Diffusion coefficient of copper, tin and copper tin alloy

2) In the conclusion, please replace “The interdiffusion coefficient in Sn95.4%Cu4.4% alloy is in good agreement with experimental measurement” by “The total calculated structure factors in the Sn95.6 % Cu4.4 % alloy are in good agreement with experimental measurements”. 3) In the formula describing ) ( 0 r w replace C eff R r r Z by C eff R r r Z . 4) In numbers in the text and in the scales of all figures, please replace commas (,) by decimal dots (.) 5) At the end of the main text, please replace Sn %Cu % by Sn %Cu % . 400 600 80


Introduction
The study of diffusion in liquid metals is of scientific importance as well as potential application in material science, physics and chemistry field [1 to 4].The traditional lead-tin solders have been widely used in the industry for a long time.The lead poisoning commonly occurs following prolonged exposure to lead or lead compounds.The damage often is induced slowly, but definitely, it is now well recognized as a health threat.The solder materials containing lead are replaced by lead free solder materials in electronic products.
To study the atomic transport properties of liquid tin and copper which are depended both on the electronic and the ionic structure, we use the pseudopotential formalism to construct an ionic effective potential (ion-ion potential screened by electrons).Molecular Dynamics is used in conjunction with the constructed effective potential to get the atomic structure factor of pure copper and tin and the Sn 95.6% Cu 4.4% alloy which is a well known lead free solder.Our methodology is to use a local pseudopotential [5], to fit the core parameter on the atomic structure of the pure metal which is a very severe criterium.For noble metals we used the concept of effective valence [6].Then we calculate consistently the velocity autocorrelation function and we deduce the self diffusion of pure metals: copper, tin and the diffusion coefficient of copper in the Sn 95.6% Cu 4.4% .We used the local Optimised Model Potential (OMP) proposed by Shaw [5] and the Ichimaru-Utsumi [7] dielectric function.We use Z=4 as chemical valence for tin, and Z eff =1.56 [6] for copper taken from the bibliography.The structure calculation was performed by molecular dynamics (MD) on a 4000 particles NVT system.Our calculated pair correlation functions and structure factors are compared to the experimental ones.We first present the calculation of the ion-ion effective potential.We show that the local Shaw pseudopotential represents very good the structure of the pure metals (what is not true for all pseudopotentials).More, it is transferable to the alloy since a very good agreement is obtained for the coppertin liquid alloy.We also discuss the coherence with the electronic transport properties such as the diffusion coefficient.Then we conclude.(Unless explicitly stated EPJ Web of Conferences otherwise, atomics units are used throughout: ħ = e = m =1).

Effective potential
The local pseudo-potential theory is clearly explained in the references [7][8][9].The expression of the ion-ion effective potential ) (r V calculated from the pseudopotential for alloys is given by: Here the quantity r is the inter-ionic distance.The index is the dielectric function, and the function ) (q g xc is the exchange and correlation term.We use the Shaw local pseudo-potential, whose expression in real space is: ).The Fourier transform ) ( 0 q w of ) ( 0 r w is expressed as:

Molecular dynamics
The structure is calculated by a simulation code.This code uses the standard Verlet [10] algorithm and processes in NVT ensemble.The number of involved particles in the cubic simulation box for pure copper is equal to 4000.More, we consider periodic conditions.The side length is equal to 44.51 Å.The time step δ t is equal to 2*10 -15 s and the number of iterations after thermalization is equal to 20000.The code computes the pair correlation function g(r) for each configuration and gives the mean value.with The partial structure factors are calculated by the Faber Ziman formalism: The total structure factor S (q) is simply obtained from:

Atomic transport properties
The same recorded atomic configurations allow us to calculate the self-diffusion constant D at a given temperature from the recorded atomic velocities at time t, by integral over the velocity autocorrelation function (VAF) Z ab (t) defined as follows [11][12][13].
Where N is the total number of particles, N a is the number of a-type particles, Is the velocity of the a-type particle l(a).Z ab (t) is defined as the time correlation function of the relative velocity of the centre of mass of species a with respect to the center of species b.It is decomposed into self-contributions, and distinct contributions : Where is Kronecker's symbol.
is the velocity autocorrelation function of a tagged a-type particle in the fluid.The time integrals of all the Z ab (t), Z 0 ab (t), Z d ab (t) and Z s ab (t) give the associated diffusion coefficients (DC), namely D ab , D 0 ab , D d ab and D s ab respectively.The D s ab is the usual self diffusion coefficients.

With
and measures the deviation from an ideal mixture, it equals zero when all species are identical.The interdiffusion coefficient is given:

Where
. For a nearly ideal mixture And therefore S cc (q) are the partial Bhatia-Thornton concentrationconcentration structure factors.

Results and discussion
For both metals the density is calculated using Lucas [14] compilation of density as a function of temperature under the form: ) , where T M is the melting temperature of the metal.These parameters are given in Table 1.[14] of the two heavy metals used for our calculations.

LAM14
The effective potentials for liquid copper and tin are calculated, from the OMP local pseudo-potential, with respectively core radius values equal to 1,57Å and 1,032 Å.For alloys, we work with the same parameters than for pure metals.(see Figure 1).We then calculate by DM the pair correlation functions which are displayed together with the Waseda's experimental ones.The structure factor is obtained by Fourier transformation.Our calculated curves are compared to experimental ones (copper: figure 2, tin: figure 3).The results are of Waseda [15], Eder (Square) [16], Alemany (Aptriangle) [17] Our results for the pair correlation are in good agreement with experimental ones.The agreement is better on the structure factor.The choice of the effective valence for copper improves structure factor calculations.

01013-p.3 EPJ Web of Conferences
The self diffusion results are presented in figure 5. Our results are in good agreement with measurements and theory calculations.The results for Sn 95.4% Cu 4.4% alloy are presented in figure 6.

a
and b refer to atoms of type a and b.Z a et Z b are the effective valence characterizing each metals.The normalized energy wave number characteristic ) ( q F N ab in reciprocal space is defined from the Fourier transform of the local electron-ion model potential: Where 0 Ω   is the mean atomic volume.The quantity ) (q ε
In te ra to m ic d is ta n c e r(Å ) M D (th is w o rk ) W a s e d a [1 6 ]