DPD model

When representing an aqueous solution of amphiphiles within soft DPD models, the first conceptual point consists in determining how many elementary chemical building blocks correspond to one DPD particle. To this end, a significant amount... 

The amphiphilic models obtained following the above concepts have been extensively used to address various phenomena in lipid and polymer vesicles. It should be mentioned, however, that by virtue of the high degree of coarse-graining invoked in DPD models, the borderline between lipid and polymeric amphiphiles within this description becomes less distinct. 

Dissipative particle dynamics (DPD) is a simulation technique initially developed for the simulation of complex fluids [18] and later extended for polymers. The DPD model consists of pointlike particles interacting with each other through a set of prescribed forces [19]. From a physical point of view, each dissipative particle is regarded not as a single atom or molecule but rather as a collection of atomic groups (molecules) that move in a coherent fashion. 

The cut-off radius rc t is defined arbitrarily and reveals the range of interaction between the fluid particles. DPD model with longer cut-off radius reproduces better dynamical properties of realistic fluids expressed in terms of velocity correlation function [80]. Simultaneously, for a shorter cut-off radius, the efficiency of DPD codes increases as 0(1 /t ut). which allows for more precise computation of thermodynamic properties of the particle system from statistical mechanics point of view. A strong background drawn from statistical mechanics has been provided to DPD [43,80,81] from which explicit formulas for transport coefficients in terms of the particle interactions can be derived. The kinetic theory for standard hydrodynamic behavior in the DPD model was developed by Marsh et al. [81] for the low-friction (small value of yin Equation (26.25)), low-density case and vanishing conservative interactions Fc. In this weak scattering theory, the interactions between the dissipative particles produce only small deflections. 

FIGURE 26.11 The schematics of coarse-graining idea realized by the top-down thermodynamically consistent DPD model. The Voronoi cell represents a fragment of continuum fluid. This fragment can be treated as a dissipative particle with variable mass, volume, temperature and entropy. The thermodynamically consistent DPD particles interact with forces dependent not only on their positions and velocities but on the current thermodynamic states of interacting particles as well. 

Unlike in NEMD models, the microstructures emerging due to competition between the breakup and coalescence processes can be studied by using DPD modeling. For example, in Figure 26.23, the four principal mechanisms, the same as those responsible for droplets breakup [ 118,119], can be observed in DPD simulation of the R-T instability. As shown in [116,119], moderately extended drops for capillary number close to a critical value, which is a function of dynamic viscosity ratio... 

The role of the thermal noise implemented as a Brownian force in DPD model is especially important factor in modeling of phase-separation process. 

FIGURE 26.26 Phase separation for binary systems realized by a thermodynamically consistent DPD model with thermal fluctuations switched off (a) and on (b) (cross-section is shown to display lamellar structures). As shown in (c) and (d) using 3D TC-DPD simulation we can obtained aU 6 = 1/3, 1/2, 1, and 2/3 regimes in a single simulation [20,84,125],... 

This field of liquid crystalline polymersis still in its infancy, and the best ways of coarse-graining to a DPD model from more atomistic descriptions are still open for discussion. None-the-less, the progress in this area is highly encouraging and these models show real promise in terms of solving the time and length scale problems that so often confront simulators in the areas of liquid crystals and polymers. 

Figure 1. A DPD model for a flexible main-chain LCP with three semi-rigid units...

The positions and the velocities of the particles are solved in accordance with the above equations by implementing Newton s equation of motion and a modified version of the velocity. While the interaction potentials in MD models are high-order polynomials of the distance ry between two particles, in DPD models the potentials are softened in order to approximate the effective potential at microscopic length scales. The form of the conservative force is chosen in particular to... 

The phase morphology obtained for a simplified DPD model of the PVBPA/PEEK copolymer depends on the way the coarse-grained model is set up. Mutliscale approach was indeed helpful to find the parameters for the mesoscale DPD simulations. 

Such a model with a Gaussian form of the interaction, v, has been utilized by Zuckermann and coworkers [105] in order to investigate polymer brushes. The DPD-model of Groot and collaborators [100-102] also shares many features with this soft coarse-grained model. In this model, the conservative force is derived from a pairwise potential of the form... 

The thermodynamic integration scheme can be appUed to different models including coarse-grained, partide-based models of amphiphihc systems and membranes [133, 134] (e.g., soft DPD-models [135-137], Lennard-Jones models [138,139], or solvent-free models [140-142] of membranes) as well as field-theoretic representations [28]. It can be implemented in Monte Carlo or molecular dynamic simulations, as well as SCMF simulations [40-42, 86], field-theoretic simulations [28], and external potential dynamics [27, 63, 64] or dynamic density functional theory [143, 144]. 

Furthermore, it is expected that DPD will play an ever-increasing role in multi-scale modeling approaches through bridging of the atomistic and continuum scales. In such approaches, atomistic simulations are performed to build the DPD models, followed by DPD simulations which provide the necessary input to the continuum codes. 

The form of DPD applied in this study is where the equations of motion for an ensemble of beads correspond to Langevin-Brownian dynamics with pair-wise central forces [43,44]. DPD conserves momentum which is essential for recovering the correct hydrodynamic behavior at sufficiently large length and timescales. In contrast with molecular dynamics, a DPD model describes a reduced number of degrees of... 

The conservative force (Equation 21.5) between the beads accounts for chemical interactions in the DPD model system and is described by a repulsive soft potential in terms of an interaction parameter and a weight function (o. The weight function is a simple linear ramp, giving rise to very soft repulsive forces between the particles. Varying the strength of changes the response of the flnid to an applied stress and spatial arranganent of different bead types. 

A silica melt was chosen as the benchmark system, as it is the principle base component of many inorganic glasses producing a validated DPD model of this glass is... 

FIGURE 21.9 Velocity profiles perpendicular to the system walls for increasing shear force showing DPD model averages and NS solutions. 

FIGURE 21.10 Normalized plot of SiOj viscosity calculated from the DPD model and... 

The DPD viscosity was initially converted to an Arrhenius form, ln(fj )/hi(fj) against TJT, where represents the experimental value of the glass transition tanperature of 1500 K. A Min-Max normalization was then applied to the viscosity data. A numerical error in the viscosities computed by DPD of 4% was estimated from the magnitude of the deviation in the mean velocity as a function of simulation time step. A first-order Arrhenius fit was then calculated to the trend. A similar normalization and Arrhenius fit was then applied to Si02 viscosity data obtained from published sources [58,59]. The resulting data can be compared with the DPD model calculations, as shown in Figure 21.10. The DPD model and the experimental data both show excellent agreement with a first-order Arrhenius model. The difference in the slopes and intercepts of the two trends was found to be less than 3% in both cases. 

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