MSE 501: Molecular Dynamics, An Introduction

I've discussed the sheer number of subspecialties in materials science before. I'd like to go into a bit of depth on my personal subspecialty, simulation. My work is in molecular dynamics (MD). The linked Wikipedia article is fairly technical. Fundamentally speaking, molecular dynamics uses Newton's First Law to determine the motion of atoms or particles according to a force equation. Particle interaction forces are described using one of many possible forms of the potential energy equation, and by integrating F=ma over a finite timestep, the motion of each particle can be determined.

Force potentials (mostly just called potentials) come in several basic forms, but typically are strongly repulsive at short distances with a low energy well at the preferential distance between different particle types. These potentials can be empirical or derived from quantum mechanical calculations, and may or may not take into account things like charge interactions or bonding, depending on your system. The majority of work in simulation goes into generating and validating these potentials for your system.

But how do we get started? There has to be an initial configuration from which positions are predicted and integrated. There's several ways, but it typically boils down to either using a crystalline structure, a random generator, or some combination of the two. These systems typically contain 10^4 to 10^6 atoms, with very few facilities capable of reaching millions of atoms(BlueGeneL, Blue Waters, etc). This is because for a number of atoms N, the number of calculations for each time step scales as roughly 3*N^2.

One million atoms isn't very big. One mole is more than 100,000,000,000,000,000 times larger than that. Even if Moore's Law were to continue indefinitely (which it probably can't, but that's another post), the sheer amount of computing time necessary to simulate a mole of material for more than a picosecond will still be staggering. For me, currently, to simulate 100,000 atoms using 8 clusters for one nanosecond would take roughly 20 hours on our fairly fancy schmancy server, or 160 hours of CPU time. To get around this, we use periodic boundary conditions. As long as our simulation box is large enough that atoms can't try and interact with themselves, we can effectively create an infinite solid. If we want to study a surface, we can chose not to implement periodic boundaries in that direction, or add a vacuum layer. However, in amorphous systems especially, periodic boundaries must be used carefully, lest you over-constrain your system.

So what is MD good for? Quantum mechanical methods are more accurate and finite element analysis can simulate much larger systems. Molecular dynamics (and its cousin, Monte Carlo) is one of the best ways to understand biomolecules and predict things like protein folding. It can be used to understand the mechanisms of radiation damage and shock wave damage. It's also growing as a tool to study interfaces, which tend to become surfaces in experimental methods, which changes certain structural details. It can also be used to study molecules which are simply too large for quantum methods (i.e., more than a few hundred atoms). MD can predict crystal structures for pressures and temperatures, which while not achievable in the lab, happen at the center of the Earth, or in space. The ultra-high tensile strength of carbon nanotubes was predicted by MD before it was verified by experiment.

Materials science is far from the only field that uses molecular dynamics as a tool. We may ask different questions of our results, but people from geophysicists to biologists use MD. And I think that's pretty spiffy.

Stress, Strain and Moduli

Since both of my major projects as well as my coursework are hitting on the subject this week, I though I'd talk about elastic moduli. However, before I can talk about that, we need to talk a bit about stress, strain and the concept of anisotropy.

Stress is fundamentally defined as force per unit area. Strain is the change in length per unit length. In a Cartesian coordinate system (which we use unless we're doing pipes or spheres), this means that to fully describe the stress and strain in three dimension, we end up with matrices.

Stress is typically written as σ, while strain is ε (we use them for conductivity and dielectric constants too...). By combining our two matrices according to the generalized Hooke's Law , we end up with a fourth-order tensor, Cijkl which we call the stiffness tensor. We can also invert the relationship and get Sijkl which is the compliance tensor. I think every mechanics of materials professor ever wishes the symbols on these two were reversed, but we're stuck with the convention of C=stiffness and S=compliance. 


Now we get to the idea of anisotropy, which is the concept that materials don't behave the same in all orientations. For a perfectly isotropic material (like liquid), the strain response will be the same regardless of direction. In this case, the number of independent elastic constants drops to two: Young's modulus and Poisson's ratio. Young's modulus is also known as the modulus of elasticity, while Poisson's ratio describes the degree to which a materials expands in the unconstrained directions when compressed in the third.  In an anisotropic material, these values are no longer constants, but instead become matrices as well. 


My current research nemesis is the bulk modulus, which is particularly challenging to determine using experiments or simulations. It can either be determined from the derivative of the pressure-volume relationship or the derivative of the pressure-density relationship. In both cases, relaxation and equilibration are critical to minimize scatter in the data. However, reality being what it is, you usually end up with at least some scatter, and can consequently get some crazy derivatives depending on the method you chose for defining the slope at each point.


Bulk modulus is also unique in terms of the anisotropy question: other moduli measure response to a linear stress, be it axial or shear. Bulk modulus describes the compressibility of a material under isotropic pressure, and therefore, the value of the bulk modulus isn't a directional dependent thing. Which is kind of cool. It still makes me want to bang my head on a wall right now, though.

MSE Fundamentals

Somewhat uniquely among graduate program with a similar undergraduate degree, a very large fraction of MSE grad students majored in something else as an undergrad (usually physics or chemistry). However, in most programs, there's not a good review class covering the things all your other classes will assume you know already.  Taking the undergraduate overview class can be frustrating for many grad students, with a lot of overlap with material they do know, and little depth in areas they don't. Consequently, those of us with a BS in MSE often end up tutoring our peers in the major-specific basics.

I'm going to try and run two different types of posts on materials science fundamentals. MSE 101 posts will be aimed towards those with a background in general chemistry and basic calculus, but not much beyond that. MSE 501 posts will be aimed at explaining topics to someone with a BS degree in science or engineering. If nothing else, I want to have these written down somewhere I can find them later when asked.

If there's a particular aspect of materials science you'd like me to address, please free to leave it in the comments.