Polymers interaction potential

Whereas in small molecule systems crystal planes are not necessarily obviously related to the molecular structure, in polymers packing of chains or helices will naturally generate layered structures and the relevance of the interplanar distance to the nature of the polymer-polymer interaction potential becomes more obvious. 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

The focus of this chapter is on an intermediate class of models, a picture of which is shown in Fig. 1. The polymer molecule is a string of beads that interact via simple site-site interaction potentials. The simplest model is the freely jointed hard-sphere chain model where each molecule consists of a pearl necklace of tangent hard spheres of diameter a. There are no additional bending or torsional potentials. The next level of complexity is when a stiffness is introduced that is a function of the bond angle. In the semiflexible chain model, each molecule consists of a string of hard spheres with an additional bending potential, EB = kBTe( 1 + cos 0), where kB is Boltzmann s constant, T is... 

If each polymer is modeled as being composed of N beads (or sites) and the interaction potential between polymers can be written as the sum of site-site interactions, then generalizations of the OZ equation to polymers are possible. One approach is the polymer reference interaction site model (PRISM) theory [90] (based on the RISM theory [91]) which results in a nonlinear integral equation given by... 

One tool for working toward this objective is molecular mechanics. In this approach, the bonds in a molecule are treated as classical objects, with continuous interaction potentials (sometimes called force fields) that can be developed empirically or calculated by quantum theory. This is a powerful method that allows the application of predictive theory to much larger systems if sufficiently accurate and robust force fields can be developed. Predicting the structures of proteins and polymers is an important objective, but at present this often requires prohibitively large calculations. Molecular mechanics with classical interaction potentials has been the principal tool in the development of molecular models of polymer dynamics. The ability to model isolated polymer molecules (in dilute solution) is well developed, but fundamental molecular mechanics models of dense systems of entangled polymers remains an important goal. 

Light scattering from macromolecules is used routinely to obtain molecular weights, radii of gyration and polymer-solvent and polymer-polymer interaction parameters. A closely related technique, electric field induced light scattering (EFLS) (13,14) has received less attention, but is also potentially useful for polymer characterization. 

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

FIG. 13.13 Interaction between polymer-coated particles. Overlap of adsorbed polymer layers on close approach of dispersed solid particles (parts a and b). The figure also illustrates the repulsive interaction energy due to the overlap of the polymer layers (dark line in part c). Depending on the nature of the particles, a strong van der Waals attraction and perhaps electrostatic repulsion may exist between the particles in the absence of polymer layers (dashed line in part c), and the steric repulsion stabilizes the dispersion against coagulation in the primary minimum in the interaction potential. 

Wong et al. [131] measured directly the interaction potential between a tethered ligand and its receptor in aqueous media. Using a surface force apparatus, the interaction force-distance profile was determined between streptavidin immobilized on a lipid bilayer and biotin tethered to the distal end of lipid-PEG. Both lipid bilayers containing streptavidin and biotin were absorbed onto the surface of mica having a specific curvature. Both cationic and anionic polymer grafted... 

Xv being a phenomenological interaction parameter for the noncombinatorial part of the solute (polymer) chemical potential, defined on a volume fraction basis. Equations similar to equation (1.9) and (1.12) serve to define x on a segment fraction basis, x -... 

The possibility of occurrence of instability of colloidal dispersions in the presence of free polymer was first predicted by Asakura and Oosawa (5), who have shown that the exclusion of the free polymer molecules from the interparticle space generates an attractive force between particles, DeHek and Vrij (1) have developed a model in which the particles and the polymer molecules are treated as hard spheres and rederived in a simple and illuminating way the interaction potential proposed by Asakura and Oosawa. Using this potential, they calculated the second virial coefficient for the particles as a function of the free polymer concentration and have shown that... 

Here a mixture of sterically stabilized colloidal particles, solvent, and free polymer molecules in solution is considered. When two particles approach one another during a Brownian collision, the interaction potential between the two depends not only on the distance of separation between them, but also on various parameters, such as the thickness and the segment density distribution of the adsorbed layer, the concentration and the molecular weight of the free polymer. The various types of forces that are expected lo contribute to the interaction potential are (i) forces due to the presence of the adsorbed polymer, (ii) forces due to the presence of the free polymer, and (iii) van der Waals forces. It is assumed here that there are no electrostatic forces. A brief account of the nature of these forces as... 

In one of the limiting cases, the free polymer is allowed to penetrate the adsorbed layer around the particles. One may note that when the free polymer and the adsorbed polymer are both present in the steric layer around the particle, the interactions between the two must be taken into account while evaluating the in-terparticle forces. However, in the absence of a detailed knowledge of the structure of the adsorbed layer, it is difficult to evaluate this contribution to the interaction potential. Then, the situation is similar to the one considered by Asakura and Oosawa (16), and the force of attraction between two bare particles of radius a in the presence of free polymer molecules of radius / can be expressed as the product of the osmotic pressure Pmm and the area of the intersection of the two overlapping volumes ... 

In order to apply the above procedure to determine the conditions of phase separation, we have chosen the system of polyisobutene-stabilized silica particles with polystyrene as the free polymer dissolved in cyclohexane. The system temperature is chosen to be the 8 temperature for the polystyrene-cyclohexane system (34.5°C), corresponding to the experimental conditions of deHek and Vrij (1). The pertinent parameters required for the calculation of the contribution of the adsorbed layers to the total interaction potential are a = 48 nm, u, =0.18 nm3, 5 = 5 nm, Xi = 0.47(32), X2 = 0.10(32), v = 0.10, and up = 2.36 nm3. It can be seen from Fig. 2 that these forces are repulsive, with very large positive values for the potential energy at small distances of separation and falling off to zero at separation distances of the order of 25, where 6 is the thickness of the adsorbed layer. At the distance of separation 5, the expressions for the interpenetration domain and the interpenetration plus compression domain give the same value for the free energy, indicating a continuous transition from one domain to the other. 

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