Progress toward a Monte Carlo Simulation of the Ice VI-VII Phase Transition

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1 Progress toward a Monte Carlo Simulation of the Ice VI-VII Phase Transition Christina Gower 2010 NSF/REU PROJECT Physics Department University of Notre Dame Advisor: Dr. Kathie E. Newman August 6, 2010

2 Abstract We study the phase transition between the self-clathrate forms of ice: Ice VI and Ice VII. This continues a project started the previous summer by REU student, Dawn King, who studied the geometry of the structural phase transition and developed a common coordinate system for the two structures. We are working to develop a phenomenological Ising-like spin model of the system to capture the energetics of the transition. The ultimate goal is to study the ice phase transition using Monte Carlo simulation methods. We have created a FORTRAN 95 program implementing the Metropolis algorithm and have tested it on the two-dimensional Ising model. We present results which show the nature of the Ising phase transition. The next step will be to extend the Ising model into an Ising-like model for describing the three-dimensional interpenetrating lattices of ice.

3 0.1 Introduction This project is a continuation from the summer 2009, where Dawn King studied the geometrical structure of the phase transition between Ice VI and Ice VII. Ice VI and VII are known as the self-clathrate forms because their structure is composed of interpenetrating sublattices with tetrahedral bonds. Our goal is to understand the mechanism that causes Ice VI to transform into Ice VII. When the phase transition occurs, hydrogen bonds are broken and the structure is distorted. We want to model the system through Monte Carlo simulations. 0.2 Monte Carlo Simulations Monte Carlo simulation was originally termed statistical sampling and was used to solve integrals of complicated functions Ref. [1]. This technique was particularly useful for functions of higher than three dimensions. Statistical sampling was not implemented in the field of physics until the 1930 s when Fermi used it on his work in neutron diffusion. Fermi did not publish his method of statistical sampling, so the Monte Carlo simulation was not acknowledged in the physics community until Nicolas Metropolis introduced it in his paper on hard-sphere gases in It is now a widely used numerical method for calculating the partition function of systems in statistical mechanics Ref. [3]. The partition function is the sum of all of the states in a system, denoted by Z = ν e βeν Ising Model The Ising Model is used in statistical mechanics as a way to mathematically explore the properties of a ferromagnet Ref. [1]. A ferromagnet is a substance that has a high level of magnetization. The model consists of two-state values, called spins, that are arranged on a

4 lattice or a graph. The spins represent dipoles in a ferromagnet. The Ising model is generally Figure 1: The Ising Model arranged in one of two initial states: The T = 0 state and the T = state. When the temperature is zero, we know that all of the spins are oriented in the same direction. This is termed an ordered state, and is illustrated in the figure above. The initial condition of T = specifies that the spins are randomly oriented. A disorded, or randomly ordered, system would consist of an equally likely chance for a given state to be either spin-up or spin-down. The Hamiltonian for the Ising Model is given by; H = J s i s j B ij i s i (1) Where s i are the spins on the lattice, and J is the interacting energy, or coupling constant, between nearest neighbor spins. Nearest neighbors are sites on the lattice that are directly touching the selected site. The first term is the sum of the selected spin multiplied by it s nearest neighbors, and is scaled by J. The second term in the Hamiltonian takes into account an external magnetic field, denoted by B. For the two-dimensional Ising model, the phase transition occurs at a temperature of T c 2.269J. Below this point, called the Critical Temperature, the orientation of the spins are completely random. Above the Critical Temperature most of the spins spontaneously align in one direction without the aid of an external magnetic field. Ref. [3]

5 0.2.2 Metropolis Algorithm The Metropolis algorithm was the very first Monte Carlo simulation done through a computer. This algorithm simulates the random fluctuations of a dynamic system as it reaches equilibrium. This is done through the following steps: Create a lattice. Select a site on the lattice. Calculate the change in energy of the system. Generate a random number r strictly between 0 and 1 If the random number r is less than e E, then flip the sign of the spin Otherwise, randomly choose the next site from nearest neighbors and repeat process 0.3 Our Project Ice VI and VII are complex structures composed of interpenetrating sublattices and are further complicated by the order-disorder of the hydrogen bonds between the molecules. Before we could intelligently study the Ice VI-VII transition, we had to first explore a simpler model to understand the nuances of such a process. Our solution was to devote this summer to simplifying our model of Ice into a two-dimensional Ising Model. With this, we were able to explore the logic behind writing a program for a phase transition. We discovered that if we divided the coupling constant by k b T, our Hamiltonian would be dimensionless and we wouldn t have to worry about the units we were working in. We plotted the energy of each state against the number of iterations. By changing the energy constant, J, we can see that

6 our state comes to equilibrium at specific energies. This is what we expect to happen, which tells us that our program is consistent and follows the correct logical steps Computer Simulation We created the Monte Carlo simulation using FORTRAN 95. Our code was based primarily on the Metropolis algorithm. We declared a two-dimensional array, assigned the value 1 to each site in the array, then randomly chose a nearest neighbor to alter. The FORTRAN code contains an IF statement that applies periodic boundary conditions to the array. Periodic boundary conditions allow us to model an infinite square lattice, because the lattice repeats itself infinitely in all directions. Once our array is made, we call a random number r and run through iterations of the Metropolis algorithm Results From our FORTRAN code, we were able to graph the energy changes in the system against the number of iterations the program went through in GNUplot. We varied our coupling constant and the number of iterations in our code in order to see the effects on the coupling constant, as well as determine the point of equilibrium of the system. We can see from the following charts that we accurately modeled a two-dimensional Ising model. By developing an Ising-like model of ice, we will be able to study the phase transition between Ice VI-VII both geometrically and through statistical mechanics. These methods will allow us to study the energetics of the transition as well as to gain a better understanding of the properties of Ice. We can see from Figure 2 that our system comes to equilibrium within the first five thousand iterations. Figure 3 shows us a more detailed description of the system coming to equilibrium.

7 3600 "J2n100000" Figure 2: Energy vs. Iterations on a 30x30 Lattice where J is set at 2 with iterations "J2n10000" Figure 3: Energy vs. Iterations on a 30x30 Lattice where J is set at 2 with iterations.

8 0.4 Future We can see from our figures that our program is successful. Our hope is to expand our program in the future into three dimensions. We have already worked on the visualization of the Ice model, and now that we have developed a FORTRAN model for a two-dimensional Ising Model we can start developing an Ising-like model in three dimensions for the Ice VI-VII transition.

9 Bibliography [1] Chandler, David. Introduction to Modern Satistical Mechanics. New York: Oxford UP, Print. [2] Allen, M. P., and D. J. Tildesley. Computer Simulation of Liquids. Oxford, England: Clarendon, Print. [3] Newman, M. E. J., and G. T. Barkema. Monte Carlo Methods in Statistical Physics. Oxford: Clarendon, Print. [4] Landau, David P., and K. Binder. A Guide to Monte Carlo Simulations in Statistical Physics. Cambridge: Cambridge UP, Print. 7