Diblock copolymers Helmholtz energy

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. 

The SSL theory regards chemically distinct blocks to be completely immiscible. The Helmholtz energy of the molten diblock copolymer system consists of two contributions one is the interfacial energy Fint and the other the elastic energy of blocks Fst. In Semenov s SSL theory (Likhtman and Semenov, 1994 Semenov, 1985), the excess Helmholtz energy per copolymer chain is simplified and expressed as ... 

By combining Equation (53) with Equation (56), the elastic Helmholtz energy of AB diblock copolymer chain can be obtained. 

By combining Equations (62), (63), and (64) with Equation (61), the elastic Helmholtz energy of AB diblock copolymer chain can be obtained. Finally, we have FAsy by adding Equation (50) to Equation (51). 

Figure 35 Helmholtz energies predicted by SSL for AB diblock copolymers under curved confinements as a function of the thickness d (f = 0.5, = 30, and N = 20) at...

Figure 36 (a) Helmholtz energy predicted by SSL for AB diblock copolymers... 

In this work, a framework of the SSL theory for diblock copolymer melts confined in ring-like curved surfaces has been proposed. When the curvature approaches to zero, it reduces to the well-known SSL theory for the parallel lamellar phases. In the case of the equal confined thickness to the exterior radius, it can also be extended to the system with a nanopore confinement. Moreover, the Helmholtz energy of the concentric cylinder barrel, sector column and CMSC phases in 2D confinements based on this SSL theoretical framework can be evaluated in the convenient manner. The calculated results show that the diblock copolymer melts exhibit a layer-type transition with a similar mechanism, regardless of ring-like curved surfaces, planar surfaces, and nanopores. 

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 order to better understand the SA behavior, it is instructive to look at entropy and enthalpy changes in the formation of an ordered state. The Helmholtz free energy of an AB diblock copolymer (di-BCP) is approximately given by ... 

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