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Langmuir 2007, 23, 8678-8680
Microphase Separation in Two-Dimensional Athermal Polymer Solutions on a Triangular Lattice Piotr Polanowski† and Jeremiasz K. Jeszka*,‡ Department of Molecular Physics, Technical UniVersity of Lodz, 90-924 Lodz, Poland, and Center of Molecular and Macromolecular Sciences, Polish Academy of Sciences, 90-363 Lodz, Poland ReceiVed April 20, 2007. In Final Form: June 29, 2007 The results of Monte Carlo simulations of 2D polymer solutions are presented. The simulations were performed under athermal conditions for long chains (up to 1024 beads) over a full range of polymer concentration φ, explicitly taking into account the solvent molecules. The results obtained for short chains (N e 256) are in good agreement with previous simulations whereas for long chains microphase separation is observed below φ ) 0.6. This phenomenon is attributed to strong excluded volume interactions in 2D systems. A sort of interpenetration of the coils is also observed.
It is generally believed that under athermal conditions, which correspond to the situation in a good solvent, polymers are soluble in low-molecular-weight solvents at all concentrations. Under such conditions, when interactions P-S, P-P, and S-S (where P stands for a polymer unit and S stands for a solvent) are equal there is no energetical contribution ∆H to the free energy of mixing ∆G ) ∆H - T∆S, so the system should be miscible at all concentrations because of an increase in entropy. This approach does not explicitly take into account the excluded volume interactions. They can be taken into account by assuming suitable potentials, thus giving a contribution to ∆G. Calculations based on the Edwards Hamiltonian were performed by Kosmas and Vlahos1,2 for chains of different lengths. These authors concluded that there should be a miscibility gap, especially in 2D systems, that is more pronounced for a larger difference in chain length. No other results supporting this conclusions have been published to date. Two-dimensional polymer systems have been a subject of interest for many years and have been studied using various theories (see, for example, ref 3 for a review) and computer simulations.3-8 Recently, elegant experiments on DNA chains adsorbed on lipid bilayers were reported,9-11 which confirmed theoretical predictions for dilute 2D solutions. The molecular masses of the studied chains were high (more than 26 500 base pairs); however, because of the large persistence length of DNA, it corresponds to a medium molecular weight of flexible polymers. Monte Carlo (MC) simulations of 2D polymer systems have been performed by different groups, and no indication of a phase separation was reported, However, in most cases the simulated chains were relatively short, and the concentration range that was studied was limited. In the present work, we report the results of MC simulations of longer chains over the full concentration range. Another * Corresponding author. E-mail:
[email protected]. † Technical University of Lodz. ‡ Polish Academy of Sciences. (1) Vlahos, C.; Kosmas, M. Polymer 2003, 44, 503-507. (2) Kosmas, M. K.; Vlahos, C. H. J. Chem. Phys. 2003, 119, 4043-4051. (3) Yethiraj, A. Macromolecules 2003, 36, 5854-5862. (4) Teraoka, I.; Wang, Y. M. Macromolecules 2000, 33, 6901-6903. (5) Wang, Y. M.; Teraoka, I. Macromolecules 2000, 33, 3478-3484. (6) Reiter, J.; Edling, T.; Pakula, T. J. Chem. Phys. 1990, 93, 837-844. (7) Nelson, P. H.; Hatton, T. A.; Rutledge, G. C. J. Chem. Phys. 1997, 107, 1269-1278. (8) Bishop, M.; Satiel, C. J. J. Chem. Phys. 1986, 85, 6728-6731. (9) Maier, B.; Radler, J. O. Macromolecules 2001, 34, 5723-5724. (10) Maier, B.; Radler, J. O. Macromolecules 2000, 33, 7185-7194. (11) Maier, B.; Radler, J. O. Phys. ReV. Lett. 1999, 82, 1911-1914.
significant difference as compared with previous simulations is that in our simulations the solvent molecules are explicitly included, which makes the system more realistic. The fact that the solvent molecules also cannot cross the polymer chains in their motions makes an important difference in two dimensions, which is more pronounced for longer chains. The solvent domains are incompressible at full occupancy, which is also particularly important in two dimensions. This effect was neglected in selfavoiding random walk simulations. We used a cooperative motion algorithm (CMA) that was successfully used for various polymer systems and is described in detail elsewhere.12-14 It allows the simulation of dense systems (at full occupancy), including polymer melts, and is very efficient, which makes simulations for large chain lengths possible in a reasonable period of time. Off-lattice simulations for similar system would be orders of magnitude longer, and the difference between our results on a triangular lattice and literature data3 decreases with increasing N and φ and is already small for N ) 256. The simulations were carried out under athermal conditions on a triangular lattice (coordination number ) 6) with periodic boundary conditions. The simulation box was 256 sites × 256 sites. Chains of beads joined by unbreakable bonds represented the macromolecules. Single beads represented the solvent molecules. Only one bead (belonging to a polymer chain or not) could occupy a lattice site at a time (strict excluded volume condition), and all of the lattice sites were occupied. Thus, only cooperative movements were allowed, which also preserved chain continuity. Polymer concentration φ is defined as the ratio of monomers forming the chains to all of the beads in the system. The mean-square average end-to-end distance 〈R2〉 was defined as
〈R2〉 ) 〈(b r1 - b r N)2〉 rN are space coordinates of chain ends, and the where b r1 and b radius of gyration 〈Rg2〉 was defined as
〈Rg2〉 )
〈∑ 1
N
N i)1
(b ri - b r cm)2
〉
where N is the total number of beads constituting the chain and b rcm is a coordinate of the chain center of mass. Averaging was (12) Pakula, T.; Jeszka, K. Macromolecules 1999, 32, 6821-6830. (13) Pakula, T.; Geyler, S. Macromolecules 1987, 20, 2909-2914. (14) Pakula, T. Macromolecules 1987, 20, 679-682.
10.1021/la701167e CCC: $37.00 © 2007 American Chemical Society Published on Web 07/24/2007
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Langmuir, Vol. 23, No. 17, 2007 8679
Figure 1. Concentration dependence of the mean-square radius of gyration for various chain lengths.
Figure 2. Snapshots of the simulated systems for φ ) 0.60 (a) and 0.40 (b). Chain length N ) 256.
performed over at least 500 relaxed chains (i.e., several independent systems when necessary). The generated systems of appropriate concentration were equilibrated until 〈R2〉 and 〈Rg2〉 reached constant values (cf. Figure A in Supporting Information). Such equilibrated systems were used as starting systems in the reported simulations. The MC time step is defined as a simulation time after which one attempt to move per bead was made (on average). The presented results were obtained from simulations carried out for at least 107 MC time steps. Figure 1 shows the results obtained for 〈Rg2〉 for several chain lengths N as a function of concentration. It can be seen that in all cases both parameters decrease with increasing polymer concentration as predicted by various theories. The following scaling prediction has been suggested in the semidilute and concentrated regimes (overlapping coils)15
〈Rg2〉 ∝ 〈R2〉 ∝ φ(1 - 2ν)/(dν - 1)
(1)
where d is the spatial dimension of the system. Thus, for the 2D system
〈Rg2〉 ∝ 〈R2〉 ∝ φ-1
(2)
Of course, in the dilute regime 〈Rg2〉 is concentrationindependent. According to eq 2, the concentration dependence should not show any irregularities. Indeed, in Figure 1 it can be seen that for short chains the concentration dependence is smooth, as also found in previous studies.3,5 However, for the longest chains it is evident that this dependence is not “smooth”. One can observe that for N ) 1024 and 512 there is a rather sharp transition between the semidilute regime and the concentrated regime around φ ) 0.6 and 0.5, respectively. Similar results were obtained for 〈R2〉. The chain dimension (strictly speaking, the end-to-end distance) is related to the chain entropy via the partition function. The ideal chain entropy is simply proportional to R. Thus, such irregular behavior is indicative of a phase transition in the system. Figures 2 and 3 show typical snapshots of the simulated systems for N ) 256 and 1024, φ ) 0.6 and 0.4, (i.e., above and below the transition). It can be seen that for shorter chains, N ) 256 (Figure 2), the polymer is homogeneously distributed on the lattice for both concentrations. In the case of N ) 1024 (Figure 3) for the concentration φ ) 0.6 (Figure 3a) and for higher (15) De Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.
Figure 3. Snapshots of the simulated systems for φ ) 0.60 (a) and 0.40 (b and c). Chain length N ) 1024. Pictures b and c present the same system (picture c is taken after an additional 107 Monte Carlo steps). The color attributed to a given chain is the same.
concentrations, the polymer distribution is also homogeneous. The situation changes dramatically for lower concentrations. The snapshots for φ ) 0.4 are shown in Figure 3b,c. One can see surprisingly large domains of almost pure solvent (black), mostly surrounded by long, almost extended parts of some chains. The solvent concentration in other regions is practically the same as for φ ) 0.6. The snapshot shown in Figure 3c shows the same system after an additional 107 MC steps. The colors of the chains are the same, so one can see that they were well mixed during this simulation time. The solvent areas are now in different places, and in many cases, they are surrounded by different chains. This proves that the observed microphase separation is not an artifact due to insufficient mixing time. Similar pictures were obtained after various simulation times in the 0.3-0.55 concentration range. Another interesting feature that can be observed in the snapshots in Figure 3 is the interpenetration of the coils. Polymer chains in two dimensions cannot cross each other, so it was argued that the other chains must be excluded from the area occupied by a given chain and the chains should be disc-like.15 Previous simulations for shorter chains have cast doubt on this idea.3 However, a careful inspection of our snapshots reveals that this
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is generally not the case. Some chains have a dumbbell shape with two bulky, densely packed “heads” joined by narrow “joints” (indicated by the arrows in Figure 3a). In such cases, it is possible that the centers of mass of the two chains are close to each other. Long-chain coils can also interpenetrate in the sense that a domain of one chain can be partially surrounded by another chain. The dotted circle in Figure 3a indicates a place in which two green chains occupy the central part of the coil of the blue chain. Other such places are marked by arrows in Figure 3c. In Figure 3c, in the solvent-rich region the blue chain penetrates the coil of the green one on its right and partially surrounds the brown chain on its left. In Figure 3b,c (phase-separated systems), one can see many elongated chains such as the blue one and the yellow one indicated by the arrows in Figure 3b. An increasing population of such chains is responsible for the increase in 〈Rg2〉 observed for long chains below φ ) 0.6 (Figure 1). The homogeneity of the solvent distribution can also be analyzed by considering the number of solvent “molecules” in the vicinity of a given bead. In particular, the bead for which all of the nearest and second-nearest neighbors (18 on the triangular lattice) are “solvent” molecules can be regarded as belonging to a domain of pure solvent. In Figure 4a, one can see the fraction of such “pure solvent” beads plotted versus concentration for various chain lengths. The data are averaged over five relaxed systems. It can be seen that the curves for short chains are similar and lay close to each other whereas for chains with N ) 512 and 1024 a significant increase in the pure solvent fraction is observed over the 0.5-0.2 and 0.6-0.2 concentration ranges, respectively. This confirms that phase separation and the formation of solvent domains take place in this case. The microstructure of multicomponent systems can also be analyzed using the scattering structure factor, S(q), defined as
S(q) )
sin(qr)
∑r g(r)
qr
(3)
where q is the scattering vector, r ) rij) ri - rj is the distance between sites i and j, and g(r) ) 1/K〈c(ri) c(rj)〉 is site-site correlation function (where K is the number of sites in the system). Contrast operators ck and cl are assumed to be equal to 1 for the polymer beads and -1 for the solvent Figure 4b shows collective structure factors for different chain lengths at φ ) 0.5. For N ) 512 and 1024 chains, one can see a strong maximum at small q indicating the appearance of domains as a result of microphase separation. In summary, we show that in the system in which solvent molecules are taken into account, for sufficiently long chains microphase separation takes place in 2D polymer solutions, even under athermal conditions. This effect is related to the strong excluded volume interactions in 2D systems where the chains cannot cross each other and the solvent molecules cannot cross the chains. However, significant interpenetration of polymer coils
Figure 4. Fraction of solvent molecules that have 18 solvent neighbors, plotted as a function of polymer concentration for chains of various lengths (top) and collective structure factors for different chain lengths at φ ) 0.5 (bottom) (values averaged over 10 independent simulated systems).
is observed in many cases. Although in lattice simulations excluded volume effects are not quite properly treated, especially on a short scale, the related differences should be less important on a large scale for long chains, and at least the qualitative picture is expected to be very similar. The simulated phenomena could be observed in the case of polymer chains intercalated in layered silicates or strongly adsorbed on a surface. Acknowledgment. Financial support by the Ministry of Science and Education (project 3 T08E 41 26) is kindly acknowledged. Supporting Information Available: Examples of the evolution in time of Rg and R and the orientational relaxation of R (autocorrelation function) in a generated system. This material is available free of charge via the Internet at http://pubs.acs.org. LA701167E
