Lennard—Jones beads

Carbon dioxide (CO2) is represented by a single Lennard-Jones bead the interactions between C02-beads are adjusted to reproduce the experimental critical point. A modified Lorentz-Berthelot mixing rule is used to describe the interactions between hexadecane and CO2 beads [82,83] ... 

Figure 2 Adsorption of the Lennard-Jones bead-spring chains in a good solvent, T/ 1.21, onto an attractive substrate. Cumulant ratio,...

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,...

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. 

Langevin 115, 120 Lennard-Jones beads/sites 73, 76, 78 Lennard-Jones chains 70 Lennard-Jones cluster 167 Lennard-Jones energy 17 Lennard-Jones fluid 21, 69 Lennard-Jones interactions 85 Lennard-Jones liquid 52... 

Finally, we briefly mention an example that combines a coarse-grained modeling ansatz with grand canonical simulations. In this particular case, hexadecane is modeled as a chain of five coarse-grained Lennard-Jones beads [Eq. (1.9)] that are connected by FENE [Eq. (1.10)] springs [48]. The solvent, carbon dioxide, is represented by a simple Lennard-Jones bead that contains an additional term to account for the (spherically averaged) quadrupolar moment of CO2 [82, 111]. Simulation parameters e, a, and q (for CO2) are derived by equating the critical temperature, the critical density, and the quadrupolar moment (in the case of CO2) of simulation and experiment for the pure components. Mixture parameters eab and... 

To illustrate these techniques, we consider the coarse-grained model for hexadecane and carbon dioxide, which we have discussed in Section 1.2. The correction of the Lorentz-Berthelot rule is set to = 0.9 and the temperature is fixed atllcBr/E = 0.92. (In this case carbon dioxide was modeled as a simple Lennard-Jones bead without quadrupolar moment.) The short- and long-ranged interactions at the polymer-solid contact are described by a 9-3-potential of the form ... 

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... 

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... 

Simulations of monolayers have focused on internal phase transitions, e.g., between the expanded phase and the condensed phases, between different tilted phases, etc. These phenomena cannot be reproduced by models with purely repulsive interactions. Therefore, Haas et al. [148,149] represent the amphiphiles as stiff Lennard-Jones chains, with one end (the head bead) confined to move in a plane. In later versions of the model [150-152], the head bead interactions differ from those of the tail beads they are taken to be purely repulsive, and the head size is variable. 

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]. 

Multiparticle collision dynamics provides an ideal way to simulate the motion of small self-propelled objects since the interaction between the solvent and the motor can be specified and hydrodynamic effects are taken into account automatically. It has been used to investigate the self-propelled motion of swimmers composed of linked beads that undergo non-time-reversible cyclic motion [116] and chemically powered nanodimers [117]. The chemically powered nanodimers can serve as models for the motions of the bimetallic nanodimers discussed earlier. The nanodimers are made from two spheres separated by a fixed distance R dissolved in a solvent of A and B molecules. One dimer sphere (C) catalyzes the irreversible reaction A + C B I C, while nonreactive interactions occur with the noncatalytic sphere (N). The nanodimer and reactive events are shown in Fig. 22. The A and B species interact with the nanodimer spheres through repulsive Lennard-Jones (LJ) potentials in Eq. (76). The MPC simulations assume that the potentials satisfy Vca = Vcb = Vna, with c.,t and Vnb with 3- The A molecules react to form B molecules when they approach the catalytic sphere within the interaction distance r < rc. The B molecules produced in the reaction interact differently with the catalytic and noncatalytic spheres. 

Figure 7 Comparison of melt structure factor and single-chain structure factor for PB (upper panel, calculated as scattering from the united atoms only) and a bead-spring melt (lower panel, in Lennard-Jones units).

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.

The inferacfions befween nonbonded uncharged beads in CGMD simula-fions are modeled by fhe Lennard-Jones (LJ) pofenfial ... 

In an early attempt to model the dynamics of the chromatin fiber, Ehrlich and Langowski [96] assumed a chain geometry similar to the one used later by Katritch et al. [89] nucleosomes were approximated as spherical beads and the linker DNA as a segmented flexible polymer with Debye-Huckel electrostatics. The interaction between nucleosomes was a steep repulsive Lennard-Jones type potential attractive interactions were not included. 

Classical molecular simulation methods such as MC and MD represent atomistic/molecular-level modeling, which discards the electronic degrees of freedom while utilizing parameters transferred from quantum level simulation as force field information. A molecule in the simulation is composed of beads representing atoms, where the interactions are described by classical potential functions. Each bead has a dispersive pair-wise interaction as described by the Lennard-Jones (LJ) potential, ULj(Ly) ... 

In our model, a PFPE molecule is composed of a finite number of beads with different physical or chemical properties [Fig. 1.41]. For simplicity, we assume that all the beads, including the endbeads, have the same radius. Lennard-Jones... 

All beads, including the endbeads in PFPE Zdol, interact with each other by a pairwise, dispersive, tmncated Lennard-Jones (LJ) potential as follows... 

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.

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