Polymer Journal, Vol. 14, No. 7, pp 579-581 (1982) SHORT COMMUNICATION Computer Simulation of Polymer Chain Statistics Takao MINATO, Kazuo IDEURA,* and Akira HATANO* Nippon Information Industry Co., Ltd., Dai-hyaku Seimei Shibuya Bldg., Shibuya, Shibuya-ku, Tokyo 150, Japan. *Department of Pure and Applied Sciences, College of General Education, The University of Tokyo, 8-1, Komaba 3-chome, Meguro-ku, Tokyo 153, Japan. (Received January 5, 1982) KEY WORDS Computer Simulation 1 Chain Statistics 1 Monte Carlo Method I Path I Globule-Coil Transition I Many methods for the Monte Carlo simulation METHOD have been developed1 in regard to the excluded volume of a flexible chain. The self-avoiding-walk We replace (f.1= I, 2, d; d is the on a lattice is one of these. The method presented dimentional number) by the Gaussian number, here is rather primitive but realistic. We have re­ (MJw This makes the contour length L=Nii which moved the constraint of walks on the lattice and has no definite value, where ii= hence can take any interactions between segments Similarly we replaceR by R., {r;} by {i;} and U({r;}) into the simulation. by U({i';}). By repeating these replacements, we When a flexible chain consists oflinearly linked N finally obtain a set of paths. This is very similar to segments of length a and interacting each other, its the concept underlying the renormalization group "partition" function ZN is given by method in which the short-range correlation is replaced by a Gaussian-type function. As a result of these replacements, i) represents ZN= exp { 1 U({r,})]} (1) -t{J a set of segments belonging to a small region, ii) L where {r;} is a set of position vectors of successive and hence R have no definite values but distribute in segments (i=O, 1, 2, · · ·, N) and Gaussian form, iii) the interaction potential 0 is not U( {r;}) is the interaction potential between seg­ bare but renormalized. Thus, our method of simu­ ments. Path {r;} stands for a set of all allowable lation may be expected to give reasonably realistic paths. results. Our special interest is the root-mean-square (R 2 ) For simplicity, we assume the interaction poten­ of the end-to-end vector R = rN- r0 , i.e., tial [; to be represented by N )2 (R2)= I ( I (3) path {rt} i= 1 with x exp { + U({r;})]} IZN for lr;-iil :::::;ro (2) for I i; -iiI >ro In this note, attention has been directed to the (4) calculation of (R2 ) by the Monte Carlo simulation. Thus, with the interaction parameter a, the effective range r0 , and the standard deviation of we can generate a path whose weight is exp [-{L(MY+ U({i';})}] and then the square of 579 T. MINATO, K. IDEURA, and A. HATANO the end-to-end vector for the path. By summing up The results are shown in Figure la-2. In these all these paths, we can evaluate eq 2. figures, the dots are the simulated results, and the lines indicate the relations between (R 2 ) and N 2 '. RESULTS We have simulated eq2 for d=2, 3, and 4, choosing, for each d, the values of r0 and rx as given 4-a) 4-b) in Table I. The standard deviation of is taken 112 1 2 as p- in every case. The scale of r0 is p- 1 • For each simulation, we have generated about 104 paths. Table I. Simulation results for 2v chosen for the best fit from the relation of (R 2 )ocN2 ', dis the dimensional number of the space, r0 and IX are the interaction parameters given in eq 4, and the last column refers to Figures la--c. r0 is scaled by p- 112 3 0.5 4 1.0 2 0.5 1\ "'0:: v ""' 10 20 30 50 100 N (c) Figure la-c. a, b, and c are log-log plots of (R 2 ) vs. N for various values of IX in 3-, 4-, and 2-dimentional space, respectively: a, IX=2.0 (small circle), 0.8 (square), 0 (triangle), and -0.8 (large circle); b, IX=2.0 (circle) and 0 (dot), and c, IX=0.5 (circle) and 0 (dot), respectively. 10 20 30 50 100 N Each straight line is plotted to make the best fit for (a) (R 2 )ocN2 ' with v given in Table I. 580 Polymer J., Vol. 14, No. 7, 1982 Computer Simulation of Polymer Chain Statistics done in this work. It should be mentioned that the collapse to zero size cannot occur in spite of the lack • • of the core part in the effective potential. This is due to the condition such that the finite deviation of the 20 •• elements is greater than the effective range r0 . Simulations were carried out at the Computer Center of the University of Tokyo . • N•60 REFERENCES 10 • I. See, for example, C. Domb, Adv. Chern. Phys., 15, 229 (1969); M. Barber and B. Ninham, "Random • and Restricted Walks," Gordon and Breach, New York, 1971; P. H. Verdier and W. H. Stockmayer, J . • Chern. Phys., 36, 227 (1962); J. Mazur and F. L. McCrackin, J. Chern. Phys., 49, 648 (1968); F. L. McCrackin, J. Mazur, and C. L. Guttman, Macro­ -2.0 -.8 0 .8 2.0 molecules, 6, 859 (1973); R. Grishman, J. Chern. Interactlm Phys., 58, 220 (1973); M. Janssens and A. Bellemans, Macromolecules, 9, 303 (1975); D. C. Rapaport, J. Figure 2. <R 2 ) vs. interaction parameter IX for N=60. Phys. A10, 637 (1977); D. Ceperley, M. H. Kalos, Certainly, the chain is obviously collapsed below IX= -0.8 because <R 2 )rx.N213 (see Figure Ia). and J. L. Lebowitz, Phys. Rev. Lett., 41, 313 (1978); A. Baumgartner, J. Chern. Phys., 72, 871 (1980); D. Richter, A. Baumgartner, K. Binder, B. Ewen, and J. The best-fit values of v are given in Table I. In B. Hayter, Phys. Rev. Lett., 13, 109 (1981). 2. H. Miyakawa and N. Saito, Polym. J., 10, 27 (1978). particular our attention was focused on the case in 3. Recently an interesting work for the collapse by the which attractive interactions are present in 3-dimen­ dynamic Monte Carlo method has been done; I. sional space (see Figure Ia and Figure 2). We can Wehman, J. L. Lebowitz, and M. H. Kalos, Macro­ see a globule-coil transition,2 •3 although a definite molecules, 14, 1495 (I 981 ). conclusion is necessary in order to carry out a much 4. T. Minato and A. Hatano, submitted to Polym. J. more extensive and detailed simulation4 than that Polymer J., Vol. 14, No. 7, 1982 581 .