Polymer systems Helmholtz energy

Equation (12) can be used to calculate the residual Helmholtz energy of mixing (AmixFrIsing) of monomer mixture of polymer system in step 2 of Figure 1. However, the mole fraction x, should be replaced by volume fraction ,. We have... 

The residual Helmholtz energy due to the dissociation of polymer chains in pure state and the association of polymer chains in mixture state can be calculate by Equation (5). The pair correlation functions of component i in the corresponding Ising lattice system are calculated by gf = 1 / fyfij (Liu al., 2007). The residual... 

Finally, by substituting Equations (7), (13), and (14) into Equation (7), we obtain the Helmholtz energy of mixing for multicomponent polymer systems... 

In the lattice representation of a polymer solution, each polymer segment or solvent molecule occupies one lattice site, while the system is regarded as a binary mixture of polymer and solvent. The Helmholtz energy of system can be expressed as... 

Many polymer blends or block polymer melts separate microscopically into complex meso-scale structures. It is a challenge to predict the multiscale structure of polymer systems including phase diagram, morphology evolution of micro-phase separation, density and composition profiles, and molecular conformations in the interfacial region between different phases. The formation mechanism of micro-phase structures for polymer blends or block copolymers essentially roots in a delicate balance between entropic and enthalpic contributions to the Helmholtz energy. Therefore, it is the key to establish a molecular thermodynamic model of the Helmholtz energy considered for those complex meso-scale structures. In this paper, we introduced a theoretical method based on a lattice model developed in this laboratory to study the multi-scale structure of polymer systems. First, a molecular thermodynamic model for uniform polymer system is presented. This model can... 

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. 

Consider a polymer system that shows phase separation into a phase with a high density of polymer in solution that coexists with a phase with a low density of polymer in solution. For this system, one can write the Helmholtz free energy of the form ... 

Doi [4,100,114,117] formulated a(E) for rodlike polymer solutions by noticing that (E> is related to the change 8(AF) in the dynamic Helmholtz free energy of the system due to a virtual deformation k St in a short time 8t by... 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

It is useful to summarise the assumptions considered for the thermodynamic analysis discussed above, and the main results obtained. Based on the stress strain relationship for the glassy system in the form of equation 3, the chemical potential of a solute component in the polymeric mixture may be calculated as the derivative of the specific non-equilibrium Helmholtz free energy with respect to the moles of solute per polymer mass Yi at constant temperature, pressure and specific volume, as expressed in eqaution 12. On the other side, under the same assumption, the nonequilibrium Helmholtz free energy has a unique value at given temperature, specific volume and composition, whatever is the pressure of the system, as stated in equation 13. 

Previously we have seen a heuristic derivation of TPTl for the special case of a pure polymer with fixed bond length. Actually, TPTl is a much more general theory which allows to describe a wide variety of systems. We will henceforth restrict our attention to the special case of multicomponent mixtures. We will assume that each of the components is a chain molecule, described as an ensemble of A,- identical beads, which interact with each other and with any other bead in the system by means of a site-site potential, Vtj. Furthermore, adjacent beads on the same molecule interact by means of a bonding potential,. The composition of the mixture may be described by means of the total molecular density, p, and the molar fractions of each component, Xj. According to TPTl, the total Helmholtz free energy of the multicomponent mixture may be described in terms of three different contributions [237, 253] ... 

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