Stereochemistry of vinyl polymers plays a large role in determining their fundamental behavior. However, atactic chains (0 < p_m < 1, where p_m denotes the probability of a meso diad), due to quenched randomness in stereochemical sequences, are fundamentally different from chains of the same monomer unit with pure stereochemistry. While stereochemically pure chains can be described by a single conformational partition function, a large number of partition functions describe chains of intermediate stereochemical compositions, even when all chains have the same number of bonds. For coarse-grained simulation, it is important to know the minimum number of states required to reach the correct asymptotic behavior of the mean square end-to-end distance of chains. Therefore, we study the influence of the quenched randomness in the stereochemical sequences of atactic chains on the mean square unperturbed end-to-end distance, <r₂>₀.

A virtual bond model is constructed between the centers of mass of the C₆ rings in polystyrene. The virtual bond model is a rotational isomeric state (RIS) model is derived from a conventional RIS model expressed in terms of the C-C bonds in the main chain. The conventional RIS model is mapped to the corresponding bond lengths, bond angles and torsion angles on the virtual bond model for a small molecule, 2,4,6,8-tetraphenylnonane (TPN) of the appropriate stereochemical sequence. Simplifications of the virtual bond model are made by including only a few of the conformations of the highest statistical weight at the specific stereochemistry. Calculations of the end-to-end distances are conducted for 200 independent stereochemical sequences of 1000 C—C bonds.

We find that a zeroth approximation virtual bond model, which retains only the most probable conformations of each tetrad, find the correct asymptotic dependence of <r₂>₀ ~ n when n → ∞ if 0 < p_m < 1. The same zeroth approximation model incorrectly yields <r₂>₀ ~ n² if p_m is either 0 or 1. At intermediate p_m, the mean square end-to-end dimensions of the zeroth approximation are surprisingly close to the full, C-C bond based RIS model. For all isotactic, syndiotactic and atactic chains, the first approximation, which includes the next most probable conformations at each tetrad, yields the correct asymptotic dependence. Rather than adding more conformations, slight changes in some of the conformations by adjustments in the soft degrees of freedom create excellent agreement between the <r₂>₀ of the first approximation virtual bond model and the full RIS model based on C-C bonds over the entire range of stereochemical composition, 0 < p_m < 1.

This achievement of the zeroth approximation model reveals the important role of the quenched randomness of the stereochemical sequences in determining the unperturbed dimensions of atactic chains. The constraints of the virtual bond model provide a method of coarse-graining for atactic polymer chains with bulky side groups. As an example, the virtual bond model developed here is modified to model polystyrene using Monte Carlo on a bond-fluctuation model cubic lattice. The ability to reproduce equilibrium properties for different stereochemical compositions is discussed.