Joy Billups Woller A THESIS. Presented to the Faculty of. The Graduate College in the University of Nebraska

Size: px

Start display at page:

Download "Joy Billups Woller A THESIS. Presented to the Faculty of. The Graduate College in the University of Nebraska"

Elwin Lang
5 years ago
Views:

1 DETERMINATION OF THE THERMODYNAMIC PROPERTIES OF FLUIDS BY GIBBS ENSEMBLE MONTE CARLO COMPUTER SIMULATIONS by Joy Billups Woller A THESIS Presented to the Faculty of The Graduate College in the University of Nebraska In Partial Fulfillment of the Requirements For the Degree of Master of Science Major: Chemistry Under the Supervision of Professor Xiao C. Zeng Lincoln, Nebraska July, 1997

2 DETERMINATION OF THE THERMODYNAMIC PROPERTIES OF FLUIDS BY GIBBS ENSEMBLE MONTE CARLO COMPUTER SIMULATIONS Adviser: Xiao C. Zeng Joy Billups Woller, M.S. University of Nebraska, 1997 Gibbs ensemble Monte Carlo (GEMC) is a useful technique for studying the thermodynamic properties of fluids. The theory and methodology of GEMC are explained, and the results of its use in two different projects are discussed. The first project involves a study of the effects of size, energy, and dipole moment parameters on the vapor liquid equilibria of binary mixtures containing Lennard-Jones (LJ) and Stockmayer (SM) components. The results of this work show that the phase coexistence properties of the mixtures are sensitive to the difference between the reduced dipole moments of the two components: an increase in this difference causes the phase envelope to widen for both the LJ SM and SM SM mixtures. Additional work which compares LJ SM and SM SM results to vapor liquid equilibria data simulated using an angleaveraged approximation of long-range dipolar interactions, the Keesom potential, suggests that the Keesom potential is not suitable to approximate the dipolar interactions for mixtures when the dipolar interaction is dominant. The second project involves

3 GEMC simulations of carbon dioxide, in which each molecule is modeled using three LJ sites with a point charge centered on each atom. Parameters for the carbon dioxide interaction potentials were generated using ab initio techniques, and the vapor liquid coexistence data produced were compared to experimental and previous simulation results. The ab initio potential parameters did not produce data consistent with either experimental or previous simulation results, which used parameters calculated from fitting to the experimental data. The disparity may be caused by the fact that ab initio calculations cannot properly account for the many interactions occurring in the liquid phase, or the fact that the higher stability of some CO 2 CO 2 orientations is not considered. While further attempts to generate potential parameters using ab initio methods may produce better results, fitting of experimental data clearly provides better LJ potential parameters for use in GEMC simulations of carbon dioxide at this point.

4 To Mom and Dad with love and thanks.

5 ACKNOWLEDGMENTS I would first like to thank my adviser, Professor Xiao C. Zeng, for the opportunity to venture into the interesting world of computer simulations and for his guidance in performing research. He has been a kind and generous teacher. I must also extend sincere thanks to Dr. Guangtu Gao of Beijing, China, who spent a year in Professor Zeng s lab as a visiting scholar. Dr. Gao s patient instruction and programming skill made our collaboration a valuable learning experience for me and a productive opportunity to explore fluid equilibria. Also, thank you to Ruben Parra, who provided the ab initio parameters for our carbon dioxide work. I would also like to acknowledge the following members of our group for their support: Hong Yuan, Kwang Jin Oh, Mark Gibson, Dr. Kenichiro Koga, and Dr. David Swanson. Thank you to Professors John J. Stezowski and William H. Braunlin, who graciously agreed to be on my committee and hear my oral defense. The work contained in this thesis was supported in part by the National Science Foundation, the Office of Naval Research, and the Petroleum Research Fund administered by the American Chemical Society. I credit (or blame, depending on my mood) two people with guiding me into this fascinating, and sometimes frustrating, world of chemistry. The first is my high school teacher, Mr. John McNeilly, who did a fantastic job of laying the solid foundation upon which I ve built all I know about chemistry. The second, Dr. Richard Shoemaker, has been my mentor for the past six years, and his encouragement and friendship helped get me through my darkest hours, i.e., organic chemistry. To my parents I owe a profound debt of gratitude for the appreciation of education they instilled in me, through their words and by example. I must also thank them for

6 teaching me the value of persistence, without which I couldn t have stuck with what has been, at times, a miserable exercise in masochism. Though Mom hasn t been able to help me with homework since I started long-division, at her knee (and sometimes across it) I learned enough about character and honesty to fill a few textbooks of my own. From Dad, I have learned the value of hard work, the power of logic, and a few choice swear words to use when a cow runs the wrong way. Thank you both for being terrific parents. I d also like to acknowledge my siblings for, well, their existence. I wouldn t be who I am today without them, so it s fortunate they re around to blame. Stephanie, you re a great kid and you have a wonderful future ahead. I hope you find something you truly love to do and pursue it with reckless abandon. Wes, you are the most honorable person I know, and I m intensely proud of you. I couldn t ask for a better brother. And Brandi, though you missed out on most of the sibling squabbles that went along with childhood in our family, I hope I made up for that deficiency while you lived with us. I m proud to call you my sister. Finally, I d like to thank my husband, Kevin, whose patient understanding and kind words are my source of strength and inspiration. While we face many uncertainties about our future (will we have to live in New Jersey?), the knowledge that we face it together makes all worries seem trivial. Thank you for being a wonderful husband and true friend. I love you.

7 CONTENTS Chapter 1. An Introduction to Computer Simulations Intermolecular Forces and Potential Models Simulation Methods Gibbs Ensemble Monte Carlo...7 References...12 Chapter 2. Gibbs Ensemble Monte Carlo of Binary Mixtures Containing Dipolar Molecules Introduction Potential Models Simulation Method Results Discussion...31 References...32 Chapter 3. Gibbs Ensemble Monte Carlo of Carbon Dioxide Using Ab Initio Size and Energy Parameters Introduction Potential Model Simulation Method Results Discussion...42 References...44 Appendix A...45 Appendix B...46

8 1 CHAPTER 1 AN INTRODUCTION TO COMPUTER SIMULATIONS The use of computers to predict thermodynamic properties has a relatively short history. However, recent rapid advances in computer power and data storage capacity combined with new simulation techniques have made computer simulations increasingly practical. The information gleaned from computer simulations of molecular systems can provide insight into the interactions of molecules and the quality of the theory describing chemical systems. Once the quality of the models and techniques used is verified, computer simulations can also provide invaluable data which may be used in predicting the thermodynamic properties of fluids, data which may be used by chemical engineers and environmental scientists, among others. In the past decade, a new computer simulation method has emerged as a useful tool in computing phase coexistence behavior of fluids. That method, Gibbs ensemble Monte Carlo, is discussed in this chapter, along with the other techniques and models needed in generating phase coexistence information by computer simulations. 1.1 Intermolecular Forces and Potential Models One of the most challenging tasks in performing computer simulations of molecular systems is coming up with a mathematical description of molecules which is both realistic and computationally feasible. While complex potential models, with

9 2 numerous adjustable parameters, can provide excellent descriptions of molecular interactions, they are often too computationally demanding to perform the Monte Carlo or molecular dynamics simulations required for calculating thermodynamic properties. On the other hand, the simplest potential models, with only a few adjustable parameters, don t necessarily describe reality. A compromise option involves putting a great deal of effort into optimizing the parameters of the simple potential models until they produce a reasonable approximation of reality. The work reported in the next two chapters is concerned primarily with two potential functions: the Lennard-Jones potential, which describes non-electrostatic attractive and repulsive interactions between pairs of molecules, and the Stockmayer potential, which describes dipolar interactions between pairs of molecules. By working with these relatively simple descriptions of complex interactions, we hope to better understand how they may be applied to predictive simulations used in chemical engineering and other fields The Lennard-Jones Potential Function The Lennard-Jones (LJ) potential is one of the most widely used intermolecular potentials, valued primarily for its computational simplicity, and is written as: σ ij Ur ( ij ) = ε ij r ij ij σ r ij

10 3 where σ ij is the LJ size parameter, ε ij is the LJ energy parameter, and r ij is the separation between particles i and j. The LJ function is plotted in Figure 1.1; both the independent and dependent variables are shown as reduced variables (see Appendix A). The LJ size parameter is equal to the intermolecular separation for which the energy is zero, which is easily shown to be σ = r, where r m m is the intermolecular separation at minimum U* energy, or U ( r) = ε. The LJ size and energy parameters are usually determined by choosing some physical property and fitting values of σ ε r* that property as calculated by the potential function to experimentally determined values. Figure 1.1 The Lennard-Jones potential energy function. Asterisks indicate reduced variables (see Appendix A) The Stockmayer Potential Function For two polar molecules, the contribution of the long-range electrostatic attraction between dipoles must be added to the description of intermolecular forces. The Stockmayer potential, used in this work to describe dipolar interactions, can be written with the LJ potential function as: 12 v v v v v v v v v σ ij σ ij µ i µ j ( µ i rij)( µ j rij) Ur ( ij, µ i, µ j ) = ε ij r 6 ij r ij rij rij

11 4 where v µ i and v µ j are the vectors for the embedded point dipoles in particles i and j; the other parameters are as in the LJ function. 1 The contribution from dipolar interactions can also be represented in scalar form as: U Dipole µ iµ j = 3 ( 2cosθicosθj sinθisinθjcos φij) 1.3 r ij where the angles representing the spatial relationships between particles i and j are shown in Figure 1.2. The simplicity of working with point dipoles makes the Stockmayer potential an attractive option in modeling polar fluids. Of course, since it ignores the effects of polarization and higher-order Figure 1.2 Spatial relationships between particles i and j used to calculate Stockmayer potentials. multipoles, it oversimplifies the interactions occurring in real fluids. However, in modeling dipolar and non-polar interactions of a polar fluid, the Stockmayer potential describes the essential aspects of a polar fluid. 2 In the work described in Chapter 2, the Stockmayer potential is examined without comparing it to real fluids.

12 5 1.2 Simulation Methods Since the Lennard-Jones function falls off rapidly as the intermolecular distance increases, the energy contribution at more than a few σ has a negligible effect on individual molecules. To reduce computational demands, a cutoff radius is set, beyond which intermolecular interactions are not explicitly calculated. In other words, when r r c, where r c is the cutoff radius, the intermolecular pair potential is set to zero. At the end of a simulation, a long-range correction term may be added to the total energy to account for the accumulated effects of the truncation. To allow a simulated system, with its limited number of molecules, to model bulk fluids, periodic boundary conditions are applied. To create periodic boundary conditions, the simulation cell (that which contains the molecules being simulated) is replicated in all dimensions around the cell. In Figure 1.3, a two-dimensional representation of a ninemolecule simulation is shown. One consequence of this replication is that molecules near the sides of the simulation cell experience intermolecular interactions with the periodic images of other molecules; without periodic boundary conditions, molecules near cell boundaries would experience very different environments than those in the center of the simulation cell. Another consequence of the periodic boundary conditions is that the transfer of a molecule (using either molecular dynamics or Monte Carlo methods) across a boundary results in an image of that molecule moving back into the simulation cell on the other side.

13 6 Figure 1.3 Two-dimensional representation of periodic boundary conditions. The center cell is the simulation cell; the gray cells hold the periodic images of the simulation molecules. Typically the box length in a simulation is longer than the cutoff radius applied to the potential function for nonpolar and uncharged molecules; as a result, these molecules do not interact with their periodic images. For long-range electrostatic interactions, however, special techniques must be applied to efficiently sum the interactions between a molecule and its periodic images. The specifics of the Ewald sum, the technique used in this research, are not explained in depth here; descriptions of the method are available

14 7 from a number of sources. 3 5 Details of the Ewald sum as they apply to the specific projects in this work are discussed in the next two chapters. 1.3 Gibbs Ensemble Monte Carlo In 1987, the procedure for Gibbs ensemble Monte Carlo (GEMC) simulations was first proposed by Panagiotopoulos 6 as a means of directly calculating the phase coexistence properties of either pure fluids or mixtures. GEMC has the advantage of being able to directly calculate two-phase equilibrium properties without requiring that the free energy be calculated as in conventional methods. 7 The goal of GEMC is to create a two-phase system, without an interface, in which the temperature, pressure, and chemical potential of all components in the two phases are equal. 8 However, there is no true constant-µpt ensemble, as extensive variables would be unbounded if only intensive parameters were specified. At least one extensive variable must be fixed, and for GEMC, that variable is the number of molecules N. 5 The effect is that the simulated system has coexisting phases in which the temperature, pressure, and chemical potential (or potentials, in the case of mixtures) are fixed. Since the actual value of µ is not specified (only the difference between different phases is set so that µ = 0) GEMC does not create a true constant-µpt ensemble, but such an ensemble is effectively simulated. The algorithm for performing a GEMC simulation involves applying three types perturbations to a system of two boxes, one containing gas and one containing liquid. The

15 8 perturbations require displacing a molecule within its current simulation box, changing the volume of the two boxes, and transferring a molecule from one box to another. There are two versions of GEMC, one which couples the volume changes so that the total volume of the vapor-liquid system remains constant (NVT) and a version in which the volume changes are not coupled but are used to ensure both boxes are at the same pressure (NPT). The NPT-GEMC method, in which the coexistence pressure is specified in advance, is better for modeling mixtures and is used in the binary mixture work described in Chapter 2, while the NVT-GEMC is used to simulate pure fluids and is adopted in the carbon dioxide work described in Chapter 3. 8 The first step in performing a GEMC simulation requires setting up two boxes, one for the liquid phase and one for the gas phase. These boxes are usually set up initially in a face-centered cubic structure, and each models a region deep within the bulk fluid. Periodic boundary conditions are applied, and no interface between the two phases exists in the simulation. After the phase boxes are set up, the three different types of perturbations are applied. Appendix B contains a flow diagram showing how these perturbations are implemented in the inner loop of a GEMC computer simulation.

16 Particle Displacement In this step, a particle is moved to some random new position within its original box. As such, the box is essentially a system at constant-nvt, and particles are moved using the normal Metropolis Monte Carlo algorithm. 9 The particle displacement is accepted or rejected based on the probability given by min(1,3 move ) where 3 move is given in Equation 1.4. Figure 1.4 Particle displacement within a box. exp( βenew) 3move = = exp( β E) 1.4 exp( βe ) old and E is the change in configurational energy for the move Volume Change In NPT-GEMC, to assure the constant pressure criterion is met, the volume of each box is varied independently, and is accepted or rejected with a probability given by min(1,3 vol ), where 3 vol is given in Equation vol = I I I II I V + V exp β E + E N kbt ln I V II II II V + V I II N k ln + ( + ) BT P V V II V 1.5 where the superscript I and II designate the properties of each individual box and k B is the Boltzmann constant.

17 10 Figure 1.5 shows a single box undergoing a volume change; note that the molecules remain in the same relative positions with respect to one another and the box boundaries. Figure 1.5 Before and after a volume change in a single box. Though it is possible to vary the volumes of each box simultaneously, convergence occurs more quickly when the volume of only one box is changed at a time, so the probability equation may be simplified to I I I I V + V I 3 vol = exp β E N kbt ln + P V I 1.6 V where the superscript I designates the box undergoing a volume perturbation. Again, E is the change in configurational energy. 8 For NVT-GEMC, the volumes of both boxes are adjusted at the same time so that the total volume is conserved; an increase in volume in one box is coupled with a decrease of the same amount in the other box. The volume perturbation in this step is either accepted or rejected based on the probability given by min(1,3 vol ), where 3 vol is given in Equation The superscripts I and II refer to the different boxes. 3 vol = N ( [ exp β E I B I V k T ln + E I II + V I V N II V k T ln B II + V II V 1.7

18 Particle Transfer In this perturbation, required to achieve coexistence between the two phases, a randomly selected particle is transferred from one phase box to the other, as shown in Figure 1.6. The picture is somewhat misleading, however, as the Figure 1.6 Particle transfer between phases. transfer is actually accomplished by creating a particle in one phase (for example, box I) and simultaneously destroying a particle in the other phase (box II). During this perturbation, the coexisting phases are representative of the grand canonical ensemble (constant-µv I T and constant-µv II T), where the two boxes are at the same temperature and each component has the same chemical potential in both boxes. The probability that the transfer will be accepted is given as 3 transfer, which may be calculated using Equations 1.8, 1.9, and 1.10, in which the particle is being transferred to box I from box II. I I I I 3 = exp[ β E + ln( zv /( N + 1))] 1.8 transfer II II II II 3 = exp[ β E + ln( N / zv )] 1.9 transfer II I I II I II V ( N + 1) 3 transfer = 3transfer 3transfer = exp β E + E + kbt ln I II 1.10 V N where the activity coefficient z is exp(βµ)/λ 3, λ is the de Broglie wavelength of constituent particles, and µ is the chemical potential. 8

19 12 REFERENCES Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces: Their Origin and Determination; Clarendon: Oxford, Van Leeuwen, M. E. Fluid Phase Equil. 1994, 99, 1. Ewald, P. P. Ann. Phys. 1921, 64, 253. Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, 1987; pp Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Academic: Boston, Panagiotopoulos, A. Z., Molec. Phys., 1987, 61, 813. Panagiotopoulos, A. Z.; Quirke, N.; Stapleton, M.; Tildesley, D. J. Mol. Phys. 1988, 63, 527. Panagiotopoulos, A. Z.; Stapleton, M. R. Fluid Phase Equil. 1989, 53, 133. Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. N.; Teller, E. J. Chem. Phys. 1953, 21, 1087.

Phase transitions of quadrupolar fluids

Phase transitions of quadrupolar fluids Seamus F. O Shea Department of Chemistry, University of Lethbridge, Lethbridge, Alberta, Canada, T1K 3M4 Girija S. Dubey Brookhaven National Laboratory, Upton, New