The Bond-Fluctuation Lattice Model

The large-scale structure of polymer chains in a good solvent is that of a self-avoiding random walk (SAW), but in melts it is that of a random walk (RW).11 The large-scale structure of these mathematical models, however, is 

When one implements an MC stochastic dynamics algorithm in this model (consisting of random-hopping moves of the monomers by one lattice constant in a randomly chosen lattice direction), the chosen set of bond vectors induces the preservation of chain connectivity as a consequence of excluded volume alone, which thus allows for efficient simulations. This class of moves 

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Figure 3 Sketch of the bond-fluctuation lattice model. The monomer units are represented by unit cubes on the simple cubic lattice connected by bonds of varying length. One example of each bond vector class is shown in the sketch.

Fig. 13a. Predicted variation of X,ff/(Axo) vs (N0/N) i Eq. (134). Dot-dash line shows the asymptotic behavior 3Urf/(Axo)oc(N/N) 1/2> while the marks indicate the scale for (N/Nc) itself, b Variation of the observed normalized critical temperature k Tc/(eN) with N1/2, from a simulation of the bond fluctuation lattice model (see Sect. 4). Marks indicate the values of N chosen. Note that for large enough N the integral equation theory of Schweizer and Curro [44-49] implies that this plot could be mapped on Fig. 13a), by multiplying the coordinate scales with suitable constants, without any other adjustable parameters being available. From Deutsch and Binder [92]...

In all of the work we will present in the following we will employ the bond-fluctuation lattice model [25-28] a rendering of which is shown in Fig. 1. 

Figure 1. Rendering of the three-dimensional version of the bond-fluctuation lattice model.

This idea that the solvent flow field can be approximated by the Brinkman equation has been used in several recent simulations of a polymer brush in simple shear flow. In these simulations, the solvent is not included explicitly but it s effect is modeled using the Brinkman equation. Lai and Binder [65] and Lai and Lai [66], using a bond fluctuation lattice model, and Miao et al. [67], using a continuum model, studied the properties of a dense polymer brush in a flow field by modifying the standard Metropolis Monte Carlo transition probability to take into account the effective force acting upon the brush chains by the moving sol-... 

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... 

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... 

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... 

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

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],... 

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. 

Here, M — kN is the number of empty sites after k chains have been placed on the lattice and constitutes the number of potential starting points for the (k + l)th chain. The factor z(z — 1)N 2 represents the number of possibilities to place the remaining N — 1 monomers of the chain after the first monomer has been placed, forbidding only the immediate back-folding of the walk. The factor which for the bond-fluctuation model counts the number of... 

Lattice Monte Carlo Model for Polymers A Comparison of Static and Dynamic Properties with the Bond-Fluctuation Model and Application to Random Media. 

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