PHY480 Report 1

 

Molecular Dynamics Study of the Effect of Uniaxial – Tensile Stress on solid-solid Phase-Transformation of β-Sn to α-Sn

 

Introduction

 

Sn-37Pb solder alloy had been used for a long time to fabricate electronic interconnects. However toxicity of Pb has lead to the replacement of this alloy with lead – free alternatives such as Sn-3.5 Ag. Most of these replacement alloys have β-Sn as the main constituent. As a result this element decides the mechanical and electrical of Sn-based solders. It is known that below 13˚C, β-Sn (Crystal structure: Body Centered Tetragonal, BCT) undergoes phase transformation to α-Sn (Crystal Structure: Diamond Cubic, DC) when kept at this temperature for a long time. Transformation is further accelerated upon the application of stresses [1]. α-Sn being brittle in nature leads to failure. This presents itself as a problem for electronic equipments which operate in subzero temperatures for long durations under stresses (e.g. aircrafts). Given the complicated nature of such stresses, nucleation of α-Sn may be accelerated under their influence. This report intends to suggest a method for calculating the time required for nucleating α-Sn from β-Sn under the influence of a uniaxial tensile stress using Molecular Dynamics coupled with the laws of Thermodynamics. It is assumed that a constant uniaxial tensile stress applied in the [001] direction of a bicrystal (of unit dimensions) of β-Sn. The two unit cells are placed on top of each other on the (001) plane with their inplane crystal axes parallel to each other. MD simulation can then be done on a cluster of 26 atoms (two unit cells) placed on the lattice sites of two BCT crystals of a β-Sn.

 

Computation technique

 

The interaction potential between the atoms is considered to be of Lennard-Jones potential type,

 

Where,

V_ij = Lennard Jones potential

r = Interatomic distance

 

The above equation can also be used to represent the internal energy of the material. However Lennard-Jones potential does not take into account of the temperature dependence. This can be resolved if we consider the Helmholtz free energy,

 

A_ij = V_ij – TS(r_ij) (2)

 

 

Where,

A_ij = Helmholtz Free Energy between to atoms

V_ij = Internal Energy represented with Lennard-Jones potential between the i^th and j^th

atom

T = Temperature (constant)

S = Configurational Entropy

r_ij = Interatomic displacement vector between the i^th and j^th atom

 

 

Configurational entropy takes into account the atomic locations on the lattices of both the phases of Sn. The following figure, Fig.1, gives the crystal structure of the two allotropic forms of Sn.

 

Figure 1a. Body Centered Tetragonal Lattice of β-Sn with lattice parameters being a = 5.813 Ǻ, and c = 3.18 Ǻ [2]

 

Figure 1b. Diamond Cubic Lattice of α-Sn with lattice parameters being a = 6.341 Ǻ [3]

The forces acting on individual atoms due Helmholtz free energy

 

 

 

F_ij¹ is the force on the atoms only due to their interaction potential and temperature dependence. Application of a uniaxial stress also contributes to the net force acting the atoms. The direction of the uniaxial stress tensor is shown in figure 2.

 

Figure 2. The above figure shows the arrangement of the two lattices of β-Sn and the direction of applied stress. It is also assumed that this stress is acting on a unit area of the actual bicrystal.

 

Therefore the net force acting on individual atoms is given by

 

F_ij = F_ij¹ + σ .1 (4)

 

According to Newton’s second Law

 

F_ij= m d²r_ij/dt² (5)

Boundary Conditions:

 

1) Coordinates of the 26 atom cluster is determined by their location of in the lattice (Figure 2) relative to a XYZ coordinate axes. Starting velocities are assumed to be zero.

 

2) DC structure contains 18 atoms, whose coordinates can be determined from the crystal structure. These locations form the required location of the atoms in the new allotropic form of Sn.

 

Coupled with the differential equation (3) and the above stated boundary conditions, time required for the allotropic transformation can be calculated.

 

 

References

 

[1] Kariya Y, Gagg C, Plumbridge WJ; “Tin pest in lead-free solders”; SOLDERING

&SURFACE MOUNT TECHNOLOGY 13 (1): 39-40 2001

 

[2] http://cst-www.nrl.navy.mil/lattice/struk.picts/a5.png

 

[3] http://www.msm.cam.ac.uk/phase-trans/2003/MP1.crystals/MP1.crystals.html