Monte Carlo Study of Patchy Nanostructures SelfAssembled from a Single Multiblock Chain
Abstract
We present a lattice Monte Carlo simulation for a multiblock copolymer chain of length N=240 and microarchitecture . The simulation was performed using the Monte Carlo method with the Metropolis algorithm. We measured average energy, heat capacity, the mean squared radius of gyration, and the histogram of cluster count distribution. Those quantities were investigated as a function of temperature and incompatibility between segments, quantified by parameter . We determined the temperature of the coilglobule transition and constructed the phase diagram exhibiting a variety of patchy nanostructures. The presented results yield a qualitative agreement with those of the offlattice Monte Carlo method reported earlier, with a significant exception for small incompatibilities, , and low temperatures, where 3cluster patchy nanostructures are observed in contrast to the 2cluster structures observed for the offlattice chain. We attribute this difference to a considerable stiffness of lattice chains in comparison to that of the offlattice chains.
I Introduction
Block copolymers are studied mainly due to a fundamental interest in the soft matter research and also in the industrial applications 200 (2004). he physical properties, such as elasticity, toughness or electrical conductivity, depend mainly on chemical composition and microarchitecture of the polymer chain. Numerous studies show that diblock polymer melts can spontaneously form variety of nanostructures such as: ordered layers, hexagonally packed cylinders, cubically ordered spheres and the celebrated gyroid structures (Leibler, 1980; Matsen and Schick, 1994). It has been shown that one can also obtain different phases by varying the chain microarchitecture(200, 2004; Binder and Paul, 2008; Banaszak et al., 2002, 2003). We expect that increasing the complexity of microarchitecture leads to more nanostructures. Indeed, triblock copolymers show new phases like lamellacylinder or lamellasphere combinations, which have been confirmed experimentally. Synthesizing more complex microarchitectures, including the cyclic and the branched ones, results in a plethora of new phases (Knoll et al., 2004). Moreover, not only polymer melts are prone to selfassembly into various phases but also a single copolymer polymer chain can selfassemble into various structures, referred to as patchy nanostructures or patchy particles (Zhang and Glotzer, 2004). Recent studies in both computer simu lation and supramolecular chemistry show that such systems are thermodynamically stable and, more importantly, can be obtained by chemical synthesis (Zhang and Glotzer, 2004; Rupar et al., 2012). Such patchy nanostructures can be used as building blocks for selfassembling nanodevices. It has been shown that due to well defined symmetry of such particles they can form aggregates with an ordered spatial structure. This behavior can be used for the bottomup approach, overcoming the size limits of producing particles of sizes from 7 to 17 nm with a defined symmetry, in one step (Rupar et al., 2012; Zhang et al., 2011). In previous studies, using an offlattice Monte Carlo method with a discontinuous (square well) potential (Lewandowski and Banaszak, 2011) and using the LennardJones (LJ) potential (Parsons and Williams, 2007), the stability of patchy nanostructures was investigated. In order to probe the free energy landscape more efficiently, the parallel tempering (PT) method (Katzgraber et al., 2006; Sikorski, 2002; Gront and Kolinski, 2007; Lewandowski et al., 2010; Beardsley and Matsen, 2010) was employed in ref. (Lewandowski and Banaszak, 2011) and WangLandau method (Wang and Landau, 2001) in ref. (Parsons and Williams, 2007). In this study we intend to study one of the chain microarchitectures, specifically , but using a lattice Monte Carlo method. While the offlattice models are more realistic, they also require more computational effort. Therefore it is significant to asses the relative merits of the lattice model. If the lattice model gives similar results, then it may be reasonable to use the lattice model rather than the offlattice models. The aim of this paper is to answer the following questions:

are the nanostructures obtained in the lattice simulation the same as those obtained in the offlattice simulation?

is the phase diagram obtained in the lattice simulation the same as that obtained in the offlattice simulation?
Ii Model and method
In this study, a coarsegrained model is used. The polymer chain is placed on the face centered cubic (FCC) lattice with coordination number z=12 and the bond length equal to . Chain bonds are not allowed to be broken or stretched. The periodic boundary conditions are applied. The size of simulation box is chosen to fit a fully extended chain. Polymersolvent interactions are included in an implicit manner in the polymerpolymer interaction potential. Polymer chain consists of two types of monomers: A and B. Interaction energy between monomers is defined as follows: and . The parameter is positive and serves here as an energy unit. We define reduced energy as
(II.1) 
and reduced temperature as
(II.2) 
where N is the number of chain monomers and is the Boltzmann constant. Negative value of interaction energy means that there is a net attraction between monomers. Reduced temperature parameter is used to control the quality of solvent, from good to bad which causes a transition from swollen state to a globular state. Dimensionless parameter is a measure of compatibility between two monomers of different type. Lower values of mean that an attraction between monomers A and B is lower than between monomers of the same type. For the copolymer chain becomes a homopolymer chain. The chain consists of 120 monomers of type A and 120 monomers of type B which form the multiblock microarchitecture . During the simulation various ’s and ’s are probed; from 2.27 to 20.0 (because below 2.27 nothing seems to change in the polymer structure), and from 0.1 to 1.0.
The simulation was performed by standard Monte Carlo simulation method with Metropolis acceptance criteria (Metropolis et al., 1953). In order to perform a Monte Carlo move we use pullmove algorithm, which employs chain movements, reminiscent of the reptation moves, by pulling the chain in a random direction as described in detail in reference (Lesh et al., 2003). We define one Monte Carlo Step (MCS) as the attempt to perform one move according to the algorithm. Each simulation consists of MCS. First the system is equilibrated athermally, and next steps are run in the thermal condition. We calculate the thermal averages from last steps.
Iii Results and discussion
We start the simulation by equilibrating the system in the athermal limit, where . After the system reaches its thermal equilibrium we start to cool it down to a variety of ’s. First, we present results for the lowest compatibility parameter . As is decreased, it is expected that the chain undergoes a transition from a swollen state to an intermediate state pearlnecklace, and after that to the globular state (Lewandowski et al., 2008). In figure (1) we show the Monte Carlo results for the energy per monomer, , mean squared radius of gyration, , and heat capacity, , of the polymer chain, as a function of . The heat capacity, , is obtained from the energy fluctuations as follows
(III.1) 
where N is the number of monomers, and denotes the thermal average. The reduced energy does not change much from high temperatures to about where it starts to decrease. A similar behavior is observed for mean squared radius of gyration which also decreases below . In figure (1)c) we show the temperature dependence for the heat capacity, . The maximum in corresponds approximately to the inflection point in the reduced energy presented in figure (1)a). From those two observations we can estimate the coiltoglobule transition temperature, . In the figure (3) we present the equilibrated structures in a coiled state (a), in a state which is close coiltoglobule transition (b) and in a globular state (c) for . As expected, upon cooling the chain collapses.
In order to better distinguish different patchy nanostructures, we use cluster count distribution as in ref. (Lewandowski and Banaszak, 2011). In figure (2), we present histogram of probability of clusters with number of segments set to 2, 3, 4, 5, and 6, for . We show that the 6cluster structure (and also ncluster structures, with ) is most stable at ’s that higher than , but from to , the 5clusters occur with the highest probability. Next from to the 4clusters prevail. It is also worthwhile to notice that within this range falls the coiltoglobule transition, . For the 3clusters are the most probable patchy nanostructures. The representative 2clusters are shown in figure (4) and they seem to be similar to the lamellar nanophase which is observed in diblock copolymer melts. Probability of observing the 3clusters gradually increases, reaching unity at low temperatures. Variations in number of clusters and, as a result, in the patchy nanostructure can also be discerned by measuring the heat capacity with higher temperature resolution (more ’s), but additional Monte Carlo simulations would be required.
Next, we describe the simulation results for ’s from 0.2 to 1.0, in more detail. In figure (5) we show the dependence of the coiltoglobule transition temperature, , which is similar to that observed in ref. (Lewandowski and Banaszak, 2011), indicating that increases upon increasing .
Similarly as for , we determine the temperature dependencies for the heat capacity, the mean squared radius of gyration and the energy per monomer. In figure (6) we present cluster count distribution for . In this case we also observe that the 3cluster structure has the highest probability at low but the probability of obtaining the 2cluster structures increases to 0.3 at . Transition between 4 and 5cluster is shifted towards higher temperatures, as expected. The range for 4clusters shrinks significantly from (4.75, 5.90) for to (5.5, 5.92) for . The transition temperature from 4cluster to 5cluster patchy structure () is close to the coiltoglobule transition temperature, . Above that temperature there is no obvious prevalence of any structure as it becomes increasingly disordered upon heating.
In figure (7) we show cluster count distribution for the representative ’s: 0.4, 0.5, 0.8 and 1.0. We observe that the probability maxima for different cluster counts (for ) are shifted towards higher temperatures as is increased. It is interesting to identify the most probable patchy nanostructures at low temperatures, as they may correspond to the native states in biopolymers (Shakhnovich, 2006). The most probable structures for 0.1, 0.2 and 0.3 are the 3clusters. On the other hand, for ’s which are equal or greater than 0.4 the 2cluster are most probable. However, it is interesting to record that the probability of the dominant nanostructures, either 2cluster or 3cluster, approaches unity at low temperatures. Sample snapshots of molecules for various ’s at are shown in figure (8)a). For lower ’s segments are strongly segregated which leads to formation of a nanostructure consisting of two hemispheres (doubledrop structure). Other structures which can be observed are the handshakelike structures which are shown in figure (8)b and 8c. Above the structures become more disordered.
In this paper as well as in the previous study (Lewandowski and Banaszak, 2011) we do not observe a single native state for the polymer chain. Structures with different number of clusters can coexists. Only for the lowest temperature the probability for a given number of clusters is approximately 1.0. Therefore we can conjecture that instead of one native state for given () we obtain numerous stable states with different spatial order but with the same energy. In order to compare the present lattice model with the offlattice model we construct a phase diagram in the () space. According to previous multiblock studies we expect that those two models should give similar results (Lewandowski et al., 2008). In particular, from ref. (Lewandowski et al., 2008) we learn that in order to match the coiltoglobule transision temperature for lattice model, , with that for the offlattice model, , we need to use a multiplicative factor, R, defined below:
(III.2) 
The value of R depends on both the chain length and microarchitecture, but to a good approximation it is equal to about 6.5 (Lewandowski et al., 2008). In figure (9) we present both the results for offlattice model (a) and the results for the lattice model (b), using the microarchitecture. In both cases most probable structure remains the same: 2, 3, 4, 5, 6clusters. Comparing maxima of probability of obtaining certain clusters we can observe that in both cases those maxima shift to highers temperatures as the compatibility increases. The most significant difference is for . In the previous work (ref. (Lewandowski and Banaszak, 2011)) for lower temperatures and for all values of the most probably structure was the 2cluster. In this paper for lower ’s and the most probable structure is the 3cluster. However, if we compare the transition temperature from the B (2cluster) region to the C (3cluster) region at then their ratio is about 7 which is close to R 6.5 from ref. (Lewandowski et al., 2008). It is reassuring to notice if we were to rescale the offlattice phase diagram by this factor, then both phase diagrams would be very similar to each other, with the exception of a region which is roughly determined by the following inequalities: and .
The reason for this discrepancy may be related to the fact that the lattice chain is stiffer than the offlattice chain with the same number of segments. Indeed, when we compare the offlattice phase diagrams for the microarchitecture with that for the microarchitecture (see ref. (Lewandowski and Banaszak, 2011)), we notice that for the microarchitecture the 3cluster is the most stable structure at low temperatures. Obviously, the 6block of the microarchitecture is stiffer than the 10block of the microarchitecture, and therefore we can conclude that the block stiffness promotes the 3cluster structure, as also observed for the lattice microstructure.
Iv Conclusions
Two different approaches, offlattice and lattice Monte Carlo simulations confirm that multiblock copolymer chain in the globular state forms variety number of different patchy nanostructures. We observed the expected behavior of phase coiltoglobule transition upon cooling. We report thermodynamic and structural properties such as energy, specific heat, mean square radius of gyration. We constructed the phase diagram in the () space which presents the same results as for the offlattice simulation in the case of . For lower compatibility we report that the most probably phase is 3cluster.
Finally we answer the questions that we posed in the introduction:

the nanostructures obtained in the lattice simulation are the same as those obtained in the offlattice simulation

the phase diagram obtained in the lattice simulation is mostly similar to that obtained in the offlattice simulation, but qualitatively different for ’s smaller than 0.4 and ’s smaller than 5 (3cluster is more probable than the 2cluster). This discrepancy, however, can be tentatively attributed to a considerable stiffness of lattice chains in comparison to that of the offlattice chains.
Acknowledgments
A significant part of the simulations was performed at the Poznan Computer and Networking Center (PCSS).
References
 Developments in Block Copolymer Science and Technology. John Wiley & Sons, Ltd, Jan 2004. ISBN http://id.crossref.org/isbn/0470093943. doi: 10.1002/0470093943. URL http://dx.doi.org/10.1002/0470093943.
 M. Banaszak, S. Woloszczuk, T. Pakula, and S. Jurga. Computer simulation of structure and microphase separation in model triblock copolymers. Phys. Rev. E, 66:031804, Sep 2002. doi: 10.1103/PhysRevE.66.031804. URL http://link.aps.org/doi/10.1103/PhysRevE.66.031804.
 M. Banaszak, S. Woloszczuk, S. Jurga, and T. Pakula. Lamellar ordering in computersimulated block copolymer melts by a variety of thermal treatments. The Journal of Chemical Physics, 119(21):11451, 2003. ISSN 00219606. doi: 10.1063/1.1622375. URL http://dx.doi.org/10.1063/1.1622375.
 T. M. Beardsley and M. W. Matsen. Monte Carlo phase diagram for diblock copolymer melts. The European Physical Journal E, 32(3):255–264, Jul 2010. ISSN 1292895X. doi: 10.1140/epje/i201010651x. URL http://dx.doi.org/10.1140/epje/i201010651x.
 K. Binder and W. Paul. Recent developments in Monte Carlo simulations of lattice models for polymer systems. Macromolecules, 41(13):4537–4550, Jul 2008. ISSN 15205835. doi: 10.1021/ma702843z. URL http://dx.doi.org/10.1021/ma702843z.
 Dominik Gront and Andrzej Kolinski. Efficient scheme for optimization of parallel tempering Monte Carlo method. J. Phys.: Condens. Matter, 19(3):036225, Jan 2007. ISSN 1361648X. doi: 10.1088/09538984/19/3/036225. URL http://dx.doi.org/10.1088/09538984/19/3/036225.
 Helmut G Katzgraber, Simon Trebst, David A Huse, and Matthias Troyer. Feedbackoptimized parallel tempering Monte Carlo. Journal of Statistical Mechanics: Theory and Experiment, 2006(03):P03018–P03018, Mar 2006. ISSN 17425468. doi: 10.1088/17425468/2006/03/p03018. URL http://dx.doi.org/10.1088/17425468/2006/03/P03018.
 Armin Knoll, Robert Magerle, and Georg Krausch. Phase behavior in thin films of cylinderforming aba block copolymers: Experiments. The Journal of Chemical Physics, 120(2):1105, 2004. ISSN 00219606. doi: 10.1063/1.1627324. URL http://dx.doi.org/10.1063/1.1627324.
 Ludwik Leibler. Theory of microphase separation in block copolymers. Macromolecules, 13(6):1602–1617, Nov 1980. ISSN 15205835. doi: 10.1021/ma60078a047. URL http://dx.doi.org/10.1021/ma60078a047.
 N. Lesh, M. Mitzenmacher, and S. Whitesides. A complete and effective move set for simplified protein folding. In 7th Annual International Conference on Research in Computational Molecular Biology (RECOMB) 2003, pages 188–195. ACM Press, 2003.
 K. Lewandowski and M. Banaszak. Intraglobular structures in multiblock copolymer chains from a monte carlo simulation. Phys. Rev. E, 84:011806, Jul 2011. doi: 10.1103/PhysRevE.84.011806. URL http://link.aps.org/doi/10.1103/PhysRevE.84.011806.
 K. Lewandowski, P. Knychala, and M. Banaszak. Proteinlike behavior of multiblock copolymer chains in a selective solvent by a variety of lattice and offlattice Monte Carlo simulations. phys. stat. sol. (b), 245(11):2524–2532, Nov 2008. ISSN 15213951. doi: 10.1002/pssb.200880252. URL http://dx.doi.org/10.1002/pssb.200880252.
 K. Lewandowski, P. Knychala, and M. Banaszak. Paralleltempering MonteCarlo simulation with feedbackoptimized algorithm applied to a coiltoglobule transition of a lattice homopolymer. Computational Methods in Science and Technology, 16(1):29–35, 2010. ISSN 15050602. doi: 10.12921/cmst.2010.16.01.2935. URL http://dx.doi.org/10.12921/cmst.2010.16.01.2935.
 M. W. Matsen and M. Schick. Lamellar phase of a symmetric triblock copolymer. Macromolecules, 27(1):187–192, Jan 1994. ISSN 15205835. doi: 10.1021/ma00079a027. URL http://dx.doi.org/10.1021/ma00079a027.
 Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller. Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6):1087, 1953. ISSN 00219606. doi: 10.1063/1.1699114. URL http://dx.doi.org/10.1063/1.1699114.
 Drew F. Parsons and David R. M. Williams. Single chains of block copolymers in poor solvents: Handshake, spiral, and lamellar globules formed by geometric frustration. Phys. Rev. Lett., 99:228302, Nov 2007. doi: 10.1103/PhysRevLett.99.228302. URL http://link.aps.org/doi/10.1103/PhysRevLett.99.228302.
 P. A. Rupar, L. Chabanne, M. A. Winnik, and I. Manners. Noncentrosymmetric cylindrical micelles by unidirectional growth. Science, 337(6094):559–562, Aug 2012. ISSN 10959203. doi: 10.1126/science.1221206. URL http://dx.doi.org/10.1126/science.1221206.
 Eugene Shakhnovich. Protein folding thermodynamics and dynamics:Â where physics, chemistry, and biology meet. Chemical Reviews, 106(5):1559–1588, May 2006. ISSN 15206890. doi: 10.1021/cr040425u. URL http://dx.doi.org/10.1021/cr040425u.
 Andrzej Sikorski. Properties of starbranched polymer chains. application of the replica exchange Monte Carlo method. Macromolecules, 35(18):7132–7137, Aug 2002. ISSN 15205835. doi: 10.1021/ma020013s. URL http://dx.doi.org/10.1021/ma020013s.
 Fugao Wang and D. P. Landau. Efficient, multiplerange random walk algorithm to calculate the density of states. Phys. Rev. Lett., 86:2050–2053, Mar 2001. doi: 10.1103/PhysRevLett.86.2050. URL http://link.aps.org/doi/10.1103/PhysRevLett.86.2050.
 Zhang and Sharon C. Glotzer. Selfassembly of patchy particles. Nano Lett., 4(8):1407–1413, Aug 2004. ISSN 15306992. doi: 10.1021/nl0493500. URL http://dx.doi.org/10.1021/nl0493500.
 Jing Zhang, ZhongYuan Lu, and ZhaoYan Sun. A possible route to fabricate patchy nanoparticles via selfassembly of a multiblock copolymer chain in one step. Soft Matter, 7(21):9944, 2011. ISSN 17446848. doi: 10.1039/c1sm05845b. URL http://dx.doi.org/10.1039/C1SM05845B.