Probabilistic Analysis and Design of HCP Nanowires: An Efficient Surrogate Based Molecular Dynamics Simulation Approach

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We investigate the dependency of strain rate, temperature and size on yield strength of hexagonal close packed (HCP) nanowires based on large-scale molecular dynamics (MD) simulation. A variance-based analysis has been proposed to quantify relative sensitivity of the three controlling factors on the yield strength of the material. One of the major drawbacks of conventional MD simulation based studies is that the simulations are computationally very intensive and economically expensive. Large scale molecular dynamics simulation needs supercomputing access and the larger the number of atoms, the longer it takes time and computational resources. For this reason it becomes practically impossible to perform a robust and comprehensive analysis that requires multiple simulations such as sensitivity analysis, uncertainty quantification and optimization. We propose a novel surrogate based molecular dynamics (SBMD) simulation approach that enables us to carry out thousands of virtual simulations for different combinations of the controlling factors in a computationally efficient way by performing only few MD simulations. Following the SBMD simulation approach an efficient optimum design scheme has been developed to predict optimized size of the nanowire to maximize the yield strength. Subsequently the effect of inevitable uncertainty associated with the controlling factors has been quantified using Monte Carlo simulation. Though we have confined our analyses in this article for Magnesium nanowires only, the proposed approach can be extended to other materials for computationally intensive nano-scale investigation involving multiple factors of influence.

Introduction

Nanoscale analysis has become immensely popular across all fields of research in the last few decades. The fundamental properties of any material such as strength, ductility, creep, fracture behavior, durability etc. can be controlled widely at nanoscale compared to its bulk specimen. Rapid advancements in electron microscopy and other devices that can manipulate materials at nano-scale along with the advancement in computing power are accelerating the progress of nanomaterials research. Numerous nanoscale simulations have been reported over the last two decades[1], [2], [3], [4], [5], [6], [7]. These studies reveal that there are several external and internal factors in a material, which affect the mechanical strength. Dislocation plasticity, dislocation movement, grain size, grain geometry and alloying elements control for the strength of a material can be considered as internal factor while strain rate (SR), temperature (T) and size (or diameter of nanowire d) can be attributed as external factors. We focus on the effect of three external factors that affect mechanical strength in a nanowire. HCP-Magnesium nanowires have been considered in this study for the purpose of analysis. A novel methodology is proposed in the realm of nano-scale research for quantifying relative sensitivity of these three factors by analyzing their relative variances. Subsequently probabilistic analysis has been carried out using Monte Carlo Simulation (MCS) to study the relative coefficient of variation considering randomness in the three factors. Large-scale multiple-simulation (~103) based statistical approach has been reported in scientific literature for nano-scale analyses that requires carrying out very large number of expensive and time consuming molecular dynamics simulations[8]. We propose a surrogate based molecular dynamics (SBMD) simulation approach where the number of actual MD simulations can be drastically reduced and thereby thousands of virtual simulations can be performed to investigate the material behavior thoroughly in a computationally efficient paradigm. Virtual simulation refers to the surrogate model predictions in this article. Application of the proposed SBMD simulation approach has enabled us to apply global optimization algorithms in the field of MD simulation based investigations of materials, wherein we have optimized the size of nanowire to maximize the yield strength for different temperatures and strain rates. By implementing the SBMD simulation approach we practically replaced the expensive MD simulations by an efficient mathematical model that facilitates us to conduct a detail and robust analysis on the strength of the nanowire.

In the context of MD simulations there is considerable amount of available literature, where mechanical properties have been studied against variables such as strain rate, temperature or size effect[9], [10], [11], [12], [13], [14], [15], [16], [17]. Both temperature and strain rate have also been taken into account together and attempts have been made to carry out quantitative study of all the factors in nanoscale specimens[18], [19], [20], [21], [22]. The temperature sensitivity analysis has been carried out by an Arrhenius expression[23] which is directly related to the activation energy of a reaction that can be mechanical, physical or a chemical reaction. Temperature dependence determines the thermal softening of a specimen. Other than temperature, there are factors like work hardening and strain rate sensitivity that also contribute and determine ductility. Explicit expression has been reported to quantify sensitivity of a single factor (strain rate) on a specific material response[24], but relative sensitivity analyses of multiple factors are still very scarce to find in nano-scale research. Few studies have attempted to correlate certain mechanical properties to cross-sectional size of nanowires by performing a regression analysis to arrive at some empirical formulae[25], [26]. Most of the studies reported so far in scientific literature present the variation of a response quantity of interest with respect to one factor, keeping all other factors constant. In reality all the factors act simultaneously in a process. The relative sensitivity of each controlling factor cannot be truly assessed unless we vary multiple factors simultaneously in an experiment/simulation. In the proposed analysis of variance approach for sensitivity quantification, we have varied all the three considered factors simultaneously to quantify their relative sensitivity on the yield strength of the material. This article hereafter is organized as follows: Section-2: description of the proposed SBMD simulation approach in details; Section-3: Large-scale simulation based results and discussion on the yield strength of HCP nanowires; Section-4: conclusion.

Section snippets

Surrogate Based Molecular Dynamics (SBMD) Simulation

In this section the proposed SBMD simulation approach is discussed along with the three analyses (sensitivity analysis, optimization and uncertainty quantification) performed for HCP nanowires to comprehensively study the behavior of this material (Fig. 1). Variance based sensitivity analysis has been carried out in conjunction with the SBMD simulation approach for quantifying relative sensitivity of the three considered factors. Variance based sensitivity analysis is a form of global

Results and Discussion

In this section we have discussed details about surrogate model formation including validation of the model with respect to original data and thereby results on the surrogate based analysis of HCP nano-wire. For surrogate model formation, we have chosen ranges of the controlling factors on the basis of available literature as: SR [107 s−1, 1010 s−1], T [200 K, 600 K] and d [4 nm, 12 nm]. From this analysis domain, we have algorithmically selected 21 efficient combinations of the three

Conclusion

A critical analysis on yield strength of HCP magnesium nano-wires following an efficient SBMD simulation approach is presented in this article. The proposed SBMD simulation approach in nano-scale has enabled us to carry out large-scale simulation based analyses in a computationally efficient way and subsequently furnish new results for deterministic as well as stochastic analyses concerning yield strength of the material. Sensitivity of the three controlling factors (strain rate, temperature

Acknowledgments

TM acknowledges the financial support from Swansea University through the award of Zienkiewicz Scholarship. SA acknowledges the financial support from The Royal Society of London through the Wolfson Research Merit award. The authors also gratefully acknowledge the valuable comments of Dr. Dibakar Datta (Stanford University) on this work during preparation of the manuscript.

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