Langmuir 2007, 23, 4809-4818
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Alignment of Lamellar Diblock Copolymer Phases under Shear: Insight from Dissipative Particle Dynamics Simulations Martin Lı´sal*,†,‡ and John K. Brennan§ E. Ha´ la Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, 165 02 Prague 6, Czech Republic, Department of Physics, Faculty of Science, J. E. Purkinje UniVersity, 400 96 Ustı´ n. Lab., Czech Republic, and U.S. Army Research Laboratory, Weapons and Materials Research Directorate, Aberdeen ProVing Ground, Maryland 21005-5066 ReceiVed October 21, 2006. In Final Form: January 29, 2007
Sheared self-assembled lamellar phases formed by symmetrical diblock copolymers are investigated through dissipative particle dynamics simulations. Our intent is to provide insight into the experimental observations that the lamellar phases adopt parallel alignment at low shear rates and perpendicular alignment at high shear rates and that it is possible to use shear to induce a transition from the parallel to perpendicular alignment. Simulations are initiated either from lamellar structures prepared under zero shear where lamellae are aligned into parallel, perpendicular, or transverse orientations with respect to the shear direction or from a disordered melt obtained by energy minimization of a random structure. We first consider the relative stability of the parallel and perpendicular phases by applying shear to lamellar structures initially aligned parallel and perpendicular to the shear direction, respectively. The perpendicular lamellar phase persists for all shear rates investigated, whereas the parallel lamellar phase is only stable at low shear rates, and it becomes unstable at high shear rates. At the high shear rates, the parallel lamellar phase first transforms into an unstable diagonal lamellar phase; and upon further increase of the shear rate, the parallel lamellar phase reorients into a perpendicular alignment. We further determine the preferential alignment of the lamellar phases at low shear rate by performing the simulations starting from either the initial transverse lamellar structure or the disordered melt. Since the low shear-rate simulations are plagued by the unstable diagonal lamellar phases, we vary the system size to achieve the natural spacing of the lamellae in the simulation box. In such cases, the unstable diagonal lamellar phases disappear and lamellar phases adopt the preferential alignment, either parallel or perpendicular. In agreement with the experimental observations, the simulations show that the lamellar phase preferentially adopts the parallel orientation at low shear rates and the perpendicular orientation at high shear rates. The simulations further reveal that the perpendicular lamellar phase has lower internal energy than the parallel lamellar phase, whereas the entropy production of the perpendicular lamellar phase is higher with respect to the parallel lamellar phase. Values of the internal energy and entropy production for the unstable diagonal lamellar phases lie between the corresponding values for the parallel and perpendicular lamellar phases. These simulation results suggest that the relative stability of the parallel and perpendicular lamellar phases at low shear rates is a result of the interplay between competing driving forces in the system: (a) the system’s drive to adopt a structure with the lowest internal energy and (b) the system’s drive to stay in a stationary nonequilibrium state with the lowest entropy production.
1. Introduction Diblock copolymers are polymers consisting of two linear blocks (A and B) of mutually insoluble polymers, chemically connected end-to-end. When a melt of these polymers is quenched, it forms a microphase separated structure, the shape of which depends on the length ratio of the two blocks and the block interactions. For example, the melt of symmetrical diblock copolymers separates into a lamellar phase. It is observed experimentally that the sheared lamellar phases adopt parallel alignment (Figure 1a) at low shear rates and perpendicular alignment (Figure 1b) at high shear rates and that it is possible to use shear to induce a transition from the parallel to perpendicular alignment.1,2 Equilibrium diblock copolymer melts have been well studied1-4 but the behavior of diblock copolymer melts in * To whom correspondence should be addressed. † Academy of Sciences of the Czech Republic. ‡ J. E. Purkinje University. § U.S. Army Research Laboratory, Weapons and Materials Research Directorate. (1) Fredrickson, G. H.; Bates, F. S. Annu. ReV. Mater. Sci. 1996, 26, 501-550. (2) Hamley, I. W. J. Phys.: Condens. Matter 2001, 13, R643-R671. (3) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29, 1091-1098. (4) Groot, R. D.; Madden, T. J. J. Chem. Phys. 1998, 108, 8713-8724.
out-of-equilibrium situations, e.g., under shear, is not fully understood. For example, several coarse-grained nonequilibrium molecular dynamics (NEMD) studies5-10 as well as a recent dissipative particle dynamics (DPD) study11 have shown that by increasing the shear rate, the alignment of the lamellar phase switches from parallel to perpendicular. These simulation studies are in agreement that the perpendicular lamellar phase is the preferential alignment at high shear rates. However, at lower shear rates more evidence of the relative stability of the parallel and perpendicular lamellar phases is needed. For example, Guo et al.5 observed the transformation of the lamellar phase from a transverse alignment (Figure 1c) into a parallel alignment at low shear rates in their NEMD studies of a simple continuum model of diblock copolymers. Using the DPD method, Liu et al.11 simulated the transformation of a diagonal lamellar phase (5) Guo, H.; Kremer, K.; Soddemann, T. Phys. ReV. E 2002, 66, 061503. (6) Guo, H. J. Chem. Phys. 2006, 125, 214902. (7) Rychkov, I.; Yoshikawa, K. J. Chem. Phys. 2004, 120, 3482-3488. (8) Rychkov, I. Macromol. Theory Simul. 2005, 14, 207-242. (9) Fraser, B.; Denniston, C.; Mu¨ser, M. H. J. Polym. Sci. Part B: Polym. Phys. 2005, 43, 970-982. (10) Fraser, B.; Denniston, C.; Mu¨ser, M. H. J. Chem. Phys. 2006, 124, 104902. (11) Liu, W.; Qian, H.-J.; Lu, Z.-Y.; Li, Z.-S.; Sun, C.-C. Phys. ReV. E 2006, 74, 021802.
10.1021/la063095c CCC: $37.00 © 2007 American Chemical Society Published on Web 03/22/2007
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Figure 1. Alignment of a lamellar phase with respect to the shear direction: (a) parallel, (b) perpendicular, (c) transverse, and (d) (unstable) diagonal alignment.
(Figure 1d) prepared under zero shear, into a parallel alignment at a very low shear rate and into a perpendicular alignment at a higher shear rate. These simulation observations are supported by the theoretical work of Fredrickson,12 who developed a stochastic model of the concentration dynamics of block copolymer melts and predicted that parallel lamellas are stable at low shear rates, whereas perpendicular lamellas are stable at high shear rates. On the other hand, the NEMD studies of Fraser et al.9,10 indicate, based on thermodynamic arguments, that the perpendicular alignment of the lamellar phase should be more stable at any finite shear rate. Fraser et al. suggested that surface effects and defects are responsible for the stability of the parallel lamellar phase observed experimentally at low shear rates.13,14,15 Unfortunately, there is no straightforward way to determine the relative stability of the various microphases that are out of equilibrium, e.g., under shear. Although the relative stability of equilibrium systems can be measured by free-energy minimization, no general quantitative criterion exits for nonequilibrium systems. In this work, we use the DPD method16 to provide deeper insight into the metastability and preferential alignment of the lamellar phases under different shear rates. The DPD method is a particle-based mesoscale simulation technique that correctly reproduces (similar to coarse-grained molecular dynamics) the hydrodynamic behavior on long length and time scales, which is critical when studying the mesoscopic structure and dynamics (12) Fredrickson, G. H. J. Rheol. 1994, 38, 1045-1067. (13) Anastasiadis, S. H.; Russell, T. P.; Satija, S. K.; Majkrzak, C. F. Phys. ReV. Lett. 1989, 62, 1852-1855. (14) Balsara, N. P.; Hammouda, B.; Kesani, P. K.; Jonnalagadda, S. V.; Straty, G. C. Macromolecules 1994, 27, 2566-2573. (15) Winey, K. I.; Patel, S. S.; Larson, R. G.; Watanabe, H. Macromolecules 1993, 26, 2542-2549. (16) Hoogerbrugge, P. J.; Koelman, J. M. V. A. Europhys. Lett. 1992, 19, 155-160.
Lı´sal and Brennan
Figure 2. Final snapshots of the system at different shear rates γ corresponding to the DPD simulations in the parallel lamellar phase.
Figure 3. Orientational order parameter O as a function of time t for different shear rates γ corresponding to the DPD simulations in the parallel lamellar phase.
of diblock copolymer melts. Our intent is to characterize the shear induced transformation and to determine the preferential alignment under different shear rates at the mesoscale level. Special focus is on providing more evidence for the relative stability of the parallel and perpendicular lamellar phases at low shear rates. The remaining sections of the paper are organized as follows. The simulation methodology is described in section 2 along with the DPD diblock copolymer model and simulation details. Results are discussed in section 3. Finally, section 4 gives our conclusions.
2. Simulation Methodology 2.1. DPD for Diblock Copolymer Systems. DPD models the diblock copolymer chain as a collection of point particles of type A and of type B, connected end-to-end, that represent lumps of
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Figure 4. Simulation snapshots at different times t corresponding to the parallel-to-perpendicular reorientation at the shear rate γ ) 0.4.
the chain containing several segments of block A or of block B.17 The collection of DPD particles can be viewed as a coarsegrained (mesoscopic) representation of an atomistic polymeric model in which atomistic details are ignored. As a result, the DPD particles behave as a soft spheres that can pass through each other allowing for larger time steps and thus longer time scales to be realized. DPD particles interact with each other through a pairwise, two-body, short-ranged force f that is written as the sum of a conservative force, fC, dissipative force, fD, and random force, fR
fij ) fCij + fDij + fRij
(1)
fC includes a soft repulsion force fCr acting between two particles and a harmonic spring force fCs acting between adjacent particles (17) Groot, R. D.; Warren, P. B. J. Chem. Phys. 1997, 107, 4423-4435.
in a diblock copolymer chain. fCrand fCs are given by
( )
fCr ij ) aij 1 -
rij rij rc rij
)0
(rij < rc) (rij g rc)
(2)
and
fijCs ) -Crij
(3)
respectively. In eqs 2 and 3, aij is the maximum repulsion between particles i and j, rij ) ri - rj, rij ) |rij|, ri is the position of a particle i, rc is the cutoff radius and C is the spring constant. Equation (2) is one of the simplest form used for fCr. More realistic expressions of fCr can be obtained by mapping the radial distribution function of a DPD model onto the radial distribution
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Figure 6. Orientational order parameter O as a function of time t corresponding to a transverse-to-parallel reorientation at the shear rate γ ) 0.005 and to a transverse-to-perpendicular reorientation at γ ) 0.5.
Figure 5. (a) Internal energy per particle u, (b) entropy production per volume, S˙ /V ≡ dis/dt, and (c) asphericity parameter As as a function of shear rate γ for parallel and perpendicular lamellar phases. Circles and triangles correspond to the DPD simulations starting from a parallel and perpendicular lamellar phase, respectively. The lines are drawn as a guide to the eye and to denote data points for which the respective phases are stable.
function of a detailed atomistic model.18 The remaining two forces fD and fR are given by
fDij ) -ηωD(rij)
( )
rij rij ‚υ rij ij rij
(4)
fRij ) σωR(rij)
σij rij x∆t rij
(5)
where ωD(r) and ωR(r) are weight functions given by
( )
ωD(r) ) [ωR(r)]2 ) 1 )0
r rc
2
(r < rc) (r grc)
(6)
η is the friction coefficient, σ is the noise amplitude, σ2 ) 2ηkBT,19 (18) Chen, L.-J.; Qian, H.-J.; Lu, Z.-Y.; Li, Z.-S.; Sun, C.-C. J. Phys. Chem. B 2006, 110, 24093-24100. (19) Espan˜ol, P.; Warren, P. B. Europhys. Lett. 1995, 30, 191-196.
kB is Boltzmann’s constant, T is the temperature, υij ) υi - υj, υi is the velocity of a particle i, ζij is the Gaussian random number, and ∆t is the time step. The evolution of DPD particles in time t is governed by Newton’s equations of motion. 2.2. Simulation Details. We studied diblock copolymers consisting of five DPD particles of type A and five DPD particles of type B, A5B5, in a cubic box of length L. We performed simulations at constant volume rather than at constant pressure (as described in ref 9) since the constant pressure simulations may suffer a collapse of the simulation cell whenever the lamellae reorient under shear.10 Using rc as the unit of length, kBT as the unit of energy, and the mass of a DPD particle as the unit of mass, we carried out DPD simulations at the particle density F ) 3.0 and kBT ) 1.0, and with σ ) 3.0 and η ≡ σ2/(2kBT) ) 4.5. Diblock copolymer model parameters used were those obtained by Groot and Warren from a mapping of DPD polymer models onto Flory-Huggins-type models,17,20 i.e.
aii ) aij ) aii +
75kBT F
1 + 3.9N-0.51 (χN)eff 0.306N
(7)
where N is the length of a DPD diblock copolymer, (χN)eff is the effective Flory-Huggins parameter, and C ) 4.0. By setting (χN)eff ) 35, well above the value of 10.5 corresponding to the order-disorder transition,1,2 we obtained from eqs 7 aAA ≡ aBB ) 25.0 and aAB ) 50.0. We integrated the DPD equation of motions using the simple modification of the velocity-Verlet algorithm proposed by Groot and Warren17 with ∆t ) 0.05. Our value of ∆t corresponds to a typical ∆t value used in DPD simulation studies. If not stated otherwise, DPD simulations were performed with the number of diblock copolymers Np ) 2400 (L ) 20) and with the length of the DPD run trun ) 50 000. Such the value of trun was sufficient to minimize the possible existence of long-living unstable phases during the simulations. Our A5B5 DPD diblock copolymer model can be considered to mimic low molecular weight diblock copolymers such as polyisoprene-polystyrene (PI-PS) with a molecular weight of approximately 8000 g/mol and with a volume fraction of PI (20) Horsch, M. A.; Zhang, Z.; Iacovella, C. R.; Glotzer, S. C. J. Chem. Phys. 2004, 121, 11455-11462.
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Figure 7. Simulation snapshots at different times t corresponding to the transverse-to-parallel reorientation at the shear rate γ ) 0.005.
approximately 0.5.21 According to a coarse-graining procedure given by Maiti et al.,22 15 PS and 23 PI segments can be mapped onto one A and one B DPD particle, respectively. Therefore, the mass mb and volume Vb of one DPD particle are approximately 1570 g/mol and 2640 Å3, respectively. Then, using rc ) (FVb)1/3 given by Maiti et al.,22 the cutoff radius is approximately 20 Å and the box length corresponding to L ) 20 is approximately 400 Å. Further, based upon experimental measurements of the PI-PS system,21 (χN)eff ) 35 corresponds to approximately 400 K. Then, by using t ) [aAAkBT/(mbrc2)]1/2 ht (where ht is the time in real units),23 we get ∆t ) 0.05 is approximately 500 fs. (21) Khandpur, A. K.; Fo¨rster, S.; Bates, F. S.; Hamley, I. W.; Ryan, A. J.; Bras, W.; Almdal, K.; Mortensen, L. Macromolecules 1995, 28, 8796-8806. (22) Maiti, A.; Wescott, J.; Goldbeck-Wood, G. Int. J. Nanotechnol. 2005, 2, 198-214. (23) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications, 2nd ed.; Academic Press: London, 2002.
Steady planar shear flow was induced by means of the LeesEdwards boundary conditions.24 In the Lees-Edwards boundary conditions, the simulation box and its images centered at (x,y) ) ((L, 0), ((2L, 0), ..., are taken to be stationary. Boxes in the layer above, (x,y) ) (0, L),((L, L), ((2L, L), ... are moving at a speed γL in the positive x direction, where γ is the shear rate. Boxes in the layer below, (x, y) ) (0, - L), ((L, - L), ( (2L, - L), ..., move at a speed γL in the negative x direction. A system under such conditions is subjected to a uniform steady shear in the xy plane. For purposes of comparison with experiments and other simulation methods, we set γ ∝ (3kBT/m)1/2, reduced by a characteristic length. For example, for the maximum value of γ used in this study, γ ) 0.6 in DPD units corresponds to 7 times (3kBT/m)1/2/L. (24) Lees, A. W.; Edwards, S. F. J. Phys. C: Solid State Phys. 1972, 5, 19211929.
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Figure 8. Simulation snapshots at different times t corresponding to the transverse-to-perpendicular reorientation at the shear rate γ ) 0.5.
During the simulations, we evaluated the non-vanishing components of the pressure tensor Pxx, Pyy, and Pzz and Pxy ) Pyx, the internal energy per particle u25 and the entropy production S˙ . The S˙ is related to the external work done on the system per unit time, i.e., shear rate times stress times volumeV.26 For our system, S˙ is thus defined in term of the pressure tensor as
S˙ ≡
diS ) -γPxyV dt
(8)
We further determined the asphericity parameter As27 given by (25) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids, 1st ed.; Clarendon Press: Oxford, 1987. (26) De Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics, 2nd ed.; Dover: Amsterdam, 1984. (27) Talsania, S. K.; Wang, Y.; Rajagopalan, R.; Mohanty, K. K. J. Colloid Interface Sci. 1997, 190, 92-103.
As )
(I1 - I2)2 + (I1 - I3)2 + (I2 - I3)2 2(I1 + I2 + I3)2
(9)
where Ii (i ) 1, 2, 3) are eigenvalues of the tensor of inertia of diblock copolymers. Generally speaking, As has a value of zero for spherical objects and unity for infinite cylindrical objects, with intermediate values representing elongated objects. In addition to the simulation snapshots, we also monitored the time evolution of the system by calculating the time-resolved orientational order parameter O which is the largest eigenvalue of the Saupe tensor28
QRβ )
3 2Np
Np
1
ei,Rei,β - δRβ ∑ 2 i)1
(10)
where ei is the unit vector along the end-to-end direction of a
Lamellar Diblock Copolymer Phases
Figure 9. (a) Internal energy per particle u and (b) entropy production per volume, S˙ /V ≡ dis/dt, as a function of shear rate γ for lamellar phases. Circles, triangles and diamonds correspond to the DPD simulations starting, respectively, from a parallel and transverse lamellar phase, and from a disordered melt. The solid and dashed lines denote the stable parallel and perpendicular lamellar phases, respectively.
diblock copolymer i and δRβ is the Kronecker delta. O is zero in a completely disordered state and it is unity if a system is perfectly aligned.
3. Results and Discussion 3.1. Relative Stability of Perpendicular and Parallel Alignment. To address the issue of the relative stability of the perpendicular and parallel lamellar phases, we carried out DPD simulations in either a perpendicular or parallel lamellar phase, and applied a steady planar shear starting from a quiescent state. The quiescent state corresponded to lamellar structures prepared under zero shear and with lamellae aligned either parallel or perpendicular to the applied shear direction. The initial lamellar structures were fully equilibrated and energetically relaxed before applying the shear to ensure Pxx ) Pyy ) Pzz within statistical uncertainty. We examined whether the initial perpendicular and parallel lamellar phase remains stable by visual inspection of the simulation snapshots. The perpendicular lamellar phase persists for all shear rates investigated (up to γ ) 0.6), whereas the parallel lamellar phase is only stable up to γ = 0.2. For 0.2 < γ < 0.4, the parallel lamellar phase transforms into a diagonal (28) De Gennes, P. G.; Prost, J. The Physics of Liquid Crystals, 1st ed.; Clarendon Press: Oxford, 1993.
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lamellar phase, whereas for γ g 0.4, the parallel phase reorients into a perpendicular lamellar phase. Figure 2 shows examples of the final snapshots at four different γ’s corresponding to the simulations in which the lamellae were initially oriented parallel to the shear direction. Note that the undulation of the parallel lamellae at γ ) 0.2 indicates an onset of the reorientation of the parallel lamellar phase into the diagonal lamellar phase. Such undulated parallel lamellae persisted even when we increased trun from 50 000 to 250 000. The perpendicular and parallel lamellar phases further showed a strengthening of the inequality Pyy > Pzz . Pxx upon increasing γ, in agreement with previous NEMD simulations.7,8 Also in agreement with these NEMD simulations, the perpendicular lamellar phase exhibited a reduction in lamellar spacing with increasing γ. Considering the collective character of the transition from the parallel to the perpendicular alignment of the lamellae, we needed to verify that the previous DPD results were not affected by the box size and the time step used. We thus performed additional DPD simulations for the larger system, Np ) 5273 (L ) 26), by starting from the parallel quiescent state where ∆t ) 0.01 and ∆t ) 0.05. We carried out these simulations for γ ) {0.1, 0.2, 0.225, 0.25, 0.4, 0.6}, and the results were in perfect agreement with those for the smaller system, Np ) 2400 (L ) 20), except with the case where γ ) 0.225 and ∆t ) 0.01. In this case, the system did not reorient into the diagonal lamellar phase, but it exhibited strongly undulated parallel lamellae. This suggests that the γ corresponding to reorientation of the lamellae out of parallel alignment may slightly depend on the box size and time step used. We thus estimate that in our DPD system, the reorientation of the lamellae out of parallel alignment takes place between γ = 0.200 and 0.225. To shed more light onto the time evolution of the systems from the quiescent state, we monitored O as a function of t. In the case of the simulations in which the lamellae were initially oriented perpendicular to the shear direction, the diblock copolymers relax rather quickly to an equilibrium orientational order parameter, Oeql, for all values of γ. Values of Oeql were between 0.2 and 0.3. For simulations in which the lamellae were initially orientated parallel to the shear (Figure 3), the diblock copolymers very quickly converged to Oeql’s when γ e0.15, and the lamellae remain intact. When the parallel lamellar phase becomes undulated (γ ) 0.2) or reorients into a diagonal lamellar phase (γ ) 0.225), the O ’s initially go through a minimum (i.e., through a partially disordered state) before reaching their Oeql’s. During the complete parallel-to-perpendicular transformation of the lamellar phase (γ ) 0.4), O passes through a plateau value of about 0.45 before approaching Oeql ≈ 0.29 which is in agreement with the value of Oeql for the perpendicular lamellar phase at γ ) 0.4. The shear induced parallel-to-perpendicular transformation for γ ) 0.4 is further elucidated in Figure 4 where we show the simulation snapshots at different times. It is evident in Figure 4 that after shear is applied the system is disordered up to t ) 200 at which time the system appears to be orientating itself into a diagonal lamellar phase. However, at t ) 300, the system becomes more disordered again as it reorients itself. At t ) 400 we begin to see the formation of lamellae, whereas at t ) 500, it becomes apparent that the system will adopt a perpendicular orientation. Next, we plot ensemble averages of u, S˙ /V, and As as a function of γ in Figure 5. Note that, for γ g0.4, the parallel to perpendicular transition has occurred so that the systems are orientated the same and correspondingly they have the same values of u, S˙ /V, and As within statistical uncertainties. We determined S˙ /V since
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Figure 10. Final snapshots of the system corresponding to the DPD simulations started from a disordered melt for different system sizes at the shear rate γ ) 0.1; Np is the number of diblock copolymers and L is the length of the simulation box.
stationary (nonstrongly) nonequilibrium states such as sheared lamellar phases at low shear rates are characterized by a minimum of the entropy production.26 We see in Figure 5 that for all values of γ the perpendicular lamellar phase has a lower u than the parallel lamellar phase, whereas S˙ /V of the perpendicular lamellar phase is higher with respect to the parallel lamellar phase. Further, the behavior of As indicates that the diblock copolymers in the parallel lamellar phases adopt, on average, more elongated conformations than in the perpendicular lamellar phases. Under zero shear, the parallel and perpendicular lamellar phases are degenerate, and they have equivalent internal energy and entropy. For low shear rates, Figure 5, panels a and b, thus suggests that the perpendicular phase is more stable with respect to the parallel lamellar phase from an energy standpoint (i.e., u of the perpendicular lamellar phases is less than u of the parallel lamellar phases), whereas the parallel phase is more stable with respect to the perpendicular lamellar phase from an entropy production standpoint (i.e., S˙ /V of the parallel lamellar phases is less than S˙ /V of the perpendicular lamellar phases). Overall, the DPD simulations of the perpendicular and parallel lamellar phases indicate that shear can induce a transition from the parallel to
perpendicular alignment and that the perpendicular alignment is the preferential orientation of the lamellar phases at high shear rates. On the other hand, these results do not clearly indicate which alignment, parallel or perpendicular, is the preferable orientation of the lamellar phases at low shear rates. 3.2. Preferential Alignment at Low Shear. To determine the preferential alignment of the lamellar phases at low shear rates, we performed similar simulations but started from either a transverse alignment of the lamellar phase or a disordered melt obtained by energy minimization of a random diblock copolymer structure. Similarly as in the cases of the simulations of the perpendicular and parallel lamellar phases, an initial lamellar structure was prepared under zero shear with lamellae oriented transverse to the shear direction and with Pxx ) Pyy ) Pzz within statistical uncertainty. As typical examples of the two behaviors that we found when shear was applied to these structures, the time evolution of the transverse lamellar phase into a parallel alignment (γ ) 0.005) and into a perpendicular alignment (γ ) 0.5) is demonstrated in Figure 6 by plotting O as a function of t, and in Figures 7 and 8 in which simulations snapshots are shown at different times. In the case of the transverse-to-parallel
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Figure 11. Final snapshots of the system corresponding to the DPD simulations started from a transverse lamellar phase for different system sizes at the shear rate γ ) 0.25; Np is the number of diblock copolymers and L is the length of the simulation box.
reorientation, the lamellae as a whole first rotate toward a parallel alignment, and this is accompanied by a decrease in O. Then, at t ) 300, the lamellae begin breaking into microdomains and O passes through a minimum. The reorientation is completed by a rotating and merging of the microdomains into parallel lamellae. In the case of a transverse-to-perpendicular reorientation, the shear flow immediately breaks the lamellae into microdomains that then reorient themselves to align perpendicular to the shear, which then later merge to form the perpendicular lamellae. The corresponding behavior of O as a function of t is that after a sharp initial increase from the quiescent state O continuously decreases slowly toward its Oeql. Similar results were found when the disordered melt was used as the starting structure. Although both starting configurations (the transverse lamellar structure and the disordered melt) rearranged themselves into perpendicular lamellar phases at high shear rates (γ > 0.3), results at low shear rates are not completely conclusive. Shown in Figure 9 are plots of the ensemble averages of u and S˙ /V as a function of γ for the various structures. As in the cases of the perpendicular and parallel lamellar phases, the simulations also exhibited a
strengthening of the inequality Pyy > Pzz . Pxx upon increasing γ, which again is in agreement with recent NEMD simulations.7,8 Although at some low shear rates such as γ ) 0.005, 0.08, and 0.1, the starting structures resulted in parallel or almost parallel lamellar phases, there were some γ’s for which the starting structures became trapped in diagonal lamellar phases (i.e., with the lamellae oriented in the shear direction but diagonally between a parallel and perpendicular alignment). As already discussed by Fraser et al.,9,10 once the diagonal lamellar phase is found, there is a very high-energy barrier to reorientation, and this energy barrier is exceedingly unlikely to be crossed in any amount of simulation time. Fraser et al. also showed that, by melting the diagonal lamellar phase around the edges of lamellae while subjecting the system to the same shear rate, the lamellae rotate into their preferential alignment (i.e., either into parallel or perpendicular orientations). Hence, a diagonal lamellar phase is not expected to be a final sheared lamellar phase structure. Figure 9 supports this instability argument since the values of u and S˙ /V for the diagonal lamellar phase are sandwiched between the corresponding values of the parallel and perpendicular lamellar
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phases and it is unlikely that the system would not select either a phase with the lowest energy or a phase with the lowest entropy production. To provide more evidence that the parallel lamellar phase is more stable than the perpendicular lamellar phase at low shear rates we varied the system size from L ) 20 (Np ) 2400) to L ) 30 (Np ) 8100) and performed DPD simulations at γ ) 0.1 starting from both a transverse lamellar phase and a disordered melt. By varying the system size, we can attain the natural spacing of the sheared lamellae (i.e., a simulation box with an integral number of lamellae), as opposed to the formation of partial lamellae which tends to distort the overall orientation of the structure. When the simulation box contains an integral number of lamellae, the simulations are less vulnerable to becoming trapped in unstable diagonal lamellar phases. Figure 10 shows the final simulation snapshots corresponding to the DPD simulations started from a disordered melt for different system sizes L (Np) at γ ) 0.1. Figure 10 provides clear proof that the lamellar phase preferentially adopts the parallel alignment at the low shear since nearly all of these final snapshots exhibit parallel or almost parallel orientation of the lamellae. An equivalent conclusion can be drawn from the simulations started from a transverse lamellar structure. In addition, we performed simulations at various system sizes starting from a transverse lamellar phase and from a disordered melt at a higher shear rate γ ) 0.25, i.e., slightly above the value of γ corresponding to the parallel-to-perpendicular transition. Recall that our simulations with a system size of L ) 20 (Np ) 2400) starting from parallel and transverse lamellar phases, and from a disordered melt became trapped in the diagonal lamellar phase. Figure 11 shows the final simulation snapshots corresponding to the DPD simulations started from a transverse lamellar phase. We can see that the final snapshots mostly display perpendicular or almost perpendicular orientation of the lamellae. An equivalent conclusion can be drawn from the simulations started from a disordered melt. These results suggest that the shear induced parallel-to-perpendicular reorientation of the lamellar phase is a first-order transition since by discarding unstable diagonal lamellar phases the parallel-to-perpendicular reorientation appears at a unique value of γ and it is accompanied by sudden changes in thermodynamic quantities such as u and S˙ /V.
Lı´sal and Brennan
Note that in some cases (e.g., L ) 28 in Figure 10 or L ) 25 in Figure 11) only partial lamellae form in the simulation box. This can be attributed to the simulation box size and the use of periodic boundary conditions. The size of the simulation cell does not permit a complete lamellae to form, and fostered by the use of periodic boundary conditions, the system reorients itself to maximize the like interactions and thereby distorts the lamellae alignment. Similar to previous DPD studies,11 sinusoidal and chevron undulations are also evident in Figures 10 and 11, L ) 29 and L ) 27, respectively, but the stability of these structures remains to be determined. If all cases in Figures 10 and 11 for which partial lamellae formed were considered as artifacts of the simulation cell and therefore neglected, then excellent agreement with experimental observations would be found; that is, at low shear rate, the lamellar phase preferentially aligns parallel to the shear direction, and at high shear rate, the lamellar phase preferentially aligns perpendicular to the shear direction.
4. Conclusions We used dissipative particle dynamics simulations to provide insight into the metastability and preferential alignment of sheared lamellar diblock copolymer phases. In agreement with experiment, the simulations showed that the lamellar phase adopts a parallel alignment under low shear and a perpendicular alignment under high shear, and that the shear is capable of inducing a transition from a parallel to a perpendicular lamellar phase. The simulation results further indicated that the relative stability of the parallel and perpendicular lamellar phases at low shear rates is a result of the interplay between competing driving forces in the system: (a) the system’s drive to adopt a structure with the lowest internal energy; and (b) the system’s drive to stay in a stationary nonequilibrium state with the lowest entropy production. Acknowledgment. This research was supported by the Grant Agency of the Czech Republic (Grant No. 203/05/0725), by the National Research Programme “Information Society” (Projects Nos. 1ET400720507 and 1ET400720409), and by the Sixth Framework Programme of the European Community (Project MULTIPRO No. 033304). LA063095C
