铝的晶格常数-体弹模量及弹性常数分子模拟.docx

Calculation of material lattice constant and bulk modulus

2012
CC

Calculation of material lattice constant
and bulk modulus
Summary：
Aluminum is one of the world's most used metals, the calculated aluminum lattice constant and bulk modulus can be used to improve the performance of the aluminum consequently make better use of aluminum. In virtue of molecular dynamics simulation software，we can solve the lattice constant . By the derivative of the lattice constant, the bulk modulus can be obtained. The elastic constants of a material display the elasticity and we can use the software material studio to simulate and get them. The simulation results match the experimental values.
Key words:
Aluminum, lattice constant, bulk modulus, elastic constant , simulation.

Introduction:
In materials science, in order to facilitate analysis about the way in which the crystal particles are arranged, the basic unit can be removed from the crystal lattice as a representative (usually the smallest parallel hexahedron) as a composition unit of dot matrix, called a cell (i.e. solid State Physics "original cell" concept); lattice constant (or so-called lattice constant) refers to the side length of the unit cell, in other words, the side length of each parallel hexahedral cells. Lattice constant is an important basic parameters of crystal structure. Figure A is the basic form of the lattice constant.

Figure A
Lattice constant is a basic structural parameter，which has a direct relationship with the bondings between the atoms ，of the crystal substance. It reflects the changes in the internal composition of the crystal of the lattice constant, force state changes, etc.
The bulk modulus (K or B) of a substance measures the substance's resistance to uniform compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. Its base unit is the pascal. Figure B describes the effect of bulk modulus.

Figure B
The bulk modulus can be formally defined by the equation:

where is pressure, is volume, and denotes the derivative of pressure with respect to volume.
Our research object is aluminum，whose atomic number is 13 and relative mass is 27.The reserves of aluminum ranks only second to ferrum compared with other metallic elements. Aluminum and aluminum alloy are considered the most economic and applicable in many application fields as a consequence of their excellent properties. What’s more，increased usage of aluminum will result from designers' increased familiarity with the metal and solution to manufacturing problems that limit some applications.
The crystal structure of aluminum is face-centered cubic. The experimental value of lattice constant and bulk modulus are 0.0.40491nm and 79.2Gpa.
Computing theory and methods：
Our simulation is on the basis that aluminum is of face-centered cubic crystal structure. We can get the exact value of lattice constant in virtue of molecular dynamics simulation software. Then by the derivation of lattice constant for energy E，we obtain bulk modulus.
To start with ，compile a script for the use of operation and simulation in lammps. We set periodic boundary conditions in the script and create an analog box，whose x ,y ,z coordinate values are all confined to [0,3].Run the script in lammps, calculating the potential energy，kinetic energy as well as the nearest neighbor atoms for each atom. Finally put out the potential energy function of aluminum.
Extract the datas under the linux system produced by lammps to continue the computation by means of matlab，from which we can get the lattice constant through several times of matching.
Figure C
Figure C——the curve shows the relationship between cohesive energy and lattice constant，which is what we get in the process of computing in matlab，points out the lattice constant corresponding with the least cohesive energy. The horizontal ordinate of the rock bottom stand for the lattice constant of aluminum which can be clearly located as 0.40500nm.
Since we have obtained the lattice constant，we simulated the visualization of aluminu’s crystal structure. Figure D is what we get through the visualization.
Figure D
The bulk modulus is defined as：

As for cubic cell，the formula can be transformed into the
following pattern：

The bulk modulus can be calculated with the formula above combined with the lattice constant. Finally，the bulk modulus is 78.1Gpa.
Besides，to enrich our research, we have calculated the elastic constants of aluminum. Because of the symmetry of face-centered cubic，aluminum only has three elastic constants. To reach the target ，we have to establish a cell of aluminum in the software material studio in the first place and then transform it into a primitive cell. Figure E and figure F are the cell and the primitive cell we have established during the simulation .
Figure E
Figure F
It comes to the CASTEP step after the geometry optimization. We managed to make Ecut=350eV and
Kpoints=16*16*16 ，both of which are extremely important and also sensitive to the calculation of elastic constant， after several trials. Figure G is the primitive cell that has been through the geometry optimization.
Figure G
The calculated results are as follows:
C11=106.2Gpa
C12=60.5Gpa
C44=28.7Gpa
Correspondingly ，the experimental range of the three elastic constants are listed below：
C11=108~112Gpa
C12=61.3~66Gpa
C44=27.9~28.5Gpa
As we can see，although a little outside of the value range of the experimental standards, our results are right within the error range.
Conclusion：
Our group has calculated the lattice constant and bulk modulus of aluminum，both of which coincide with the experimental value，by means of lammps and matlab. Moreover ，we have found out that bulk modulus has a close relationship with temperature. As lattice constant haven’t made any change under small change of temperature while the energy of the material have changed，so we concluded that temperature change can influence bulk modulus as a consequence of the change of cohesive energy change resulting from temperature change.
There are still problems in our research as you can see that the three elastic constants are a little out of the value range. But this group is the one closest to the experimental value. Our group have concluded that the errors result from the script which can affect the accuracy of the simulation.
Besides，the most valuable thing we have learned is that we must seek the solutions and never give up in face of difficulties.
References：
[1]材料科学基础（胡赓祥、蔡珣、戎咏华 上海交通大学出版社）
[2] Ayton, Gary; Smondyrev, Alexander M; Bardenhagen, Scott G; McMurtry, Patrick; Voth, Gregory A. “Calculating the bulk modulus for a lipid bilayer with none quilibrium molecular dynamics simulation". Biophysical Society. 2002.
[3] Cohen, Marvin (1985).
"Calculation of bulk modulus of diamond and zinc-blende solids". Phys. Rev. B 32: 7988–7991.
[4] Watson, I G; Lee, P D; Dashwood, R J; Young, P. Simulation of the Mechanical Properties of an Aluminum Matrix Composite using X-ray Microtomography: Physical Metallurgical and Materials Science. Springer Science & Business Media. 2006.
[6] Ashley, Steven. Aluminum vehicle breaks new ground. Engineering--Mechanical Engineering. Feb 1994.
[7] Sanders, Robert E, Jr; Farnsworth, David M. Trends in Aluminum Materials Usage for Electronics. Metallurgy. Oct 2011.

附文：lammps脚本

units metal
boundary p p p
atom_style atomic
variable i loop 20
variable x equal 4.0+0.01*$i
lattice fcc $x

region box block 0 3 0 3 0 3
create_box 1 box
create_atoms 1 box
pair_style adp
pair_coeff * * AlCu.adp Al
mass 1 27
neighbor 1 bin
neigh_modify every 1 delay 5 check yes
variable p equal pe/108
variable r equal 108/($x*3)^3
timestep 0.005
thermo 10
compute 3 all pe/atom
compute 4 all ke/atom
compute 5 all coord/atom 3
dump 1 all custom 100 dump.atom id xs ys zs c_3 c_4 c_5
dump_modify 1 format"%d %16.9g %16.9g %16.9g %16.9g %16.9g %g"
min_style sd
minimize 1.0e-12 1.0e-12 1000 1000
print "@@@@ (lattice parameter,rho,energy per atom):$x $r $p"
clear
next i
jump in.al
至于代码的含义，可参考其它资料或查询lammps官网manual。至于势函数，可在这个网站下载：http://www.ctcms.nist.gov/potentials/
附言：整个过程大致如下：
1. 晶格常数：先安装lammps软件。ubuntu版的比较陌生，且安装比较复杂。建议安装windows版。编写脚本。附文即是一例，编写完后保存为in.---(后面自定义)，同时要去除后缀名，以便lammps软件识别。然后在软件中输入：jump in.--- 。lammps即运行，运算结果以文件输出。
2. 体弹模量：用matlab处理，计算即可得到。
3. Vmd可视化：因为这是个静态过程，可视化显得可有可无。执行也比较简单，用上面输出的log.lammps文件，导入vmd软件中即可实现可视化。
4. 弹性常数：先安装materials studio。这个过程可能有点复杂，可参考其它资料。至于如何实现，也可查找分子模拟论坛。
总结：以上是我们的经验之谈，希望能帮到用得着的朋友，圆满完成任务。