- Email: [email protected]
Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures
Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures 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 19 September 2016 Received in revised form 27 March 2017 Accepted 28 March 2017 Available online xxxx Keywords: Diblock Triblock Melt Copolymer Diffusion Dynamic Monte Carlo Order-disorder transition Strong segregation limit Low temperatures CMA
a b s t r a c t Static and dynamic properties of diblock and triblock copolymer melts are simulated over a wide range of temperatures in vicinity of the order-disorder transition and also in the strong segregation regime. Dynamic Monte Carlo method, known as the Cooperative Motion Algorithm, is used with a parallel tempering scheme in order to determine a variety of physical properties. Structure factors are of particular interest since they can be directly related to Small Angle X-ray Scattering data. We also report energy, specific heat, mean-squared end-to-end distance, and the translational diffusion coefficient. Moreover, we determine order-disorder temperatures and investigate the behavior of the melts at low temperatures. We show that the finite size effects can be associated with the spatial reorientations of lamellae. Furthermore, we confirm the existence of a sharp low-temperature peak in specific heat, which we relate to the transition from a state with the diffused domain interface to a state with the sharp domain interface. The chain length dependence of the order-disorder temperatures and the interfacial smoothing temperatures are also presented. Below order-disorder transition temperature a significant loss of the chain mobility is observed, as indicated by an abrupt decrease of the diffusion coefficient. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Incompatibility of the covalently bonded blocks of the copolymers and their inability to segregate on a macroscopic scale lead to a process of order-disorder transition (ODT). As a result, block copolymers can form a variety of ordered nanophases [1–25] that show a significant promise in technological applications [26]. This process is mostly governed by a competition between enthalpic interfacial and entropic stretching energies. A-B-C triblock copolymers are known for their ability to form tens of different phases [26]. On the other hand, A-B diblock and A-B-A triblock copolymers organize into only a few stable nanophases such as layers, hexagonally packed cylinders, gyroid with the Ia3d symmetry, cubic or face-centered micelles, and the O70 -phase [27,28]. While the phase behavior of diblocks and triblocks is similar, their mechanical properties differ significantly. Triblocks in ordered phase can form looped and bridged configurations. Bridges, capable of connecting neighboring domains, can promote the formation of molecular network. Recently we found [29,30] that strongly asymmetric (with
⇑ Corresponding author. E-mail address: [email protected] (M. Banaszak). URL: http://simgroup.amu.edu.pl (M. Banaszak).
respect to A1/A2 composition) A1-B-A2 triblocks with very short one A-block can form a novel kind of molecular network. In this case short terminal blocks organize, at low temperatures, into a subnetwork of A-micelles within the B-domain. In previous papers [31,32] we reported the simulation of structural and static properties of the 7-16-7 triblock and 8-8 diblock copolymer melts over a wide range of temperatures. Both systems self-assembled into a lamellar nanophase, exhibiting two characteristic peaks in the specific heat. The first peak, recorded at a high-temperature, was interpreted as corresponding to the orderdisorder transition, whereas the second one, recorded at lowtemperature, was associated with an abrupt transition from a state with slightly diffused domain interfaces to a state with almost perfectly smooth interdomain surface. We described that effect quantitatively by introducing a new parameter, K, which was a measure of the spatial distribution of the junction points within a given interface in a direction normal to layers. The junction point was defined as a connection between two different blocks within a single chain. The effect of smoothing was also accompanied by a significant energy drop and chain stretching described by the rise of mean-squared end-to-end distance. More recently, Yang et al. [33] studied the interfacial ordering in a symmetric diblock copolymers below the ODT temperature. In order to prevent periodicity trappings they also used quenching
http://dx.doi.org/10.1016/j.nimb.2017.03.147 0168-583X/Ó 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
2
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
method rather than slow cooling. Their results confirmed our findings [31,32] that the interface thickness becomes narrower as the temperature is decreased. However, they did not report the second C V peak because they did not probe at sufficiently low temperatures. While the quenching method works quite well for higher temperatures, it generates long relaxation times in the strong segregation limit. This can lead to unreliable estimates of the sampled properties. To overcome this problem the parallel tempering (PT) method was proposed [34–37]. This method allows exchange of configurations between neighboring temperatures and thus the energy barriers between the local free energy minima can be overcome. As a result we have a more efficient equilibration, sharp peaks in specific heat, and much better accuracy in the localization of both the order-disorder and order-order transitions [36–39]. The goal of this paper is to simulate the dynamic and static properties of the lamellar phase, as shown in Fig. 1, for a series of symmetric diblock and triblock copolymer melts of varying chain length. 2. Model and method The simulations are performed using the Cooperative Motion Algorithm (CMA) for a face-centered cubic lattice with a coordination number z ¼ 12 and lattice constant a ¼ 2. Chain bonds of pffiffiffi length 2a=2 cannot be broken or stretched. Standard periodic boundary conditions are applied. Lattice is completely filled with the monomers and the chains satisfy excluded volume condition. The segments of type i and j interact by ij , where AB ¼ , and AA ¼ BB ¼ 0. This interaction is limited to the nearest neighbors. The parameter is used as an energy unit. We define the reduced energy per lattice site as E ¼ E=, and the reduced temperature as T ¼ kB T=, where kB is the Boltzmann constant. The Metropolis method is not used because the dynamics of the system is of interest in this paper. At a given temperature, T , the Boltzmann factor p ¼ expðEfinal =kB T Þ is compared with a random
Fig. 1. Lamellar nanophase from the simulation of 16-16 diblock copolymer melt in a weak segregation regime.
number r 2 h0; 1i, and if p > r the move is accepted. The CMA algorithm was developed by Tadeusz Pakula and successfully applied to study the dynamics of homopolymer melts [40,41] and diblock copolymer melts [12], yielding results that were in agreement with a dielectric spectroscopy measurements for polystyrenepolyisoprene systems. We use a parallel tempering method, where M replicas of the system are simulated in parallel, each at different temperature, T i , where i ¼ 1; . . . ; M. An attempt to move a segment defines 1=na Monte Carlo step (MCS), where na is the number of lattice sites. After 20000 MCS we attempt to exchange replicas with neighboring T i ’s in random order with a probability
p T i $ T iþ1 ¼ min½1; exp½ðbi biþ1 ÞðU iþ1 U i Þ;
ð1Þ
where bi ¼ 1=kB T i , and U i is the potential energy of the replica at T i . This method allows efficient equilibration, especially at low temperatures. The initial 8 106 MCS are used to equilibrate the system using PT method and the following 2 106 MCS are used, without replica exchange, to collect the data. We repeat this experiment three times, starting from different initial configurations. In each starting configuration the polymer chains are uniformly distributed within the simulation box, assuming statistical conformations and random orientations. We can relate T to the Flory v parameter by the following approximate relation:
v¼
7:5 ; T
ð2Þ
as demonstrated earlier[42]. This equation can be used to relate the experimental v’s with the theoretical T ’s.
Fig. 2. Snapshots of the simulated the 16-16 diblock melt (a, b, c), and the 8-16-8 triblock melt (d, e , f); (a) and (d) present the state above order-disorder transition temperature; (b) and (e) are taken at T ¼ 3:0; (c) and (f) show lamellae at T ¼ 1:0. Only the B segments are shown.
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
3
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
In this paper, we report energy per lattice site, E , mean-squared end-to-end distance, R2 , and the specific heat, C V , calculated from the fluctuations of energy as follows:
hðE hE iÞ i 2
CV ¼
nc T 2
;
where nc is the number of chains in a simulation box.
ð3Þ
We also calculate the structure factor, SðkÞ, by averaging over statistically independent configurations using the following equation: !
1 Sðk Þ ¼ na
*
na X n¼1
! !
cosðk r m Þ
!2 þ
na X n¼1
!2 +
! ! sinðk r m Þ
; thermalav erage
ð4Þ
Fig. 3. Simulation results for the 16-16 diblock copolymer melt: (a) energy per lattice site, E =na ; (b) specific heat, C V ; (c) squared end-to-end distance, R2 ; and 8-16-8 triblock copolymer melt: (d) energy per lattice site, E =na ; (e) specific heat, C V ; (f) squared end-to-end distance, R2 .
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
4
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx !
where na is the number of segments of type a and r m is the position of the mth segment of type a. The magnitude of wavevector, k, varies from kmin ¼ 2p=L to kmax ¼ 2p=b, where L is the lattice size and b is the minimum distance between the monomers. The wavevectors !
k are commensurate with the simulation box size and this constraint limits their possible lengths. In order to describe the domain interface, for the lamellar phase, we calculate the K parameter defined by the following equation:
PM
K¼
r i Þ2 ; M
i¼1 ðr
ð5Þ
where M is the number of junction points within a given interface, r P M is i¼1 r i =M, and r i is calculated as ! !
ri ¼ ri n
ð6Þ
Fig. 4. Simulation results for the 24-24 diblock copolymer melt: (a) energy per lattice site, E =na ; (b) specific heat, C V ; (c) squared end-to-end distance, R2 ; and 12-24-12 triblock copolymer melt: (d) energy per lattice site, E =na ; (e) specific heat, C V ; (f) squared end-to-end distance, R2 .
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx !
with r i denoting the position of i-th junction point in the simulation box ði ¼ 1; 2; . . . ; MÞ. The last term in Eq. (6) is the normalized vec!
tor, n , perpendicular to the layers, defined as
ðx; y; zÞ n ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 x þ y2 þ z2
!
ð7Þ
We determine this vector by calculating the monomer concentration profiles in a large number of directions [x, y, z], where x; y; z ¼ m; m þ 1; . . . ; m, and m is an integer; in our case
5
m ¼ 16. The direction in which the variations of the squared difference between the concentration profiles, ð/A /B Þ2 , reach their maximum is the desired direction perpendicular to the layers. The parameter K is averaged over all interfaces in the simulation box with weights proportional to the number of junction points within a given interface. Finally, we report translational diffusion using the following relation
Fig. 5. Simulation results for the 32-32 diblock copolymer melt: (a) energy per lattice site, E =na ; (b) specific heat, C V ; (c) squared end-to-end distance, R2 ; and 16-32-16 triblock copolymer melt: (d) energy per lattice site, E =na ; (e) specific heat, C V ; (f) squared end-to-end distance, R2 .
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
6
D¼
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
hsðtÞ2 i ; 6t
ð8Þ
where D is the diffusion coefficient, hsðtÞ2 i is the mean-square displacement of the center of mass of polymer chains after a time t. 3. Results In this paper we simulate the monodisperse A-B diblock melts (with the 16-16, 24-24, and 32-32 chain sequences) and A-B-A triblock melts (with the 8-16-8, 12-24-12, and 16-32-16 chain sequences). We use different sizes of the simulation box for a different sequences: the 32 32 32 box for 16-16 and 8-16-8, the 48 48 48 box for 24-24 and 12-24-12, and finally the 64 64 64 box for 32-32 and 16-32-16. The usual periodic boundary conditions are applied. To illustrate the order-disorder transition, we show Fig. 2 which contains selected snapshots from simulations of both A-B (a, b and c) and A-B-A melts (d, e and f). In Fig. 2a and d we can observe the disordered state. As the temperature is lowered the lamellae are formed at T ¼ 3:0 for both diblocks (Fig. 2b) and triblocks
(Fig. 2e). These lamellae are well formed but their interfaces are clearly diffused. However, as the temperature is further decreased to T ¼ 1:0, we can observe perfectly smooth interfaces (with very few defects), as shown in Fig. 2c and f. Therefore, this Figure also illustrates the transition from a state with the diffused interface to a state with the sharp interface. The temperature which corresponds to this crossover will be denoted as T LT . In Figs. 3–5 we report the interaction energy per lattice site, the specific heat, and mean-squared end-to-end distance as a function of the reduced temperature, T , for three simulated copolymer chain lengths: N ¼ 32; N ¼ 48, and N ¼ 64. Each Figure contains data calculated for the diblock melt (a, b, c) and the corresponding triblock melt (d, e, f) of a given chain length, N. In particular, in Fig. 3b we can see the first peak in specific heat at T ¼ 12:2 which is accompanied by a significant energy decrease (see Fig. 3a). We identify this temperature as the order-disorder temperature for the 16-16 diblock melt. Comparing Fig. 3e and d we similarly identify T ¼ 7:8 as the order-disorder temperature for the 8-16-8 triblock melt. According to the mean-field theory for pure diblock melts [43] the T ODT scales linearly with the chain length. We also observe that the order-disorder transition temperatures for diblock
Fig. 6. The interfacial parameter, K, as a function of the reduced temperature, T , for: (a) the 16-16 diblock melt, (b) the 8-16-8 triblock melt.
Fig. 7. The relation between copolymer chain length and: (a) order-disorder temperature, T ODT ; (b) low-temperature interfacial ordering temperature, T LT . Squares represent diblocks while circles denote triblocks.
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
and triblock melts differ approximately by a factor two. This is not surprising since one can consider the 8-16-8 triblock as two 8-8 diblocks connected by their B-ends. Next, we focus on the low-temperature peak in specific heat and the related effects. The maximum of the peak is located between T ¼ 2:4 and T ¼ 2:5 for diblock melt (Fig. 3b) and between T ¼ 1:9 and T ¼ 2:0 for triblock melt (Fig. 3e). In Fig. 3a and d we observe an abrupt decrease of the energy at these temperatures, and accompanying significant chain stretching of diblock (Fig. 3c) and triblock (Fig. 3f) melt. The copolymer chains assume orientations which are mostly normal to the layers (this was also reported by Yang and coworkers [33] for the 35-35 diblock melt). The energy decrease indicates a reduction of unfavorable contacts between segments. This effect, observed previously [31,32], is related to the transition from the state with diffused interfaces (Fig. 2b and e) to the state with sharp surface between the domains (Fig. 2c and f). Thus we focus on the A/B interface thickness, described by the parameter K (see Eq. 5). When all junction points, defined as a connection points between the blocks, fit into a perfect two-dimensional plane, the parameter K is zero. In Fig. 6 we report the interface thickness for the 16-16 diblock melt (6a), and the 8-16-8 triblock melt (6b) as a function of the reduced temperature, T . We can see that above T ’ 2:4 for diblock and above T ’ 1:9 for triblock the parameter K is higher than unity and increases slightly with increasing temperature. Both temperatures correspond to the low-temperature C V peaks for the 16-16 diblock melt (Fig. 3b) and the 8-16-8 triblock
7
melt (Fig. 3e). Below these temperatures, on the other hand, we observe a significant drop of the parameter K to the value close to zero in the vicinity of T ¼ 1:0. This is accompanied by sharp and smooth interfaces (Fig. 2c and f) at T ¼ 1:0. Now, we explore the N effects by considering longer chains with N ¼ 48 and N ¼ 64. In Fig. 4 we report the energy per lattice site (4a and d), the specific heat (4b and e), and mean squared endto-end distance (4c and f) for the 24-24 diblock and 12-24-12 triblock melt. In Fig. 5 we show the same quantities for the 32-32 diblocks and the 16-32-16 triblocks. We notice that the behavior of longer chains is similar to that for the chains with N ¼ 32. In particular, we observe both the ODT and low-temperature C V peaks are accompanied by matching behavior of the energy and the end-to-end distance. Moreover, in Figs. 3–5, for temperatures between ODT and the low-temperature C V peak, we can observe several discontinuities in energies accompanied by the corresponding discontinuities in R2 . These effects are related to the rearrangements of the lamellar structure within the simulation box, from a state slightly below T ODT with three nonzero components of a vector normal to the layers, towards having two nonzero components, and finally reaching the state with one nonzero component at the lowest temperatures. These discontinuities are not related to any significant changes in the corresponding specific heat. In Fig. 7a we present T ODT ’s as a function of N for diblocks (open squares) and triblocks (open circles). We can observe that T ODT ’s scale linearly with N, which agrees with the mean-field theory. Moreover, the corresponding slopes are described by 0.37 and
Fig. 8. Structure factor, SðkÞ, calculated for the 16-16 diblock (a, b) and the 8-16-8 triblock (c, d) at the following temperatures: (a) and (c) T ¼ 1:0; (b) and (d) T ¼ 3:0.
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
8
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
0.17, respectively, as expected because the A-B-A triblock chain consists effectively of two shorter A-B diblock chains (thus the approximate factor of two). Similar data for T LT ’s, as a function of N, are presented in Fig. 7b. In contrast to a significant increase of T ODT with increasing N; T LT remains largely unchanged with increasing N. While T LT ’s of the diblocks are slightly larger than
those of corresponding triblocks, we do not observe a shift by a factor of two. Next, we turn our attention to Fig. 8, where we report structure factors for the 16-16 diblock melt (8a and b) and the 8-16-8 triblock melt (8c and d), as defined by Eq. 4. Fig. 8a and c show structure factors calculated at T ¼ 1:0, whereas Fig. 8b and d at
Fig. 9. Translational diffusion coefficient, D, as function of the reduced temperature, T , for the following diblock melts: (a) 16-16, (b) 24-24, (c) 32-32; and triblock melts: (d) 8-16-8, (e) 12-24-12, (f) 16-32-16.
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
T ¼ 3:0. We can clearly observe several peaks, Sðki Þ, where i ¼ 0; 1; . . ., for which the ratios ki =k0 are integers. These ratios clearly indicate the lamellar nanostructure. Moreover, for T ¼ 1:0 we observe well defined (sharp) lamellae and for T ¼ 3:0 interfaces are diffused. This is in agreement with previous results presented in this paper. Therefore, we think that the Monte Carlo structure factors should be compared to the Small Angle Xray Scattering data in order to verify our findings experimentally. Finally, we present some insight into the dynamics of the simulated systems. In Fig. 9 we show the translational diffusion coefficient, D (calculated from Eq. 8) as a function of the reduced temperature, T , for the simulated series of diblock and triblock melts. For all systems, we observe a significant decrease of the translational diffusion in the vicinity of the order-disorder transition temperature. However, below T ODT the rate of D decrease upon cooling is significantly smaller than that above T ODT . This is clearly related to the formation of lamellar nanostructure, at T ODT , which has to slow down the chain mobility. Below T LT copolymer chains significantly lose their mobility and D drops almost to zero. We can conjecture that the sharp interfaces can lead to the chain trappings. The effect was observed at T ¼ 1:94 for the 16-16 melt (Fig. 9a), at T ¼ 1:91 for the 24-24 melt (Fig. 9b), at T ¼ 1:94 for the 32-32 melt (Fig. 9c), at T ¼ 2:41 for the 8-16-8 melt (Fig. 9d), at T ¼ 2:38 for the 12-24-12 melt (Fig. 9e), and at T ¼ 2:43 for the 16-32-16 melt (Fig. 9f). 4. Conclusions Static and dynamic properties of diblock and triblock copolymer melts were simulated over a wide range of temperatures in a vicinity of order-disorder transition and in the strong segregation regime. We confirmed the existence of sharp low-temperature C V peaks, which we associated with transition from a state with diffused interfaces to a state with sharp interdomain surfaces. Our results are in agreement with the mean-field prediction that the order-disorder temperature scales linearly with the chain length. We found, however, that this prediction did not apply to the low-temperature interfacial ordering temperature. We also found that the calculated diffusion coefficients decrease upon cooling. Moreover, the low-temperature C V peaks corresponded to those temperatures which marked the onset of an almost total loss of the chain mobility. Whether this effect is the reflection of the underlying lattice or whether it is a universal phenomenon remains an unresolved issue. Therefore the Small Angle X-ray Scattering data for copolymer melts would be valuable to resolve this problem.
9
lations was performed at the Poznan Computer and Networking Center (PCSS). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
H. Fried, K. Binder, J. Chem. Phys. 94 (1991) 8349. A. Weyersberg, T.A. Vilgis, Phys. Rev. E 48 (1993) 377. M.W. Matsen, M. Schick, Macromolecules 27 (1994) 187. H. Watanabe, Macromolecules 28 (1995) 5006. A.K. Khandpur, S. Forster, F.S. Bates, I.W. Hamley, A.J. Ryan, W. Bras, K. Almdal, K. Mortensen, Macromolecules 28 (1995) 8796. U. Micka, K. Binder, Macromol. Theory Simul. 4 (1995) 419. M.W. Matsen, J. Chem. Phys. 102 (1995) 3884. T. Pakula, J. Comput. Aided Mater. Des. 3 (1996) 351. G.S. Grest, M.-D. Lacasse, K. Binder, A.M. Gupta, J. Chem. Phys. 105 (1996) 10583. M.W. Matsen, F.S. Bates, Macromolecules 29 (1996) 1091. T. Dotera, H. Hatano, J. Chem. Phys. 105 (1996) 8413. T. Pakula, K. Karatasos, S.H. Anastasiadis, G. Fytas, Macromolecules 26 (1997) 8463. M.W. Matsen, J. Chem. Phys. 108 (1997) 785. M. Laradji, A.-C. Shi, J. Noolandi, R.C. Desai, Macromolecules 30 (1997) 3242. G. Floudas, T. Pakula, G. Velis, S. Sioula, N. Nadjichristidis, J. Chem. Phys. 108 (1998) 6498. S. Phan, G.H. Fredrickson, Macromolecules 31 (1998) 59. M.W. Matsen, R.B. Thomson, J. Chem. Phys. 111 (1999) 7139. C.Y. Ryu, T.P. Lodge, Macromolecules 32 (1999) 7190. K. Karatassos, S.H. Anastasiadis, T. Pakula, H. Watanabe, Macromolecules 33 (2000) 523. S.-M. Mai, W. Mingvanish, S.C. Turner, C. Chaibundit, J.P.A. Fairclough, F. Heatley, M.W. Matsen, A.J. Ryan, C. Booth, Macromolecules 33 (2000) 5124. F. Drolet, G.H. Fredrickson, Macromolecules 34 (2001) 5317. G.H. Fredrickson, V. Ganesan, F. Drolet, Macromolecules 35 (2002) 2002. J. Huh, W.H. Jo, G. ten Brinke, Macromolecules 35 (2002) 2413. T. Aoyagi, T. Honda, M. Doi, J. Chem. Phys. 117 (2002) 8153. O.N. Vassiliev, M.W. Matsen, J. Chem. Phys. 118 (2003) 7700. F.S. Bates, G.H. Fredrickson, Phys. Today 52 (1999) 32. T.S. Bailey, C.M. Hardy, T.H. Epps, F.S. Bates, Macromolecules 35 (1999) 7007. M. Takenaka, T. Wakada, S. Akasaka, S. Nishisuji, K. Saijo, H. Shimizu, M.I. Kim, H. Hasegawa, Macromolecules 40 (2007) 4399. S. Woloszczuk, K.P. Mineart, R.J. Spontak, M. Banaszak, Phys. Rev. E 91 (2015) 010601. S. Woloszczuk, S. Jurga, M. Banaszak, Phys. Rev. E 94 (2016) 022502. M. Banaszak, S. Wołoszczuk, S. Jurga, T. Pakula, J. Chem. Phys. 119 (2003) 11451. S. Woloszczuk, M. Banaszak, S. Jurga, T. Pakula, M. Radosz, J. Chem. Phys. 121 (2004) 12044. J. Yang, M.A. Winnik, T. Pakula, Macromol. Theory Simul. 14 (2005) 9. R.H. Swendsen, J.S. Wang, Phys. Rev. Lett. 57 (1986) 2607. D.J. Earl, M.W. Deem, Phys. Chem. Chem. Phys 7 (2005) 3910. K. Lewandowski, P. Knychala, M. Banaszak, Comp. Meth. Sci. Tech. 16 (1) (2010) 29. T.M. Beardsley, M.W. Matsen, Eur. Phys. J. E 32 (2010) 255. S. Woloszczuk, M. Banaszak, R.J. Spontak, J. Polym. Sci. B 51 (2013) 343. S. Woloszczuk, M. Banaszak, Fluid Phase Equilibr. 358 (2013) 156. K.U. Kirst, F. Kremer, T. Pakula, and J. Hollingshurst, J. Colloid Polym. Sci. 272, 1420 (1994). T. Pakula, S. Geyler, T. Edling, D. Boese, Rheol. Acta 35 (1996) 631. S. Woloszczuk, M. Banaszak, Eur. Phys. J. E 33 (2010) 343. L. Leibler, Macromolecules 13 (1980) 1602.
Acknowledgements Grant No. DEC-2012/07/B/ST5/00647 (S.W. and M.B.) of the Polish NCN is gratefully acknowledged. A significant part of the simu-
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures 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 19 September 2016 Received in revised form 27 March 2017 Accepted 28 March 2017 Available online xxxx Keywords: Diblock Triblock Melt Copolymer Diffusion Dynamic Monte Carlo Order-disorder transition Strong segregation limit Low temperatures CMA
a b s t r a c t Static and dynamic properties of diblock and triblock copolymer melts are simulated over a wide range of temperatures in vicinity of the order-disorder transition and also in the strong segregation regime. Dynamic Monte Carlo method, known as the Cooperative Motion Algorithm, is used with a parallel tempering scheme in order to determine a variety of physical properties. Structure factors are of particular interest since they can be directly related to Small Angle X-ray Scattering data. We also report energy, specific heat, mean-squared end-to-end distance, and the translational diffusion coefficient. Moreover, we determine order-disorder temperatures and investigate the behavior of the melts at low temperatures. We show that the finite size effects can be associated with the spatial reorientations of lamellae. Furthermore, we confirm the existence of a sharp low-temperature peak in specific heat, which we relate to the transition from a state with the diffused domain interface to a state with the sharp domain interface. The chain length dependence of the order-disorder temperatures and the interfacial smoothing temperatures are also presented. Below order-disorder transition temperature a significant loss of the chain mobility is observed, as indicated by an abrupt decrease of the diffusion coefficient. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Incompatibility of the covalently bonded blocks of the copolymers and their inability to segregate on a macroscopic scale lead to a process of order-disorder transition (ODT). As a result, block copolymers can form a variety of ordered nanophases [1–25] that show a significant promise in technological applications [26]. This process is mostly governed by a competition between enthalpic interfacial and entropic stretching energies. A-B-C triblock copolymers are known for their ability to form tens of different phases [26]. On the other hand, A-B diblock and A-B-A triblock copolymers organize into only a few stable nanophases such as layers, hexagonally packed cylinders, gyroid with the Ia3d symmetry, cubic or face-centered micelles, and the O70 -phase [27,28]. While the phase behavior of diblocks and triblocks is similar, their mechanical properties differ significantly. Triblocks in ordered phase can form looped and bridged configurations. Bridges, capable of connecting neighboring domains, can promote the formation of molecular network. Recently we found [29,30] that strongly asymmetric (with
⇑ Corresponding author. E-mail address: [email protected] (M. Banaszak). URL: http://simgroup.amu.edu.pl (M. Banaszak).
respect to A1/A2 composition) A1-B-A2 triblocks with very short one A-block can form a novel kind of molecular network. In this case short terminal blocks organize, at low temperatures, into a subnetwork of A-micelles within the B-domain. In previous papers [31,32] we reported the simulation of structural and static properties of the 7-16-7 triblock and 8-8 diblock copolymer melts over a wide range of temperatures. Both systems self-assembled into a lamellar nanophase, exhibiting two characteristic peaks in the specific heat. The first peak, recorded at a high-temperature, was interpreted as corresponding to the orderdisorder transition, whereas the second one, recorded at lowtemperature, was associated with an abrupt transition from a state with slightly diffused domain interfaces to a state with almost perfectly smooth interdomain surface. We described that effect quantitatively by introducing a new parameter, K, which was a measure of the spatial distribution of the junction points within a given interface in a direction normal to layers. The junction point was defined as a connection between two different blocks within a single chain. The effect of smoothing was also accompanied by a significant energy drop and chain stretching described by the rise of mean-squared end-to-end distance. More recently, Yang et al. [33] studied the interfacial ordering in a symmetric diblock copolymers below the ODT temperature. In order to prevent periodicity trappings they also used quenching
http://dx.doi.org/10.1016/j.nimb.2017.03.147 0168-583X/Ó 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
2
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
method rather than slow cooling. Their results confirmed our findings [31,32] that the interface thickness becomes narrower as the temperature is decreased. However, they did not report the second C V peak because they did not probe at sufficiently low temperatures. While the quenching method works quite well for higher temperatures, it generates long relaxation times in the strong segregation limit. This can lead to unreliable estimates of the sampled properties. To overcome this problem the parallel tempering (PT) method was proposed [34–37]. This method allows exchange of configurations between neighboring temperatures and thus the energy barriers between the local free energy minima can be overcome. As a result we have a more efficient equilibration, sharp peaks in specific heat, and much better accuracy in the localization of both the order-disorder and order-order transitions [36–39]. The goal of this paper is to simulate the dynamic and static properties of the lamellar phase, as shown in Fig. 1, for a series of symmetric diblock and triblock copolymer melts of varying chain length. 2. Model and method The simulations are performed using the Cooperative Motion Algorithm (CMA) for a face-centered cubic lattice with a coordination number z ¼ 12 and lattice constant a ¼ 2. Chain bonds of pffiffiffi length 2a=2 cannot be broken or stretched. Standard periodic boundary conditions are applied. Lattice is completely filled with the monomers and the chains satisfy excluded volume condition. The segments of type i and j interact by ij , where AB ¼ , and AA ¼ BB ¼ 0. This interaction is limited to the nearest neighbors. The parameter is used as an energy unit. We define the reduced energy per lattice site as E ¼ E=, and the reduced temperature as T ¼ kB T=, where kB is the Boltzmann constant. The Metropolis method is not used because the dynamics of the system is of interest in this paper. At a given temperature, T , the Boltzmann factor p ¼ expðEfinal =kB T Þ is compared with a random
Fig. 1. Lamellar nanophase from the simulation of 16-16 diblock copolymer melt in a weak segregation regime.
number r 2 h0; 1i, and if p > r the move is accepted. The CMA algorithm was developed by Tadeusz Pakula and successfully applied to study the dynamics of homopolymer melts [40,41] and diblock copolymer melts [12], yielding results that were in agreement with a dielectric spectroscopy measurements for polystyrenepolyisoprene systems. We use a parallel tempering method, where M replicas of the system are simulated in parallel, each at different temperature, T i , where i ¼ 1; . . . ; M. An attempt to move a segment defines 1=na Monte Carlo step (MCS), where na is the number of lattice sites. After 20000 MCS we attempt to exchange replicas with neighboring T i ’s in random order with a probability
p T i $ T iþ1 ¼ min½1; exp½ðbi biþ1 ÞðU iþ1 U i Þ;
ð1Þ
where bi ¼ 1=kB T i , and U i is the potential energy of the replica at T i . This method allows efficient equilibration, especially at low temperatures. The initial 8 106 MCS are used to equilibrate the system using PT method and the following 2 106 MCS are used, without replica exchange, to collect the data. We repeat this experiment three times, starting from different initial configurations. In each starting configuration the polymer chains are uniformly distributed within the simulation box, assuming statistical conformations and random orientations. We can relate T to the Flory v parameter by the following approximate relation:
v¼
7:5 ; T
ð2Þ
as demonstrated earlier[42]. This equation can be used to relate the experimental v’s with the theoretical T ’s.
Fig. 2. Snapshots of the simulated the 16-16 diblock melt (a, b, c), and the 8-16-8 triblock melt (d, e , f); (a) and (d) present the state above order-disorder transition temperature; (b) and (e) are taken at T ¼ 3:0; (c) and (f) show lamellae at T ¼ 1:0. Only the B segments are shown.
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
3
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
In this paper, we report energy per lattice site, E , mean-squared end-to-end distance, R2 , and the specific heat, C V , calculated from the fluctuations of energy as follows:
hðE hE iÞ i 2
CV ¼
nc T 2
;
where nc is the number of chains in a simulation box.
ð3Þ
We also calculate the structure factor, SðkÞ, by averaging over statistically independent configurations using the following equation: !
1 Sðk Þ ¼ na
*
na X n¼1
! !
cosðk r m Þ
!2 þ
na X n¼1
!2 +
! ! sinðk r m Þ
; thermalav erage
ð4Þ
Fig. 3. Simulation results for the 16-16 diblock copolymer melt: (a) energy per lattice site, E =na ; (b) specific heat, C V ; (c) squared end-to-end distance, R2 ; and 8-16-8 triblock copolymer melt: (d) energy per lattice site, E =na ; (e) specific heat, C V ; (f) squared end-to-end distance, R2 .
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
4
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx !
where na is the number of segments of type a and r m is the position of the mth segment of type a. The magnitude of wavevector, k, varies from kmin ¼ 2p=L to kmax ¼ 2p=b, where L is the lattice size and b is the minimum distance between the monomers. The wavevectors !
k are commensurate with the simulation box size and this constraint limits their possible lengths. In order to describe the domain interface, for the lamellar phase, we calculate the K parameter defined by the following equation:
PM
K¼
r i Þ2 ; M
i¼1 ðr
ð5Þ
where M is the number of junction points within a given interface, r P M is i¼1 r i =M, and r i is calculated as ! !
ri ¼ ri n
ð6Þ
Fig. 4. Simulation results for the 24-24 diblock copolymer melt: (a) energy per lattice site, E =na ; (b) specific heat, C V ; (c) squared end-to-end distance, R2 ; and 12-24-12 triblock copolymer melt: (d) energy per lattice site, E =na ; (e) specific heat, C V ; (f) squared end-to-end distance, R2 .
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx !
with r i denoting the position of i-th junction point in the simulation box ði ¼ 1; 2; . . . ; MÞ. The last term in Eq. (6) is the normalized vec!
tor, n , perpendicular to the layers, defined as
ðx; y; zÞ n ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 x þ y2 þ z2
!
ð7Þ
We determine this vector by calculating the monomer concentration profiles in a large number of directions [x, y, z], where x; y; z ¼ m; m þ 1; . . . ; m, and m is an integer; in our case
5
m ¼ 16. The direction in which the variations of the squared difference between the concentration profiles, ð/A /B Þ2 , reach their maximum is the desired direction perpendicular to the layers. The parameter K is averaged over all interfaces in the simulation box with weights proportional to the number of junction points within a given interface. Finally, we report translational diffusion using the following relation
Fig. 5. Simulation results for the 32-32 diblock copolymer melt: (a) energy per lattice site, E =na ; (b) specific heat, C V ; (c) squared end-to-end distance, R2 ; and 16-32-16 triblock copolymer melt: (d) energy per lattice site, E =na ; (e) specific heat, C V ; (f) squared end-to-end distance, R2 .
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
6
D¼
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
hsðtÞ2 i ; 6t
ð8Þ
where D is the diffusion coefficient, hsðtÞ2 i is the mean-square displacement of the center of mass of polymer chains after a time t. 3. Results In this paper we simulate the monodisperse A-B diblock melts (with the 16-16, 24-24, and 32-32 chain sequences) and A-B-A triblock melts (with the 8-16-8, 12-24-12, and 16-32-16 chain sequences). We use different sizes of the simulation box for a different sequences: the 32 32 32 box for 16-16 and 8-16-8, the 48 48 48 box for 24-24 and 12-24-12, and finally the 64 64 64 box for 32-32 and 16-32-16. The usual periodic boundary conditions are applied. To illustrate the order-disorder transition, we show Fig. 2 which contains selected snapshots from simulations of both A-B (a, b and c) and A-B-A melts (d, e and f). In Fig. 2a and d we can observe the disordered state. As the temperature is lowered the lamellae are formed at T ¼ 3:0 for both diblocks (Fig. 2b) and triblocks
(Fig. 2e). These lamellae are well formed but their interfaces are clearly diffused. However, as the temperature is further decreased to T ¼ 1:0, we can observe perfectly smooth interfaces (with very few defects), as shown in Fig. 2c and f. Therefore, this Figure also illustrates the transition from a state with the diffused interface to a state with the sharp interface. The temperature which corresponds to this crossover will be denoted as T LT . In Figs. 3–5 we report the interaction energy per lattice site, the specific heat, and mean-squared end-to-end distance as a function of the reduced temperature, T , for three simulated copolymer chain lengths: N ¼ 32; N ¼ 48, and N ¼ 64. Each Figure contains data calculated for the diblock melt (a, b, c) and the corresponding triblock melt (d, e, f) of a given chain length, N. In particular, in Fig. 3b we can see the first peak in specific heat at T ¼ 12:2 which is accompanied by a significant energy decrease (see Fig. 3a). We identify this temperature as the order-disorder temperature for the 16-16 diblock melt. Comparing Fig. 3e and d we similarly identify T ¼ 7:8 as the order-disorder temperature for the 8-16-8 triblock melt. According to the mean-field theory for pure diblock melts [43] the T ODT scales linearly with the chain length. We also observe that the order-disorder transition temperatures for diblock
Fig. 6. The interfacial parameter, K, as a function of the reduced temperature, T , for: (a) the 16-16 diblock melt, (b) the 8-16-8 triblock melt.
Fig. 7. The relation between copolymer chain length and: (a) order-disorder temperature, T ODT ; (b) low-temperature interfacial ordering temperature, T LT . Squares represent diblocks while circles denote triblocks.
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
and triblock melts differ approximately by a factor two. This is not surprising since one can consider the 8-16-8 triblock as two 8-8 diblocks connected by their B-ends. Next, we focus on the low-temperature peak in specific heat and the related effects. The maximum of the peak is located between T ¼ 2:4 and T ¼ 2:5 for diblock melt (Fig. 3b) and between T ¼ 1:9 and T ¼ 2:0 for triblock melt (Fig. 3e). In Fig. 3a and d we observe an abrupt decrease of the energy at these temperatures, and accompanying significant chain stretching of diblock (Fig. 3c) and triblock (Fig. 3f) melt. The copolymer chains assume orientations which are mostly normal to the layers (this was also reported by Yang and coworkers [33] for the 35-35 diblock melt). The energy decrease indicates a reduction of unfavorable contacts between segments. This effect, observed previously [31,32], is related to the transition from the state with diffused interfaces (Fig. 2b and e) to the state with sharp surface between the domains (Fig. 2c and f). Thus we focus on the A/B interface thickness, described by the parameter K (see Eq. 5). When all junction points, defined as a connection points between the blocks, fit into a perfect two-dimensional plane, the parameter K is zero. In Fig. 6 we report the interface thickness for the 16-16 diblock melt (6a), and the 8-16-8 triblock melt (6b) as a function of the reduced temperature, T . We can see that above T ’ 2:4 for diblock and above T ’ 1:9 for triblock the parameter K is higher than unity and increases slightly with increasing temperature. Both temperatures correspond to the low-temperature C V peaks for the 16-16 diblock melt (Fig. 3b) and the 8-16-8 triblock
7
melt (Fig. 3e). Below these temperatures, on the other hand, we observe a significant drop of the parameter K to the value close to zero in the vicinity of T ¼ 1:0. This is accompanied by sharp and smooth interfaces (Fig. 2c and f) at T ¼ 1:0. Now, we explore the N effects by considering longer chains with N ¼ 48 and N ¼ 64. In Fig. 4 we report the energy per lattice site (4a and d), the specific heat (4b and e), and mean squared endto-end distance (4c and f) for the 24-24 diblock and 12-24-12 triblock melt. In Fig. 5 we show the same quantities for the 32-32 diblocks and the 16-32-16 triblocks. We notice that the behavior of longer chains is similar to that for the chains with N ¼ 32. In particular, we observe both the ODT and low-temperature C V peaks are accompanied by matching behavior of the energy and the end-to-end distance. Moreover, in Figs. 3–5, for temperatures between ODT and the low-temperature C V peak, we can observe several discontinuities in energies accompanied by the corresponding discontinuities in R2 . These effects are related to the rearrangements of the lamellar structure within the simulation box, from a state slightly below T ODT with three nonzero components of a vector normal to the layers, towards having two nonzero components, and finally reaching the state with one nonzero component at the lowest temperatures. These discontinuities are not related to any significant changes in the corresponding specific heat. In Fig. 7a we present T ODT ’s as a function of N for diblocks (open squares) and triblocks (open circles). We can observe that T ODT ’s scale linearly with N, which agrees with the mean-field theory. Moreover, the corresponding slopes are described by 0.37 and
Fig. 8. Structure factor, SðkÞ, calculated for the 16-16 diblock (a, b) and the 8-16-8 triblock (c, d) at the following temperatures: (a) and (c) T ¼ 1:0; (b) and (d) T ¼ 3:0.
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
8
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
0.17, respectively, as expected because the A-B-A triblock chain consists effectively of two shorter A-B diblock chains (thus the approximate factor of two). Similar data for T LT ’s, as a function of N, are presented in Fig. 7b. In contrast to a significant increase of T ODT with increasing N; T LT remains largely unchanged with increasing N. While T LT ’s of the diblocks are slightly larger than
those of corresponding triblocks, we do not observe a shift by a factor of two. Next, we turn our attention to Fig. 8, where we report structure factors for the 16-16 diblock melt (8a and b) and the 8-16-8 triblock melt (8c and d), as defined by Eq. 4. Fig. 8a and c show structure factors calculated at T ¼ 1:0, whereas Fig. 8b and d at
Fig. 9. Translational diffusion coefficient, D, as function of the reduced temperature, T , for the following diblock melts: (a) 16-16, (b) 24-24, (c) 32-32; and triblock melts: (d) 8-16-8, (e) 12-24-12, (f) 16-32-16.
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147
S. Wołoszczuk, M. Banaszak / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
T ¼ 3:0. We can clearly observe several peaks, Sðki Þ, where i ¼ 0; 1; . . ., for which the ratios ki =k0 are integers. These ratios clearly indicate the lamellar nanostructure. Moreover, for T ¼ 1:0 we observe well defined (sharp) lamellae and for T ¼ 3:0 interfaces are diffused. This is in agreement with previous results presented in this paper. Therefore, we think that the Monte Carlo structure factors should be compared to the Small Angle Xray Scattering data in order to verify our findings experimentally. Finally, we present some insight into the dynamics of the simulated systems. In Fig. 9 we show the translational diffusion coefficient, D (calculated from Eq. 8) as a function of the reduced temperature, T , for the simulated series of diblock and triblock melts. For all systems, we observe a significant decrease of the translational diffusion in the vicinity of the order-disorder transition temperature. However, below T ODT the rate of D decrease upon cooling is significantly smaller than that above T ODT . This is clearly related to the formation of lamellar nanostructure, at T ODT , which has to slow down the chain mobility. Below T LT copolymer chains significantly lose their mobility and D drops almost to zero. We can conjecture that the sharp interfaces can lead to the chain trappings. The effect was observed at T ¼ 1:94 for the 16-16 melt (Fig. 9a), at T ¼ 1:91 for the 24-24 melt (Fig. 9b), at T ¼ 1:94 for the 32-32 melt (Fig. 9c), at T ¼ 2:41 for the 8-16-8 melt (Fig. 9d), at T ¼ 2:38 for the 12-24-12 melt (Fig. 9e), and at T ¼ 2:43 for the 16-32-16 melt (Fig. 9f). 4. Conclusions Static and dynamic properties of diblock and triblock copolymer melts were simulated over a wide range of temperatures in a vicinity of order-disorder transition and in the strong segregation regime. We confirmed the existence of sharp low-temperature C V peaks, which we associated with transition from a state with diffused interfaces to a state with sharp interdomain surfaces. Our results are in agreement with the mean-field prediction that the order-disorder temperature scales linearly with the chain length. We found, however, that this prediction did not apply to the low-temperature interfacial ordering temperature. We also found that the calculated diffusion coefficients decrease upon cooling. Moreover, the low-temperature C V peaks corresponded to those temperatures which marked the onset of an almost total loss of the chain mobility. Whether this effect is the reflection of the underlying lattice or whether it is a universal phenomenon remains an unresolved issue. Therefore the Small Angle X-ray Scattering data for copolymer melts would be valuable to resolve this problem.
9
lations was performed at the Poznan Computer and Networking Center (PCSS). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
H. Fried, K. Binder, J. Chem. Phys. 94 (1991) 8349. A. Weyersberg, T.A. Vilgis, Phys. Rev. E 48 (1993) 377. M.W. Matsen, M. Schick, Macromolecules 27 (1994) 187. H. Watanabe, Macromolecules 28 (1995) 5006. A.K. Khandpur, S. Forster, F.S. Bates, I.W. Hamley, A.J. Ryan, W. Bras, K. Almdal, K. Mortensen, Macromolecules 28 (1995) 8796. U. Micka, K. Binder, Macromol. Theory Simul. 4 (1995) 419. M.W. Matsen, J. Chem. Phys. 102 (1995) 3884. T. Pakula, J. Comput. Aided Mater. Des. 3 (1996) 351. G.S. Grest, M.-D. Lacasse, K. Binder, A.M. Gupta, J. Chem. Phys. 105 (1996) 10583. M.W. Matsen, F.S. Bates, Macromolecules 29 (1996) 1091. T. Dotera, H. Hatano, J. Chem. Phys. 105 (1996) 8413. T. Pakula, K. Karatasos, S.H. Anastasiadis, G. Fytas, Macromolecules 26 (1997) 8463. M.W. Matsen, J. Chem. Phys. 108 (1997) 785. M. Laradji, A.-C. Shi, J. Noolandi, R.C. Desai, Macromolecules 30 (1997) 3242. G. Floudas, T. Pakula, G. Velis, S. Sioula, N. Nadjichristidis, J. Chem. Phys. 108 (1998) 6498. S. Phan, G.H. Fredrickson, Macromolecules 31 (1998) 59. M.W. Matsen, R.B. Thomson, J. Chem. Phys. 111 (1999) 7139. C.Y. Ryu, T.P. Lodge, Macromolecules 32 (1999) 7190. K. Karatassos, S.H. Anastasiadis, T. Pakula, H. Watanabe, Macromolecules 33 (2000) 523. S.-M. Mai, W. Mingvanish, S.C. Turner, C. Chaibundit, J.P.A. Fairclough, F. Heatley, M.W. Matsen, A.J. Ryan, C. Booth, Macromolecules 33 (2000) 5124. F. Drolet, G.H. Fredrickson, Macromolecules 34 (2001) 5317. G.H. Fredrickson, V. Ganesan, F. Drolet, Macromolecules 35 (2002) 2002. J. Huh, W.H. Jo, G. ten Brinke, Macromolecules 35 (2002) 2413. T. Aoyagi, T. Honda, M. Doi, J. Chem. Phys. 117 (2002) 8153. O.N. Vassiliev, M.W. Matsen, J. Chem. Phys. 118 (2003) 7700. F.S. Bates, G.H. Fredrickson, Phys. Today 52 (1999) 32. T.S. Bailey, C.M. Hardy, T.H. Epps, F.S. Bates, Macromolecules 35 (1999) 7007. M. Takenaka, T. Wakada, S. Akasaka, S. Nishisuji, K. Saijo, H. Shimizu, M.I. Kim, H. Hasegawa, Macromolecules 40 (2007) 4399. S. Woloszczuk, K.P. Mineart, R.J. Spontak, M. Banaszak, Phys. Rev. E 91 (2015) 010601. S. Woloszczuk, S. Jurga, M. Banaszak, Phys. Rev. E 94 (2016) 022502. M. Banaszak, S. Wołoszczuk, S. Jurga, T. Pakula, J. Chem. Phys. 119 (2003) 11451. S. Woloszczuk, M. Banaszak, S. Jurga, T. Pakula, M. Radosz, J. Chem. Phys. 121 (2004) 12044. J. Yang, M.A. Winnik, T. Pakula, Macromol. Theory Simul. 14 (2005) 9. R.H. Swendsen, J.S. Wang, Phys. Rev. Lett. 57 (1986) 2607. D.J. Earl, M.W. Deem, Phys. Chem. Chem. Phys 7 (2005) 3910. K. Lewandowski, P. Knychala, M. Banaszak, Comp. Meth. Sci. Tech. 16 (1) (2010) 29. T.M. Beardsley, M.W. Matsen, Eur. Phys. J. E 32 (2010) 255. S. Woloszczuk, M. Banaszak, R.J. Spontak, J. Polym. Sci. B 51 (2013) 343. S. Woloszczuk, M. Banaszak, Fluid Phase Equilibr. 358 (2013) 156. K.U. Kirst, F. Kremer, T. Pakula, and J. Hollingshurst, J. Colloid Polym. Sci. 272, 1420 (1994). T. Pakula, S. Geyler, T. Edling, D. Boese, Rheol. Acta 35 (1996) 631. S. Woloszczuk, M. Banaszak, Eur. Phys. J. E 33 (2010) 343. L. Leibler, Macromolecules 13 (1980) 1602.
Acknowledgements Grant No. DEC-2012/07/B/ST5/00647 (S.W. and M.B.) of the Polish NCN is gratefully acknowledged. A significant part of the simu-
Please cite this article in press as: S. Wołoszczuk, M. Banaszak, Monte Carlo structure factors and selected physical properties of symmetric copolymer melts at low temperatures, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.03.147