Diffusions_Part_1 - Mobility of atoms and diffusion....

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University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei Mobility of atoms and diffusion. Einstein relation. = Δ N 1 i 2 i i 2 )) 0 ( r ) t ( r ( N 1 ) t ( r MSD r r r This expression is called Einstein relation since it was first derived by Albert Einstein in his Ph.D. thesis in 1905 (see note below) This expression relates macroscopic transport coefficient D with microscopic information on the mean square distance of molecular migration The 6 in this formula becomes 4 for a two-dimensional system and 2 for a one- dimensional system (see next page) This equation is suitable for calculation of D in MD simulation only for sufficiently high temperatures, when D > 10 12 m 2 /s Time t cannot be too large for a finite system (D drops to 0 when MSD approaches the size of the system) For periodic boundaries “true” atomic displacements should be used Derivation of this equation is given on pages 78-79 of the textbook by D. Frenkel and B. Smit In MD simulation we can describe the mobility of atoms through the mean square displacement that can be calculated as The MSD contains information on the diffusion coefficient D, ns fluctuatio Dt 6 A ) t ( r MSD 2 + + = Δ = r Historic note:
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This note was uploaded on 02/14/2012 for the course MSE 4270 taught by Professor Zhigilei during the Fall '11 term at UVA.

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Diffusions_Part_1 - Mobility of atoms and diffusion....

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