Volume interaction occupied

FIQ. 1 Sketch of the BFM of polymer chains on the three-dimensional simple cubic lattice. Each repeat unit or effective monomer occupies eight lattice points. Elementary motions consist of random moves of the repeat unit by one lattice spacing in one lattice direction. These moves are accepted only if they satisfy the constraints that no lattice site is occupied more than once (excluded volume interaction) and that the bonds belong to a prescribed set of bonds. This set is chosen such that the model cannot lead to any moves where bonds should intersect, and thus it automatically satisfies entanglement constraints [51],... 

Models that, either naturally or through approximation, can be discretized are suitable for study using Monte Carlo simulations. As an example, we give a brief outline below of the simulations of drug release from cylinders assuming Fickian diffusion of drug and excluded volume interactions. This means that each molecule occupies a volume V where no other molecule can be at the same time. 

If the new site is an empty site then the move is allowed and the particle is moved to this new site. If the new site is already occupied, the move is rejected (since excluded volume interactions are assumed). 

To obtain this equation, the ideal gas law—which ignores interactions between molecules—requires two modifications to describe the effects of the forces between molecules, which are repulsive at short distances and attractive at large distances. We know from Section 9.5 that pressure is determined by the product of the momentum transferred per collision with the walls of the container times the number of collisions per second. So, it is necessary to see how repulsive and attractive forces modify the collision rate away from the value it would have in the ideal gas. Because of repulsive forces, molecules cannot occupy the same space at the same time. They exclude other molecules from the volumes they occupy in this way, the effective volume available to a given molecule is not Y, but V — nb, where is a... 

A specification of two interaction potentials is necessary to describe the model system (1) an excluded volume interaction preventing two particles from occupying the same position in space and (2) the long-range Coulomb potential. 

The parameters a and b are characteristic of the substance, and represent corrections to the ideal gas law due to the attractive (dispersion) interactions between the atoms and the volume they occupy due to their repulsive cores. We will discuss van der Waals equation in some detail as a typical example of a mean-field theory. 

The second interaction can be considered as follows, for two monomer units separated by a large number of monomers, the probability of an overlap between the two is non-zero. If the balance of the interactions, monomer-solvent and monomer-monomer leads to a repulsive interaction, as happens in a good solvent, these two distant monomer units repel each other. This excluded volume interaction increases the volume occupied by the macromolecule compared to that of an ideal chain the chain is swollen in a good solvent. A new statistical description of tlie chain conformation is needed, but the key point is that the scaling law still applies and we have R=N, but with v = 0.588 instead of 0.5. The normalised scattering intensity in the asymptotic range becomes ... 

One way to understand this is by considering the sucrose molecules (which are significantly larger than water molecules) as particles in suspension in the water. Einstein showed that the viscosity of a dilute suspension of non-interacting spherical particles (i/) in a liquid of viscosity r] increases with the fraction of the volume they occupy ( ). 

Peek and Hill [1950] used a close-packed lattice with zo = 12 and nearest-neighbor distance fl. The Amolecules were distributed between B lattice sites hence, the volume fraction occupied was y = N/B, and the holes fraction h = l —y. The interaction energy was introduced via the quasichemical equation. Unfortunately, the /i-fraction did not enter explicitly either the partition function or derived from it free energy. Instead, the authors introduced an expandable lattice parameter a = where [Pg.235]

Figure 1. Schematic illustration of the construction of a coarse-grained model for a macromolecule such as polyethylene. In the example shown here, the subchain formed by the three C-C bonds labeled 1,2,3 is represented by the effective bond labeled as I, the subchain formed by the three bonds 4,5,6 is represented by the effective bond labeled as II, etc. In the bond-fluctuation model the length b of the effective bond is allowed to fluctuate in a certain range 6min

Phase diagram for a polymer-solvent system, x is the Flory interaction parameter, and is the volume fraction occupied by the polymer. The condition x = I /2 defines the Flory 6 temperature, in usual cases such as polystyrene-cyclohexane,... 

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