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Properties of Two-Dimensional Polymers Macromolecules 1982,15, 549-553 549 The squared frequencies uA2are obtained from the ei- (4) Flory, P. J. "Statistical Mechanics of Chain Molecules"; In- genvalues Of this matrix described in the text* terscience: New York, 1969; p 315. (5) Volkenstein,M. V. &Configurational Statistics of polymeric for N = 3 and 4 will be found in ref 7. Chains" (translated from the Russian edition by S. N. Ti- masheff and M. J. Timasheff);Interscience: New York, 1963; References and Notes p 450. (1) Weiner, J. H.; Perchak, D. Macromolecules 1981, 14, 1590. (6) Frenkel, J. "Kinetic Theory of Liquids"; Oxford University (2) Weiner, J. H., Macromolecules, preceding paper in this issue. Press: London, 1946, Dover reprint, 1955; pp 474-476. (3) G6, N.; Scheraga, H. A. Macromolecules 1976, 9, 535. (7) Perchak, D. Doctoral Dissertation, Brown University, 1981. Properties of Two-Dimensional Polymers Jan Tobochnik and Itzhak Webman* Department of Mathematics, Rutgers, The State University, New Brunswick, New Jersey 08903 Joel L. Lebowitzt Institute for Advanced Study, Princeton, New Jersey 08540 M. H. Kalos Courant Institute of Mathematical Sciences, New York university, New York, New York 10012. Received August 3, 1981 ABSTRACT: We have performed a series of Monte Carlo simulations for a two-dimensional polymer chain with monomers interacting via a Lennard-Jones potential. An analysis of a mean field theory, based on approximating the free energy as the sum of an elastic part and a fluid part, shows that in two dimensions there is a sharp collapse transition, at T = 0, but that there is no ideal or quasi-ideal behavior at the transition as there is in three dimensions. The simulations and mean field theory agree very well. While the simulations are not sensitive enough to extract precise values for the exponent v, it is clear that it is close to the Flory values: v = 3/4, T > 0; v = T < 0. The mean field theory also gives v = *I3at T = 0. Introduction simulation studies, and other theories also give numbers The large-scale configurational properties of a polymer for v close to the~e.~-~These numbers reflect the more chain in a good solvent are determined by excluded volume pronounced excluded volume effect in lower dimensions. interactions between distant segments of the chain, which In three dimensions a polymer segment can pass over or cause the chain to swell relative to an ideal coil, where there under another segment, but in two dimensions it is ener- are no interactions. In a poor solvent the net interaction getically unfavorable for two segments to cross. between polymer segments is attractive, which causes the In a poor solvent the polymer will confine itself to as chain to contract relative to an ideal coil. These effects small a region of space as is consistent with the excluded can be modeled by an effective interaction potential be- volume interaction. Assuming the density within this tween polymer segments which has a strong short-range region is roughly uniform, then N c: Rd, where d is the repulsive part and a weak attractive part. Then by varying dimension of the system. Thus, we expect the power law the temperature, one can simulate the effects of a good dependence of the size of the chain R on the number of solvent (high temperature) and a poor solvent (low tem- units N to be described by an exponent v = l/d. For an perature), just as in ordinary fluids where the repulsive ideal coil the power law dependence also holds with v = part of the pair potential is most important at high tem- l/*. Thus, in three dimensions the size of the polymer is peratures and the attractive part is most important at low a smaller power of N than the ideal coil, while in two temperatures. In this work we shall use a truncated dimensions the exponent v is the same for the ideal coil Lennard-Jones potential to represent the interaction be- and the collapsed coil. The "collapse" of a polymer in two tween effectively independent polymer segments. Each dimensions may therefore be qualitatively different from segment then represents the average properties of many that observed in three dimensions. monomers. Monte Carlo simulations in three dimensions show that Both theory and experiment indicate that in a good the change in R as the temperature is lowered near the solvent the mean end-to-end distance of a polymer chain, collapse transition becomes steeper with increasing N.5 R, depends on the number of polymer units, N, as R = This suggests that for very long chains the collapse occurs AN", in the limit of large N. Here A is a constant that over a very narrow temperature range. Experiments by depends on the temperature or properties of the solvent. Sun et al. indeed show this kind of phase transition for The exponent Y depends on the dimensionality of the dilute solutions of polystyrene macromolecules in a cy- system. Flory's theory1 predicts that in three dimensions clohexane solvent.6 Recently, Vilanove and Rondelez7have Y = 0.6 and in two dimensions v = 0.75. Experiments, measured the surface pressure for two-dimensional poly- mer chains and found values of v given by 0.79 f 0.01 and 0.56 f 0.01, respectively, for two different polymer solu- *Current address: Exxon Research and Engineering, Linden, N.J. 07036. tions. The first value is close to the Flory value predicted 'Permanent address: Departments of Mathematics and Physics, theoretically for swollen chains and the second is more Rutgers, The State University, New Brunswick, N.J. 08903. typical of a collapsed chain. If the error bars quoted are 0024-9297/ 82 / 2215-0549$01.25/ 0 1982 American Chemical Society 550 Tobochnik et al. Macromolecules meaningful, then they suggest that the excluded volume exponent of Flory may be too small and that a very dif- ferent theory is needed to describe a collapsed chain. N+1 Thus, investigation of two-dimensional polymers at this 80 0 time takes on greater import. 40 0 The purpose of this paper is to obtain a better under- 20 0 standing of the properties of two-dimensional polymers by means of Monte Carlo simulations and an analysis based on Flory's theory and extensions by de Genness and oth- er~.~ Computational Procedure Our model chain is made up of N + 1 beads (i.e., N links) located at positions (ri)connected by rigid rods. The distance between nearest-neighbor beads is taken to be the unit of length. Any two beads along the chain interact via I I , 1 I I I a Lennard-Jones potential V(rij) with a cutoff rc shifted 05 IO 15 20 25 30 so that V(rJ = 0; i.e. E" Figure 1. a2= R2/Nfor N + 1 beads from Monte Carlo simu- V(rij) = 4e[(u/rij)12 - (u/rij)6 + (~/r,)~- (u/rC)l2] lation. The curves are from the Flory theory: (-) liquid part rij < rc (la) of free energy based on van der Waals approximation; (-) (e* < 2.0) and (---) (e* > 2.0),Flory theory up to third order in virial V(rij) = 0 rij > rc Ob) expansion; (- -) Flory theory up to second order in virial expansion. For most of our work rc = 2 and u = 0.7. to cancel out for the most part so that Flory's theory gives The ensemble of configurations was generated by a quite accurate results. Our philosophy in using this theory reptation Monte Carlo dynamics as explained in ref 5. is to assume that it is correct except for possibly small Briefly, the method involves the usual Metropolis Monte corrections to the exponent v which are undetectable in Carlo scheme where a move is defined as taking a bead our simulations and most experimental situations.) This from one end of the chain and placing it at the other end, free energy is then minimized with respect to the expansion in a position such that the angle between the old last link factor CY = R/IW2. This leads to the result, in d dimen- and the new link is determined by the Boltzmann factor. sions, that This procedure works very well at high and medium tem- peratures, where the ends of the polymer have freedom to move around. At lower temperatures the procedure is less efficient since the end can get trapped in a position such where f(p,T) is the (excess) free energy per particle of a that a bead cannot be added to it. This leads to a very fluid interacting with interparticle potential V(r) at a low acceptance ratio which prevents us from going to very density p = (c/d)(N/Rd),where c N 1/r for d = 2, c N low temperatures. 3/4r for d = 3; i.e., the monomers are assumed to be Equilibrium averages are obtained by dividing the total distributed uniformly in a spherical volume of radius R. sequence of Monte Carlo steps into 30 blocks, where each In the expanded state, p << 1 and f(p,T) - '/JJ2(T)p2, block contains 500(N + 1) attempted moves. The final where U2(T)is the second virial coefficient. In the simple average and standard deviation is obtained by averaging Flory theory, therefore, one replaces the right side of (2) over the block averages after discarding the first couple by IPdi2U2( T) , giving of blocks to allow the chain to come to equilibrium. We &+2 - cyd = P-df2(J2(T) used values of N + 1 equal to 20, 40, and 80. (3) A variety of size and shape properties of the polymer We then identify the 8 temperature, corresponding to the was studied.
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