The atomic mean square displacement (MSD, Â¯(Ïƒ_i^2 ) ) is often used in computational materials science studies to calculate measurable properties from the atomic trajectories of simulations; for example, the diffusion coefficient, which according to Einstein relations (Einstein 1905) on the random walk is 1/6 of the slope of the trend of Â¯(Ïƒ_i^2 ) vs. time (Chandler 1987). Equally relevant is the mean square relative displacement (MSRD, Â¯(Ïƒ_ij^2 )), used in Xray Spectroscopies, mainly EXAFS, to describe the atomic disorder in solids (Calvin 2013) (Fornasini 2014). Less known is the relevance of the MSRD in Xray scattering from nanoparticles. In particular, in Total Scattering methods (Pair Distribution Function and Debye Scattering Equation), which rely on an atomistic description of the nanoparticles, the MSRD is the key to distinguish dynamic (thermal) and static disorder (Krivoglaz 1969) (Kuhs 2006). Interestingly, the trend of the MRSD with the distance is characteristic of the nanoparticle shape, an aspect investigated in some detail in this Thesis work. More generally it can be shown that beyond the expected effect of nanocrystal size, the shape alters the contribution of the surface, which is quite relevant for the MSRD. The importance of the shape and of the surface region holds also in case of clusters of nanoparticles, not only in isolated particles. Besides the MSRD, the atomic configurations simulated by molecular dynamics (MD) can also be used to calculate the socalled Warren plot (or diagram), originally introduced in the seminal work of Warren & Averbach of the â€˜50s to describe the effects of plastic deformation in metals (Warren B.E. 1950). Recent work has shown how to obtain Warren plots from the analysis of the diffraction line profiles according to the Whole Powder Pattern Modelling (WPPM) (L. M. Scardi P. 2002) (Scardi P 2017) (P. E.W. Scardi 2018), in particular from the analysis of the strain component of the diffraction peak profile broadening. As proposed in this work, If the Warren plot can be calculated directly from MD simulations, then it is possible to proceed backwards, and construct more reliable strain functions from an atomistic knowledge of the local atomic displacement caused by static and dynamic disorder components. This thesis is divided in two main parts, discussing two different but complementary topics: atomistic modelling and calculations of displacement quantities, application of the above results to experimental case studies, based on the modelling of diffraction data from nanocrystalline systems. We start by describing the atomistic simulations and vibrational properties calculated for several atomic configurations. The main case study concerns Palladium nanoparticles of different sizes and shapes, for which we show that vibrational properties and correlation properties between atoms pairs are greatly influenced by the geometric shape of the nanoparticle and to a lesser extent by their size. The interest is on truncated cubes, i.e. cubes whose edges and corners are progressively removed, as in the series of socalled Wulff solids, ranging from the cubic to the octahedral shape (Wulff G. 1901). As shown in (ii), these are the object of several experimental studies. The developed methodologies are nevertheless applicable to other cases, like the clusters of nanocrystals observed in powders produced by highenergy ball milling, which is also a topic discussed in (ii). The work aims to show a general approach to atomistic modelling, both for isolated nanoparticles with definite shapes, and grains of unspecified shape in plastically deformed polycrystalline materials. We then use the values for displacement quantities (e.g., MSD, MSRD) calculated for the simulated systems to compare them to the experimental results. An underlying fact that seems to hold in all the different cases is that the surface behaviour of nanomaterials has the largest influence on the displacement quantities. For isolated particles we observe strong correlation between displacement quantities and the shape; whereas in the case of a nanocrystalline grain clusters (Figure 1 1) we see that no matter the defects inside the grain, the main contribution to MSRD is given by the grain boundary.
Atomic Modelling of Disorder in Metal Nanocrystals / Flor, Alberto.  (2019), pp. 1171.
Atomic Modelling of Disorder in Metal Nanocrystals
Flor, Alberto
20190101
Abstract
The atomic mean square displacement (MSD, Â¯(Ïƒ_i^2 ) ) is often used in computational materials science studies to calculate measurable properties from the atomic trajectories of simulations; for example, the diffusion coefficient, which according to Einstein relations (Einstein 1905) on the random walk is 1/6 of the slope of the trend of Â¯(Ïƒ_i^2 ) vs. time (Chandler 1987). Equally relevant is the mean square relative displacement (MSRD, Â¯(Ïƒ_ij^2 )), used in Xray Spectroscopies, mainly EXAFS, to describe the atomic disorder in solids (Calvin 2013) (Fornasini 2014). Less known is the relevance of the MSRD in Xray scattering from nanoparticles. In particular, in Total Scattering methods (Pair Distribution Function and Debye Scattering Equation), which rely on an atomistic description of the nanoparticles, the MSRD is the key to distinguish dynamic (thermal) and static disorder (Krivoglaz 1969) (Kuhs 2006). Interestingly, the trend of the MRSD with the distance is characteristic of the nanoparticle shape, an aspect investigated in some detail in this Thesis work. More generally it can be shown that beyond the expected effect of nanocrystal size, the shape alters the contribution of the surface, which is quite relevant for the MSRD. The importance of the shape and of the surface region holds also in case of clusters of nanoparticles, not only in isolated particles. Besides the MSRD, the atomic configurations simulated by molecular dynamics (MD) can also be used to calculate the socalled Warren plot (or diagram), originally introduced in the seminal work of Warren & Averbach of the â€˜50s to describe the effects of plastic deformation in metals (Warren B.E. 1950). Recent work has shown how to obtain Warren plots from the analysis of the diffraction line profiles according to the Whole Powder Pattern Modelling (WPPM) (L. M. Scardi P. 2002) (Scardi P 2017) (P. E.W. Scardi 2018), in particular from the analysis of the strain component of the diffraction peak profile broadening. As proposed in this work, If the Warren plot can be calculated directly from MD simulations, then it is possible to proceed backwards, and construct more reliable strain functions from an atomistic knowledge of the local atomic displacement caused by static and dynamic disorder components. This thesis is divided in two main parts, discussing two different but complementary topics: atomistic modelling and calculations of displacement quantities, application of the above results to experimental case studies, based on the modelling of diffraction data from nanocrystalline systems. We start by describing the atomistic simulations and vibrational properties calculated for several atomic configurations. The main case study concerns Palladium nanoparticles of different sizes and shapes, for which we show that vibrational properties and correlation properties between atoms pairs are greatly influenced by the geometric shape of the nanoparticle and to a lesser extent by their size. The interest is on truncated cubes, i.e. cubes whose edges and corners are progressively removed, as in the series of socalled Wulff solids, ranging from the cubic to the octahedral shape (Wulff G. 1901). As shown in (ii), these are the object of several experimental studies. The developed methodologies are nevertheless applicable to other cases, like the clusters of nanocrystals observed in powders produced by highenergy ball milling, which is also a topic discussed in (ii). The work aims to show a general approach to atomistic modelling, both for isolated nanoparticles with definite shapes, and grains of unspecified shape in plastically deformed polycrystalline materials. We then use the values for displacement quantities (e.g., MSD, MSRD) calculated for the simulated systems to compare them to the experimental results. An underlying fact that seems to hold in all the different cases is that the surface behaviour of nanomaterials has the largest influence on the displacement quantities. For isolated particles we observe strong correlation between displacement quantities and the shape; whereas in the case of a nanocrystalline grain clusters (Figure 1 1) we see that no matter the defects inside the grain, the main contribution to MSRD is given by the grain boundary.File  Dimensione  Formato  

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