11 12 13 14 15 a linear function of N-2 at any temperature, give a N-2 relationship similar to that obtained for much longer Figure I. The plots of the configuration entropy per mol S versus 7 chain using Monte Carlo method by Collet : N- 2 at different temperature, R =I .987 calK- 1 mol- 1 . S=a(N-2)+b (6) Table II. Values of a and b at different temperature The reason we use N- 2 instead of chain length N here T a b is that the internal rotations of N- 2 bonds contribute to the conformational entropy. 50K 8.432 X J0- 3 0.7441 a and bin eq 6 are dependent on the temperature, the lOOK 0.2088 0.8225 values of a and b at different temperature are listed in 150K 0.4880 0.6989 Table II. a and b can be expressed as the function of 200K 0.6689 0.5535 250K 0.7694 0.4509 a temperature. Using the least square method, we get 300K 0.8257 0.3889 and b which are good for the temperature from 50 K to 350K 0.8590 0.3518 400K, 400K 0.8799 0.3290 a=2.380 x l0- 2 -0.6745r+7.229r 2 -4.448r 3 -7.602r 4 +6.748r 5 it seems reasonable to consider that the relations are still (7) valid for long chain. Let Sm stand for the entropy per 2 3 b= 0.6980 + 0.5737r +4.624r - 29.20r bond per mol of an infinity chain, then +42.53r 4 -19.2Ir 5 Sm= lim Sj(N-2) where r has a relationship to temperature T, r = N--+ro (8) exp(ajkT). a is minus, the magnitude of a is arbitrary, =a we choose a=E"= -0.278kcalmol- 1 here. One merit of choosing r instead of T in eq 7 is that a and b will In Figure 2, we show Sj(N- 2) versus temperature T converge when temperature tends infinity. Also, the mean for chain length N = 7, 12, and 17, Sm versus T is also square deviation of a and b are largely decreased when shown. It indicates that Sj(N- 2) decreases as N increase, we use r, especially at high temperature. The value of a and converges rapidly when N increases. Since the in­ and b under 50 K or above 400 K can be estimated from ternal rotation of bonds contributes the conformational the tendency of them near 50 K or 400 K. entropy, when the chain length increases, the space near The linear relations of S versus N- 2 are so good that the bond is gradually occupied by other bonds, the Polym. J., Vol. 28, No.3, 1996 247 M. Luo and J. Xu large at very low temperature. Therefore the calculation of the entropy of very low temperature becomes difficult 0.9 and even impossible. Thus, T 2 must be determined by extrapolating from high temperature. 0.8 The curve of Sm versus T is almost straight in the 0.7 region between 80 K and 140 K where the entropy Sm varies rapidliest. Extrapolating to the lower temperature 0.6 from this linear region, one will find that the crossover c::J I temperature between sm = 0 and sm for high temperature 3 0 5 range is about 65 K.
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