Model bond fluctuation

Fig. 8.8 The bond fluctuation model. In this example three bcmds in the polymer arc incorporated into a singk effecti bond between effective moncmers . (Figure adapted from Baschnagel J, K Binder, W Paul, M Laso, U Sutcr, I Batouli [N ]ilge and T Burger 1991. On the Construction of Coarse-Grained Models for Linear Flexible Polymer-Chains -Distribution-Functions for Groups of Consecutive Monomers. Journal of Chemical Physics 93 6014-6025.)...

In the case of the bond fluctuation model [36,37], the polymer is confined to a simple cubic lattice. Each monomer occupies a unit cube of the system and the bond length between the monomers can fluctuate. On the other... 

FIG. 9 Typical ground states for a simple glass forming bond fluctuation model [42,47],... 

The bond fluctuation model (BFM) [51] has proved to be a very efficient computational method for Monte Carlo simulations of linear polymers during the last decade. This is a coarse-grained model of polymer chains, in which an effective monomer consists of an elementary cube whose eight sites on a hypothetical cubic lattice are blocked for further occupation (see... 

P. Y. Lai. Statics and dynamics of a polymer chain adsorbed on a surface Monte Carlo simulation using the bond fluctuation model. Phys Rev E 49 5420-5430, 1994. 

I. Gerroff, A. Milchev, W. Paul, K. Binder. A new off-lattice Monte Carlo model for polymers A comparison of static and dynamic properties with the bond fluctuation model and application to random media. J Chem Phys 95 6526-6539, 1993. 

A particularly simple lattice model has been utilized by Harris and Rice [129] and subsequently by Stettin et al. [130] to simulate Langmuir mono-layers at the air/water interface chains on a cubic lattice which are confined to a plane at one end. Haas et al. have used the bond-fluctuation model, a more sophisticated chain model which is common in polymer simulations, to study the same system [131]. Amphiphiles are modeled as short chains of monomers which occupy a cube of eight sites on a cubic lattice and are connected by bonds of variable length [132], At high surface coverage, Haas et al. report various lattice artefacts. They conclude that the study... 

FIG. 7 Snapshot of a bilayer conformation with a pore in the bond-fluctuation model. The dark spheres represent head particles, the light spheres tail particles. Around the pore, the amphiphiles rearrange so as to shield the bilayer interior from the solvent. (From Muller and Schick [133].)... 

In fact, the variable x /Gi controls the "crossover" from one "universality class" " to the other. I.e., there exists a crossover scaling description where data for various Gi (i.e., various N) can be collapsed on a master curve Evidence for this crossover scaling has been seen both in experiments and in Monte Carlo simulations for the bond fluctuation model of symmetric polymer mixtures, e.g Fig. 1. One expects a scaling of the form... 

With these techniques it became possible to check Eq. (3) for the bond-fluctuation model.It was found that the critical temperature indeed scales with the expected... 

Monte-Carlo Simulation of the Bond-Fluctuation Model. 108... 

The Bond Fluctuation Model of Dense Polymer Melts ... 

Mapping Atomistically Detailed Models of Flexible Polymer Chains in Melts to Coarse-Grained Lattice Descriptions Monte Carlo Simulation of the Bond Fluctuation Model... 

Although this athermal bond fluctuation model is clearly not yet a model for any specific polymeric material, it is nevertheless a useful starting point from which a more detailed chemical description can be built. This fact already becomes apparent, when we study suitably rescaled quantities, such that, on this level, a comparison with experiment is already possible. As an example, we can consider the crossover of the self-diffusion constant from Rouse-like behavior for short chains to entangled behavior for longer chains. 

Fig. 5.3. Log-log plot of the self-diffusion constant D of polymer melts vs. chain length N. D is normalized by the diffusion constant of the Rouse limit, DRoUse> which is reached for short chain lengths. N is normalized by Ne, which is estimated from the kink in the log-log plot of the mean-square displacement of inner monomers vs. time [gi (t) vs. t]. Molecular dynamics results [177] and experimental data on PE [178] are compared with the MC results [40] for the athermal bond fluctuation model. From [40]...

The bond fluctuation model not only provides a good description of the diffusion of polymer chains as a whole, but also the internal dynamics of chains on length scales in between the coil size and the length of effective bonds. This is seen from an analysis of the normalized intermediate coherent scattering function S(q,t)/S(q,0) of single chains ... 

Figures 5.3 and 5.4 are just two pieces of evidence that the bond fluctuation model is a reasonable starting point for describing the properties of polymer melts. Thus the next step has to be to incorporate suitable information about the chemical structure and the energetics of specific polymers into the model.

The Mapping Between Specific Polymers and the Bond Fluctuation Model... 

We now attempt to map distributions such as those shown in Fig. 5.6 over a wide range of temperatures (actually a range from T = 250 to 800 K was chosen [32]) by the bond fluctuation model, by equating m subsequent effective bonds of the bond fluctuation model to the n chemical bonds of the atomistic... 

At this point, we will comment on how this procedure generalizes to other polymers. The other case that was considered by us [28,30,32,175,176] was concerned with bisphenol-A-polycarbonate (BPA-PC) (cf. Fig. 5.1). While for PE we had a correspondence that five chemical repeat units correspond to one effective bond of the bond fluctuation model, for BPA-PC the mapping ratio was inverse - one chemical repeat unit was mapped onto three effective bonds One must consider, however, the very different sizes of the chemical repeat units while for PE this is a single CH2 group, in BPA-PC the repeat unit involves 12 C-C or C-0 bonds along the backbone, and the end-to-end distance of the repeat unit is of the order of 10 A. Thus in this case also one effective bond corresponds to a group of four successive covalent bonds along the backbone of the chain, and a lattice unit corresponds to about 2.03 A [175],... 

We now ask how well does the bond fluctuation model with these bond lengths and bond angle potentials reproduce the properties of real polymer melts quantitatively. First of all, it must be admitted that the model yields a qualitatively reasonable picture of the amorphous structure, as exemplified by... 

Fig. 5.9. Plot of the static collective structure factor S(q) for BPA-PC at T = 570 K, as obtained from mapping to the bond fluctuation model. From [175]...

Fig. 5.15. Self-diffusion constant for PE chains (Cioo) plotted vs. temperature, as predicted from the coarse-grained bond fluctuation model. From [32].

In this section, the state of the art of the lattice description of real polymers in terms of the bond fluctuation model augmented with bond length and bond angle potentials has been discussed. It has been shown that the approach has both merits and weaknesses. 

A rather crude, but nevertheless efficient and successful, approach is the bond fluctuation model with potentials constructed from atomistic input (Sect. 5). Despite the lattice structure, it has been demonstrated that a rather reasonable description of many static and dynamic properties of dense polymer melts (polyethylene, polycarbonate) can be obtained. If the effective potentials are known, the implementation of the simulation method is rather straightforward, and also the simulation data analysis presents no particular problems. Indeed, a wealth of results has already been obtained, as briefly reviewed in this section. However, even this conceptually rather simple approach of coarse-graining (which historically was also the first to be tried out among the methods described in this article) suffers from severe bottlenecks - the construction of the effective potential is neither unique nor easy, and still suffers from the important defect that it lacks an intermolecular part, thus allowing only simulations at a given constant density. 

With the Monte Carlo technique, a very large number of membrane problems have been worked on. We have insufficient space to review all the data available. However, the formation of pores is of relevance for permeation. The formation of perforations in a polymeric bilayer has been studied by Muller by using Monte Carlo simulation [67] within the bond fluctuation model. In this particular MC technique, realistic moves are incorporated, such that the number of MC steps can be linked to a simulated time. 

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