Lennard bead-spring model

Concluding this section, one should mention also the method of molecular dynamics (MD) in which one employs again a bead-spring model [33,70,71] of a polymer chain where each monomer is coupled to a heat bath. Monomers which are connected along the backbone of a chain interact via Eq. (8) whereas non-bonded monomers are assumed usually to exert Lennard-Jones forces on each other. Then the time evolution of the system is obtained by integrating numerically the equation of motion for each monomer i... 

Figure 14 Master curve generated from mean-square displacements at different temperatures, plotting them against the diffusion coefficient at that temperature times time. Shown are only the envelopes of this procedure for the monomer displacement in the bead-spring model and for the atom displacement in a binary Lennard-Jones mixture. Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusive regime for the bead-spring model ( ° 63), the MCT von Schweidler description of the plateau regime, and typical length scales R2 and R2e of the bead-spring model.

Figure 1.41. Potential energies for the bead-spring model LJ1—Lennard-Jones potential LJ2—van der Waals potential EXP1, EXP2—short-range polar potential FENE—finitely extensible nonlinear elastic potential.

One model which has been extensively used to model polymers in the continuum is the bead-spring model. In this model a polymer chain consists of Nbeads (mers) connected by a spring. The easiest way to include excluded volume interactions is to represent the beads as spheres centered at each connection point on the chain. The spheres can either be hard or soft. For soft spheres, a Lennard-Jones interaction is often used, where the interaction between monomers is... 

A simple generic bead spring model of chains can be used to study universal polymer properties that do not depend on specific chemical details. Bonds between neighbouring Lennard-Jones particles in a chain can be represented by the finite extension non-linear elastic (FENE) potential. 

In the simulation study of fluorescence anisotropy decay, a generic bead-spring model of the polymer was used. It is schematically shown in Fig. 19. Each bead can represent one or several monomer units in a real polymer. The degree of dissociation, a, is defined as the fraction of monomer units carrying electric charges. The interaction between monomer units of the polymer is modeled by the Lennard-Jones potential and the solvent quality is controlled by the depth of this potential, e. As shown by Micka, Holm and Kremer, 0.34 corresponds to the theta state [146]. The simulation study was performed for several values of > 0.33, i.e., under poor solvent conditions. The simulation technique used was MD coupled to a Langevin thermostat, i.e., the polymer was simulated in an implicit solvent. The counterions were simulated explicitly. A more detailed description of the polymer model can be found in the original paper [87]. 

Figure 9 Molecular dynamics simulation of a Lennard-Jones, bead-spring model, (a) Slip length, 8, as a function of the strength, [ an, of attraction between a hard, corrugated substrate and liquid for temperature, kgT/ [ =. 2. The solid line with circles is obtained from the Couette and Poiseuille profiles (NEMO) according toeqn [37], whereas the dashed line with squares, from the Green-Kubo (GK) relation, eqn [39]. The curve marks the behavior 1/ [ jii in accord with eqn [40]. The inset illustrates the velocity profiles of the Couette and Poiseuille flows, from which the slip length has been estimated for [mii = 0.6, measured in units of the Lennard-Jones parameter, [. Adapted from Servantie, J. Muller, M. Phys. Rev. Lett. 2008, 101,...

Fig. 1.8 (a) Log-log plot of the reduced mean square gyration radius, / g(0))/(.Rg(O)). vs. rescaled chain lenrth N - l)((/>f ) where f is the root mean-square bond length, and the theoretical value for the exponent i/(i/ = 0.59) is used. All data refer to a bead-spring model with stiff repulsive Lennard-Jones interaction, as described in Section 1.2.1 (Eq. [1.7]). (b) Same as (a) but for the bond fluctuation model. In both (a) and (b) the straight line indicates the asymptotic slope of the crossover scaling function, resulting from eq. (1.15). (From Gerroff et al.)... 

Fig. 1.10 Log-log plot of relaxation time n vs. chain length N, for the bead-spring model with soft Lennard-Jones repulsion and the bond fluctuation model. Open circles (and left ordinate scale) refer to off-lattice model at = 0.0625, full dots (and right scale) to the bond fluctuation model at < = 0.05 (data taken from Ref. 128). Straight lines indicated the power laws Ti oc N, where the exponent z = 2.3 or 2.24, respectively, is reasonably compatible with the theoretical prediction 2 = 2i/ + 1 2.18. Insert shows the ratio ts/tj. It is seen that both...

The bead-spring model of polymer chains utilized for molecular dynamics calculations in Ref. 19 consists of beads interacting through a truncated and shifted 12-6 Lennard-Jones potential... 

More importantly, since the single-chain dynamics obeys Rouse behavior, only x/Xo = segment movements are required to relax a chain conformation. Thus, the total effort to simulate the system amounts to 4 10 segment motions that are more than 10 orders of magnitude less than for models like the bond fluctuation model or Lennard-Jones bead-spring models. 

An advantage of a soft, coarse-grained, off-lattice model is the ability to simultaneously and accurately calculate the pressure, p, and the chemical potential, p. Abandoning the lattice-description allows a precise calculation of the pressure, p, and simulations at constant pressure or tension. This is also possible in off-lattice models with harsh excluded volume interactions (e.g., a Lennard-Jones bead-spring model). The accurate calculation of the chemical potential by particle insertion methods. 

Coarse-grained bead-spring models presented here are similar to those used in earlier studies of polymer melts and networks [49, 50] and tethered chains [51]. In this model, each chain consists of N beads, which are referred to as monomers, connected to form a linear chain. The interaction potential V (r) between two monomers separated by a distance r is taken as a Lennard Jones 6 12 potential, Eq. 1. In most... 

Fig. 7. The PRISM (curves) and MD (points) generated direct correlation functions [59] C(r) for a bead-spring model liquid with repulsive Lennard-Jones interactions, o- = 3.93 A, e = 0.092 kcal mol . The PRISM results were determined using the Q(k) from MD as input to PRISM. The inset shows the corresponding results for polyethylene (N = 24 sites, p =...

We start by discussing the structural phase behavior of symmetric diblock copolymers in selective solvent, which will be used as a point of comparison. Figure 13(a) plots the approximate structural phase behavior for the simple Lennard-Jones bead-spring model commonly used to study amphiphilic sys-tems as a function of volume fraction, temperature state point below the order-disorder temperature. Specifically, simulations were performed for a symmetric hStS... 

The system used in the simulations usually consists of solid walls and lubricant molecules, but the specific arrangement of the system depends on the problem under investigation. In early studies, hard spherical molecules, interacting with each other through the Lennard-Jones (L-J) potential, were adopted to model the lubricant [27], but recently we tend to take more realistic models for describing the lubricant molecules. The alkane molecules with flexible linear chains [28,29] and bead-spring chains [7,30] are the examples for the most commonly used molecular architectures. The inter- and intra-molecular potentials, as well as the interactions between the lubricant molecule and solid wall, have to be properly defined in order to get reliable results. Readers who intend to learn more about the specific techniques of the simulations are referred to Refs. [27-29]. 

Fig. 12. Equation of state (a) and phase diagram (b) of a bead-spring polymer model. Monomers interact via a truncated and shifted Lennard-Jones potential as in Fig. 6 and neighboring monomers along a molecule are bonded together via a finitely extensible non-linear elastic potential of the form iJpENE(r) = — 15e(iJo/

Potentials and Geometry. We will describe results for films containing linear short-chain molecules. These are modeled using a simple bead-spring potential that has been used extensively in studies of polymer structure and dynamics (24). Each spherical monomer within the molecule interacts with all other monomers through a Lennard-Jones (LJ) potential Vy that is truncated beyond rc. For monomers separated by a distance r[Pg.92]

A simple and very popular example of a coarse-grained off-lattice model is given by the bead-spring chain of Kremer and Grest [70,7 Ij. In this model monomers interact via a Lennard-Jones potential ... 

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