Coarse-graining

As a matter of fact, the CG model is physically different from the reference atomistic one and cannot describe all of its features instead, systematic procedures to derive CG potentials are actually based on the reproduction of selected target properties (e.g., structure, thermodynamic data, or inter-molecular potential and forces) obtained from a reference atomistic model or, less frequently, from experimental data. 

More general procedures to derive CG potentials have been developed, mainly in the context of thermoplastic polymers [44—48] and biological systems [49-51] research. CG models for polymers are mainly to be credited to the group of Polymer Theory of the Max Planck Institute in Mainz (Kremer, MilUer-Plate, and co-workers). These models, which require a reference atomistic simulation for the derivation of the CG potential, can be classified in terms of the properties of the reference model they aim to reproduce. Conversely, Force-based approaches parameterize the CG potential by reproducing the instantaneous forces, sampled along simulation trajectories, between CG beads from atomistic models [52]. 

The obvious advantage of CG MD simulations is that accessible time and length scales are far larger than those achievable with atomistic models. This speedup is the result of three different factors (1) the smaller number of particles enormously reduces the computational cost of evaluation of pair interactions, characterized by the well known scaling with the number of particles N (2) the smoother and softer CG potentials allow longer time-steps to be used in the numerical integration of the equations of motion (3) the dynamics of CG models is intrinsically faster than in atomistic ones (see below), allowing for a reduction of equilibration times. 

Besides these fundamental issues, other practical limitations hamper the applications of CG models. One is that the development of a CG potential relies on a proper atomistic FF, whose derivation is itself not a straightforward task 

Consider the local density, p(r, t), of particles distributed and moving in space. We explicitly indicate the position and time dependence of this quantity in order to express the fact the system may be non-homogeneous and out of equilibrium. To define the local density we count the number of particles n(r, t) in a volume AQ about position r at time t (for definiteness we may think of a spherical volume centered about r). The density 

We can make these statements more quantitative by defining the dynamical density variable (see Section 1.2.1) according to 

Coarse graining in time is similarly useful. It converts a function that is spiky (or has other irregularities) in time to a function that is smooth on timescales shorter than Az, but reproduces the relevant slower variations of this function. This serves to achieve a mathematically simpler description of a physical system on the timescale of interest. The attribute of interest may be determined by the experiment—it is 

Another systematic way to coarse grain a function/(r) is to express it as a truncated version of its Fourier transform 

In fact, if we were interested on variations on such length scales, we should have replaced the 

It should be noticed that coarse graining is a reduction process We effectively reduce the number of random variables used to describe a system. This statement may appear contradictory at first glance. In (1.179) we convert the function p r,t), Eq. (1.178), which is completely specified by 3 N position variables (of A/ particles) to the function t) that appears to depend on any point in continuous space. 

However, spatial variations in the latter exists only over length scales larger than so the actual number of independent variables in the coarse-grained system is of order where is the system volume. This number is much smaller 

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The second example shows results obtained with an angle beam probe for transverse waves in coarse grained grey cast iron. Two commercially available probes are compared the composite design SWK 60-2 and the standard design SWB 60-2. The reflector in this example is a side-drilled hole of 5 mm diameter. The A-scans displayed below in Fig. 5 and 6 show that the composite probe has a higher sensitivity by 12 dB and that the signal to noise ratio is improved by more than 6 dB. 

Another example shows a 4 MHz longitudinal wave probe WSY70-4 normally used for testing of coarse grained austenitic material. In this application a high pulse amplitude is... 

As reported before [Ref. 1], there are some essential parameters that influence the results of the testing, such as the thickness of the expired specimen, the quality and coarse grain of the built-in concrete, and the properties of the specimen-surface for the transducer s coupling. At the onset of testings none of tlrese parameters were available. As a result, we had to carry out preliminary investigations in order to prove the applicability of our testing-technique "in situ". 

A Technique of Ultrasonic Testing without Dead Zone for Coarse-Grained TC4 Extrusion Pipe. - The Development of Single Crystal Creeping Wave Prohe. 

Because these pipes are key components used for airplanes, their ultrasonic testing quality must be guranteed. Therefore, the author has conducted studies about the flaw detection methods for coarse-grained TC4P extrusion pipes. 

Fig. 7 The longitudinal microstructure( x 100) of coarse-grained TC4(Ti-6Al-4V)P extrusion pipe of artificial reference test pipe...

The ensemble density p g(p d ) of a mixing system does not approach its equilibrium limit in die pointwise sense. It is only in a coarse-grained sense that the average of p g(p,. d ) over a region i in. S approaches a limit to the equilibrium ensemble density as t —> oo for each fixed i . 

A fiirther theme is the development of teclmiques to bridge the length and time scales between truly molecular-scale simulations and more coarse-grained descriptions. Typical examples are dissipative particle dynamics [226] and the lattice-Boltzmaim method [227]. Part of the motivation for this is the recognition that... 

Coarse-grained models have a longstanding history in polymer science. Long-chain molecules share many common mesoscopic characteristics which are independent of the atomistic stmcture of the chemical repeat units [4, 5 and 6]. The self-similar stmcture [7, 8, 9 and 10] on large length scales is only characterized by a single length scale, the chain extension R. 

On short length scales the coarse-grained description breaks down, because the fluctuations which build up the (smooth) intrinsic profile and the fluctuations of the local interface position are strongly coupled and camiot be distinguished. The effective interface Flamiltonian can describe the properties only on length scales large compared with the width w of the intrinsic profile. The absolute value of the cut-off is difficult... 

Monte Carlo simulations, which include fluctuations, then yields Simulations of a coarse-grained polymer blend by Wemer et al find = 1 [49] in the strong segregation limit, in rather good... 

Figure B3.6.3. Sketch of the coarse-grained description of a binary blend in contact with a wall, (a) Composition profile at the wall, (b) Effective interaction g(l) between the interface and the wall. The different potentials correspond to complete wettmg, a first-order wetting transition and the non-wet state (from above to below). In case of a second-order transition there is no double-well structure close to the transition, but g(l) exhibits a single minimum which moves to larger distances as the wetting transition temperature is approached from below, (c) Temperature dependence of the thickness / of the enriclnnent layer at the wall. The jump of the layer thickness indicates a first-order wetting transition. In the case of a conthuious transition the layer thickness would diverge continuously upon approaching from below.

These chain models are well suited to investigate the dependence of tire phase behaviour on the molecular architecture and to explore the local properties (e.g., enriclnnent of amphiphiles at interfaces, molecular confonnations at interfaces). In order to investigate the effect of fluctuations on large length scales or the shapes of vesicles, more coarse-grained descriptions have to be explored. 

A fiirther step in coarse graining is accomplished by representing the amphiphiles not as chain molecules but as single site/bond entities on a lattice. The characteristic architecture of the amphiphile—the hydrophilic head and hydrophobic tail—is lost in this representation. Instead, the interaction between the different lattice sites, which represent the oil, the water and the amphiphile, have to be carefiilly constmcted in order to bring about the amphiphilic behaviour. 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

The coarse-graining approach is commonly used for thermodynamic properties whereas the systematic or random sampling methods are appropriate for static structural properties such as the radial distribution function. 

Fig. 8.8 The bond fluctuation model. In this example three bcmds in the polymer arc incorporated into a singk effecti bond between effective moncmers . (Figure adapted from Baschnagel J, K Binder, W Paul, M Laso, U Sutcr, I Batouli [N ]ilge and T Burger 1991. On the Construction of Coarse-Grained Models for Linear Flexible Polymer-Chains -Distribution-Functions for Groups of Consecutive Monomers. Journal of Chemical Physics 93 6014-6025.)...

In order to dry the crystals, the Buchner funnel is inverted over two or three thicknesses of drying paper (i.e., coarse-grained, smooth surfaced Alter paper) resting upon a pad of newspaper, and the crystalline cake is removed with the aid of a clean spatula several sheets of drying paper are placed on top and the crystals are pressed flrmly. If the sheets become too soiled by the mother liquor absorbed, the crystals should be transferred to fresh paper. The disadvantage of this method of rapid drying is that the recrystallised product is liable to become contaminated with the Alter paper flbre. 

The dynamic mean-field density functional method is similar to DPD in practice, but not in its mathematical formulation. This method is built around the density functional theory of coarse-grained systems. The actual simulation is a... 

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