J. Phys. Chem. 1992, 96, 3931-3934
3931
Concentration Dependence of Shape Fluctuations of Uniform Star Polymerst Shu-Jun Su and Jeffrey Kovac* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996- 1600 (Received: September 5, 1991)
The equilibrium and dynamic properties of a single star polymer immersed in a bath of linear molecules were studied using a Monte Carlo computer simulation technique. Attention was focused on the shape fluctuations of the stars as described by the eigenvalues of the radius-of-gyration tensor. We also studied the center-of-mass diffusion and the rigid body rotation of the stars. We find that the star molecules contract and change shape as the concentration of linear bath molecules increases. The diffusion, rotation, and shape relaxation are all slowed by the presence of the bath molecules.
Introduction In a recent paper] we investigated the shape fluctuations of isolated uniform star polymers using a Monte Carlo computer simulation technique. In this paper we report a modest extension of this work in which the dynamics of shape fluctuations of a single three- or four-arm star polymer immersed in a bath of linear polymers of varying concentration are studied. For comparison, we also studied a linear molecule, which can be regarded as a two-arm star. As in our earlier work, the computer simulation model chosen is the bond fluctuation model originated by Carmesin and Kremer.2 This is the only lattice model which is appropriate for the study of the dynamics of nonlinear polymers. While there have been a number of computer simulation studies of both the static and dynamic properties of star polymers? there is only one previous attempt to investigate the effects of concentration on the motions of these interesting nonlinear molecules! Smit and co-workers have studied the effect of concentration of monomers on some of the static and dynamic properties of a single, small, three-arm star using molecular dynamics. Since this is an exploratory study, we have also limited ourselves to molecules of rather modest size. We have studied three- and four-arm stars with arms containing 20 “monomers”. The overall molecular sizes are therefore 61 and 81 units. The linear probe molecule, or two-arm star, contains 41 units. The linear bath molecules were chosen to be the same length as the arms of the stars, that is, 20 units. Three nonzero concentrations were studied, as well as isolated molecules which represent the zero concentration limit. Shape fluctuations were monitored by diagonalizing the instantaneous radius-of-gyration tensor of the tar.^,^ The eigenvalues represent the extensions of segment density along the three principal axes of an ellipsoid which describes the instantaneous shape of the molecule. Autocorrelation functions of these eigenvalues are a probe of the dynamics of shape fluctuations of the star. From the eigenvectors we can extract the two rotational angles which the principal axes make with respect to a fixed set of laboratory axes. Autocorrelation functions of the spherical harmonics of these angles monitor the rigid body rotation of the star molecule. These two types of functions provide a good description of the internal motions of nonlinear molecules. Although there are no known experimental probes of these motions, we do feel that the study of shape fluctuations has great heuristic value because it gives a useful picture of the molecular motion. We have also monitored the center-of-mass diffusion of the stars as a function of the concentration of the linear bath molecules. The concentration dependence of the diffusion of stars dissolved in a matrix of linear chains has been measured experimentally by Lodge and co-workers using light scattering.’ These experimental measurements provide a convenient comparison for our simulation results. We find, as expected, that the stars contract slightly as the concentration of linear molecules increases. In addition, we find that the shape changes slightly in that the smallest eigenvalue
becomes relatively smaller as the concentration increases, making the probe molecule more aspherical. We also find that the diffusion, the shape relaxation, and the rotation of the molecule slow down as the concentration increases.
Model The bond fluctuation model was described in our previous paper to which the reader is referred for details. It is an extension of the original work of Carmesin and Kremer to three dimensions. Briefly, each monomer is represented by the eight vertices of a simple cubic lattice which form a unit cube. These cubes are connected by bonds which have a maximum length of v‘16. Randomly chosen monomers are moved by sliding one lattice unit in the Ax, *y, or fz directions subject to three conditions. First, the bonds can be extended no longer than the specified maximum length. Second, no two monomers can share any lattice site. Third, there can be no bond crossings. This latter condition is the most difficult to implement numerically. It requires checking each movement to make sure that the area swept out by a bond in a prospective movement does not cut any other bond. There are no restrictions on bond angles other than those which derive from these three conditions. Beginning with a completely equilibrated configuration, we have simulated the motion of a single star immersed in a bath of linear molecules of varying concentrations. The linear molecules all have a length of 20 monomers, which is also the length of the arms of the stars. Stars with two, three, and four arms were studied. The two-arm star is a linear molecule of size N = 41. The two stars have sizes N = 61 (three arms) and N = 81 (four arms). The concentrations studied were c = 0.0 (isolated molecule), c = 0.19, c = 0.24, and c = 0.29 in units of fraction of lattice sites occupied. The concentration range studied was limited by the amount of computer time required to perform the simulations. For each simulation we monitored the chain dimensions by computing the eigenvalues of the radius-of-gyration tensor. The equilibrium averages of these quantities describe the shape of the molecule. We also computed the mean-square radius of gyration, ( R G 2 ) ,and the mean square center-to-end distance, ( R e Z ) ,for the arms of the star. We used the instantaneous values of the eigenvalues and eigenvectors of the radius-of-gyration tensor Sij
= N-’EXimXjm- N-2EXimX,m m
m
(1)
(1) Su,S.; Denny, M. S.; Kovac, J. Macromolecules 1991, 24, 917. (2) Carmesin, I.; Kremer, K. Macromolecules 1988, 21, 2819. (3) For example, see: Grest, G. S.; Kremer, K.; Witten, T. A. Macromolecules 1987, 20, 1376. Bishop, M.; Clarke, J. H. R.J . Chem. Phys. 1989, 90,6647. Barrett, A. J.; Tremain, D. L. Macromolecules 1987, 20, 1687 and references therein. (4) Smit, B.; van der Put, A.; Peters, C. J.; de Swann Arons, J.; Michels, J. P . J. J . Chem. Phys. 1988,88, 1988. ( 5 ) Koyama, R.J . Phys. SOC.Jpn. 1967, 22, 973; 1968, 24, 580. ( 6 ) Solc, K.; Stockmayer, W. H. J. Chem. Phys. 1971,54,2756. Solc, K. J . Chem. Phys. 1971,55. 3354. Gobush, W.; Solc, K.; Stockmayer, W. H.
Dedicated to Marshall Fixman on the occasion of his 60th birthday. *Author to whom correspondence should be addressed.
0022-365419212096-393 1%03.00/0 0 1992 American Chemical Society
3932 The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 TABLE I: Equilibrium Properties of the Star Molecules as a Function of Molecular Size, N, and Concentration, c N 41
61
81
c
0.00 0.19 0.24 0.29 0.00 0.19 0.24 0.29 0.00 0.19 0.24 0.29
(RE2) 310 299 288 279 302 290 279 270 297 286 276 265
(RG’) 97.7 96.6 95.0 93.5 114 113 110 11 I
126 125 121 119
(AI) 74.8 74.1 73.4 72.7 78.0 77.5 74.8 76.0 78.3 78.8 77.0 75.1
(A,)
16.8 16.7 16.0 15.3 26.8 26.1 26.3 26.4 33.7 32.5 32.2 32.0
(A,)
6.12 5.74 5.63 5.56 9.73 9.48 8.91 8.80 14.2 13.5 13.3 12.1
TABLE II: Center-of-Mw Mffusion Constant, 0 / 6 , of the Star Polymer as a Function of Molecular Size, N,and Concentration, c
N
(A~):(Az):(A~)
12.2:2.7:1 12.9:2.9:1 13.0:2.8:1 13.1:2.7:1 8.0:2.8:1 8.2:2.8:1 8.4:2.9:1 8.6:3.0:1 5.5:2.4:1 5.8:2.4:1 5.8:2.4:1 6.2:2.6:1
C
41
61
81
0.00
0.421 0.302 0.225 0.179
0.240 0.189 0.164 0.140
0.177 0.142 0.133 0.122
0.19 0.24 0.29
0 N=41 A
where Ximdenotes the ith coordinate of the mth monomer, to compute the autocorrelation functions CAW = ( ( A ( t )4 0 ) )
Su and Kovac
N=61 N=81
0 (o
0.250
0
- ( A ) ~ ) / ( ( A- ~w2) )
0
A
where Ai are the three eigenvalues of the tensor and Y, are the second-order spherical harmonics of the angles which the principal axes make with respect to a fixed set of laboratory axes. As usual, the ensemble averages were approximated by time averages. The time unit is proportional to the total number of monomers in the system. Typically, 800 000 time steps were required for equilibration, and an additional 1000000 time steps were used to compute the averages. The center-of-mass diffusion constant, D, was calculated by monitoring the mean square displacement of the center of mass, CM(t) = ( (Km(t)- R,,(0))2), as a function of time. The slope of a graph of C M ( t ) vs t is 6 0 in three dimensions. The autocorrelation functions of the eigenvalues, Cb(t),showed a simple exponential decay at long tiniis so we were able to extract a relaxation time, T A j ,by fitting a least-squares line to the long time region of semilog plot of C(t) vs t. As in our previous work, we find that the autocorrelation functions of the spherical harmonics do not exhibit an exponential decay. Therefore, we have fit them to the Kohlrausch-Williams-Watts (KWW) function (3) The parameter T~~is the KWW relaxation time, and p is usually regarded as a measure of the width of the spectrum of relaxation times.
Results and Discussion The equilibrium properties of the probe molecules are listed in Table I for the various concentrations studied. There we have collected the values of the mean square center-to-end distance for the arms, ( k 2 the)mean , square radius of gyration, (h2), and the average value of the three eigenvalues, (Ai). In the final column we have computed the ratio (A,):( A2):(A3) which indicates the degree of asphericity of the chain. There are several points to note. First, these results are generally consistent with those in our previous paper, although in this study we find the threeand four-arm stars to be slightly more asymmetric due to a larger value of ( A l ) . This, in turn, results in slightly larger values of (h2). (Unfortunately, there is an error in Table I of ref 1. The arm lengths of the four-arm stars are listed incorrectly. In order, they should be 9, 12, 15, 18, and 21. An interpolation gives a value of ( R G 2of ) 136 for the four-arm star with arms of length 20. This is quite close to the value obtained here.) We feel that these results are within the statistical error of the simulations, which is usually 5-lo%, particularly since we are studying such small molecules. Second, the overall size of the molecule as ) slightly as the concentration inmeasured by ( R G 2 decreases creases. This agrees with the experimental results of Lodge and co-workers’ and with the previous simulation results of Smit and co-~orkers.~ Third, as expected, the asphericity of the molecules decreases as the number of arms increases. Fourth, for a given
A
A
4
0.009 0.009
03w
0.lW
0 W
C F i i 1. Plot of the center-of-massdiffusion constant, D/6, as a function of concentration, c, for the three probe molecules.
molecular topology, the asphericity increases slightly as the concentration increases. The increase in asphericity is mainly due to a decrease in the smallest eigenvalue, A3, as the concentration increases. The molecules are becoming more “platelike” at higher concentrations. This is somewhat unexpected. Intuitively, we would expect that the intramolecular excluded-volume forces would prevent a contraction along this highly compact direction. Instead, it appears that the molecule is relieving the excludedvolume stresses by becoming more two-dimensional. This trend is not seen in the data of Smit and co-workers. Instead, their data show a small decrease in the relative size of the largest eigenvalue which indicates that the stars were becoming slightly more spherical. Their simulation, however, used a much smaller star with arms containing only six monomers and showed a much different asphericity (10:4:1) at zero concentration. They also used a solvent of monomers rather than linear polymers. The differences between the two studies could certainly be due to the differences in the models or to the differences in molecular sizes. Since the increases in asphericity are small, it is also possible that they are insignificant compared to the statistical error. It is therefore difficult to assess the generality of our observation. We are also unable to answer the interesting question of whether this trend will continue at much higher concentrations of bath molecules or in star melts. We plan to investigate these questions in future studies. Because of the limited number of molecular sizes that we have been able to study, it was not possible to explore any of the equilibrium scaling properties of these molecules. This is something that we hope to pursue in the future. The values of the center-of-mass diffusion constant, D, are collected in Table I1 and plotted as a function of concentration in Figure 1. In qualitative agreement with the experimental results of Lodge and co-workers,’ we find a decrease in D as the concentration increases. This trend was also seen by Smit and c o - ~ o r k e r s .The ~ four-arm star shows a much weaker concentration dependence than the three-arm star, even though it is a much larger molecule. Because of its more compact structure, the four-arm star is probably less intertwined with the bath molecules and, therefore, is less affected by the increase in bath concentration. For larger molecules with longer arms, however, the concentration dependence might well be different. The eigenvalue relaxation times for the three different probe molecules are listed as a function of concentration in Table 111. These relaxation times characterize the dynamics of the shape fluctuations of the molecules. Again, the results obtained here
The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 3933
Shape Fluctuations of Uniform Star Polymers TABLE III: Eigenvalue Relaxation Times, Q,, as a Function of Molecular Size, N, and Concentration, c; Mean Segment Densities, p, of the Stars and the Products i k / p for AU Three Relaxation Times
N 41
c
0.00 0.19 0.24 0.29 61 0.00 0.19 0.24 0.29 81 0.00 0.19 0.24 0.29
TA,
TA1
TA)
5090 5960 6750 7020 5370 6270 6850 7190 5590 6870 7720 8540
2980 4000 4590 5030 3320 4300 4890 5520 3530 4750 5230 5960
1170 2040 2690 2860 1460 2260 2580 2930 2130 3410 4030 4560
P
lo3
1.27 1.38 1.49 1.59 0.717 0.758 0.831 0.823 0.514 0.561 0.585 0.665
TAlIP
TAl/p
TAjlP
6.48 8.19 10.02 11.15 3.85 4.75 5.69 5.92 2.88 3.85 4.51 5.68
3.79 5.50 6.81 7.98 2.38 3.08 4.06 4.54 1.82 2.66 3.06 3.97
1.49 2.81 3.99 4.54 1.05 1.62 2.15 2.41 1.09 1.91 2.36 3.03
0 j
e
e
0
e
A
-f- "
A A
N=41
Ai
1000,
0
TABLE I V Rotational Relaxation Times, i Y p and 8 Values, Obtained from the Fits to the KWW Function, as a Function of Molecular Size, N, and Concentration, c N C TY,lBI 7Y2lB2 TYjIB3 15911.24 17211.44 41 0.00 13611.25 0.19 i49ji.29 168) I .19 183ji .46 0.24 17611.11 18711.21 19511.38 19711.15 0.29 19311.19 20611.42 19311.56 15011.40 17911.34 61 0.00 16411.31 19711.49 0.19 18811.29 20611.49 18711.34 20111.15 0.24 21711.47 0.29 20311.33 208J1.18 21611.61 81 0.00 17711.43 18711.48 22011.57 0.19 18311.46 19511.36 23111.58 0.24 19811.45 20711.35 24211.56 0.29 21011.41 21611.34
are consistent with those obtained in our previous work. As expected, the largest eigenvalue has the longest relaxation time and the smallest eigenvalue, the shortest. These times become larger as the concentration increases. The concentration dependence is shown graphically in Figure 2 where we plot in rAi vs c. The concentration dependence of In qiis approximately linear, but there is a suggestion that the curves are leveling off at the highest concentrations. It is possible that the increase in relaxation time as a function of concentration is simply due to the contraction of the coil. To test this, we have computed the mean internal segment densities, p, of the test molecules assuming that the molecular shape is ellipsoidal. These segment densities are also given in Table 111. If the increase in relaxation time is due only to increasing segment density, the quotient T ~ ~ / which P , is listed in the final columns of Table 111, should be independent of concentration. Clearly, rAi is not proportional to p in general. For the four-arm star at the two highest concentrations, it is possible that such a relationship is beginning to emerge. While the compression of the coil does have some effect on the relaxation time, it appears that interactions with the bath molecules are a more important factor in the increase in relaxation time. Since the molecules studied here are small, this cannot be called an entanglement effect. It is, however, a direct consequence of the excluded-volume condition which reduces the number of allowed motions as the molecules become intertwined. Table IV lists the rotational relaxation times, ryi,and j3 parameters extracted from the fits of the rotational autocorrelation functions to the KWW function. While there is some scatter, the j3 parameters are all in the range from 1.1 to 1.6 with most values near 1.4. The values of f Y , and j3 obtained in this study are internally consistent but differ significantly from those obtained in our earlier paper. This is due to the fact that in the present work we used longer simulation runs to calculate the correlation functions. Therefore, the KWW function was fit over a much larger range in time, approximately twice that used in ref 1. This results in a larger j3 parameter and consequently a smaller T ~ This difference indicates that one must be careful in using the KWW function to fit data and in interpreting the resulting parameters. Since all the fits were done consistently in this work, we feel safe in making some qualitative observations about the
0
0
e
e
0 e
A A A
N=61 10004 0.000
0.1 00
0300
0.
C
0
0 e
0
e A
x
41
w
0
e A
A
-k
1h
10001
N=B1
concentration dependence of the rotational relaxation. First, as expected, the rotations slow down with increasing concentration. It is interesting that the largest molecule, the four-arm star, has the smallest fractional increase in its rotational relaxation times over the concentration range studied, while the linear molecule has the largest. This is probably due to the difference in shape. The four-arm star is less aspherical and, therefore, will encounter less resistance to rotation from neighboring chains. (It should be remembered that there is no solvent friction in this model.) It will be interesting to see whether this is also true for stars with much longer arms. Conclusion
~
.
In this paper we have studied the concentration dependence of the equilibrium and dynamic properties of three- and four-arm star polymers. We have found that the instantaneous shape is a particularly revealing way of understanding these nonlinear molecules. For example, increasing the concentration of the linear bath molecules changes not only the size but also the shape of the molecules. The degree of asphericity then affects the rotational motion of the molecule and its concentration dependence. While the shape of the molecule is not experimentally accessible, at least at present, it seems to be a very useful theoretical concept for understanding nonlinear polymers. Our simulations have provided an interesting picture of the molecular shape of both linea8 and nonlinear polymers and an alternate view of the internal
J . Phys. Chem. 1992, 96, 3934-3942
3934
motions of these molecules as a combination of shape fluctuations and rigid body rotations. The advantage of this view is that it can be applied to molecules of any molecular topology and therefore can lead to a unified picture of polymer motions. We hope these simple simulation results will provide stimulation for (8) Hahn, T. D.; Ryan, E. T.; Kovac, J. Macromolecules 1991,24, 1205.
the development of this unusual perspective on polymer structure and dynamics. Acknowledgment. We thank the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences, for financial support of this work. We also thank the University Of Tennessee Computer Center for their continuing support.
The Poisson-Boltzmann Equation for Aqueous Solutions of Strong Polyelectrolytes without Added Salt: The Cell Model Revisited Michel Mandel Department of Physical and Macromolecular Chemistry, Gorlaeus Laboratories, Leiden University, Leiden, The Netherlands (Received: September 9, 1991)
An extensive discussion of the analytical solution for the Poisson-Boltzmann equation in cylindrical symmetry for strong polyelectrolytes in the cell model is presented. The reduced mean electrostatic potential u at finite dilutions is discussed in terms of its dependence on the polyelectrolyte equivalent concentration C,, its charge density parameter [,and the distance of closest approach a of the counterions to the polyion. It is shown that in the limit a 0 counterion condensation is expected. For more realistic nonzero values of a, the reduced potential u at a given relative position r / R in the cell with radius R is practically independent of the linear charge density for [ > 2, but its value depends on the product aZCc.The value u(a) of the reduced potential near the surface of the polyion is (-dependent, however, under the same conditions. A large fraction
-
of all the counterions in the cell accumulate, on the average, in the neighborhood of the polyion, this fraction being larger the higher [ is and the lower the product a2Ccis. The fraction of ions accumulated between the polyion surface at a and a distance from the polyion axis equal to the screening length 1/x is high, reaching values exceeding 80% and being higher the smaller a2Ccis. This fraction of counterions (the “associated” counterions) occupies a smaller part of the total cell volume than the counterions situated between l / x and R, which are characterized by a relatively low electrostatic interaction energy with the polyion, u < 1 (the “free” counterions).
Introduction
Solutions containing highly charged macromolecules (polyelectrolytes) and no added low molar mass salt are characterized by very specific equilibrium and dynamic properties, which are far from being quantitatively understood.’ One of the reasons for a behavior which differs considerably from that of polyelectrolyte solutions containing an abundant amount of added salt is, probably, the much weaker screening of the electrostatic interactions. In the latter the small ions supplied by the salt can to a large extent shield off the charges located on the macromolecular chain, forming ion atmospheres around each of them. In the absence of salt, the number of ions in the solution is determined by the electroneutrality condition only and single ion atmospheres around each charged site on the polyion are not likely to occur. The screening of the electrostatic forces between all charged particles involvedan essential condition for the stability of the system-must arise in a different way, but it is not yet well established how this proceeds. This is at the origin of the difficulties underlying the theoretical treatment of polyelectrolyte solutions without added salt which, at present, still challenges polymer scientists. Already many years ago, an attempt to treat this theoretical problem by an analogous procedure as used with other systems containing charged particles has been tried. Starting from the idea that, at least at sufficiently low concentrations, the weakly screened charged macromolecular chains must have an extended conformation, the Poisson-Boltzmann equation for the mean electrostatic potential around a cylindrical polyion surrounded by counterions was s o l ~ e d . ~The - ~ electrostatic contribution to the (1) Mandel, M. Polyelectrolytes. In Encyclopedia ofPolymer Science and Engineering, 2nd. ed.; Mark, F. H., Bikales, N. M., Overberger, C. G . , Menges, G., Eds.; Wiley: New York, 1988; Vol. 11, 739. (2) Alfrey, T.; Berg, P. W.; Morawetz, H . J. Polym. Sci. 1951, 7, 543.
0022-365419212096-3934%03.00/0
solution free energy could thus be estimated: In particular the solution of the Poisson-Boltzmann (PB) equation in the cell model4,’ made it possible to explain semiquantitatively some important thermodynamic properties of these systems at high dilutions, such as the osmotic coefficient. Nevertheless, in more recent years this solution of the PB equation for polyelectrolyte systems without added salt has received less attention than its counterpart for solutions containing an excess of salt, notwithstanding the fact that for the latter only numerical or approximate solutions are available. In particular, no extensive study has been attempted to confront results derived from the solution of the PoissonBoltzmann equation, with predictions based on the condensation concept. The concept of condensation, first explicitly introduced by Imai,6-7was later fully exploited by Manning8-lo to derive his rather successful limiting laws for polyelectrolytes represented as “line charges”. For polyelectrolyte solutions with an excess of added salt, both theoretical approaches have been extensively discussed and compared (see,e.g., in the few examples (3) Fuoss, R. M.; Katchalsky, A.; Lifson, S.Proc. N a f l . Acad. Sci. U S A 1951, 37, 579.
(4) Lifson, S.;Katchalsky, A. J. Polym. Sci. 1953, 13, 43. (5) Katchalsky, A. Pure Appl. Chem. 1971, 26, 327. (6) Imai, N.; Ohinshi, T. J. Chem. Phys. 1959, 30, 1115. (7) Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1970. (8) Manning, G. S. J. Chem. Phys. 1969, 51, 924, 934, 3249. (9) Manning, G. S.Annu. Rev. Phys. Chem. 1972, 23, 117. (10) Manning, G. S. In Polyelectrolytes; Selegny, E., Mandel, M., Straws, U. P., Eds.: Reidel: Dordrecht, 1974; p 9. (1 1) Stigter, D. Prog. Colloid Polym. Sci. 1978, 65, 45. (12) Westra, S.W. T.; Leyte, J. C. Ber. Bunsen-Ges. Phys. Chem. 1979, 83, 672. (13) Anderson, C. F.; Record, M. T., Jr. Biophys. Chem. 1980, / I , 3 5 3 . (14) Kure Klein, B.; Anderson, C. F.; Record, M. T., Jr. Biopolymers 1981, 20, 2263. (15) Ramanathan, G . V.; Woodbury, C. P., Jr. J. Chem. Phys. 1982, 77, 4133.
0 1992 American Chemical Society
