Model Flory lattice

It is worthy to recall here that in the Flory lattice model, the solvent molecule occupies one cell of the lattice and the polymer molecule occupies p successive cells... 

Meanwhile, Flory predicts a higher critical volume fraction. It is thought that the Flory lattice model is comparatively simple, even though the addition of the soft interactions does not change this shortcoming. 

The lattice model, as put forth by Flory [84, 85], has been proved successful in the treatments of the liquid crystallinity in polymeric systems, despite its artificiality. In our series of work, the lattice model has been extended to the treatment of biopolypeptide systems. The relationship between the polypeptide ordering nature and the LC phase structure is well established. Recently, by taking advantage of the lattice model, we formulated a lattice theory of polypeptide-based diblock copolymer in solution [86]. The polypeptide-based diblock copolymer exhibits lyotropic phases with lamellar, cylindrical, and spherical structures when the copolymer concentration is above a critical value. The tendency of the rodlike block (polypeptide block) to form orientational order plays an important role in the formation of lyotropic phases. This theory is applicable for examining the ordering nature of polypeptide blocks in polypeptide block copolymer solutions. More work on polypeptide ordering and microstructure based on the Flory lattice model is expected. 

Abe A, Ballauff M (1991) The Flory lattice model. In Cifetri A (ed) Liquid crystallinity in polymers principles and fundamental properties. Wiley-VCH, Weinheim, Chap. 4... 

Figure 2.24. A schematic diagram of the Flory lattice model with varying cell sizes near the liquid/vapour interface.

As predicted by Flory, the critical aspect ratio of rod molecules for thermotropic liquid crystallinity is 6.4 [2,3]. However, in this prediction, the influence of temperature was not considered. The temperature effect was included in the form of an orientationally dependent energy function in more recent development of the Flory lattice model [3]. For LMWLCs, the aspect ratio is defined as Ljd, in which L is the length and d is the diameter of the molecule. For LCPs, the aspect ratio is replaced by persistence ratio through replacing molecule length L by persistence length q [3],... 

Generally, polymer chains exhibit a variety of flexibility, from flexible to rigid rodlike polymers. Polymer blends containing a liquid crystalline ordering have received considerable attention. The volume of literature on the theories and experiments of liquid crystallinity in polymer blends is now very extensive. There are many theoretical models for liquid crystalline polymers, such as the Flory lattice model [3-8], wormlike chain model [9-11], and Onsager virial theory [12] for rigid rods. An example of a recent review of liquid crystalline polymers is the text book of Donald and Windle [13[. 

The Onsager theory, or second virial expansion, is very successful in predicting the qualitative behavior of the isotropic-nematic phase transition of hard rods. H owever, it is an exact theory in the limit ofI/D—>oo. Straley has estimated that, for L/D < 20, the contribution of the third virial coefficient to the free energy in the nematic phase should be at least comparable to that of the second virial coefficient [22]. For more concentrated solutions, an alternative approach such as the Flory lattice model [3, 5] is required. The full phase diagram of rod-like colloidal systems has been obtained in computer simulations [23, 24]. The effects of polydispersity of rods [19, 25] and charged rods [10] are also important in the phase transitions. The comparison between Onsager theory and experimental results has been summarized by Vroege and Lekkerkerker [26]. 

Polymer simulations can be mapped onto the Flory-Huggins lattice model. For this purpose, DPD can be considered an off-lattice version of the Flory-Huggins simulation. It uses a Flory-Huggins x (chi) parameter. The best way to obtain % is from vapor pressure data. Molecular modeling can be used to determine x, but it is less reliable. In order to run a simulation, a bead size for each bead type and a x parameter for each pair of beads must be known. 

Figure 8.1 The entropy of mixing (in units of R) as a function of mole fraction solute for ideal mixing and for the Flory-Huggins lattice model with n = 50, 100, and 500. Values are calculated in Example 8.1.

The lattice model that served as the basis for calculating ASj in the last section continues to characterize the Flory-Huggins theory in the development of an expression for AHj . Specifically, we are concerned with the change in enthalpy which occurs when one species is replaced by another in adjacent lattice sites. The situation can be represented in the notation of a chemical reaction ... 

In this section and the last, we have examined the lattice model of the Flory-Huggins theory for general expressions relating AHj and ASj to the composition of the mixture. The separate components can therefore be put together to give an expression for AGj as a function of temperature and composition ... 

Both Flory [143] and Huggins [144] in 1941 addressed themselves to this problem with the initial aim of describing solutions of linear polymers in low molecular weight solvents. Both used lattice models, and their initial derivations considered only polymer length (rather than shape, i.e. branching, etc.) The derivation given here will also limit itself to differences in molecular size, but will be based on an available volume approach. 

From a thermodynamic point of view, the heteropolymer globule in hand represents a subsystem which is composed of a macromolecule involving lu l2 units Mi, M2 and molecules of monomers Mi, M2 whose numbers are Mi,M2. Among these variables and volume fractions a in the framework of the simplest Flory-Huggins lattice model there are obvious stoichiometric relationships... 

Any parameter that influences either the enthalpy or the entropy of the system might change the free energy in such a manner that phase separation occurs. For a system like those studied here, being composed of a polymer and a solvent, the entropy of mixing, AS, has been calculated independently by Flory [53,55,56] and Huggins [57,58] based on the lattice model. It is assumed that each solvent molecule occupies one cell of the lattice. The volume of the smallest unit serves as reference volume. It is stated that the macromolecule can be composed of ad-... 

The addition of water to solutions of PBT dissolved in a strong acid (MSA) causes phase separation in qualitative accord with that predicted by the lattice model of Flory (17). In particular, with the addition of a sufficient amount of water the phase separation produces a state that appears to be a mixture of a concentrated ordered phase and a dilute disordered phase. If the amount of water has not led to deprotonation (marked by a color change) then the birefringent ordered phase may be reversibly transformed to an isotropic disordered phase by increased temperature. This behavior is in accord with phase separation in the wide biphasic gap predicted theoretically (e.g., see Figure 8). The phase separation appears to occur spinodally, with the formation of an ordered, concentrated phase that would exist with a fibrillar morphology. This tendency may be related to the appearance of fibrillar morphology in fibers and films of such polymers prepared by solution processing. 

Solution of long-chain molecules When two liquids mix to form a mixture, the entropy change is similar to that of the volume expansion, as long as the solute molecules have the same size as the solvent molecules and are randomly distributed. But when the solute forms long-chain molecules, the correct method of calculating the entropy was given by Flory. First consider a lattice model where the solvent and the solute molecules have the same volume. Let i and 2 be the number of solvent and... 

Gibbs and DiMarzio [47] (GD) first developed a systematic statistical mechanical theory of glass formation in polymer fluids, based on experimental observations and on lattice model calculations by Meyer, Flory, Huggins, and... 

Simulations [73] have recently provided some insights into the formal 5c —> 0 limit predicted by mean field lattice model theories of glass formation. While Monte Carlo estimates of x for a Flory-Huggins (FH) lattice model of a semifiexible polymer melt extrapolate to infinity near the ideal glass transition temperature Tq, where 5c extrapolates to zero, the values of 5c computed from GD theory are too low by roughly a constant compared to the simulation estimates, and this constant shift is suggested to be sufficient to prevent 5c from strictly vanishing [73, 74]. Hence, we can reasonably infer that 5 approaches a small, but nonzero asymptotic low temperature limit and that 5c similarly becomes critically small near Tq. The possibility of a constant... 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

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