Validating Structure Models from Simulations

Comparing structural information from X-ray and neutron diffraction provides a very valuable way to validate MD simulation results of glasses. In some simple systems, the partial pair distribution function or partial structure factors of all atom pairs can be determined experimentally and they provide excellent validations for simulated structures. However, as the composition becomes more complicated and more elements included, larger number of pair contributions will complicate the comparison and the validation becomes more and more difficult in multicomponent glass systems. For example, for binary oxides, e.g. sodium silicate, there are six partial pair distribution functions, but for a four component systems, for example the bioactive glass composition, there are a total of fifteen partials contributions. The overlap between partial contributions makes it very challenging to assign the peaks and to determine the quality of comparison and hence the validation of the simulated structure models. 

The Rx factor proposed by Wright [50] is commonly used to quantify the difference of the simulated and experimental total correlation function T(r), which is a form of pair distribution function and used in comparison with experiments due to symmetric broadening in experiments [51, 52]. The Rx is defined as [50], 

The total correlation function after broadening for the neutron diffraction case is then expressed as. 

It is also possible to compare the reciprocal space structure factors, either from neutron or X-ray diffraction measiu ements. The partial structure factors are first calculated through Fourier transformation the partial pair distribution functions using 

There are several site specific experimental techniques such as solids state NMR, EXAFS and Raman spectroscopy that can give additional structural information to be compared directly with simulation results and thus are able to provide further validations. For example, NMR results not only provide Qn distributions but also how the Qn species are linked together through double or multi-quantum experiments. This kind of site specific experimental methods is an additional opportunity for detailed structure comparison and validation. 

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This paper is structured as follows in section 2, we recall the statement of the forward problem. We remind the numerical model which relates the contrast function with the observed data. Then, we compare the measurements performed with the experimental probe with predictive data which come from the model. This comparison is used, firstly, to validate the forward problem. In section 4, the solution of the associated inverse problem is described through a Bayesian approach. We derive, in particular, an appropriate criteria which must be optimized in order to reconstruct simulated flaws. Some results of flaw reconstructions from simulated data are presented. These results confirm the capability of the inversion method. The section 5 ends with giving some tasks we have already thought of. 

The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. It is necessary to decide what equation of state to use for the vapor phase (ideal gas, Redlich-Kwong-Soave, Peng-Robinson, etc.) and what model to use for liquid activity coefficients [ideal solutions, solubility parameters, Wilson equation, nonrandom two liquid (NRTL), UNIFAC, etc.]. See Sec. 4, Thermodynamics. It is necessary to consider mixtures of chemicals, and the interaction parameters must be predictable. The best case is to determine them from data, and the next-best case is to use correlations based on the molecular weight, structure, and normal boiling point. To validate the model, the computer results of vapor-liquid equilibria could be checked against experimental data to ensure their validity before the data are used in more complicated computer calculations. 

However, it has turned out that the most accurate way of fixing these parameters is through matching of simulated phase equilibria to those derived from experiment.33 As a final step, the potential, regardless of its source, should be validated through extensive comparison with available experimental data for structural, thermodynamic, and dynamic properties obtained from simulations of the material of interest, closely related materials, and model compounds used in the parameterization. The importance of potential function validation in simulation of real materials cannot be overemphasized. 

MD simulations of model membrane systems have provided a unique view of lipid interactions at a molecular level of resolution [21], Due to the inherent fluidity and heterogeneity in lipid membranes, computer simulation is an attractive tool. MD simulations allow us to obtain structural, dynamic, and energetic information about model lipid membranes. Comparing calculated structural properties from our simulations to experimental values, such as areas and volumes per lipid, and electron density profiles, allows validation of our models. With molecular resolution, we are able to probe lipid-lipid interactions at a level difficult to achieve experimentally. 

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

Computational modeling is a powerful tool to predict toxicity of drugs and environmental toxins. However, all the in silico models, from the chemical structure-related QSAR method to the systemic PBPK models, would beneht from a second system to improve and validate their predictions. The accuracy of PBPK modeling, for example, depends on precise description of physiological mechanisms and kinetic parameters applied to the model. The PBPK method has primary limitations that it can only predict responses based on assumed mechanisms, without considerations on secondary and unexpected effects. Incomplete understanding of the biological mechanism and inappropriate simplification of the model can easily introduce errors into the PBPK predictions. In addition values of parameters required for the model are often unavailable, especially those for new drugs and environmental toxins. Thus a second validation system is critical to complement computational simulations and to provide a rational basis to improve mathematical models. 

The traffic-based stochastic simulation of the crossing event used here has been developed in an internal project at BMW Group and was summarized here with respect to its structure and functions. The basic idea is a stochastic modeling of all processes from the pedestrian s decision to cross a road in the given scenario to the (avoided) accident. Each process is linked with appropriate probability distributions (mainly from literature). The pedestrian and the driver of the vehicle are implemented with respect to their individual attributes (e.g., age). The simulation results in uncritical crossings and in this specific scenario in about 0.2% collisions. Overall effects in the accident events have been compared to data available in accident data bases to make a step towards validation of the model. The simulation described includes the whole process chain. 

This final section of this chapter is devoted to the interplay between theory and the model-driven experimental techniques, such as diffraction methods. Gas-phase electron diffraction and powder diffraction methods are more heavily reliant on direct input from simulation, for example in providing starting structures for least-squares refinement analyses (Section 2.11.3), whereas in the spectroscopic techniques discussed above the need is more for validation and assignment. 

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