United atom group model

Fig. 18.10 (a) United atom group models for monomers of PE, PP, poly(ethyl ethylene) (PEE), poly(hexene-l) (PHI), poly(ethylene propylene) (PEP), PIB, and head-to-head PP (hhPP). Circles designate CH groups, solid lines represent C-C bonds inside the monomer, and dotted lines indicate the C-C bonds linking the monomer to its neighbors along the chain, (b) United atom models of iPP, hhPP, and PEP homopolymer chains constructed by linking monomers of (a) (Freed and Dudowicz 2005)... 

Fig. 10 United atom group model (a) of the actual norbornene monomer and (b) the simplified norbornene structure used in the BLCT calculations. The dashed lines represent the bonds linking the monomer to its neighbors along the chain...

Fig. 12 United atom group models for 1,2 and 1,4 poly(butadiene) (PB) monomers. The models do not distinguish between saturated and unsaturated PB or between cis and trans 1,4 units...

The constraint in Eq. (38) that enables the direct computation of Tg is obtained by the extension of the Lindemann criterion to the softening transformation in glass-forming liquids [42, 56, 129, 130], and the details of this relation are explained in Section VI. Within the schematic model for glass formation (with specified e, Es, and monomer structure), all calculated thermodynamic properties depend only on temperamre T, on pressure P, and on molar mass Mmoi (which is proportional to the number M of united atom groups in single chains). The present section summarizes the calculations for To, Tg, Ti, and Ta as functions of M for a constant pressure of P = 0.101325 MPa (1 atm). 

Yet another way to minimize the number of parameters required is to adopt a so-called united-atom (UA) model. That is, instead of defining only atoms as the fundamental units of the force field, one also defines certain functional groups, usually hydrocarbon groups, e.g., methyl, methylene, aryl CH, etc. The group has its own single set of non-bonded and other parameters - effectively, this reduces the total number of atoms by one less than the total number incorporated into the united atom group. 

The theory is based on an extended lattice model where monomers have molecular structures illustrated in Fig. 18.15. Individual groups [CH n = 0-3] occupy single lattice sites, and the bonds between united atom groups correspond to the... 

The approximate treatment of the bonding constraints in Eq. 3 may be motivated by recourse to the Flory theorem [3,4], which states that in polymer melts it is impossible to discern whether a pair of nonbonded nearest-neighbor united atom groups belongs to different polymer chains or to distant portions of the same polymer molecule. Thus, in the lattice model description of polymer systems, the excluded volume prohibition of multiple occupancy of a site is more important than the consequences of long-range chain connectivity. Based on the Flory theorem, we introduce the zeroth-order mean-field average... 

Due to restrictions imposed by the geometry of a simple cubic lattice and due to the inability [ 18,20] of the current ECT to treat polymer systems whose monomers contain closed rings, the cycHc olefin norbornene monomer is modeled by an approximate compact structure (Fig. 10) that is composed of seven united atom groups (5a = 7). The ethylene monomer is represented by a two-bead dimer with two CH2 groups, i.e., 5b = 2. Thus, this simple model incorporates the actual size disparity between these two types of monomers, a feature that influences the magnitude of the entropic component /s of the interaction parameter xsans> and it also affects the dependence of the energetic portion of xsans on the compositions x and y of the two blend components. 

In a united atomforce field the van der Waals centre of the united atom is usually associated v ilh the position of the heavy (i.e. non-hydrogen) atom. Thus, for a united CH3 or CH2 group the vem der Waals centre would be located at the carbon atom. It would be more accurate to associate the van der Waals centre with a position that was offset slightly from the carbon position, in order to reflect the presence of the hydrogen atoms. Toxvaerd has developed such a model that gives superior performance for alkemes than do the simple united atom models, particularly for simulations at high pressures [Toxvaerd 1990]. In... 

The mapping procedure provides an explicit connection between an atomistic model of a polymer chain and the corresponding coarse-grained model. For PE, we use an united atom description for the CH2 groups, resulting in a potential of the following type [146,182,183] ... 

The wall-PRISM theory has been extended to multiple site models [95], A simple example of a multiple-site model is a vinyl polymer (e.g., polypropylene) where there are three types of united-atom sites corresponding to CH2, CH, and CH3 groups. Ignoring end effects as before the PRISM equations take the form... 

The OPLS model is an example of pair potential where non-bonded interactions are represented through Coulomb and Lennard-Jones terms interacting between sites centred on nuclei (equation (51). Within this model, each atomic nucleus has an interaction site, except CH groups that are treated as united atoms centered on the carbon. It is important to note that no special functions were found to be needed to describe hydrogen bonding and there are no additional interaction sites for lone pairs. Another important point is that standard combining rules are used for the Lennard-Jones interactions such that An = (Ai As )1/2 and Cu = (C Cy)1/2. The A and C parameters may also be expressed in terms of Lennard-Jones o s and e s as A = 4ei Oi and C ... 

---