Simple-cubic lattice polymers

First insights to freezing and collapse of flexible polymers 

Representative conformations ofa64-mer in the different pseudophases (a) Excitation from the perfect 4 x 4 x 4 cubic ground state (not shown, = -81) to the first excited crystal state, (b) transition toward globular states, and (c) dissolution into random-coil conformations. From [137]. 

Map of specific-heat maxima for several chain lengths taken from the interval H e [8,125]. Circles (O) symbolize the peaks (if any) identified as signals of the collapse 1). The low-temperature peaks (+) belong to the 

Therefore, a reasonable scaling analysis for T( (N) and C (N) could be performed only for very long chains on the sc lattice, for which, however, a precise analysis of the low-temperature behavior is extremely difficult 

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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... 

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],... 

Figure 1. Crossover scaling plot for tlie order parameter ( m > = ( ( ia - Bl / (<1>a + B)> of a symmetrical polymer mixture simulated by tlie bond fluctiiatioii model on tlie simple cubic lattice, with a concentration (jiv = 0.5 of vacant sites. Here N " ( m > is plotted vs. N t, and chain lengths from N = 32 to N = 512 are...

We consider a simple cubic lattice with a coordination number 2 = 6. For an incompressible polymer solution, each lattice site is occupied by a solvent molecule or by a segment of polymer chain. The attraction interactions between the nearest-neighbor sites are characterized by a reduced exchange energy = fi( pp + ss 2eps) between a segment p and a solvent s, where is the attractive energy of an i—j pair. 

Constants A l7 A0, and Ai are determined by the choice of the lattice in our calculations, A i =Ai =0.25 and A0 = 0.5. This choice corresponds to the simple cubic lattice, which is commonly used in the application of lattice SCFT to polymer-clay composites. To relate the lattice coordinate to "real" dimensions, we must also specify the value of the lattice size a here, we set a = 0.4 nm. (Later, in Section 3.5, when describing our results, we will automatically convert all distances from lattice units to nanometers.)... 

We begin by reviewing apphcations of the proposed HPTMC method to polymer solutions and blends. For pure polymer solutions, we simulate chains consisting of up to 16,000 sites for simple-cubic lattice models and 500 sites... 

Fig. 15. Approximate mapping of a chemically realistic polymer (polyethylene in this example) to the bond fluctuation model on the (simple cubic) lattice. In this coarse-graining one integrates n successive chemical monomers (e.g. n = 3) into one effective monomer which blocks 8 adjacent sites on the simple cubic lattice (or 4 on the square lattice in d = 2 dimensions) from occupation by other monomers. The chemical bonds 1, 2, 3 then correspond to effective bond I, bonds 4, 5, 6 to effective bond II. Some information on the chemical structure can be kept indirectly by using suitable distributions P (9) for the angle between subsequent effective bonds, but so far this has been done for homopolymer melts only [94-99]. In the simplest version of the bond fluctuation model [84-88] studied for blends in d = 3 dimensions [88, 91, 92, 99], bond lengths t are allowed to fluctuate freely from i = 2 to t = v/l0, with t = being excluded to maintain that chains do not cut through each other in the course of the random hops of the effective monomers. From Binder [95]...

Fig. 16. Moves used to equilibrate coil configurations for the self-avoiding walk model of polymer chains on the simple cubic lattice (upper party end rotations, kinkjump motions and crankshaft rotations f 107]. From time to time these local moves alternate with a move (lower pan) where one attempts to replace an A-chain by a B-chain in an identical coil configuration, or vice versa. In the transition probability of this move, the chemical potential difference Ap as well as the energy change SjF enter. From Binder [2S8]...

FIGURE 1.7 Plots of viscomelric branching parameter, g, versus branch functionahty, p, for star chains on a simple cubic lattice (unfilled circles), together with experimental data for star polymers in theta solvents , polystyrene in cyclohexane , polyisoprene in dioxane. Solid and dashed lines represent calculated values via Eqs. (1.70) and (1.71), respectively. (Adapted... 

In our computer studies of the conformational behavior of the shell-forming chains, we used MC simulations [91, 95] on a simple cubic lattice and studied the shell behavior of a single micelle only. Because we modeled the behavior of shells of kinetically frozen micelles, we simulated a spherical polymer brush tethered to the surface of a hydrophobic spherical core. The association number was taken from the experiment. The size of the core, lattice constant (i.e., the size of the lattice Kuhn segment ) and the effective chain length were recalculated from experimental values on the basis of the coarse graining parameterization [95]. 

Shaffer s bond fluctuation model (Shaffer, J. S., 1994. Effects of chain topology on polymer dynamics—Bulk melts, J. Chem. Phys., 101 4205 13). Polymers are grown as random walks on a simple cubic lattice, subjected to the excluded volume, chain connectivity, and chain uncrossability constraints described in the text. 

Fig. 13. Effect of solvent characteristics on precipitated poisoner morphologies. The figure is for a blend precipitated in (a) the average solvent and (b) a mixture of 80% good solvent (s = 1) with 20% bad solvent (e = 41). Total polsmier concentration is 3.3% and quenching temperature is set at T = 40 (in ks imits). The representation is for a simple cubic lattice of 70 X 70 X 70 sites. Courtesy of Journal of Polymer Science.

Chidsey and Murray picture the electroactive polymer film as a simple cubic lattice of oxidized sites, each with a charge charge-compensating counterions of charge z,e, and electrons of charge —e. The number density of sites is designated n, and the number density of... 

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