Static and dynamic properties of two-dimensional polymer melts

Transcription

1 Static and dynamic properties of twodimensional polymer melts I. Carmesin, Kurt Kremer To cite this version: I. Carmesin, Kurt Kremer. Static and dynamic properties of twodimensional polymer melts. Journal de Physique, 1990, 51 (10), pp < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1990 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 We 1 Phys. France 51 (1990) MAI 1990, 915 Classification Physics Abstracts c Static and dynamic properties of twodimensional polymer melts I. Carmesin (1) and Kurt Kremer (2) (1)Max PlanckInstitut für Polymerforschung, D6500 Mainz, F.R.G. (2)Institut für FestkOperforschung, KFA Jülich, D5170 Jülich, F.R.G. (Reçu le 8 novembre 1989, accepté le 7 février 1990) 2014 Abstract. present a detailed analysis of the properties of dense linear twodimensional polymer chains. The systems are simulated by the recently developed bond fluctuation method. We investigate systems of up to 80 % density. The chain length and densities considered cover the crossover from the expanded single chain limit to a dense polymer melt. The chains completely segregate and follow a power law of the chain extension R2 >~ N203BD, N being the number of bonds of the chains and 03BD 1/2. While for single = two dimensional chains the Rouse modes are not eigenmodes they are eignemodes for the chains in a 2d melt. Relaxation functions and mean square displacements display typical Rouselike behavior, with the distinction, that the prefactors for the various relaxation processes differ. 1. Introduction. Most experimental investigations of polymers consider chains in three dimensions. This is the most natural situation, however, there are many exact theoretical results for d = 2 [13]. Thus, a twodimensional polymer system would be an ideal testing case for many modern theoretical concepts, such as renormalization group and conformal invariance methods. Besides this there is not much known about the dynamics of such systems [4]. Again a twodimensional system should be ideal to investigate the Rouse model experimentally. For d 3 hydrodynamic = interactions always dominate the behavior of thé chains in solution. For d = 2, e.g. chains near a surface, this probably is not the case since long range hydrodynamic interactions may be completely screened out. For two dimensional melts the situation is especially interesting. The topological interaction is completely different compared to the d = 3 case. Following recent theoretical investigations the chains should collapse and segregate [1]. However, how does the shape of the chains look like? Do they appear as ideal random walks, although the topological aspect of the excluded volume interaction is a dominant aspect for d = 2? Recent numerical simulations were not able to discuss this question in detail [68]. Since they use bond breaking simulation methods their samples are,polydisperse. As a consequence there is a mixing of long and short chains leading to an effective expansion of the longer chains. The classical papers of Wall et al. [9] were not able to address the above questions. Baumgârtner [10] used the reptation algorithm to analyse the Article published online by EDP Sciences and available at

3 916 differences between the ideal random walk structure and the 2d melt chains. There the analysis was confined to the ratio of the radius of gÿration and the end to end distance. Bishop et al [11] use a Brownian Dynamics method in order to analyse 2d melts at rather low densities and for short chains ( N 50, p 0.50). = In addition not much is known about the dynamics of dense 2d systems. A recent investigation of the Rouse/Zimm model of 2d chains [12], of course, could not take into account the topological aspects of the problem. Experimentally these systems are also not completely out of reach. Modern technics in producing various polymers could e.g. construct chains with small side branches preferring two different solvents. If these solvents are immiscable, such polymers would be located at the interface between the two solvents. One can also think of chains between lipid monolayers. Also recent developments in the surface coating of mica sheets [13] may lead to stricly 2d systems. Finally, of course, the classical approach of spreading chains out of a volatile good solvent onto a liquid air interface may be used. After some equilibration time one then can compress the film towards a semidilute or dense layer. Of course, there one ltas to take care not to get double or multi layers. Considering these developments, we think that it is very desirable to have a better understanding of such systems. Thus we performed an extensive numerical study of a 2d manychain system. To do this we made use of a recently developed Monte Carlo algorithm [8, 15]. This bond fluctuation method was used to analyse the properties of an isolated 2d chain. The way the moves are constructed also allows to study the dynamics of 2d polymers [15]. We use this method to study systems with up to 80 % density and chain length of up to N = 100 monomers. A short preliminary account of this work is given in reference [16]. Chapter two gives a short description of the algorithm and the systems considered. Then we describe in some detail the équilibration process. Chapter three gives our results for the statics, while chapter four gives the dynamics. Finally chapter five contains the conclusions and a short outlook. 2. Model and methods. In order to investigate the dynamical properties of a 2d polymer system by mean of Monte Carlo, we have to assure that the algorithm displays Rouse dynamics for the non reversal random walk [8]. This random walk assures that consecutive monomers are not allowed to sit on top of each other. Besides this short range repulsion along the chains no additional interaction is taken into account. For such a system one can use the complete set of moves as for chains with full excluded volume. This then gives a good and dependable check on the static and dynamic properties of an algorithm. For dynamic Monte Carlo algorithms the typical requirements for the moves are, that they create new bond vectors within the chain. Standard lattice methods for d = 2 only create new bond vectors at the ends, which then diffuse into the chain. This enhances the relaxation times artificially. To overcome this problem the bond fluctuation method was introduced recently [15]. It combines the advantage of a lattice simulation, which means e.g. fast algorithms on both scalar and vector machines, with continuum approaches, so that new bond vectors are created in successful moves. Figure 1 illustrates the method. Each monomer occupies four lattice sites. The bond length 1 is restricted to be In addition, a lattice site can never be occupied by more than one monomer. For a move, a monomer is selected at random, then it jumps at random by one lattice distance into one of the four lattice directions. If the new position complies with both, the bond length restriction and the excluded volume restriction, the move is accepted and otherwise rejected. this leads to a natural time scale ro of one attempted move per monomer of the system. This algorithm was shown to give Rouse dynamics for the single random walk chain case [15].

4 Illustration 917 Fig. 1. of the bond fluctuation method. Ibe typical moves are indicated. Note that the initial state must not contain crossed bonds. Since the bond length constraint is set to prevent crossing of bonds such a configuration would be preserved forever. For dense systems the first blocked configuration, which is a situation the system cannot reach or leave, occurs at density exceeding 80% for infinite chains. Since this configuration is ordered, its probability is vanishing compared to the other random coil states. in the subsequent investigation we only use densities of up to 80%. Since our chains are finite (N 100) there will be no problem of unresolvable blocked configurations. (There is a very small probability that a random initial configuration of a chain is blocked by itself. These configurations have a local density of one. The initial conditions employed in the present work excluded such configurations automatically). The simulations were performed on a 100 x 100 square lattice with periodic boundary conditions and densities with 20%, 40%, 60%, 80% of the lattice sites,occupied. We analysed chains with 20 N 100. The N = 100 and 80% density system then constains 20 chains. The calculations were performed on a Fujitsu VP 100wector processor. typically 100 statistically independent systems of thé same kind were simulated in parallel The vectorization was set in a way that one monomer in each system was moved simultaneously. By this approach, even during the standard simulations part, we were able to vectorize the program efficiently [18]. lb start our systems, we filled the lattices with ordered arrays of chains of length N = 100 monomers and 80% density. The chains all had a Ulike form and were pairwise nested into each other. Figure 2a gives a snapshot picture of such a system shortly after the simulation started. One easily identifies from this the initial type of configuration. Figure 2b gives a typical snapshot picture of an almost equilibrated structure of the same sample. The three different marked chains give the typical cases. 1Bvo chains have a fairly smooth round surface while the third is still strongly stretched. The stretched one (which is the extreme of that sample) displays a rather unfavourable configuration indicating that the system is not yet in complete equilibrium. During the actual calculations, we made sure that both the x and the y component of the Rouse modes (see Chap. 3)

5 Typical 918 Fig. 2. (a) Snapshot configuration of a p = 0.8, N = 100 system a short time after the initial state. The internested structure is clearly displayed. (b) Snapshot configuration of the structure of figure 2a, where the system was almost equilibrated. Fig. 3. equilibrated structures for a) p = 0.2, N = 100; b) p = 0.4, N = 20. has the same amplitudes while in the very beginning they were very asymmetric due to the initial condition. This provided a sensitive test of equilibration. The short chain/low density systems then were made from the N = 100, p = 0.80 sample simply by cutting bonds and eliminating chains. As it turned out, the computing time for equilibration then was much shorter than if we

6 values 919 had started the systems from scratch again. The special initial condition was used to make sure that no memory was kept during our simulation and to check the proposed segregation [1]. The initial internested structure is specific enough to follow its decay in détail. After equilibrating the systems we followed the motion of the chains at least up to a distance of their own diameter. Considering that we always ran 100 systems in parallel this gives even for the p 0.80, = N = 100 system 20 x 100 independent diffusion paths, which is enough to provide data of good accuracy. Figure 3 gives two other typical configurations at density p 0.20 = and p = Static properties. In polymeric melts the excluded volume effect, which causes the expansion of chains in good solvent [2, 19, 20], is expected to be effectively screened out by the interaction among the different polymers. This is known theoretically to hold also for d = 2 [1, 2]. The major différence between d = 2 and d > 2, however, is that, due to the topological constraints in d 2, = the chains cannot entangle. If they want to interpenetrate, this only is possible by partial alignment of the chains. This certainly causes a reduction of entropy. The consequence is, that the chains are going to segregate. One of the interesting questions now is, how strong is the segregation depending on chainlength and concentration. The equilibrium structure of chain is supposed to be like a Hamiltonian walk. In order to investigate the structure of such a polymer melt, we first check the mean squared end to end distance R2(N) > and the radius of gyration R 2 (N) >. With rcm being the center of mass of a polymer and ri the position vector of the ith monomer they are given by Table 1 gives the results for the various densities and chain lengths considered, while figure 4 shows the data for p = 0.40 and p = 0.80 for several chain lengths. For ideal chains one expects làble I. for RG > and R2 > for the various cases considered. The error bars given for R2 > are estimated from the fluctuation between the 100 statistically independent systems, which ran simultaneously.

7 Logplot 920 Fig. 4. of R2 > and RG > vs. N for p = 0.80 (a) and p 0.40 (b). The indicated slopes = of 2v =1 show that the expected behavior is reached quite clearly. R2 > / 4 > = 6 and R2 > o:n211, V 1/2 [2, 20]. = As can be seen from the data, for N = 100 compared to the expected ratio R 2> is about 10% too small. Sjrnilar but stronger effects were also found in [10]. This certainly is a consequence of the fact that the chains rather homogenously fill a little fluctuating disk, while the random walk is a fractal object displaying self similarity [2]. The expected power law with v 1/2, however, = is fulfilled very well. From scaling for d = 2 one would expect R2(N) >1/2= Nvf (N/p2) (here v = 3/4 the single isolated chain value [2] has to be used). làldng the data from table 1 we find that the data for f (x) reasonably well collapse on a single curve (besides N 20, 25), however, = the slope is not the expected one. Ideal scaling gives a slope of 1/2 for f (x) in a log log plot, while here the slope is too large. This can be explained by the following: Scaling is only valid for the limits x = N / p2 " const but N + oo, p 0. Hère we certainly are out of this regime. Similar effects were also seen for d = 3 [21]. There is another interesting aspect about how dense the systems really are. For regions where the chains already interact it makes sense to write where all the excluded volume effects are taken into the variable density dependent persistence length p2 p (p) > 1/2. Using (2) we get land Ip as function of density as given in table II. This decrease of lp and also, as also shown by figure 5, certainly is an effect of high density. Up to about p = 0.20 there seems to be no effect on 1. For higher densities the overall chain compression reduces the average bond length.

8 (a) Persistence 921 7àble II. length and mean bond Iength for various densities following the definition of equation (2). For p = 0 equation (2) requires Ip = 00 since there the exponent is v = 3 /4 instead of 1/2, as is described by the abovementioned " scaling function. Fig. 5. Plot of the persistence length lp vs. density p due to equation (2). (b) length Î vs. density v. Plot of the mean bond This leads us back to the above discussion of the scaling of R2 ( N ) > and lîâ (N) >. Scaling assumes that the internal structure of the chains do not change. From the data, however, we see that not only the persistence length varies with density, but also the average bond length itself. The compression of the chains is so strong, that there is no space for a selfavoiding walk like blob, which is required for the validity of density scaling. Indeed, if we normalize R2 > not only by N2" but also by 2 > at least for N 100 = between p = 40% and p = 80% the expected slope shows up. Altogether this shows that the systems are in the high density limit. To look more into the details of the internal structure of the chains we analysed both, the structure function of the

9 922 chains and the Rouse modes. First, we turn to the static scattering function of the individual chain where the index 1 k denotes the spherical average over the orientation of k. Following the standard scaling theories S(k) should, for d = 2 and v = 4 for the isolated chain [2], display scattering law the fractal here e is the typical screening length on which the chain still is expanded. From scaling e vl(lvd) _ 3/2, however as one could see from the data of R2 >, R 2 > this screening length does not follow that power law for the high densities investigated hère. Thus we hardly can expect the occurrence of a k 4/3 regime for the highest densities considered. Figure 6a shows S(k) for N = 100 and p = 0.20 leading to 03BE ~ 30. Going back to figure 5 this is a reasonable number, giving n = (l212p/e2) ~ 110 monomers per screening distance. It means for p = 0.20 even the largest chains considered just interact with each other. This also explains, why there is no distinct k2 regime for p = 0.20 rather than the typical overshooting effect of the single chain. The overshooting, coming from the fluctuating ends was discussed earlier[15] and can also be found for d = 3. For d = 3, since the chain can penetrate through itself this effect is much weaker. With increasing density this effect is more and more reduced. Simultaneously, even the onset of the k4/3 region disappears, indicating that e is significantly reduced to about one or two bond lengths for p = The sequence of figures 6ac nicely displays this. For p = 0.80 we give a scaling plot of S(k) vs. k for various N with the normalization that S(k 0, N 100) = = = 1. The data reasonably collapse onto a single curve. However, following the snapshot pictures of the introduction this is somewhat surprising. There the cliains seem to homogeneously fill diskshaped regions. Such disks, however, would display the (d = 2) Porod scattering of S(k) k3. Obviously the shape fluctuations are still strong enough to prevent such a scattering behavior. In order to see this we have to consider the overall structure of the system. With R2 >~ R 2 >N N the average density throughout the chains approaches a constant for distances Ar» e. Thus for d = 2 a given chain only directly can interact with a constant number of neighbors. For a dense solution of disks typically one would expect 6 neighbors giving a triangular lattice like short range order. If we assume the chains to form s herical objects of radius r with homogeneous density they would cover a disk of radius r = 2 12 RG. Using this, we can estimate the space which would be occupied by such chains. Paking the data of table 1 the chains typically would need about 220% for (N = 100, p 0.80) = to 150% for (N 20, = p 0.20) = of the space available. However, this is simply impossible. Therefore, there must be strong fluctuations of the chain shape. typically we find the chains to have between 5 and 7 neighbors averaging to 6. This leads to a characteristic packing pictured in figure 7, as expected by figure 2. Although their overall density is relatively constant there must be strong shape fluctuations. For ideal random walks one has (d 2) = For the radius of gyration one typically gets a fluctuation reduced by a factor of two [22]. find Here we

10 (a,b,c) 923 Fig. 6. Scattering function S(k) of the individual chains for various densities and N = 100 as indicated in the figures. The data are averaged over randomly taken orientations of the scattering vector k. (d) Scaling plot S(k, N)/ (1002/N2) vs. k for N = 25, 50, 100 and p = 0.8.

11 924 Fig. 7. Illustration of the average local packing structure. which is close to the ideal random walk value. Although the chains are densely packed, this suggest that these fluctuations allow for a relatively high mobility of the chains. As we shall explain in the next chapter this is the case. These fluctuations which indicate that the chains almost look like ideal chains, also propose that it might be useful to analyse the Rouse normal modes of a random walk. For a discrete system they are given by [23] Iÿpically, this one dimensional Fourier transform along the sequence of the chain is done with periodic boundary conditions, corresponding to a cyclic chain. By subtracting the second term we cut the ring. Note that we only can expect the Rouse modes to be eigenmodes for ideal chains. In principle this would mean that the monomers arbitrarily can pass trhough each other. However, we find the Rouse modes are eigehmodes of the chains. Within the error bars the cross correlations Xp(t)Xq(t) >p#q are zero. Even the amplitudes of the modes follow the prediction of the Rouse model As figure 8 shows they follow a common curve for p = 80 and p = 0.40 for the various chain lengths. The expected slope is reproduced very well and only for small n/p2 values and p = 0.40 significant deviations can be found which is consistent with the results of S(k). Forp large the length scale is reduced to the region where the selfavoiding walk behavior starts to dominate. There deviations are expected and were also seen for d = 3 [24]. This also shows that for p = 0.80 already N = 25 reaches the asymptotic regime. In addition the ratio of the amplitudes for the two densities pictured is the same as for the squared radii of gyration. Similar to the fluctuation of the radius of gyration we can calculate 6z: = z X4 > X2 > 2)1/2 / X2 >. Since p p p Xpp mesures a typical distance in space along the chain we expect these amplitudes to be about twice

12 Scaling 925 as large as the one for RG. This is essentially similar to the end to end distance R2 >. For all densities and chain lengths and pmodes which follow the N/p2 power of the amplitudes we find 0.8 b., 1.0 in very good agreement to the above line of arguments, demonstrating the overall consistency of the picture. Fig. 8. of the mode amplitudes Xp (N, p) > N/p2 vs. for p =0.80 and p =0.40. The ratio of the two sets of data agree with the ratios of R2 > and R2 > as shown in table 1 for N/p2 > 5. The date give the average over the Cartesian coordinates. The absolute value is somewhat smaller than expected from ideal chains. 4. Dynamics. After finding that the static properties of dense polymers very well compare to ideal chains, it is especially interesting to check the chain dynamics. Since the chain cannot entangle, the reptation model certainly is not an appropriate description. On the other hand in order to move around the chains as well as the monomers have to get along each other. This puts strong constraints onto the motions of the individual monomers. However, how does this affect the overall dynamical behavior? To investigate these questions, we first analyse the mean square displacement monomers and the diffusion constants of the individual chains. of the individual Figure 9 shows the data of the diffusion constant D for p = 0.40 and p 0.80, = as given in table III. Following the Rouse model we expect D KBTIN(, = where kbt is the temperature and ( the bead friction. Since there 1 o temperature in our system but only the acceptance rate, we can deüuce the bead friction from thr acceptance rate by cônsidering the ratio KBT/(. We shall discuss

13 Diffusion 926 Fig. 9. constant 4D = lim g3(t) t>oo vs. number of monomers N. this later in the context of the modes. What is surprising is that the diffusion constant seems to display a weaker Ndependence than the Rouse model suggests. Any crossover to reptationlike behavior would cause the inverse effect, namely an increasingndependence. Certainly we would need a more extensive analysis of a larger system. As is showm later, such an acceleration does not show up in the meansquare displacements of the monomers. Thus since for this estimate of D large time intervals were used, figure 9 probably shows a residue of the equilibration. Thus Table III. Diffusion constants 4D for various N and density p = 0.4 and p =0.8.

14 Mean 927 for the motion of individual monomers N = 50 and p = 0.40, 0.80.For N = 100 the data show the same behavior. we think that the data are consistent with D N1. Figure 10 gives examples Fig. 10. square displacement gl(t) and g2(t) and the diffusion of the center of mass of the chain g3(t) for p 0.80 (a) and = p 0.40 (b) and N 50. The = = two sets of data for Yl/ Y2 give the results for middle (lower data) and end monomers (upper data). The difference in the motion between inner and the outer monomers decreases with increasing density. What is especially striking is that the difference between the mean square displacements gl, 92 of the total chain, given by and corresponding data of gl, 92 for individual inner monomers is much smaller than for d = 3. For d = 3 this effect is quite dramatic since this shifts this visibility of reptation in these quatities far beyond the entanglement length [2325]. However, similar to the chain confined to a straight tube, g2forinner monomers saturates much earlier than the mean square displacement gl. Such an effect was first observed for the chain in the straight tube [26] and seems to be typical for such constrained geometries. This certainly must be an effect of the segregation of the chains and the confinement due to the surrounding compared to the 3d case. For both cases we find a nice t112 behavior for gl, 92 up to the longest relaxation time of the chain. After that time the diffusion of the whole chain takes over. Table IV gives a collection of relaxation times. So far all data seem to reasonably agree to the Rouse model. The most crucial test for the dynamic properties is the analysis of relaxation rates of the individual modes. For this we calculated both, the relaxation

15 Relaxation 928 Thble IV. times Tp for the first three modes p for chains at densities of p The error bar of the data is around 15%. = 0.4 and 0.8. rates of the Rouse modes Xp as well as the autocorrelation function of the radius of gyration squared. Figure 11 shows the relaxation function for N = 50, p = 0.40 as an example for the first three modes. They display a good single exponential behavior for longer times, which enables us to calculate the relaxation times which enables us to calculate the relaxation times Tp. For the Rouse model one expects Figure 12 gives a scaling plot of the relaxation times 7p from table IV. They all follow the single curves and roughly follow a slope of (N/p)2. From these data as well as from the diffusion constants D we directly estimate the normalized bead friction (IKBT. Using the data of table II for l and lp and table III for D the two friction constants do not agree very well. We find that the from the diffu relaxation of the individual modes is considerably slower than one would expect sion. For d = 3 melts it was found [25] that they agree very well. Taking e.g. N 50, = p = 0.8 (here the statistic is better than for N 100) = the modes give (IKBT 6.4 x 102 while D gives 1.5 x 102, giving a ratio of ~ 4.3. For p = 40% this ratio is reduced to a value of less than 3. Thus with increasing density this deviation increases as well. However, this reflects that the monomers cannot interpenetrate freely, as required by the Rouse model. Thus, although the Rouse modes are eigenmodes and 91 does not show significant deviations from Rouse, the diffusion seems to be governed by shape fluctuations, which are faster by a constant factor than the internal reorganization which would allow the Rouse modes to relax. Similar distinctions in time scale have been proposed for star polymers [29]. Finally, let us check the overall relaxation of the radius of gyration. The autocorrélation function shows a single exponential decay similar to the modes. A more detailed inspection shows that rouphlv In order to obtain the precise N and p dependence much more estensive simulations are needed. The relaxation times are given in table V. Since the fluctuation of RG are the shape fluctuations they also dominate the diffusion. Following Doi and Edwards [4] this is governed by the slowest

16 Examples Relaxation 929 Fig. 11. of a typical mode relaxation plot for N 50, p = = 0.4. The time is given Carlo steps (mcs). in Monte Fig. 12. chàin lengths, times of the first three Rouse modes vs. N/p for p = 0.40 and p = 0.80 and different as indicated.

17 Acceptance 930 Thble V. Longest relaxation times of the autocorrelalion function R$(t)Rl(0) (Rô)2 >) for various chain lengths and densifies p = 0.4 and (flõ(0»)2» /( RG > The N2 dependence is rougihlyfulfiued. The error bar of the data is expected and 15%. to be between 10% Fig. 13. rate A(S) of attempted moves vs. density p. The insert gives the normalized deviation of the acceptance rate A(S) with respect to the acceptance rate  = A( p 0) of an = inner monomer of a single isolated chain. mode. Using again equation (11) with p = 1 shows that we arrive at a bead friction which is in very good agreement to the diffusion data, while the modes give a different result. This rather regular behavior indicates that there are no precursors of the glass transition yet in agreement to the mode analysis. The density dependence is directly seen also from the acceptance rate of the moves. For lattice gases it is known that due to back/hopping corrélations the acceptance rate decays slower than the increase of relaxation times suggest [27]. Figure 13 shows the relative acceptance rate of attempted moves compared to a single free chain. Here we find the same

18 931 power law with density as for THe; indicating that back hopping only enters in the prefactor dominated by the intrachain correlation of monomers nearby along the chemical sequence polymers. and is of the 5. Conclusion. We presented a detailed study of the properties of a melt of twodimensional polymer chains. By the use of the bond fluctuation method we were able to study both static and dynamic properties of the chains. For statics we found that the chains, although they cannot cross each other, display typical random walk behavior for many uantities like R2 >, RG >, S( k) and the Rouse modes. As function of density the decay of R, Rb shows that we are explonng the hmit of high density melts. The chains segregate comptetety and the ends seem only to ftuctuate on small scales. Nevertheless the overall shape fluctuations turn out to be sufficient to result in a scattering law S(k) V k2 instead of the Porod scattering. The amplitude of these shape fluctuations is of the same order as for ordinary random walks. For the lowest density considered we nicely find the inner k4/3 regime for short internai distances. This is important with respect to recent ideas of the ecollapse transition of polymers [28]. Considering the present result for these concepts the disorder of the defects in the lattice, althougt they are annealed, is crucial. Otherwise the scattering function of our systems would show a very different behavior. Although the overall properties of the chains are the same as for ideal Gaussian chains there are some distinct differences with respect to the inner structure, such as the uniform density in the chains or probability of monomers approaching each other. The latter quantities have been calculated [1] and a future investigation should be able to check this. The most unexpected results were found for the dynamical properties. The Rouse model does not consider any other interaction betweeen the monomers than the chain connectivity. Considering that this requires monomers crossing each other freely, we found it rather surprising that the 2melt almost perfectly reproduces the Rouse model. There is no sign of decrease of mobility due to topological constraints. Ibe only,, but significant deviation is given in the time scales. Although the mode relaxation as well as the diffusion give the same N dependence of the relaxation time, there is a difference in the prefactor of about 4 for p = 0.8. It shows that diffusion via shape fluctuations is faster than internal structure relaxation. This suggests that the similarity to the Rouse model is rather accidental since the 2d exponents for the dense system and the related quantities exhibit the same power law as the ordinary random walks. We therefore think that it is especially interesting to investigate the chain dynamics of a collapsed 3d chain. There many interesting deviations might occur, while for the single collapsed 2d chain we do not expect different behavior compared to our present findings. Acknowledgements. K.K. acknowledges support by Nato travel grant 68/860 and the hospitality of the material department of the university of California, Santa Barbara, where most of this paper was written. We are grateful to K Binder, B. Dünweg, R. Hilfer and R Pincus for interesting and helpful discussions.

19 932 References [1] DUPLANTIER B., SALEUR H., NucL Phys. B290 (1987) 291 and references therein ; DUPLANTIER B., J. Phys A19 (1986) [2] de GENNES P.G.,Scaling Concepts in Polymer Physics, (Cornell Univ. Press, Ithaca, N.Y.) [3] NIENHUIS B., Phys. Rev Lett. 49 (1982) [4] DOI M., EDWARDS S.F., The Theory of polymer Dynamics (Clarendon Press, Oxford) [5] MANSFIELD M., J. Chem. Phys. 77 (1982) [6] TUTHILL G.F., J. Chem. Phys. 90 (1989) [7] REITER J., ZIFFERE G., OLAJ O.F., preprint (1989). [8] For a general overview about lattice simulations methods for poymers see KREMER K., BINDER K., Comp. Phys. Rept. 7 (1988) 259. [9] WALL F.T., CHIN J.C., J. Chem. Phys. 66 (1977) 3143; WALL F.T., SEITZ W.A., J. Chem. Phys. 67 (1977) [10] BAUMGARTNER A., Polymer 23 (1982) 334. [11] BISHOP M., CEPERLEY D., FRISCH H.L., KALOS M.H., J. Chem Phys. 75 (1983) [12] MUTHUKUMAR M. J Chem. Phys. 82 (1985) [13] GRANICK S., private communication. [14] Such a method is the standard method to measure the 2d exponents. see e.g. VILANOVE R., POUPINET D., RONDELEZ F., Macromolecules 21 (1988) [15] CARMESIN I., KREMER K., Macromolecules 21 (1988) [16] CARMESIN I., KREMER K., in proceedings of ILL Workshop on Polymer Dynamics, Proc. in Physics, Eds. D. Richter, T. Springer (Springer Heidelberg) [17] for a detailed discussion of such ergodicity problems see also: MADRAS N., SOKAL A.D., J. Stat. Phys. 47 (1987) 573. [18] More than 90% of the code ran on the vector unit of the VP100. Besides this the VP100 does not give further information regarding the effectivity of the vector unit being used. [19] There is an extensive experimental and numerical literature devoted to 3d melts. For simulations see references [8,25] and K. Binder preprint (1989). [20] FLORY P.J., Principle of Polymer Chemistry (Cornell Univ. Press, Ithaca, N.Y.) [21] See e.g. KREMER K., Macromolecules 16 (1983) [22] LYKLEMA J.W., KREMER K., J. Phys. A19 (1988) 279. [23] KREMER K., GREST G.S., CARMESIN I., Phys. Rev. Lett. 61 (1988) 566. [24] DIAL M., CRABB S., CRABB C.C., KOVAC J., Macromolecules 18 (1985) [25] KREMER K., GREST G.S., preprint J. Chem. Phys., in press (1990). [26] KREMER K., BINDER K., J. Chem. Phys. 81 (1984) [27] KEHR K.W, BINDER K., Application of the Monte Carlo Methods in Stat. Physics, Topics Current Phys. 36 and references therein, Ed. K. Binder (Springer Verlag, Heidelberg) [28] DUPLANTlER B., Phys. Rev. A 38 (1988) 3647; DUPLANTIER B., SALEUR H., Phys. Rev. Lett. 59 (1987) 539. [29] GREST G.S., KREMER K., MILNER S., WITTEN T.A., Macromolecules 22 (1989) 1904.