Diblock copolymers simulation

Figure 7.24 (and on cover) from Groot R D and T J Madden 1998. Dynamic simulation of diblock copolymer microphase separation. The Journal of Chemical Physics 108 8713-8724. Americcm Institute of Physics. 

W. Gozdz, R. Holyst. From the plateau problem to minimal surfaces in lipids, surfactants and diblock copolymer systems. Macromol Theory Simul 5 321-332, 1996. 

The viscoelastic effects on the morphology and dynamics of microphase separation of diblock copolymers was simulated by Huo et al. [ 126] based on Tanaka s viscoelastic model [127] in the presence and absence of additional thermal noise. Their results indicate that for

bulk modulus of both blocks, the area fraction of the A-rich phase remains constant during the microphase separation process. For each block randomly oriented lamellae are preferred. 

These concentration fluctuations are pivotal to the phase transitions in block copolymer melts and are dynamic in nature. They lead to a renormahzation of the relevant interaction parameters and are thought to be responsible for the induction of the first-order nature of the phase transition [264,265]. Such fluctuations are better studied in dynamic experiments. Thus, one can observe an increasing interest in diblock copolymer dynamics. These dynamic properties are being analysed through experimental, theoretical [266,267] and computer simulation approaches [268,269] with the aim of determining the main featirres of diblock copolymer dynamics in comparison to homopolymer dynamics. There are three main issues ... 

Fig. 2.45 Illustrating the formation of a dumb-bell conformation in an AB diblock copolymer, computer simulation results show that the r.m.s. separation between A and B blocks, increases faster than the radius of gyration, Rs, with increasing segregation eN (r is a monomeric interaction energy) (Fried and Binder 1991a).The ODT is estimated to occur at eN 7.5-9.

Computer simulations of a range of properties of block copolymer micelles have been performed by Mattice and co-workers.These simulations have been based on bead models for copolymer chains on a cubic lattice. Types of allowed moves for bead chains are illustrated in Fig. 3.27. The formation of micelles by diblock copolymers under weak segregation conditions was simulated with pairwise interactions between A and B beads and between the A bead and vacant sites occupied by solvent, S (Wang et al. 19936). This leads to the formation of micelles with a B core. The cmc was found to depend strongly on fVB and % = x.w = %AS. In the range 3 < (xlz)N < 6, where z is the lattice constant, the cmc was found to be exponentially dependent onIt was found than in the micelles the insoluble block is slightly collapsed, and that the soluble block becomes stretched as Na increases, with [Pg.178]

Xu T, Wang J, Russell TP (2007) Electric field alignment of diblock copolymer thin films. In Zvelindovsky AV (ed) Nanostructured soft matter experiment, theory, simulation and perspectives. Springer, Berlin, pp 171-198... 

Fig. 1 Phase diagram of self-assembled structures in AB diblock copolymer melt, predicted by self-consistent mean field theory [31] and confirmed experimentally [33]. The MesoDyn simulations [34, 35] demonstrate morphologies that are predicted theoretically and observed experimentally in thin films of cylinder-forming block copolymers under surface fields or thickness constraints dis disordered phase with no distinct morphology, C perpendicular-oriented and Cy parallel-oriented cylinders, L lamella, PS polystyrene, PL hexagonally perforated lamella phase. Dots with related labels within the areal of the cylinder phase indicate the bulk parameters of the model AB and ABA block copolymers discussed in this work (Table 1). Reprinted from [36], with permission. Copyright 2008 American Chemical Society...

The experimental studies on phase behavior and pattern formation reviewed here have been done on substrate-supported films of cylinder-forming polystyrene- foc -polybutadiene diblock (SB) [36, 43, 51, 111-114] and triblock (SBS) [49, 62, 115-117] copolymers (Table 1), lamella-forming polystyrene- /ocfc-poly(2-vinyl pyridine) diblock copolymers (SV) [118, 119] and ABC block terpolymers of various compositions [53, 63, 120-131], In simulation studies, a spring and bid model of ABA Gaussian chains has been used (see Sect. 2) [36,42, 58, 59],... 

In this review, we introduce another approach to study the multiscale structures of polymer materials based on a lattice model. We first show the development of a Helmholtz energy model of mixing for polymers based on close-packed lattice model by combining molecular simulation with statistical mechanics. Then, holes are introduced to account for the effect of pressure. Combined with WDA, this model of Helmholtz energy is further applied to develop a new lattice DFT to calculate the adsorption of polymers at solid-liquid interface. Finally, we develop a framework based on the strong segregation limit (SSL) theory to predict the morphologies of micro-phase separation of diblock copolymers confined in curved surfaces. 

Monte Carlo simulation for diblock copolymers confined in curved surfaces... 

In order to develop a theoretical method for describing the meso-structures of diblock copolymer confined in curved surfaces, MC simulation was first used to find the possible phase separation structures of diblock copolymer melt. 

Weak preference to block copolymer. Figure 26 shows the MC simulated morphologies of A5B5 diblock copolymers confined in cylindrical nanopores with the different exterior radii Rex and eAS = bs = 0. It is shown that the lamellar structure forms parallel to the pore axis. However, there is a little difference between small and large Rex. The lamellar structure is not bended at small Rex as shown in Figure 26a, while it is bended like the wave-shape at large Rex as... 

When the confined surfaces suffer from a weak interaction with block copolymer, either parallel or vertical lamellar structures for AB diblock copolymer systems under flat and curved confinements could exhibit, as shown in Figure 27. From theoretical predictions (Turner, 1992 Walton et al., 1994) and simulations (Wang et al., 2000 Yin et al., 2004), the frustration between d and L0 could result in the alternative appearance of parallel lamellar and vertical lamellar structures under flat confinements. A question is naturally arisen can both concentric cylinder barrel and sector column structures appear under the curved confinement ... 

The morphologies of AB diblock copolymers confined between two concentric curved surfaces with exterior radii i eX and interior radii Rin have been investigated via MC simulations in our previous work... 

According to the MC simulation results mentioned above, if the diblock copolymer melts are confined in cylindrical pore or between two concentric curved surfaces with strong preference to one of the blocks, the... 

In this work, we have focused on strong surface preference with only mutual transition between the symmetrical and asymmetrical layer-type structures. SSL theory and MC simulation are further applied to investigate the self-assembled morphology of diblock copolymers confined in the nanopore. MC simulation shows that the Niayer of the concentric cylinder barrel changes with respect to the extent of frustration between the exterior radius Rex and the bulk lamellar period L0. Simultaneously, the predictions of SSL theory also show that both... 

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

In some cases, one is interested in the structures of complex fluids only at the continuum level, and the detailed molecular structure is not important. For example, long polymer molecules, especially block copolymers, can form phases whose microstructure has length scales ranging from nanometers almost up to microns. Computer simulations of such structures at the level of atoms is not feasible. However, composition field equations can be written that account for the dynamics of some slow variable such as 0 (x), the concentration of one species in a binary polymer blend, or of one block of a diblock copolymer. If an expression for the free energy / of the mixture exists, then a Ginzburg-Landau type of equation can sometimes be written for the time evolution of the variable 0 with or without flow. An example of such an equation is (Ohta et al. 1990 Tanaka 1994 Kodama and Doi 1996)... 

Fig. 3. Computer simulation results using a time-dependent Ginzburg-Landau approach, showing the microstructural evolution after a temperature jump from the lamellar phase to the hexagonal cylinder phase for a moderately asymmetric diblock copolymer. The time units are arbitrary. (Reprinted with permission from Polymer 39, S. Y. Qi and Z.-G. Zheng, Weakly segregated block copolymers Anisotropic fluctuations and kinetics of order-order and order-disorder transitions, 4639-4648, copyright 1998, with permission of Excerpta Medica Inc.)...

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