Fluid Phase Equilibria 358 (2013) 156–160
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Monte Carlo binodals for the order–disorder transition in A-B-A copolymer melts S. Wołoszczuk, M. Banaszak ∗ Faculty of Physics, A. Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland
a r t i c l e
i n f o
Article history: Received 2 July 2013 Received in revised form 6 August 2013 Accepted 8 August 2013 Available online 28 August 2013 Keywords: Triblock copolymer melt Binodals Monte Carlo Parallel tempering
a b s t r a c t The ABA triblock copolymer melts are simple and illuminating prototypes for soft self-assembling systems because they exhibit a remarkable richness of nanostructures and useful molecular features, such as B-bridges connecting the neighboring A-nanodomains. The transition from disordered phase to ordered phase is of particular interest and therefore we determine binodals for the order–disorder transition (in terms of the thermodynamic incompatibility and the triblock asymmetry), using lattice Monte Carlo method, known as cooperative motion algorithm, and also employing the parallel tempering method which is known to be efficient at lower temperatures. The simulated binodals are presented as a 3-dimensional phase diagram and confronted with earlier mean-field (self-consistent field theory) calculations. While the Monte Carlo binodals show qualitatively similar trends to those observed in the mean-field calculations, there is a pronounced numerical difference between them which indicates the significant role of fluctuations in the vicinity of the order–disorder transition. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Triblock copolymer melts can self-assemble into a plethora of nanostructures (nanophases) [1]. This self-assembly is mostly governed by a competition of entropic stretching energy and enthalpic interfacial energy. A linear triblock ABC copolymer chain consists, in general, of three the distinct blocks, A, B, and C, which are connected sequentially. Terminal blocks A and C are often built from the same type of segments, resulting in a triblock copolymer ABA which has only 2 types of segments, A and B, as the diblock. In this paper we focus on such ABA triblocks. While AB-diblock copolymer melts are known to form only a few stable nanophases (such as layers, L, hexagonally packed cylinders, C, gyroid nanostructures, G, with the Ia3d symmetry, cubically packed spherical cells S, and the O70 -phase [2,3]), the triblock melts form tens of different phases [1]. In case of a diblock AB copolymer melt, the phase behavior is controlled by the chain composition, f (volume fraction of segments of type A), degree of polymerization (total number of segments), N, and the temperature-related parameter [4,5]. The ordered nanophase can be dissolved into a disordered phase, for example, upon heating. Phase diagrams of such melts exhibiting order–disorder transition (ODT) lines, also referred to as binodals, and order–order transition (OOT) lines are known from experiment [6] and are successfully predicted by mean-field (MF) theories [7,8],
∗ Corresponding author. Tel.: +48 618295065. E-mail address:
[email protected] (M. Banaszak). 0378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2013.08.010
such as self-consistent field theory (SCFT) which is based on the standard Gaussian chain model [9], or theories including fluctuations [10,11]. Because, in the MF theories, it is sufficient to know the composition, f, and the product N (known also as thermodynamic incompatibility parameter) in order to determine the nanophase [7,12,13]. Similarly, the MF phase behavior of the ABA triblock melt is governed by N and the triblock composition, which can be parametrized by 2 convenient numbers, ˛ and ˇ, as follows f1 =
ˇ 1 − − ˛, 4 2
(1)
f2 =
1 + ˇ, 2
(2)
1 ˇ (3) − + ˛. 4 2 where fi (i = 1, 2 or 3) is the volume fraction of the block of type i (labels 1 and 3 correspond to the terminal A-blocks, and label 2 corresponds to the middle B-block); f1 + f2 + f3 = 1. In most previous studies [14–17] a different parametrization was used, as shown below f3 =
f1 = fA ,
(4)
f2 = 1 − fA ,
(5)
f3 = (1 − )fA .
(6)
The first parameter, ˛, provides a measure of asymmetry between terminal A-blocks (f1 and f3 ); ˛ = 0 corresponds to a symmetric triblock with equal terminal blocks. The second parameter,
S. Wołoszczuk, M. Banaszak / Fluid Phase Equilibria 358 (2013) 156–160
ˇ, quantifies the volume fraction of the middle block; ˇ = 0 corresponds to volume fraction equal to 1/2. The ABA triblock phase diagram can be mapped in the MF approximation, as shown for symmetric (˛ = 0) [16] and asymmetric (˛ = / 0) [17] cases by Matsen and Thompson who calculated ODT and OOT lines as a function of and fA for 3 selected degrees of thermodynamic incompatibility (N=20, 30 and 40). This phase diagram exhibited a continuous change from the A-B diblocks to the A-B-A triblocks of varying asymmetry, but the binodals were deflected for some critical degree of asymmetry. This effect was accounted for by the localization of short terminal blocks in the middle block B-domain. This picture is supported by the experimental work of Hamersky et al. [18] in which the phase behavior of PS-PI-PS triblocks (2 series: 9-46-A2 , and 9-17-A2 with the A-block grown into A-B diblock from the B-block side) was reported, and also by the corresponding Monte Carlo simulations of Woloszczuk et al. [19]. While TODT was expected to increase with increasing copolymer molecular weight due to greater thermodynamic incompatibility, it was depressed as the length of the grown A-block became comparable with the other A-block, and then the trend reversed giving rise to a minimum in TODT . While the SCFT is both efficient and successful in elucidating the phase behavior of block copolymer melts, it is a mean-field approach, and therefore it offers only approximate solutions to the Gaussian models of block copolymer melts. Therefore Monte Carlo (and also molecular dynamics [20]) simulations offer a better tool to study the copolymer melts. However, there many problems with obtaining the exact results, such as finite size effects, competition of many relevant length scales and prohibitively long relaxation times. In addition, there are also effects of the underlying lattice. The purpose of this work is verify the relation between the SCFT data and the Monte Carlo data, as established in Ref. [21], by • performing the Monte Carlo simulation over a wider range of parameters ˛ and ˇ, • using the parallel tempering method (at selected conditions) which is superior to the standard method used in previous work [21]. 2. Model and method The simulations are performed using Cooperative Motion Algorithm [22] for a face-centered cubic lattice with the coordination number z = 12 and lattice constant, a = 1. Chain bonds of length √ 2 can not be stretched or broken. The usual periodic boundary conditions are applied. The lattice sites are completely filled with chain segments and the box size is chosen to fit the chain. Since all lattice sites are occupied, a chain segment can move if other segments move simultaneously. An attempt to move a segment defines 1/na of the Monte Carlo step, where na is the number of lattice sites. The segments of type i and j interact by ij , where AB = , and AA = BB = 0. This interaction is limited to the nearest neighbors (z = 12), and the parameter, , can be related to the Flory parameter by the following equation: =
(z − 2) . kT
(7)
Since serves also as an energy unit, we can define the reduced dimensionless interaction parameter ∗ij = ij /, as well as the reduced energy per lattice site, E* /na = (E/)/na , and the reduced temperature, T* = (kT)/. Because we intend to relate the lattice simulation results to those of the continuum SCFT we also need to establish a relation between the corresponding parameters. While in Eq. (7) we use
157
(z − 2) = 10 as the number of nonbonded nearest neighbors, a more refined approach is needed, such as that of Muller and Binder [23], and Morse and Chung [24]. In particular, Morse and Chung have shown that the effective parameter, e , (in order to correspond to that used in the SCFT) should be extracted from lattice simulations as follows: • the effective number of nearest intermolecular contacts must be calculated in the athermal state for different chain lengths, N, where this number must be extrapolated to N→ ∞, yielding ze , • this ze should be used in an equation relating to T* . Thus an alternative definition of , which is in correspondence with the SCFT, can be used: e =
ze . kT
(8)
where we found (see Ref. [21]) that ze ≈ 7.5; therefore e ≈ 0.75. We start the simulation by equilibrating the system in the athermal limit, that is, where /(kT) is zero. When the system reaches its equilibrium, the polymer chains assume statistical conformations, random orientations, and become uniformly distributed within the simulation box. In order to estimate the simulation time scale we record the translational diffusion. In the athermal melt the 8-168 copolymer chain diffuse at a distance of the order of the radius of gyration in about 1.4 × 104 MC timesteps. We equilibrate the athermal melt for 1 × 107 MC timesteps, and from this state the system is quenched to a required temperature. The verification of the quality of thermal equilibration is done also by heating the system up and cooling it down again, as described in detail in [25]. For each temperature we perform 5 × 106 MC timesteps. First 2 × 106 are to equilibrate the system, and latter 3 × 106 to sample the data. For a given temperature, T* , we repeat the experiment ten times starting with different initial states. While this method (referred to as “quenching”) works quite well for higher temperatures, it tends to generate long relaxation times for lower temperatures. This results in unreliable estimates of the sampled properties. To solve this problem many modifications, as parallel tempering (PT) method were proposed, where the energy barriers of the local free energy minima can be overcome [26,27]. Therefore, we run thermal PT simulation, starting each replica with independent athermal states. In the PT method M replicas of system are simulated in parallel, each in different temperature Ti∗ , with i ranging from 1 to M. After a number of MCS (in our case it was 3000 MCS) we try to exchange replicas with neighboring Ti∗ in random order with probability: ∗ p(Ti∗ ↔ Ti+1 ) = min[1, exp(−(ˇi − ˇi+1 )(Ui+1 − Ui ))]
1/Ti∗
(9) Ti∗ .
where ˇi = and Ui is potential energy of replica at All PT rounds are started with geometric series of temperatures. For a given state point, all runs yield the same type of nanostructure, and the results are averaged over all such runs. Usually, we also assume that the first half of run is needed to equilibrate the system, and the second half is used to collect the data. In this paper we use both quenching and parallel tempering protocols. We measured such quantities as energy per lattice site, En∗ = E ∗ /na , specific heat given by following relation: CV =
(E ∗ − E ∗ )2 , nc T ∗2
(10)
where nc = na /N is the number of chains, and the variations of the end-to-end distance of triblock chain, R2 , as a function of the
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Table 1 Monte-Carlo and SCFT binodals for various designs grouped into 12 series. Series
Monte-Carlo (N)ODT
SCFT (N)ODT from Eq. (12)
Nanophase
1 1
A-B-A design 1-28-3 2-28-2
˛ 0.03125 0.0
ˇ 0.3755 0.3755
76.2 91.4
46.6 49.6
A-spheres A-spheres
2 2 2
1-26-5 2-26-4 3-26-3
0.0625 0.03125 0.0
0.3125 0.3125 0.3125
50.8 64.0 71.1
32.2 38.0 39.9
A-spheres A-spheres A-spheres
3 3 3 3 3
1-22-9 2-22-8 3-22-7 4-22-6 5-22-5
0.125 0.09375 0.0625 0.03125 0.0
0.1875 0.1875 0.1875 0.1875 0.1875
30.5 38.1 42.5 45.7 47.1
14.2 18.8 22.3 24.6 25.3
A-cylinders A-cylinders A-cylinders A-cylinders A-cylinders
4 4 4 4 4
1-16-15 2-16-14 4-16-12 6-16-10 8-16-8
0.21875 0.1875 0.125 0.0625 0.0
0.0 0.0 0.0 0.0 0.0
28.8 34.4 40.0 41.1 44.4
11.3 12.4 15.0 17.2 18.0
Layers Layers Layers Layers Layers
5 5 5 5 5 5
1-12-19 2-12-18 4-12-16 6-12-14 8-12-12 10-12-10
0.28125 0.25 0.1875 0.125 0.0625 0.0
−0.125 −0.125 −0.125 −0.125 −0.125 −0.125
29.6 33.3 38.5 42.1 44.4 45.1
13.2 14.3 16.3 18.0 19.1 19.6
Perforated layers Perforated layers Perforated layers Perforated layers Perforated layers Perforated layers
6 6 6 6 6
1-8-23 3-8-21 6-8-18 9-8-15 12-8-12
0.34375 0.28125 0.1875 0.09375 0.0
−0.25 −0.25 −0.25 −0.25 −0.25
51.3 52.1 53.3 54.5 55.0
18.6 21.80 23.9 24.8 25.0
B-cylinders B-cylinders B-cylinders B-cylinders B-cylinders
7 7 7 7 7 7
1-4-27 2-4-26 5-4-23 8-4-20 11-4-17 14-4-14
0.40625 0.375 0.28125 0.1875 0.09375 0.0
−0.375 −0.375 −0.375 −0.375 −0.375 −0.375
93.7 94.1 94.9 95.7 96.2 96.9
41.4 43.7 47.8 50.3 51.9 52.6
B-spheres B-spheres B-spheres B-spheres B-spheres B-spheres
8 8 8 8 8
3-25-3 5-22-5 8-16-8 10-12-10 11-10-11
0.0 0.0 0.0 0.0 0.0
0.3125 0.1875 0.0 −0.125 −0.1875
71.1 45.1 41.1 44.5 48.5
39.9 25.3 18.0 19.6 21.4
A-spheres A-cylinders Layers Perforated layers Bicontinuous
10 10 10 10 10
12-20-0 14-16-2 16-12-4 18-8-6 20-4-8
−0.1875 −0.1875 −0.1875 −0.1875 −0.1875
0.125 0.0 −0.125 −0.25 −0.375
29.1 32.0 38.1 53.3 97.0
8.24 12.4 16.3 23.9 50.3
Perforated layers Layers Layers B-cylinders B-spheres
11 11 11 11 11 11
6-7-19 6-10-16 6-13-13 6-16-10 6-19-7 6-22-4
0.203175 0.15625 0.109375 0.0625 0.015625 −0.03125
−0.28125 −0.1875 −0.09375 0.0 0.9375 0.1875
61.6 46.4 41.5 41.5 43.0 45.7
27.4 19.9 17.5 17.2 19.5 24.6
B-cylinders B-cylinders Layers Layers Layers A-cylinders
12 12 12 12 12 12
19-7-6 16-10-6 13-13-6 10-16-6 7-19-6 4-22-6
−0.203175 −0.15625 −0.109375 −0.0625 −0.015625 0.03125
−0.28125 −0.1875 −0.09375 0.0 0.9375 0.1875
61.6 46.4 41.5 41.5 43.0 45.7
27.4 19.9 17.5 17.2 19.5 24.6
B-cylinders B-cylinders Layers Layers Layers A-cylinders
reduced temperature. Moreover we also calculate the structure factor, S(k), by averaging over statistically independent configurations using the following equation:
n A
= 1 S(k) nA
m=1
2
cos(k · r m )
+
n A
2
sin(k · r m )
thermal average ,
m=1
√ monomers, i . e . 2a. Moreover, k vectors are commensurate with the simulation box size, and this constraint limits their number and allowed lengths. Because the system may not be isotropic, S(k) is
such that |k|
is equal to k. We calculated by averaging over all S(k) take the emergence of multiple peaks in S(k) as a signature of the order–disorder transition.
(11) where nA denotes the number of segments of type A and r m denotes the position of the m-th segment of type A. The magnitude of wave vector, k, varies from kmin = 2/L to kmax = 2/b, where L is a size of the cubic lattice whereas b is the distance between nearest
3. Results In a previous paper [21] we focused on simulation of ABA triblock copolymer melt using the CMA method, probing various degrees
S. Wołoszczuk, M. Banaszak / Fluid Phase Equilibria 358 (2013) 156–160
159
of compositional asymmetry expressed by the parameters ˛ and ˇ. We reported Monte Carlo binodals along three different ˛ − ˇ paths (series), showing three different behavior, i . e . concave-like, convex-like, and monotonic-like. We marked them as MIN, MAX, and MON respectively. These selected binodals were compared to the SCFT binodals as reported by Matsen [17]. The Matsen’s results were fitted to the below polynomial: u(˛, ˇ) = c60 ˛6 + c42 ˛4 ˇ2 + c41 ˛4 ˇ + c40 ˛4 + c24 ˛2 ˇ4 + c23 ˛2 ˇ3 + c22 ˛2 ˇ2 + c21 ˛2 ˇ + c20 ˛2 + c06 ˇ6 + c05 ˇ5 + c04 ˇ4 +c03 ˇ3 + c02 ˇ2 + c01 ˇ + c00
(12)
with coefficients cij ’s, first published in Ref. [21]. As reported in Ref. [21] the Monte Carlo binodals follow the same trend as the SCFT binodals with a shift towards lower values of incompatibility parameter, expressed by N. Such shift is expected by the Fredrickson–Helfand theory, but its magnitude is not easy to estimate. The SCFT approach is based on Gaussian chain model while the Monte Carlo method is based on the lattice model, and therefore the results for those models may be difficult to relate, especially for relatively short chains and blocks which are not Gaussian. It was also verified that for a highly asymmetric ABA triblock melt the short A-block is localized inside the B-domain as predicted by the mean-field theory [17]. In this paper we present an extensive and significantly expanded set of simulation data obtained by the CMA algorithm. Our previous simulations show that the lattice size do not affect significantly the order–disorder temperature [21,19], and thus we decided to perform simulation for the 32 × 32 × 32 lattice, and only for selected designs we simulate in larger boxes, such as 64 × 64 × 64. We also use the chains of length N = 32. The simulated binodals (order–disorder transition temperatures) expressed in terms of N are shown in Table 1, and in order to fascilitate the presentation of results we group the simulation results into 12 series. We also show the SCFT binodals calculated from eq 12 for comparison. In the last column the self-assembled nanophase below the order–disorder transition temperature (above (N)ODT ) is indicated. The Monte Carlo binodals from Table 1 are fitted to the same form of polynomial as that which was used for the SCFT binodals in Eq. (12), yielding: w(˛, ˇ) = d60 ˛6 + d42 ˛4 ˇ2 + d41 ˛4 ˇ + d40 ˛4 + d24 ˛2 ˇ4 +d23 ˛2 ˇ3 + d22 ˛2 ˇ2 + d21 ˛2 ˇ + d20 ˛2 d06 ˇ6 + d05 ˇ5 +d04 ˇ4 + d03 ˇ3 + d02 ˇ2 + d01 ˇ + d00
Fig. 1. Polynomial fit (to Eq (13)) of the Monte Carlo binodals, (N)ODT , plotted as a function of ˛ and ˇ, where the blue spheres represent the Monte Carlo binodals. This is the 3-dimensional phase diagram. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
The Monte Carlo binodals and the corresponding polynomial fit are shown in Fig. 1, summarizing the work presented in this paper in the form of 3-dimentional phase diagram. Among the twelve series, presented in Table 1, seven of them (from 1 to 7) are characterized by designs with fixed ˇ’s, three of them (from 8 to 10)
2.4 2.2 2 1.8
E *n
= = = = = = = = = = = = = = = =
1345.97 32901.2 18203.5 1944.8 −159701 −75641.7 −4369.98 −234.802 −317.003 2559.48 −4888.42 1487.46 778.735 93.2009 −24.9674 42.6623
1.4 1.2 1 0.8
0.25 0.2
Cv
(13)
0.15
b)
0.1 0.05 0
with d60 d42 d41 d40 d24 d23 d22 d21 d20 d06 d05 d04 d03 d02 d01 d00
a)
1.6
115 110 105
R2
100
c)
95 90 85 80
5
10
15
T*
20
25
30
Fig. 2. Simulation results for the 8-16-8 triblock copolymer melt simulated on the 32 × 32 × 32 lattice; (a) energy per lattice site, E* /na ; (b) specific heat, CV ; (c) meansquared end-to-end distance, R2 .
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S. Wołoszczuk, M. Banaszak / Fluid Phase Equilibria 358 (2013) 156–160
Fig. 3. Snapshot of the well developed lamellar phase for the 8-16-8 copolymer melt at T* = 4.0.
by fixed ˛’s, and two of them (11 and 12) by designs varying both in ˛ and ˇ. The ODT temperatures and related (N)ODT ’s can be determined from the reduced energy per lattice site, E* /na , specific heat, CV ,and mean-squared end-to-end distance, R2 , as shown for the 8-16-8 design in Fig. 2. In particular, the spike in the heat ∗ ≈ 7.2 which correspods capacity, CV , can be associated with TODT to N = 44.4 (see Table 1). The structure factor, S(k), is calculated from Eq. (11), as demonstrated in Ref. [28]. At high temperatures we observe characteristic for a disordered phase single broad peak, but as the temperature is lowered the multiple and narrow peaks develop, which indicates the onset of the order–disorder transition. Those peaks can be used to identify the nanostructure, and this identification can be confirmed by a visual examination of the snapshots from the simulation, such as shown in Fig. 3. In the previous paper [21] we showed a qualitative agreement between the MC and SCFT binodals, but quantitatively the simulation binodals are shifted by a factor of about 2 in terms of N. In this paper we can confirm this observation from a significantly larger set of Monte Carlo data, which was presented in Table 1 and shown in Fig. 1. Therefore we conclude that the fluctuations are significant in the vicinity of Monte Carlo order–disorder transition, and the binodals seem to follows similar qualitative trends to those observed by SCFT. 4. Conclusions We presented extensive simulations of ABA triblock copolymer melt using a lattice Monte Carlo method, employing the parallel tempering method, and probing various degrees of compositional asymmetry. Binodals for the order–disorder transition were determined in terms of the thermodynamic incompatibility between segments of different type, quantified by product N, and the triblock asymmetry parameters, ˛ and ˇ. We also provided a polynomial fit to the binodal dependency on ˛ and ˇ, and showed the 3-dimensional phase diagram. The set of results presented in Table 1 can be useful for future studies in the area of triblock
self-assembly. In particular, we indetified the nanostructures below the ODT temperatures for all cases considered, which are a convenient starting point for the construction of a full phase diagram which will contain not only ODT lines (as in this paper), but also the OOT lines. The main result of this work is the 3D phase diagram which shows that the Monte Carlo binodals exhibit qualitatively similar trends to those observed in the SCFT calculations, but there is a pronounced numerical difference between them which indicates the significant role of fluctuations in the vicinity of the order–disorder transition. It is widely recognized that fluctuations are weak in the dense polymer melts and the mean-field approaches, such as SCFT, work remarkably well [5], but they fail in the vicinity of the order–disorder transition. Therefore this work offers an insight into the area which appears to be beyond the scope the mean-field theory. Finally, it is worthwhile to reiterate that while the fluctuations are significant in the vicinity of Monte Carlo order–disorder transition, the binodals seem to follow qualitative trends similar to those observed by SCFT. Acknowledgments Grant DEC-2012/07/B/ST5/00647 of the Polish NCN is gratefully acknowledged. A significant part of the simulations was performed at the Poznan Computer and Networking Center (PCSS). References [1] F.S. Bates, G.H. Fredrickson, Phys. Today 52 (1999) 16. [2] T.S. Bailey, C.M. Hardy, T.H. Epps, F.S. Bates, Macromolecules 35 (2002) 7007. [3] M. Takenaka, T. Wakada, S. Akasaka, S. Nishisuji, K. Saijo, H. Shimizu, M.I. Kim, H. Hasegawa, Macromolecules 40 (2007) 4399. [4] I.W. Hamley, Developments in Block Copolymer Science and Technology, John Wiley & Sons, Berlin, 2004. [5] G.H. Fredrickson, The Equlibrium Theory of Inhomogeneous Polymers, Clarendon Press, Oxford, 2006. [6] A.K. Khandpur, S. Forster, F.S. Bates, I.W. Hamley, A.J. Ryan, W. Bras, K. Almdal, K. Mortensen, Macromolecules 28 (1995) 8796. [7] L. Leibler, Macromolecules 13 (1980) 1602. [8] M.W. Matsen, M. Schick, Macromolecules 27 (1994) 187. [9] M.W. Matsen, J. Phys.: Condens. Matter 14 (2002) R21. [10] G.H. Fredrickson, E. Helfand, J. Chem. Phys. 87 (1987) 697. [11] E.M. Lennon, K. Katsov, G.H. Fredrickson, Phys. Rev. Lett. 101 (2008) 138302. [12] E. Cochran, C. Garcia-Cervera, G. Fredrickson, Macromolecules 39 (2006) 2449. [13] M.W. Matsen, Eur. Phys. J. E 30 (2009) 361. [14] A.M. Mayes, M.O. de la Cruz, J. Chem. Phys. 91 (1989) 7228. [15] A.M. Mayes, M.O. de la Cruz, J. Chem. Phys. 95 (1991) 4670. [16] M.W. Matsen, R.B. Thompson, J. Chem. Phys. 111 (1999) 7139. [17] M.W. Matsen, J. Chem. Phys. 113 (2000) 5539. [18] M.W. Hamersky, S.D. Smith, A.O. Gozen, R.J. Spontak, Phys. Rev. Lett. 95 (2005) 168306. [19] S. Woloszczuk, M. Banaszak, R. Spontak, J. Pol. Sci. B: Pol. Phys. 51 (2013) 343. [20] M. Banaszak, J.H.R. Clarke, Phys. Rev. E 60 (1999) 5753. [21] S. Woloszczuk, M. Banaszak, Eur. Phys. J. E 33 (2010) 343. [22] T. Pakula, in: M.J. Kotelyanskii, D.N. Thedorou, Simulation Methods for Polymers, Marcel-Dekker, New York, 2004, Chap. 5. [23] M. Muller, K. Binder, Macromolecules 28 (1995) 1825. [24] D.C. Morse, J.K. Chung, J. Chem. Phys. 130 (2009) 224901. [25] M. Banaszak, S. Wo loszczuk, S. Jurga, T. Pakula, J. Chem. Phys. 119 (2003) 11451. [26] T.M. Beardley, M.W. Matsen, Eur. Phys. J. E 32 (2009) 255. [27] K. Lewandowski, P. Knychala, M. Banaszak, Comput. Methods Sci. Technol. 16 (2010) 29. [28] P. Knychala, M. Banaszak, M.J. Park, N.P. Balsara, Macromolecules 42 (2009) 8925.