Mobility of atoms and diffusion. Einstein elation. In M simulation we can descibe the mobility of atoms though the mean squae displacement that can be calculated as N 1 MS ( t ( i ( t i ( 0 N The MS contains infomation on the diffusion coefficient, MS (t A 6t fluctuations This epession is called Einstein elation since it was fist deived by Albet Einstein in his Ph.. thesis in 1905 (see note below This epession elates macoscopic tanspot coefficient with micoscopic infomation on the mean squae distance of molecula migation The 6 in this fomula becomes 4 fo a two-dimensional system and fo a onedimensional system (see net page This equation is suitable fo calculation of in M simulation only fo sufficiently high tempeatues, when > 10 1 m /s Time t cannot be too lage fo a finite system ( dops to 0 when MS appoaches the size of the system Fo peiodic boundaies tue atomic displacements should be used eivation of this equation is given on pages 78-79 of the tetbook by. Fenkel and B. Smit i 1 Histoic note: Befoe Albet Einstein tuned his attention to fundamental questions of elative velocity and acceleation (the Special and Geneal Theoies of Relativity, he published a seies of papes on diffusion, viscosity, and the photoelectic effect. His papes on diffusion came fom his Ph.. thesis. Einstein's contibutions wee 1. to popose that Bownian motion of paticles was basically the same pocess as diffusion;. a fomula fo the aveage distance moved in a given time duing Bownian motion; 3. a fomula fo the diffusion coefficient in tems of the adius of the diffusing paticles and othe known paametes. Thus, Einstein connected the macoscopic pocess of diffusion with the micoscopic concept of themal motion of individual molecules. Not bad fo a Ph.. thesis! Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
iffusion equation: eivation of Einstein elation fo 1 case Let s conside diffusion of paticles that ae initially concentated at the oigin of ou coodination fame, C,0 - iac delta function C, C, t C, t t Let s multiply the diffusion equation by and integate ove space: C, t C, t d d t t t Solution: πt 4t 1 ep t t C, t C, t 0 C d C, td C, td d, t C, t C, t d, t C, td C d t t A fo 1 diffusion Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
eivation of Einstein elation Let s conside diffusion of paticles that ae initially concentated at the oigin of ou coodination fame, C,0 - iac delta function C, t iffusion equation: C, t t Let s multiply the diffusion equation by and integate ove space: t Solution: C, t C, t d C, t d t 1 ep d / t t t t C C d, t d C, t, td C, tds C, td 0 C, td C, td t C, tds d C, td d dt A =1 fo diffusion in d-dimensional space Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei Review of calculus used in the pevious page ẑ z ŷ y ˆ Gadient: Gadient of a scala function: ẑ z ŷ y ˆ ( ivegence of a vecto function: z F y F F ( F z y F ( ( F ( ( F ( ( The divegence theoem: S V d S F ( d F ( z y Laplacian: L Hopital s ule: if lim g(/f( esult in the indeteminate fom 0/0 o inf/inf, then d df d dg lim ( f g ( lim
Mobility of atoms and diffusion (I d equilibium d inteatomic distance 50 Atomic paths of two atoms in FCC lattice at tempeatue below the melting tempeatue. Figues ae fom M simulations by E. H. Bandt, J. Phys: Condens. Matte 1, 1000-10014 (1989. Can we use the Einstein elation to calculate the iffusion coefficient fom these atomic tajectoies? Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
Mobility of atoms and diffusion (II d equilibium inteatomic distance Atomic paths of two atoms in amophous and liquid systems. Figues ae fom molecula dynamics simulations by E. H. Bandt, J. Phys: Condens. Matte 1, 1000-10014 (1989. Two longest atomic paths fo each simulation ae shown. Can we use the Einstein elation to calculate the iffusion coefficient fom these atomic tajectoies? Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
Using Einstein elation. Eample. Changes in atomic mobility duing solidification fom the melt. ( t A 6t fluctuations Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
Geen-Kubo fomula fo diffusion coefficient An altenative way to define in M simulation is though Velocity Autocoelation Function (Geen-Kubo epession: 1 N 1 v ( t v (0 dt vi t v N ( ( 3 3 0 0 i 1 i (0 dt Fo eliable calculation of tajectoies should be computed fo as long as the velocities emain coelated Geen-Kubo and Einstein epessions fo ae equivalent We will discuss the meaning of the Velocity Autocoelation Function late eivation of this equation is given on page 80 of the tetbook by. Fenkel and B. Smit To get most fom you M tajectoies you can use aveaging ove stating times. Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
Tempeatue dependence of diffusion Assuming Ahenius behavio fo the jump-fequency one can etact a vacancy migation enegy o an aveage activation enegy fo atomic migation in a disodeed system, T ( 0 E ep a k B T Ahenius plot fo an amophous alloy, by E. H. Bandt, J. Phys.:Condens. Matte 1, 1000-10014 (1989. Note the deviation of the diffusivities fom the Ahenius behavio at high tempeatues, when k B T becomes compaable o lage than the activation enegies. This indicates a change in the mechanism of atomic mobility (change in E a, collective motions..?. Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
Tempeatue dependence of diffusion iffusion of a cluste of 10 atoms on a substate. Shugaev et al., PRB 91, 35450, 015]. E ( T ep a E 0 kbt a = 0.41ε Lennad-Jones potential with paametes σ and used fo the substate A cutoff function defined in [Phys. Rev. A 8, 1504, 1973] is applied at 3σ LJ paametes fo atoms in the cluste ae σ cc = 0.60σ and cc = 3.7ε LJ paametes fo cluste-substate inteactions ae σ cs = σ and cs = 0.5ε The mass of an atom in a cluste is m c = 1.74m, whee m is the mass of an atom in the substate. Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
Tempeatue dependence of diffusion: Fast diffusion paths iffusion coefficient along a defect (e.g. gain bounday can also be descibed by an Ahenius equation, G.B. G.B. 0 ε ep k T with the activation enegy fo gain bounday diffusion significantly lowe than the one fo the bulk. Howeve, the effective coss-sectional aea of the boundaies is only a small faction of the total aea of the bulk (an effective thickness of a gain bounday is ~0.5 nm. The gain bounday diffusion is less sensitive to the tempeatue change becomes impotant at low T. m G.B. B Self-diffusion coefficients fo Ag. The diffusivity if geate though less estictive stuctual egions gain boundaies, dislocation coes, etenal sufaces. Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
Patial diffusivities. iffusion in heteogeneous systems. Patial diffusivities in multicomponent systems can be calculated by aveaging ove atoms of one type only. Heteogeneous diffusion. In calculation of the mean squae displacement one can aveage not ove all the atoms, but ove a cetain subclass. Fo eample, the plots below show the diffeence between atomic mobility in the bulk cystal and in the gain bounday egion. Mean-squae displacement of all atoms in the system (B, atoms in the gain bounday egion (C, and bulk egion of the system (A. The plots ae fom the compute simulation by T. Kwok, P. S. Ho, and S. Yip. Initial atomic positions ae shown by the cicles, tajectoies of atoms ae shown by lines. We can see the diffeence between atomic mobility in the bulk cystal and in the gain bounday egion. Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
Estimation of the diffusion coefficient fom the jump-fequency infomation If diffusion poceeds though vacancy migation, then one can obtain a diffusion coefficient fom the jump-fequency infomation. Assuming Ahenius behavio fo the jump-fequency (R one can etact a vacancy migation enegy (E m. R v R 0 ep E m v k B T v ~ a N v R v T. Kwok, P. S. Ho, and S. Yip, M studies of gainbounday diffusion, Phys. Rev. B 9, 5354 (1984. R v - R 0 - N v - a - the effective vacancy jump fequency pe-eponential facto ~ effective coodination numbe attempt fequency equilibium vacancy concentation that is ~ ep(-e vf /kt whee E vf is the effective vacancy fomation enegy effective squaed jump distance Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
iffusion in nanocystalline mateials: Eamples image by Zhibin Lin et al. J. Phys. Chem. C 114, 5686, 010 Ahenius plots fo 59 Fe diffusivities in nanocystalline Fe and othe alloys compaed to the cystalline Fe (feite. [Wuschum et al. Adv. Eng. Mat. 5, 365, 003] Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei
Spatially heteogeneous mobility of atoms. Eample. Changes in atomic mobility duing cystallization of amophous metal. Atomic mobility is much moe active at the font of cystallization. is not eally a diffusion coefficient in statistical themodynamics sense, but athe a quantity that eflects an aveage mobility in this mateial undegoing phase tansfomation. Univesity of Viginia, MSE 470/670: Intoduction to Atomistic Simulations, Leonid Zhigilei