Shape Phase Transition of Polyampholytes in. Two Dimensions. M. R. Ejtehadi and S. Rouhani. Department of Physics, Sharif University of Technology,

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1 IPM # 1996 Shape Phase Transition of Polyampholytes in Two Dimensions M. R. Ejtehadi and S. Rohani Department of Physics, Sharif University of Technology, Tehran P. O. Box: , Iran. and Institte for Stdies in Theoretical Physics and Mathematics Tehran P. O. Box: , Iran. Abstract We have stdied the transition in shape of two dimensional polyampholytes sing Monte Carlo simlation. We observe that polymers with randomly charged monomers get into a globlar shape at lower temperatres, provided that their total charge is below a critical vale Q c. Collapse into globlar form happens for all forms of force law, bt the critical charge depends on the form of the force law, inversely dependent on the range of interaction. The vale of the critical charge is proportional to N, the size of the polymer, in the thermodynamic limit. reza@netware2.ipm.ac.ir

2 I. Introdction The shape of polymers in eqilibrim, determines some of its physical, chemical and biological properties, ths predicting the shape for a given chemical composition of a polymer is a challenging problem of polymer science. Some of the interesting analytical approaches to this problem come from the theory of spin glasses [1,2], the replica trick [3{5], the random energy model [6{8], and the mean eld approximation [8{10]. De to extremely complicated natre of forces involved, compter simlations have been the more sccessfl tool in this stdy [11]. Modeling polymers on basis of a self avoiding random walk (SAW) sing compter simlations can predict some elementary geometrical qantities sch as the end to end distance (R e ) or the radis of gyration (R g ). For some large polymers these two qantities have the same scaling with respect to N, the nmber of monomers, < R 2 e >< R2 g > N : (1) The niversal scaling index, can be obtained throgh Monte Carlo simlations, which agrees well with Flory's calclation of for dierent dimensions, for a netral SAW [12{14] (table 1). Flory's calclations of are exact expect in d = 3, which never the less is in good agreement with simlation (table 2). Real polymers however have interactions among their monomers and between the monomers and the solvent. Ths they behave dierently to the predictions of table 1. However all polymers in a good solvent, independently of their monomer interaction, behave like a SAW at high temperatres. Ths observed vales of at high temperatres shold correspond to those of table 1 [13]. In low temperatres the eect of forces become important, ths we expect dierent vales of. 2

3 In low temperatres the polymer shold get into congrations which minimize the potential at the same time as satisfying the self avoiding constraint. For short range attractive forces, sch as the Van der Waals forces, the polymer gets into a d dimensional shape, where d need not be an integer. The index in this case is 1=d. This phase transition, which qalitatively happens at the point where the thermal energy is comparable with the potential energy, is referred to as the -point. This is the transition of a polymer from a free globlar shape to a collapsed globlar shape, determined by the minima of the potential. In this paper we stdy the transition for two dimensional polyampholytes, polymers with randomly charged monomers. In a random distribtion of charge the total charge of the polymer need not be zero. If the net total charge becomes proportional to the size of polymer, we then expect a behavior similar to polyelectrolytes, polymers with singly charged monomers. Higgs and Joany [19] have shown that polyampholytes in a good solvent, which have total vanishing charge have a smaller radis of gyration than ncharged polymers. They have estimated the size of the polymer assming screening of the long range electrostatic force between monomers and sing the Debye-Hckel estimation of the free energy. Polyampholytes with electrostatic interaction and withot the screening eect, were stdied by Kantor et al. [20,21], in three spatial dimensions. Using scaling argments and Monte Carlo simlations, they showed that polyampholytes with zero net total charge, collapse into a globlar form below a critical temperatre (T c ). They also showed that a critical net charge Q c exists below which the polymer collapses, jst like ncharged polyampholytes, where as above Q c the polymer gets into a rod shape jst like a polyelectrolyte. Of corse for T > T c both types, irrespective of their total charge, behave like a netral SAW. We have addressed this problem for polyampholytes in two dimensions. This qestion may be relevant when considering polymers in a thin lm solvent, adsorbed onto a srface. The restriction of movement in two dimensions, do give dierent reslts for a SAW in two 3

4 dimensions. Bt we do not expect qalitatively dierent behavior with regards to temperatre or total charge. Indeed the same qalitative behavior is observed, below a critical temperatre collapse into globlar form happens. There also exists a critical total charge Q c, below which, the polymer collapses jst like a zero net charge polyampholytes, and above which a rod shape is taken, similar to polyelectrolytes. However the interesting qestion is the dependence of Q c on the form of the potential. For short range potentials Q c is proportional to N, the size of the polymer. We performed or simlations for a few dierent forms of the potential, and nd agreement with an approximate expression for Q c Q 2 c ' (an 2 + bn)q 2 0 : (2) Where q 0 is the charge of the monomer, N is the size of the polymer, a and b are potential dependent constants. We give an approximate derivation of eq. (2), which is not dependent on the dimension of space, so it shold also hold in higher spatial dimensions. II. Simlations To simlate the motion of a polymer we have adapted Rose's model [22]. In this model a polymer, made p of N monomers is replaced with a SAW with N? 1 steps on a sqare lattice. Every vertex of the lattice throgh which the walk has passed, represents a monomer. Since two monomers can not occpy the same location in space, a SAW is a representation of a congration of the polymer. To allow for eects of interaction, each site of the lattice on the random walk is allowed to take a random charge, which we restrict to be q 0 for simplicity. To calclate qantities sch as the radis of gyration one needs to perform averages over 4

5 ensembles of congrations. We sed the Metropolis [11] method of importance sampling to generate these ensembles. Let s assme C is the set of all possible congrations. We then constrct a set of maps f which transform the members of C into each other: f : C! C f [s i ] = s j s i ; s j 2 C: (3) At any given moment, a weight factor determines the probability of the action of f, sing the interaction hamiltonian. Ths starting with a given congration, the action of f takes s throgh the phase space C with the passage of time, generating the more probable congrations rst. An interesting qestion is what is the most ecient set of f, sch that any two members of C can be transformed to each other with a nite nmber of actions of f. A very large set of f is not desirable since the compter has to rn throgh the set at each time step, which slows down the algorithm considerably [22]. On the other hand a very small set of f leaves some congrations nreachable. Consider the following set of moves: i - Standard move - ii - End moves (a) - (b) - 5

6 iii - Crank shaft move - iv - End reection - Withot the moves (ii.b) and (iv), the following congration: can not be reached. However, the other moves may be scient for simlation of SAW, at high temperatres becase the compact congrations sch as the above have very small entropy. Bt for or problem, where we seek compact congrations, at low temperatre we need the fll set of above moves. Apparently this set of moves seems to be scient for or problem, however we don't have a proof of this. For a given polymer of size N, the algorithm rnning time grows exponentially with N. For each congration N attempts at changing the congration is made. The nmber of possible congrations grows like N 2 in two dimensions. Note that the algorithm has to check the illegal moves as well. Ths the time to reach eqilibrim grows like N 3. In practice we observed that approximately 100N 3 was necessary to reach eqilibrim. On the other hand, calclating the change in energy, going from one congration to another, reqires N calclations de to the long range natre of the force law. Ths rn time grows like N 4, hence very long polymers are 6

7 not easy to simlate. We were ths restricted to a size N 32, with once exception where we scceeded in getting reslt for N = 48, too. Attribting a xed charge q 0 to monomers at random, leaves s with a polymer with a total random charge, between +Nq 0, and?nq 0. We then choose a monomer at random and change its charge ntil the desired total charge is achieved. This gives the initial congration to start with. We then allow this congration reach eqilibrim for a given temperatre following the Metropolis method [11], applying the moves already discssed. After reaching eqilibrim the radis of gyration is calclated, and this is then repeated 30 times at large time intervals. The whole procedre is then repeated with dierent charge congrations. Averaging over charge congration a nal vale for < R g > is obtained. Repeating the above for dierent vale of N gives a scaling relationship between < R g > and N, giving a vale for. III. Reslts We simlated polyampholytes with size N = f32; 16; 8; 4g in two spatial dimensions, sing dierent force laws acting between monomers. We assmed an electrostatic interaction of form log(r), 1 r, 1 r 2, and a short range interaction U(r) = 0 for r l, and constant otherwise. To begin with we simlated polymers with total charge zero, and observed that there exist a phase transition as in three dimensions, where the polymer gets into a globlar shape at low temperatres. This phase transition is observed for all types of potentials sed. In gs. (1.a) and (1.b) the radis of gyration is plotted vs. = ( 1 ), for the two potentials T log(r) and 1 r. Where we observe a redction in R g as increases. This redction is better seen for large polymers. We can observe the changes in as a fnction of for these two potentials in tables (3.a) and (3.b) and gs. (2.a) and (2.b). We observe that this two types of polymers have 7

8 similar behavior and the vale of for high temperatres is consistent with the two dimensional netral SAW. The interesting point is that lower temperatres, these polymers cannot collapse completely, ths > 0:5, which is the index for a two dimensional shape. This may have been cased by considering an incomplete set of moves, which can not access a collapsed shape. However increasing the nmber of moves and repeating the simlations and consistency of or reslts sggest otherwise. In two dimensions holes can form which ore srronded by like charges, ths, movement to a more compact congration becomes energetically inaccessible. Probability of having a length l of like charged monomers is proportional to a power of l [23], ths the distribtion and size of these holes obeys a scaling law. Therefore the existence of sch holes aects the scaling dimension of globlar collapse in two dimensions and forces above 0:5. With prely attractive forces these holes wold not exist and this obstrction is removed. Repeating simlations for polymers with total nonzero charge, showed that the phase transition observed by Kantor and Kardar [21] in three dimensions, exists in two dimensions as well. For a total charge greater than Q c, rod shape is observed where as for Q < Q c, globlar form, jst like a zero net charge is observed. The interesting point is the dependence of Q c on the potential and the size of the polymer. In gs. (3a-3c) we observe the reslts for < R g >, for a polymer of length N = 32, sing dierent potentials, and in gs. (4a-4c) reslts for N = 48; 32; 16, sing the potential of the eq. (12). Althogh a critical Q c dose exist for each potential, bt the vale of Q c depends inversely on the range of the potential. IV. The Critical Charge The charge on monomers is a random variable of the problem bt withot dynamics, ths 8

9 it shold be treated as a qenched variable when analyzing this systems. This is the root of similarity between this problem and spin glass systems. The appropriate qenched average is to be eected as the free energy F = log Z (4) where bar means average over qenched variable, in or case the dierent charge congration. The calclation of qantities sch as eq. (4) is riddled with diclties, one way is the replica method [3,4], bt there is a lot of controversy associated with it. We were nable to perform a direct calclation of eq. (4) for or system. Let s instead often a compromise qantity the weighted average potential energy: << U >>= Z Z D r P (r)d q P (q) X i;j U(r i ; r j )e?u (r i;r j ) : (5) In this expression D r P (r) integrates over all possible self avoiding congrations inclding the appropriate distribtion in space, D q P (q) like wise averages over all charge congrations, the factor e?u allows the more probable energy congrations to happen more freqently in the averaging. The expression of eq. (5) lacks a normalizing factor which can be added later. The sm over i and j, sms over all monomer pairs in interaction. Taking ot the weight factor e?u we end p with the expression sed by Kantor and Kardar [21]. In this case the calclation of eq. (5) leads to: << U >> (Q 2? Nq 2 0 ) (6) which indicates that the average potential changes sign at the point of total charge Q eqaling q 0 p N. Ths a phase transition is expected at critical total charge of q 0 pn. This was observed by Kantor and Kardar bt is inconsistent with or observations. 9

10 Not inclding the Boltzman factor implies that two spatially similar congration, with dierent charge congrations are eqally probable. Where as clearly when opposite charges are near each other, a greater probability is achieved de to lower energy. Let s now keep the Boltzman factor in eq. (5). We can write eq. (5) as : << U >>= n + Z D r P (r)u(r)e?u (r)? n? Z D r P (r)(r)e?u (r) (7) where n + and n? are the nmber of pairs of monomers with like and nlike charges respectively: n + = N(N? 2)q2 0 + Q2 4 n? = N 2 q 2 0? Q2 4 (8) which reslts in: << U >> (Q 2? 2 2N + (? 1)N q ) (9) where = R U (r) Dr P (r)e R : (10)?U (r) Dr P (r)e The parameter, can only be determined if the shape of the potential is given, bt is positive and larger than 1. We now observe that the average potential energy changes sign at the critical charge: Q 2 c = [(? )N N]q2 0 : (11) Ths the critical charge is proportional to N, rather than p N. This is in better agreement with or reslts. We can calclate for a simple potential: U(r) = 8 >< >: V r l 0 r > l (12) 10

11 We have = e 2V. Comparison with g. (4.b) indicates 16 Q c 20 for N = 32, which is consistent with eq. (11). This reslt indicates that in the thermodynamic limit where N! 1, critical charge scales with N. If the factor calclates to be almost near 1, then we get a dierent scaling behavior where Q c scales like N 1 2. This is of corse tre for high temperatres, i.e. = 1, bt in this region, interactions don't play a role and all polymers behave like netral self avoiding random walks. V. Discssion Simlating polyampholytes in two dimensions we observed the qalitative featre that a globlar form is taken at low temperatres if the total charge of the polymer is less than a critical vale Q c. The scaling index, of this globlar form is above 1=2, this is conjectred to be de to formation of areas srronded by parts of the polymer made p of monomers of like charge. Where sch an area is formed other parts of polymer are forbidden to enter it, de to the self avoiding natre of this system. Ths the globlar form is more swollen than a two dimensional shape, hence > 1=2. We were nable to predict the deviation from two dimensions by an analytic calclation, althogh this seems possible. Clearly a potential which is only attractive sch as the Van der Waals potential wold not give rise to this eect. The other interesting observation concerns the scaling of Q c with size. Or analysis indicates that Q c scales with N. This becomes more stark when the forces are short range. It wold also be experimentally observable if the eects of screening are made more intense, for example by changing the solvent [24,25]. It appears that this eect shold exist independent of dimension, althogh or simlations were restricted to two dimensions. For a long range force the dependence of Q c on N may become smaller, bt shold be important for large enogh N. 11

12 It mst be said that or calclations were not based on a tre qenched average. Since performing a tre qenched average seems impossible, therefore some more convincing calclation is necessary before the qestion may be settled. Acknowledgments: We are indebted to Mehran Kardar for valable discssions and sggesting this problem to s. 12

13 REFERENCES [1] K. Binder and A. P. Yong, Rev. Mod. Phys. 58, 801 (1986). [2] M. Mezard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond (Word Scientic, Singapore, 1986). [3] E. I. Shakhnovich and A. M. Gtin, Biophys. chem. 34, 187 (1989). [4] E. I. Shakhnovich and A. M. Gtin, Erophys. Lett. 8, 327 (1989). [5] V. S. Pande, A. Y. Grosberg and T. Tanaka, Cond-mat/ [6] J. D. Bryngelson and P. G. Wolynes, Proc. Natl. Acad. Sci. USA 84, 7524 (1987). [7] J. D. Bryngelson, J. N. Onchic, N. D. Socci and P. G. Wolynes, Chem-ph/ [8] T. Garel and H. Orland, Erophys. Lett. 6, 307 (1988). [9] K. A. Dill and D. Stigter, Adv. Protein Chem. 46, 59 (1995). [10] C. D. Sfatos, A. M. Gtin and E. I. Shakhnovich, Phys. Rev. E 51, 4727 (1995). [11] K. Binder, Application of the Monte Carlo Method in Statistical Physics (Springer, Berlin, 1984). [12] P. J. Flory, Statistical Mechanics of Chain Molecles (Wiley Interscience, New York, 1969). [13] P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University, Ithaca, 1979). [14] E. P. Raposo, S. M. de Oliveira, A. M. Memirosky and M. D. Cotinho-Filho, Am. J. Phys. 59, 633 (1991). [15] A. J. Gttman, Phys. Rev. B 33, 5089 (1986). [16] Ph. de Forcrand et al. J. Stat. Phys. 49, 223 (1984). 13

14 [17] J. C. Le Gillo and J. Zinn-Jstin, Phys. Rev. B 21, 3976 (1980). [18] K. A. Dill et al. Protein Science 4, 561 (1995). [19] P. G. Higgs and Jean-Francois Joanny, J. Chem. Phys. 94, 1543 (1991). [20] Y. Kantor, H. Li and M. Kardar, Phys. Rev. Lett. 69, 61 (1992); Phys. Rev. E 49, 1383 (1994). [21] Y. Kantor and M. Kardar, Cond-mat/ ; Cond-mat/ ; Phys. Rev. E 51, 1299 (1995). [22] K. Kremer, Macromolecles 16, 1632 (1983). [23] D. Ertas and Y. Kantor, Cond-mat/ [24] See the reference [19] and references therein. [25] H. Morawets, Macromolecles in Soltion (Interscience, New York, 1965). 14

15 Tables d Flory's =4 3 3=5 4 1=2 Monte Carlo Simlations for d = 3 0: :002 [15] 0:5745 0:008 0:0056 [16] 0:588 0:001 [17] table 1. table 2. table 1. Flory's calclation of for dierent dimensions. All are exact except d = 3. table 2. Monte Carlo simlation estimates of for d = 3. U(r) = log(r) U(r) = 1 r table (3.a) table (3.b). Table 3. The dependence of on for polymers with zero net charge for the potentials log(r) (3.a), and 1 r (3.b). 15

16 Figre Captions Figre 1. The radis of gyration of the polyampholyes with zero net charge and dierent nmbers of monomers vs. = 1=T for the potentials log(r) (1.a), and 1 r (1.b). Figre 2. The dependence of on for polymers with zero net charge for the potentials log(r) (3.a), and 1 r (3.b). Figre 3. The radis of gyration of the polyampholyes with net charge Q and N = 32 vs. for the potentials log(r) (3.a), 1 r (3.b), and 1 r 2 (3.c). Figre 4. The radis of gyration of the polyampholyes with net charge Q for the potential of eq. 12 vs. for N = 16 (4.a), N = 32 (4.b), and N = 48 (4.c). 16