MOLECULAR DYNAMICS SIMULATIONS OF HEMATITE NANOPARTICLE DEPOSITION ONTO A SUBSTRATE

Transcription

1 MOLECULAR DYNAMICS SIMULATIONS OF HEMATITE NANOPARTICLE DEPOSITION ONTO A SUBSTRATE By DANA R. LOUIE B.S., Mehanial Engineering, Ohio University, 1989 M.S., Aerospae Engineering, University of Florida, 1992 A thesis submitted to the Graduate Faulty of the University of Colorado at Colorado Springs In partial fulfillment of the Requirements for the degree of Master of Siene Department of Physis 2014

2 This thesis for the Master of Siene degree by Dana R. Louie Has been approved for the Department of Physis By Dr. Karen Livesey, Chair Dr. Anatoliy Pinhuk Dr. Kathrin Spendier Date

3 Louie, Dana R. (M.S., Applied Siene--Physis) Moleular Dynamis Simulations of Hematite Nanopartile Deposition onto a Substrate Thesis direted by Assistant Professor Karen Livesey Hematite nanopartile deposition onto a silion dioxide substrate in hexane solution was examined to asertain whether deposition time and rate vary depending on the number of partiles in the system, and to determine whether ordered patterns form on the substrate surfae. Average deposition time for all partiles to reah the substrate was higher for systems with greater numbers of partiles. Deposition rate was established by fitting the average height above the substrate for all partiles versus time with an exponential urve. From the urve fit, time onstant was determined. The time onstant was slightly higher for systems with greater numbers of partiles. Ordered patterns formed on the substrate surfae after the partiles had settled, and the patterns were better ordered for systems with larger numbers of partiles. A Fortran simulation program was developed to support this study. The program inluded visous and flutuating fores due to partile immersion in the hexane fluid, Van der Waals and steri fores between partiles, gravity, and the Van der Waals fore ating between eah partile and the substrate. Interpartile Van der Waals and steri fores were modeled with the Lennard- Jones fore. The Lennard-Jones model was more omputationally effiient than employing individual interpartile Van der Waals and steri fores. However, for partiles at lose distanes, the magnitude of the Lennard-Jones repulsive fore was less than that of the sum of the Van der Waals and steri fores, and thus ould have slightly affeted the pattern formed on the substrate. Future studies should investigate employing individual Van der Waals and steri fores in plae of the Lennard-Jones approximation. iii

4 DEDICATION I dediate this thesis to my husband Jeff. Thank you always for your steadfast, unwavering support. Being deeply loved by someone gives you strength, while loving someone deeply gives you ourage. Lao Tzu

5 ACKNOWLEDGEMENTS I would like to aknowledge Dr. Karen Livesey and Dr. Bob Camley for their support throughout my graduate program at University of Colorado, Colorado Springs. Dr. Livesey, as my ommittee hair and thesis advisor, devoted ountless hours in guiding me through this projet and teahing me skills and knowledge essential in onduting researh. Dr. Camley, as graduate student advisor, first met with me when I applied to the program and led me to the best path of suess in aomplishing my goals. After ompleting this program at UCCS, I will ontinue in a PhD program in Astronomy this Fall. I thank you both for helping me to begin this new journey in my life towards beoming a professional Astronomer.

6 TABLE OF CONTENTS CHAPTER INTRODUCTION CHAPTER BACKGROUND ON COLLOIDS AND COLLOIDAL FORCES SIMULATED FOR THIS PROJECT 2.1. Van der Waals Fores Steri Fores Lennard-Jones Potential... 8 CHAPTER SIMULATION DEVELOPMENT 3.1. Moleular Dynamis Periodi Boundary Conditions Random Number Generation Initial Conditions Langevin Equation Inorporating the Langevin Equation into a Moleular Dynamis Routine Determination of Best Time Step Simulation Compared to Equipartition Theorem Random Walk Motion... 25

7 3.7. Euler Method Used for Stohasti Dynamis Effets of Gravity on Colloidal Partiles CHAPTER APPLICATION OF MOLECULAR DYNAMICS SIMULATION TO HEMATITE NANOPARTICLE DEPOSITION ONTO A SUBSTRATE 4.1. Introdution Deposition Time Deposition Rate Pattern Formation CHAPTER SUMMARY, FUTURE WORK, AND CONCLUDING REMARKS 5.1. Summary Future Work Improve the Fore Model Develop a More Quantitative Pattern Analysis Tehnique Simulate a More Realisti Substrate Vary Parameters to Determine Effet on Deposition Pattern Optimize the Fortran algorithm Conluding Remarks APPENDIX EQUATIONS FOR VAN DER WAALS ENERGIES AND FORCES APPENDIX REPRESENTATIVE MAGNITUDES OF FORCES USED IN THIS STUDY vii

8 APPENDIX NUMERICAL INTEGRATION SCHEME DESCRIPTIONS A3.1. Euler Method A3.2. Verlet Algorithm A3.3. Leapfrog Algorithm A3.4. Veloity Verlet Algorithm A3.5. Veloity Verlet/Leapfrog Algorithm A3.6. Leap-Frog Algorithm for Stohasti Dynamis APPENDIX RANDOM NUMBER GENERATION A4.1. Uniform Random Number Generators A4.2. Gaussian Random Number Generators APPENDIX SIMULATION ALGORITHMS A5.1. Three-Dimensional Stohasti Dynamis Algorithm to Model Hematite Nanopartile Deposition onto a Substrate A5.2. Two-Dimensional Stohasti Dynamis Algorithm to Model Partile Motion on the Substrate APPENDIX ADDITIONAL FIGURES A6.1. Initial Partile Positions for Systems with Various Numbers of Partiles A6.2. Additional Figures Supporting Deposition Time and Rate Analysis A6.3. Additional Figures Supporting Pattern Formation Analysis viii

9 APPENDIX DATA SUPPORTING DEPOSITION TIME AND RATE ANALYSIS BIBLIOGRAPHY ix

10 CHAPTER 1 INTRODUCTION The goal of this projet was to simulate the deposition of hematite nanopartiles onto a silion dioxide substrate in order to ompare partile deposition times and rates for systems with different numbers of partiles, and to determine whether patterns were formed one partiles had settled onto the substrate surfae. Figure 1 shows a simplified depition of the system under study. The nanopartiles studied were oated with a surfatant and immersed in a hexane solution. A moleular dynamis Fortran program was developed to simulate the system under study, and deposition times and loations were reorded for all partiles as they settled onto the substrate surfae. In order to isolate the effets of the number of system partiles on the results, only partiles of one size and with one surfatant oating were onsidered. In addition, fores assumed to play a minor role in deposition rate and pattern formation suh as magneti interations between the partiles were ignored. The fores that were taken into aount inluded visous and flutuating fores, Van der Waals and steri fores between partiles, gravity, and the Van Figure 1. Visualization of system under study. Nanopartiles represented as spheres, and surfatant oating or ligand shell represented with urly strands.

11 2 Der Waals fore ating between eah partile and the substrate. Van der Waals and steri potentials between the partiles were simulated using a Lennard-Jones potential. Magneti iron oxide nanopartiles suh as hematite, magnetite, and maghemite have unique properties that an be applied to a variety of situations. (Teja and Koh) Hematite deposited onto a substrate has diret appliations as a atalyst in reations important to industry. For example, hematite is a relatively low-ost alternative that an be used in atalyti ombustion to remove volatile organi ompounds from industrial waste produts, and thus is important in proteting the environment. (Piasso, Quintilla and Pina) To plae iron oxide nanopartiles onto a substrate, some proesses form the nanopartiles in-situ as the substrate itself is made, but forming the nanopartiles separately and then depositing them onto the substrate as modeled in this study offers advantages in that there is more ontrol over partile size and omposition, and partiles tend to be more dispersed on the substrate. (Hondow and Fuller) Sine deposition of pre-formed partiles onto a substrate is a relatively new researh area, questions remain about the assemblage rate and formation of patterns on the substrate. This study takes some initial steps toward examining these questions. Results showed that mean deposition time for all system partiles to settle onto a substrate was longer for systems with greater numbers of partiles. However, the last few partiles to reah the substrate took onsiderably longer in larger systems, partially aounting for the higher mean deposition times. Deposition rate was examined by determining the time onstant orresponding to systems with various numbers of partiles. The time onstant was determined for all simulation runs by examining the average height above the substrate for all partiles over time, and fitting the data with an exponential urve. Comparison of the time onstant between systems with different numbers of partiles showed that the fall time was again longer for systems with greater numbers of partiles. However, the differene between systems with different numbers of partiles was not as pronouned as the result for mean deposition time. After partiles had deposited onto the substrate, qualitative and quantitative analyses showed that patterns did indeed form on the

12 3 substrate surfae. Patterns were better ordered or more orrelated for systems with larger numbers of partiles. The results indiate that when depositing hematite nanopartiles onto a substrate, a solution with a greater number of partiles should be hosen to attain the best partile distribution on the substrate in the optimum amount of time. The following hapters disuss the detailed proedures and results for this study. Chapter 2 presents an overview on olloids and the fores that at on a olloidal system. The hapter also disusses the Lennard-Jones fore used to model the Van der Waals and steri interpartile fores. Chapter 3 desribes development of the Fortran simulation program used in this study, to inlude various tests used to verify proper funtion of the algorithm. Chapter 4 presents the detailed results of the hematite nanopartile deposition simulations. Finally, Chapter 5 summarizes this study, and presents an overview of work to be aomplished in the future. The appendies provide additional supporting material and are referened where appliable throughout this report.

13 CHAPTER 2 BACKGROUND ON COLLOIDS AND COLLOIDAL FORCES SIMULATED FOR THIS PROJECT A olloid is a dispersion of small partiles in some medium. (Israelahvili p. xxix) Common everyday examples of olloids inlude paint, ink, and milk. The fous of this study was to examine hematite nanopartiles dispersed in hexane solution. While developing the Fortran simulation program used in this study, a dispersion of magnetite nanopartiles in blood was also examined. a In general, the term nanopartile is used to refer to partiles ranging in size from 1 to 100 nm. In this projet, a olloidal system of nanopartiles loated above a substrate was simulated as the partiles settled onto the substrate. In general, a olloid may be desribed as being either stable or unstable, depending upon whether the partiles in suspension are well dispersed or form lumps. If we look at the ommon everyday examples of olloids mentioned previously, we know that milk is generally stable beause the olloidal partiles do not settle out of solution. However, if rennet is added to milk, the olloidal partiles lump together to form a gel, whih an be ut into urds and then ompressed together to make the deliious produt alled heese. The reason the olloidal partiles lump when rennet is added is that the interpartile fores have been affeted so that the attrative fores between partiles are stronger than the repulsive fores. (Rennet Coagulation of Milk) In general, the degree of stability is affeted by the fores ating between dispersed partiles. (Valle-Delgado) In this study, if stable patterns were to form as the a Magnetite has biomedial appliations due to its bioompatibility with the human body. (Teja and Koh)

14 5 nanopartiles settled onto the substrate, a ertain balane had to be ahieved between attrative and repulsive fores that ated between partiles. Fores that an typially at on partiles in solution inlude Van der Waals fores, eletri double-layer fores, magneti fores, solvation fores, and steri fores. (Israelahvili p. 284) This study foused on Van der Waals fores between partiles, Van der Waals fores between eah partile and the substrate, and steri fores between partiles b. Eah fore is disussed in the following setions. This hapter onludes with a desription of how the Lennard-Jones fore was used to model the fores between partiles Van der Waals Fores Van der Waals fores ating between hematite partiles in hexane are attrative, as are the Van der Waals fores ating between the hematite nanopartiles and the silion dioxide substrate. A general form for both Van der Waals energy and fore is provided by Hamaker (1937). Appendix 1 shows a derivation of the form of Hamaker s equation used in the Fortran algorithm of this study. The energy between a spherial partile and a flat substrate is given by V Vps = A { r z r + r z+r z r + ln }, (Energy between partile and substrate) ( 1 ) z+r where A 132 is the Hamaker onstant between partile and substrate, r is the radius of the partile, and z is the distane from the substrate to the enter of the partile. The fore on the partile due to the substrate an be found by differentiating and is F Vps = A { 1 r 1 z r (z r) 2 (z + r) r (z + r) 2}. (Fore between partile and substrate) ( 2 ) The energy between two spherial partiles of the same size is given by V Vpp = A { 2r 2 D 2 4r 2 + 2r2 D 2 + ln D2 4r 2 D 2 }, (Energy between two partiles) ( 3 ) b Gravity and Drag fores were also modeled by the moleular dynamis algorithm. Beause these fores are not speifi to olloidal systems, they are disussed in Chapter 3.

15 6 where A 131 is the Hamaker onstant between two partiles, and D refers to the distane between the enters of two partiles. By differentiating, we find the fore between the two partiles to be F Vpp = A { 2D D 2 +4r 2 4r 2 D 2 4r2 (D 2 +4r 2 ) 2 D D3 }. (Fore between two partiles) ( 4 ) In the Lifshitz Theory of Van der Waals fores, Hamaker s onstant A for objets interating aross a medium is alulated using bulk properties of the materials involved, suh as dieletri onstants and refrative indies. (Israelahvili pp ) The value of Hamaker s onstant between the two spherial hematite partiles in hexane solution denoted A 131 differs from that between a hematite partile and the silion dioxide substrate in hexane solution denoted A 132. Values of Hamaker s onstant alulated using the Lifshitz theory an be found in the literature and are given by (Faure, Salazar-Alvarez and Bergstrom) A 131 = Joules = 0.18 ev, and ( 5 ) A 132 = Joules = ev. ( 6 ) 2.2. Steri Fores While Van der Waals fores attrat partiles together, steri fores push the partiles apart. Steri fores arise due to the surfatant oating on the partiles and serve to stabilize the olloidal system by preventing the partiles from oagulating. As depited in Figure 2, the surfatant oating is made up of many polymer strands known as ligands that are attahed to the hematite partile surfae. When two polymer-oated partiles approah eah other, a repulsive fore results as the ligand shells begin to overlap. (Israelahvili p. 387) The steri potential for two spherial partiles an be alulated using V Spp = 100rL 3 (D 2r)πh π(d 2r) 3 ktexp ( ), ( 7 ) L where L is the thikness of the ligand shell, h is the distane between two ligand headgroups on the surfae of the partile, k is Boltzmann s onstant, and T is the temperature of the solution.

16 7 (Weddemann, Ennen and Regtmeier p. 82) The fore between partiles aused by the steri potential an be found by taking the derivative of the steri potential. The result is F Spp = V D Equations (7) and (8) = 100rL2 kt[l+π(d 2r)] (D 2r) 2 πh 3 exp ( π(d 2r) ). ( 8 ) L demonstrate several noteworthy features of the steri potential and fore. First of all, note that when the distane between enters is equal to twie the radius of the nanopartile in other words when the nanopartiles at the enters of the ligand shells are touhing the values of both the potential Figure 2. Cross setion of two spherial nanopartiles oated in a polymer ligand shell. When partiles approah eah other, a repulsive fore results as the ligand shells begin to overlap. and fore equations tend to infinity. We would expet, therefore, that the nanopartiles should never touh, although it is possible for the ligand shells to touh. Note that the fore will always be repulsive, but due to the exponential deay term, the repulsive fore is strongest when the ligand shells are lose to eah other. The repulsive fores that result when ligand shells approah depend on the variables L and h. If we assume two partiles have a given ligand shell thikness and we fix their separation distane, we see that higher repulsive fores result for smaller values of h, or for headgroups that are plaed loser together. This makes sense beause when h is smaller, there are more polymer strands that separate the two partiles. Similarly, for a given headgroup separation and partiles held fixed at a given distane, we see that higher repulsive fores result for larger

17 8 values of L, or a thiker ligand shell. Thus, in general, higher repulsive fores result for higher values of L and lower values of h. Surfatant oatings an be reated with a wide range of values for L and h. In this study, a ligand thikness of L = 5 nm and a distane between headgroups of h = 5 nm were hosen. In future studies, values for L and h an be varied to determine the effets of surfatant oatings on settling time and formation of patterns on the substrate surfae, but this study was limited to a single hoie of ligand shell thikness and distane between headgroups in order to study only the role of system partile number on deposition rate and formation of patterns on the substrate Lennard-Jones Potential The Lennard-Jones potential (Israelahvili p. 136), V LJ = 4ε [ ( σ D )12 ( σ D )6 ] ( 9 ) an be used to model the summation of the Van der Waals attrative potential and the steri repulsive potential between two partiles. Here, D represents the distane between enters of the partiles, ε represents the magnitude of minimum energy, and σ represents the distane D where the potential is equal to zero. The fore between partiles orresponding to the Lennard-Jones potential an be found by taking the derivative of equation (9), whih is F LJ = V D = D 24ε D 2 (σ D )6 [1 2 ( σ D )6 ]. ( 10 ) In equation (10), the fore will be negative (repulsive) when D is less than 2 1/6 σ, and will be positive (attrative) where D is greater than 2 1/6 σ. For a given partile, the diretion in whih the fore ats is determined by the vetor D. This an most easily be seen by imagining the onedimensional ase of a fore ating between two partiles in the x diretion only. Imagine that one partile is loated to the right of another partile, and we wish to onsider the diretion of the fore

18 9 aused by the rightmost partile on the leftmost partile. In this ase, the vetor D will be point to the right. If the value of the fore alulated by equation (10) is repulsive (quantity in brakets negative), then the fore on the leftmost partile ats to the left. In ontrast, if the value of the fore alulated by equation (10) is attrative (quantity in brakets positive), then the fore on the leftmost partile ats to the right. In ontrast, if we onsider the fore ating on the rightmost partile, we see that the vetor D will point to the left, and one an learly see that a repulsive interpartile fore will at to the right, while an attrative interpartile fore will at to the left. Thus, Newton s third law is evident when we losely examine the Lennard-Jones fore. The sum of the Van der Waals and steri potentials between partiles was plotted in Mathematia and is shown in Figure 3 (dashed line) for a partile radius r = 10 nm, ligand shell thikness L = 5 nm, and distane between ligand headgroups h = 5 nm. The magnitude of minimum energy and the value of D where potential was equal to zero were read from the graph. These values were equated to ε and σ, respetively, to determine the best fit Lennard-Jones potential and are ε = Joules, ( 11 ) σ = m. ( 12 ) Figure 3 ompares a plot of the summation of the Van der Waals potential and steri potential (dashed line) to the best fit Lennard-Jones potential (solid line). The value of distane at zero potential energy and the value of minimum potential energy are the same on the two plots sine these are the points that were mathed in the urve fit. However, the distane between partile enters where the minimum potential ours differs slightly between the two urves, indiating a slight differene between where the two potentials will transition between repulsive and attrative fores. In addition, at distanes less than the minimum potential, the urve of the summation of the Van der Waals and steri potentials is steeper than that of the Lennard-Jones potential. The same an be said of the slopes between minimum potential and up to a distane of approximately 40 nm.

19 10 Sine the fore is proportional to the slope, this indiates that the fores given by the Lennard-Jones model are slightly less than those of the Van der Waals and Steri fores in these regions. In partiular, if we alulate the fores for the lose distane of 21 nm, we note a marked differene between the magnitude of the Lennard-Jones fore, on the order of N, and the sum of the Van der Waals and steri fores, on the order of 10-9 N. This differene will be important when we look at the formation of patterns on the substrate in Chapter 4. The Lennard-Jones potential depited in Figure 3 was used to model the Van der Waals and steri potentials between partiles in this projet. Although the Lennard-Jones model is not perfet, the fore requires fewer omputations, whih allowed the omputer program to run more quikly, and that in turn allowed more simulations to be ompleted. Figure 3. Comparison of Lennard-Jones potential to sum of Van der Waals and steri potentials. The partiles are 20 nm in diameter and have a 5 nm oating of ligands. See Appendix 2 for representative magnitudes of fores appliable to this study.

20 CHAPTER 3 SIMULATION DEVELOPMENT The Fortran moleular dynamis program was developed using an inremental approah. Following seletion of the ore routine and development of the initial algorithm, the various fores ating on the partiles were inorporated one-by-one, and the results were heked to ensure that the fores had been orretly implemented. At times, subroutines were developed separately so that speifi outputs ould be isolated and verified for auray. The following setions desribe eah of the major deision points in program development, and then present important heks performed to verify suess at eah stage of development Moleular Dynamis A moleular dynamis algorithm was hosen as the ore algorithm in this study due to its ability to model behavior over a speified time period. The Monte Carlo method is another simulation routine that an be employed to model partile motion in a liquid. One advantage of the Monte Carlo tehnique is that it is omputationally less intensive, so simulations an be done more quikly. However, while the Monte Carlo method is useful for studying the equilibrium properties of a system, it is not well suited for answering questions related to the time evolution of a system. (Giordano and Nakanishi p. 270) For this study, we wished to simulate the amount of time required for partiles to deposit onto a substrate. Moleular dynamis is more appropriate for investigating suh questions. In its most basi form, a moleular dynamis algorithm applies Newton s seond law to eah partile in a given system, alulating fores ating on eah partile as funtions of the

21 12 positions and veloities of the partiles at a given point in time. Then, the aeleration of eah partile is alulated from the fores. In a moleular dynamis routine, time is disretized into a time step. From the aeleration at one point in time, the position and veloity of a given partile an be alulated at the next time step. The routine is repeated over a great number of time steps to model the time evolution of a system. In more preise mathematial form, the moleular dynamis algorithm is used to solve x (t) = 1 F [x (t), x (t), t], ( 13 ) m a seond order differential equation whih may be broken into six oupled first order differential equations, namely, x (t) = v (t), and ( 14 ) v (t) = 1 F [x (t), v (t), t]. ( 15 ) m In these equations, x refers to the position, v the veloity, t the time, m the mass, and F the fore ating on a partile. Note in equations (13) and (15) that the fore F is represented as a funtion of position, veloity, and time. In order to solve the two oupled first order differential equations, a numerial integration sheme must be hosen whereby the equations are disretized. Many numerial integration shemes are available, but some are better suited for moleular dynamis routines. Two desirable qualities for an integration sheme are that it allows seletion of a relatively long time step, and that it onserves energy and be time reversible. (Allen and Tildesley p. 76) Seletion of a relatively long time step allows modeling of a longer period of real time at less ost in terms of omputational resoures. In general, we wish to hoose the longest time step possible while still retaining realisti output values for position, veloity, and energy. Appendix 3 provides more detailed desriptions of the integration methods disussed in the following paragraphs. For eah integration method, at time t = 0 the partiles are imparted with some initial positions and veloities, as desribed in Setion 3.4.

22 13 The Euler method is one of the simplest integration shemes. Given the positions and veloities at the urrent time step n, the positions and veloities at the next time step n+1 are given by x n+1 = x n + v n t, and ( 16 ) v n+1 = v n + 1 m F [x n, v n, t] t = v n + a n t. ( 17 ) Here, Δt refers to the time step, while the variable a represents the aeleration, alulated by dividing the fore by the mass. A disadvantage of the Euler method is that errors aumulate during eah step that are of order (Δt) 2, and the final error in our solution, aumulated over all time steps, is of order Δt. (Giordano and Nakanishi p. 457) Thus, exeedingly small time steps would have to be hosen in order for the Euler method to yield aeptable output values for energy. However, sine moleular dynamis routines must be exeuted over a large number of time steps to simulate the system under study, the Euler routine is not as effiient as other methods. A method more suited to moleular dynamis routines is the Veloity Verlet/Leapfrog algorithm (Strobl and Bannerman p. 59), given by x n+1 = x n + v n t + a n 2 ( t)2, ( 18 ) v n+ 1 2 = v n + a t n, and ( 19 ) 2 v n+1 = v n a n+1 t. ( 20 ) 2 Note that this method inludes a half-time step, denoted by the subsript n+½. After the position at the next time step and the veloity at the half-time step are alulated, the fores are realulated to generate the aeleration at the next time step. From this, the veloity at the next time step an be found using the half-step veloity and the next time step aeleration. The Veloity Verlet/Leapfrog algorithm is an improvement to Verlet s original method (Verlet) that allows alulation of the veloity at the same time step as the position, whih in turn permits easier alulation of the energies at a given time step. In ontrast, the basi Verlet algorithm

23 14 alls for only position steps to be alulated, using the algorithm (Allen and Tildesley p. 78) x n+1 = 2x n x n 1 + a n Δt 2. ( 21 ) If veloity is required, it must be alulated from stored values of position, suh as v n = x n+1 x n 1, ( 22 ) 2Δt an equation subjet to errors two orders of magnitude higher than the equation for positions. (Allen and Tildesley p. 78). The leapfrog algorithm was proposed to alulate veloities at every halfstep, with errors of the same order of magnitude as the positions, and may be employed using (Allen and Tildesley p. 80) x n+1 = x n + v 1 n+ Δt, and ( 23 ) 2 v 1 n+ = v 1 n + a n Δt. ( 24 ) 2 2 The leapfrog algorithm is so named beause we leap over the oordinates when we alulate the next value of veloity at the half time step. One minor drawbak to the leapfrog method is that if we wish to alulate kineti energy at an integer value of time step, we must perform the additional alulation v n = 1 (v 2 n+ 1 + v 1 n ). ( 25 ) 2 2 A benefit of the Veloity Verlet/Leapfrog algorithm is that positions and veloities are both alulated at integer values of time step. The Veloity Verlet/Leapfrog algorithm was used early in this projet when only Lennard-Jones fores were ating between partiles. Energy was onserved and animated plots of the partiles showed that they behaved as expeted under the influene of a Lennard-Jones fore. Speifially, partiles appeared to be repelled from eah other when they were within the value of 2 1/6 σ, but attrated to eah other from larger distanes. Unfortunately, the Veloity Verlet/Leapfrog algorithm in the form given by equations (18), (19), and (20) is not well-suited to a system where drag and flutuating fores are ating. In this ase, some fores are funtions of veloity, and we annot alulate these veloity-dependent fores (and

24 15 thus annot alulate aelerations) at the new time step using the half-step veloity rather, the new time step veloity, whih is unknown, would be required to alulate the veloity-dependent fores. Thus, an alternative leapfrog method developed by Van Gunsteren and Berendsen was used to inorporate frition and flutuating fores into the algorithm. This method is disussed in detail in Appendix 3 and is summarized later in this hapter. First, we take a brief interlude into boundary onditions, random number generation, and initial onditions, as well as a desription of the Langevin equation Periodi Boundary Conditions The number of partiles simulated in a olloidal system may be pratially limited by onsiderations suh as the omputer proessor speed and the time required to run a simulation. For example, in this study, the largest olloidal system studied onsisted of 100 partiles, and it took almost a full day to run a simulation on a laptop omputer. Sine resoures limit the number of partiles we an simulate, we should hoose a relatively small volume to ontain those partiles so that they an interat with one another. In this simulation, the 20 nm diameter partiles were ontained within a ubi box with 300 nm side lengths. One problem with suh a small box is that the partiles will hit the walls of the box frequently, so surfae effets suh as rebounding off the walls of the box will ome into play. These surfae effets will affet the results of the simulation. In addition, if we want to ahieve a statistially signifiant outome then we should study a system of infinite size. Although resoures may limit the size of the system simulated, periodi boundary onditions may be employed to extend the results of a simulation to an infinite system. (Allen and Tildesley pp ) To employ periodi boundary onditions, we imagine an infinite number of idential boxes lined up in all diretions next to the box under study. Figure 4 illustrates a twodimensional representation of suh periodi boundary onditions by using nine boxes. The ative

25 16 box or, the box we simulate using moleular dynamis is in the enter, and the eight boxes surrounding it represent eight of the image boxes. Eah box ontains the same number of partiles, loated in equivalent loations and with the same veloities within their respetive boxes. In the ative box, when a partile rosses a boundary of the box on one side, then a partile will also Figure 4. Illustration of two dimensional periodi boundary onditions. Eight image boxes surround the ative box in the enter. As a partile leaves one side of the ative box for example, the right another partile enters from the opposite side. In alulating fores between partiles, the fore should be alulated using the losest image, whih may not be in the ative box. ross on the opposite side. For example, as depited in Figure 4, if a partile exits the box on the right, a partile will also enter the box from the adjaent box to the left. Thus, any time a partile enters or exits one side of the box, a orresponding partile will exit or enter the box from a orresponding position on the opposite side. These so-alled periodi boundary onditions allow us to simulate an infinite system by modeling only one box in the infinite series of boxes. For a fluid subjeted to Lennard-Jones fores, a ubi box of size 6σ should be large enough that a given partile annot sense its own images in adjaent boxes. (Allen and Tildesley p. 25) Reall that σ is a parameter used in the Lennard-Jones fore and is equal to about 30 nm for this study. When using periodi boundary onditions, the Lennard-Jones fore and potential between one partile and another should be alulated only between the losest images. Thus, if we onsider a partile on the lower left side of the ative box and would like to know the potential generated

26 17 between this partile and a partile at the upper right of the simulation box, then we must onsider the potential between that partile and the losest image partile, as depited in Figure 4. For the olloidal system studied in this projet, we plaed a substrate at z = 0. Thus, periodi boundary onditions were implemented only in the x and y diretions. In the vertial diretion, one a given partile reahed the substrate, the value of z was held onstant. In addition, partiles were initially plaed no higher than 50 nm below the top of the box. However, if random fores or interpartile fores aused the partile to rise above the top of the box, then the partile height was readjusted so that it was at the top of the box. Figure 5 illustrates periodi boundary onditions with a substrate simulated at z = 0. Figure 5. Periodi boundary onditions with a substrate at z = 0. When a partile hits the substrate, the z value is fixed. Partiles an limb no higher than z = 300 nm Random Number Generation. Prodution of random numbers is vital to the suess of a stohasti dynamis algorithm. True random number generators must rely upon some unpreditable physial proess, suh as rolling die or flipping oins, to produe random numbers. (Katzgraber) However, suh proesses are inonvenient for omputer algorithms that must produe a large number of random numbers in a short period of time. Thus, most omputer simulations rely upon pseudo random number generators, whih are desribed in some detail in Appendix 4.

27 18 This projet required both uniform random numbers falling on the interval from 0 to 1, and random numbers falling on a Gaussian distribution of zero mean and some speified standard deviation. Uniform random numbers were used in assigning initial positions and veloities to partiles in those ases where the initial onditions were assigned randomly. The uniform random numbers were also required in the routine used to produe Gaussian random numbers. Gaussian random numbers were then used to generate values for the random flutuating fore, desribed in Setion 3.5. The pseudo random number generator was initialized using a negative integer as a random seed. In pseudo random number generators, when the same initializing seed is used, the same sequene of random numbers is produed. Thus, the same seed was only used while debugging the software algorithm. For the simulations desribed in Chapter 4, different random seeds were used for eah simulation run to ensure statistially signifiant results ould be attained Initial Conditions Prior to initiating a moleular dynamis simulation, the partiles must be imparted with some initial positions and veloities. Initial positions may be assigned randomly, or they may be assigned in an ordered pattern. Random initial positions were used for the results reported in this hapter. In assigning initial positions, an algorithm was implemented to ensure that partiles were not so lose that unrealistially large fores resulted. Speifially, random initial positions were reassigned if partile enters were loser than the value of σ used in the Lennard-Jones fore, whih ould result in extremely large repulsive fores. For the results reported in hapter 4, partiles were assigned to an ordered array of initial positions, whih is desribed thoroughly in the next hapter. Initial speeds were imparted randomly to the partiles in eah of the three dimensions and varied between + 1 m/se, whih is lose to the root mean square veloity that would be attained applying the equipartition theorem to the three translational degrees of freedom of the system.

28 19 When the stohasti dynamis algorithm desribed later in this hapter was implemented for flutuating and visous fores, it was found that the veloities reahed equilibrium values in agreement with the equipartition theorem within a time step, so these initial veloities were quite reasonable Langevin Equation Beause the nanopartiles were interating through a fluid at finite temperature, visous fores and flutuating fores had to be taken into aount. Both fores arise from impats with the moleules of the fluid and an be desribed using the Langevin Equation. (Lemons and Gythiel) The flutuating fores are present even when the system has reahed thermal equilibrium, and result from random thermal motion of the fluid moleules buffeting the olloidal partiles. The drag fore an be understood by imagining that some external fore for example, eletromagneti or gravitational is applied to the partiles. Then, the partiles will tend to move in a given diretion due to this external fore, but motion will be impeded in that diretion due to impats with the moleules of the fluid. Thus, the drag fore is proportional to the veloity. (Kubo) The Langevin equation for a partile ated upon only by visous and flutuating fores is (Kubo) mv (t) = mγv + R (t), ( 26 ) where the first term on the right-hand side represents the visous or drag fore that is linear in veloity, and the seond term R(t) represents the flutuating fore. The value of the drag oeffiient an be found from Stoke s formula (Pathria and Beale p. 593), mγ = 6πμr, ( 27 ) where μ is the visosity of the fluid and r is the hydrodynami radius of the partile.

29 20 The random fore is usually assumed to fall on a random Gaussian distribution, satisfying the onditions (Kubo) R (t) = 0, and ( 28 ) R (t)r (t + τ) = g(τ), ( 29 ) where g(τ) is the autoorrelation funtion of the random fore. The autoorrelation funtion is an even funtion of τ that dereases over a harateristi time alled the orrelation time. The autoorrelation funtion is related to the diffusion onstant D through an integration over all time (Pottier p. 237) g(τ)dτ = 2Dm 2. ( 30 ) The orrelation time is of the same order of magnitude as the time period between ollisions of fluid moleules with the olloidal partiles. Assuming the orrelation time is short ompared to other harateristi times of the system, suh as the time sale on whih Brownian motion ours, the autoorrelation funtion an be represented by a delta funtion δ(τ), aording to g(τ) = 2Dm 2 δ(τ). ( 31 ) In addition, Kubo has shown that the diffusion onstant is related to the visosity of a fluid through (Kubo) D = ktγ m, ( 32 ) where k is Boltzmann s onstant and T is the temperature of the system. Substituting equations (31) and (32) into equation (29), we find R (t)r (t + τ) = 2mγkTδ(τ), ( 33 ) an alternative expression for the orrelation of the random fore. Equation (33) tells us that the random fores at two different times are unorrelated. Sine the average value of the random fore is zero, we an relate this funtion to the variane of the random fore σ R as σ R 2 = R 2 R 2 = R 2 0 = R (t)r (t + τ) = 2mγkTδ(τ). ( 34 )

30 21 Equation (34) relates the variane of the random fore to the delta funtion, an infinitely high funtion of infinitesimal width. Whereas the delta funtion an be integrated over a ontinuous integral as δ(τ)dτ = 1, ( 35 ) a moleular dynamis routine requires a finite element version of the delta funtion. As desribed in Setion 3.1, numerial integration involves breaking an integral into time steps of finite but extremely small duration. Here, we wish to disretize the delta funtion suh that its integral still enloses an area of one. We an do so by estimating the delta funtion with a step funtion that is of width Δt and of height 1/Δt, as depited in Figure (6). Thus, the variane used for the random fore in a moleular dynamis algorithm, whih uses numerial integration, is given by σ 2 R = 2mγkT, ( 36 ) Δt where mγ is given by Stoke s formula, equation (27). Sine we know the mean of the random fore is zero and the variane is given by equation (36), we an selet values of the random fore at eah time step in a moleular dynamis algorithm by sampling from a Gaussian distribution or by generating random numbers on a Gaussian distribution as desribed in Appendix 4. Figure 6. A step funtion is used to estimate the delta funtion in the numerial integration routine used in moleular dynamis. Here, the delta funtion is depited as an infinitely high arrow of infinitesimal width. The step funtion is of width equal to the time step and height equal to the inverse of the time step. The time step in this diagram is equal to 0.1 ns.

31 Inorporating the Langevin Equation into a Moleular Dynamis Routine Modeling the speifi dynamis of the liquid solvent moleules in a olloidal solution would be omputationally intensive, requiring simulation of a large number of moleules ating over extremely short time steps, sine ollisions between liquid moleules take plae on a muh shorter time sale than that of Brownian motion. Fortunately, we do not need to know the speifi motion of the solvent moleules, but an aount for the effets of their motion by inorporating the visous and flutuating fores from Langevin s equation into Newton s seond law for the olloidal solute partiles, and then solving that equation numerially. (Van Gunsteren and Berendsen) Thus, for eah partile, we must solve the oupled equations x (t) = v (t), and ( 37 ) mv (t) = F [x (t)] mγv + R (t), ( 38 ) where F is the system fore on the partile due to various interations, suh as the Lennard-Jones interation between partiles, or gravity ating in the vertial diretion. An algorithm developed by Van Gunsteren and Berendsen (Van Gunsteren and Berendsen), desribed in detail in Appendix 3, was used to numerially integrate equations (37) and (38). The algorithm is similar to the leapfrog algorithm presented as equations (23) and (24) in that position is updated at eah integer value of time step, whereas veloity is updated at half-step values. Speifially, to update position and veloity, we use x n+1 = x n + v n+ 1 v n+ 1 2 Δt 2 (γδt) γδt γδt [exp (+ ) exp ( 2 2 )] + X n+ 1 ( Δt 2 2 ) X n+ 1 2 ( Δt 2 ) + O[(Δt) 3 ], and ( 39 ) = v 1 n exp( γδt) + 2 F n Δt [1 exp( γδt)] + V m(γδt) n ( Δt ) 2 exp( γδt) V n ( Δt 2 ) + O[(Δt)3 ], ( 40 ) where γ is alulated using equation (27), and where X 1 n+ ( Δt 2 2 ), X n+ 1 2 ( Δt 2 ), V n ( Δt 2 ), and

32 23 V n ( Δt 2 ) eah represent different integrals of the random fore R (t) multiplied by various quantities evaluated over ertain integrals of time. The random fore integrals are evaluated by sampling Gaussian random numbers from distributions of zero mean and speified width, as desribed in detail in Appendix 3. Equations (39) and (40) are similar in form to the equations for the leapfrog algorithm in equations (23) and (24). In fat, in the limit of infinitesimal frition oeffiient γ, the Van Gunsteren/Berendsen method redues to the leapfrog algorithm. This an be verified by substituting series expansions for the exponential terms into equations (39) and (40) and then simplifying the equations. Also like the leapfrog algorithm, note that the errors in position and veloity for eah time step are of order (Δt) 3. In general, algorithms of even higher order would be impratial for stohasti dynamis routines sine the gains made in auray due to the higher order algorithm would be offset due to the unertainties arising from the ation of the random fore R(t). (Van Gunsteren and Berendsen) On the other hand, we have seen that lower order algorithms suh as the Euler routine are not effiient for moleular dynamis simulations. Thus, the Van Gunsteren/Berendsen algorithm presents an appropriate method for modeling the olloidal system. After inorporating the Van Gunsteren/Berendsen algorithm into the Fortran simulation program, several tests were performed to ensure that the method produed reasonable results. Table 1 provides the properties of the nanopartile system used for these heks, whih are desribed in the remainder of this setion Determination of Best Time Step When initially implementing the Van Gunsteren/Berendsen algorithm, the optimum time step had to be determined. As mentioned previously, a larger time step allows simulation of a larger period of real time over a shorter period of omputational time. However, if the time step is too large, the algorithm will provide unrealisti values for position, veloity, and energy. Thus, time step was varied from 1 x to 1 x 10-8 seonds during separate simulation runs for a system

33 24 Table 1. Physial properties of nanopartile system. Physial Property Value Density of Magnetite kg/m 3 Visosity of Blood x 10-3 Pa-s Radius of Hematite Nanopartile 10 nm Hydrodynami Radius 3 10 nm Temperature (Human Body Temperature) 310K kt x J Lennard-Jones Fore ε 1.00 x J Parameters σ 30.0 nm 1 (Magnetite) 2 (Dynami Visosity of Some Common Liquids) 3 Assumes no ligand shell attahed. 4 Boltzmann s onstant times temperature of 10 partiles and a real time period of 10 million ns, or 10 ms, and output parameters for eah data run were ompared. The Lennard-Jones fore was not ative during these data runs, so the only fores ating on the partiles were visous and flutuating fores. Output parameters inluded average kineti energy of the system, as well as positions of all the partiles. The average kineti energy was ompared to the equipartition theorem. For a system whih has three translational degrees of freedom, like the one under study, the equipartition theorem says that the average kineti energy K is K = 3 NkT, ( 41 ) 2 where N refers to the number of partiles simulated in the box, k is Boltzmann s onstant, and T is the system temperature. Average values of kineti energy mathed well with the equipartition theorem for all trial values of time step. However, for time steps of 1 x 10-9 and 1 x 10-8 seonds, the values for position eventually blew up to exessively large numbers, indiating that these time steps were too large to be used to simulate the system. Ultimately, the best hoie of time step was found to be 1 x seonds, sine it gave energies onsistent with the equipartition theorem, but values for position did not grow exessively over time.

34 Simulation Compared to Equipartition Theorem After determining the optimum time step, the temperature of the system was varied and the average kineti energy ompared to the equipartition theorem. No system fores were ating on the partiles. Results appear in Table 2 and are in exellent agreement with the equipartition theorem. For example, simulations at 310K differ from the expeted equipartition value by less than 0.1%. Table 2. Average kineti energy ompared to equipartition theorem at four temperatures. Temperature (K) Average Kineti Energy Over Time (Joules) Perent Differene Between Simulation and Duration of data run (nse) Equipartition Simulation 2 Equipartition Theorem Theorem % 1 million x x % 1, x x % 1 million x x % 10,000 1 Simulations were all run for 10 partiles, using time steps of 1 x se. 2 Average kineti energy for the simulation was found by averaging the kineti energy for all 10 partiles over all time steps where data was reorded. No system fores were ating on the partiles Random Walk Motion If the movement generated by the Van Gunsteren/Berendsen routine is truly random, then the motion of the partiles under the influene of solely the visous and flutuating fores (i.e. no system fores suh as Lennard-Jones fores) should follow a random walk. Two methods were devised to hek the random motion of the system partiles. First, the mean square displaement of the partiles was alulated eah time that data snapshots were taken. Diffusion oeffiient an be alulated using both the mean square displaement and the physial properties of the system. Agreement between these two alulation methods would indiate that the algorithm is operating orretly. Seondly, the motion of a single partile ated upon by visous and flutuating fores was plotted in 3 dimensions.

35 26 For spherial partiles, the Diffusion oeffiient an be alulated by using Einstein s relation (Pathria and Beale p. 595) D = kt = ( x )(310K) m2 = x 6πμr 6π(3 x 10 3 )(10 x ) s ( 42 ) where k is Boltzmann s onstant, T is the temperature of the system, μ is the visosity of the fluid, and r is the hydrodynami radius of the partile. Substituting values for these quantities, the diffusion oeffiient for the nanopartile system is found to be x m 2 /s. In three dimensions, the relationship between the diffusion oeffiient D and the mean square displaement r N 2 is given by (Pathria and Beale p. 593) or, solving for D, r N 2 = 6Dt, ( 43 ) D = r N 2 6t, ( 44 ) where t represents the time. Mean square displaement for N partiles at a speifi time t an be alulated using r 2 N = N (x i (t) x i (0))2 +(y i (t) y i (0)) 2 +(z i (t) z i (0)) 2 i=1, ( 45 ) N where x i(t), y i(t), and z i(t) represent the total displaement of the partiles at time t, while x i(0), y i(0), and z i(0) represent the starting positions of the partiles. The Van Gunsteren/Berendsen algorithm was tested by simulating a system of 10 partiles over a time period of 10 million ns (10 ms), or 100,000,000 yles, with data taken every 10,000 yles. Figure 7 shows a plot of the variation in mean square displaement over time. A linear urve fit to the data, found using Mathematia, is given by r N 2 = t ( 46 ) The urve fit is good with an R-squared value of Sine the slope of this urve is r N 2 /t, we an divide the slope by six to find a value of diffusion oeffiient of D = x m 2 /s, whih differs from the value alulated using Einstein s relation by 5.09%.

36 27 Figure 7. Mean square displaement for a 10 partile system ated upon by visous and flutuating fores. The algorithm also alulated values of diffusion oeffiient using equation (44) eah time a data snapshot was taken. Averaging these values of diffusion oeffiient over all snapshots, we find an average value of D = x m 2 /s, whih differs from the Einstein relation value by 1.69%. The agreement of the diffusion oeffiient alulated from data generated by the Van Figure 8. Random walk movement of one partile over time due to visous and flutuating fores. Data taken over 100,000 ns in 10 ns intervals.

37 28 Gunsteren/Berendsen routine and the Einstein relation indiates that the algorithm is working orretly. The seond method used to determine whether partiles simulated using the Van Gunsteren/Berendsen algorithm exhibited random walk behavior was to plot the motion of a single partile as modeled by the algorithm. Figures (8) and (9) show 10,000 representative points of motion of two separate partiles ated upon solely by visous and flutuating fores. In Figure (8), data snapshots were taken every 10 ns for a total of 100,000 ns, while in Figure (9) snapshots were taken every 100 ns for a total of 1 million ns. In both plots, the motion appears to flutuate within the box, rather than following some straight path. In some loations of Figure (8), we an see where the periodi boundary onditions have brought the partile from one side of the box (bottom) to the other (top). Although data points in Figure (9) are muh further apart, we an learly see that the partile moves Figure 9. Random walk movement of one partile over time due to visous and flutuating fores. Data taken over 1 million ns in 100 ns intervals. throughout the entire box.

38 Euler Method Used for Stohasti Dynamis For omparison, an Euler algorithm, using the Euler integration sheme given by equations (16) and (17), was developed and applied to the olloidal system under study. The aeleration in equation (17) an be alulated using a n = 1 [ F m n (x n ) mγv n + R n ], ( 47 ) where F represents system fores suh as the Lennard-Jones fore, γ is alulated using equation (27), and R is the random flutuating fore, determined by generating a random Gaussian number from a distribution of zero mean and width σ R, with σ R given by equation (36). A time step of 1 x se was required to attain aeptable output values of position, veloity, and energy using the Euler routine. Thus, if we onsider the time step alone, it would take approximately 100 times longer to simulate a olloidal system over a given real time period using the Euler routine in plae of the Van Gunsteren/Berendsen algorithm. d For this projet, a trial simulation was onduted for a system of 10 partiles and a time period of 10,000 ns (10 million yles), with data snapshots taken every ns (or every 1,000 yles). Average kineti energy for all partiles over time was 6.5 x Joules, differing by about 1.2% from the expeted equipartition value of 6.42 x Joules. A glane at Table 2 will verify that the Van Gunsteren/Berendsen algorithm produes results more in line with the expeted equipartition values, with a perent differene less than 0.1% at 310K. Although the Euler routine is a simple method that ould be used to model a olloidal system, the aompanying larger omputation times and lower auray make it muh less desirable than the Van Gunsteren/Berendsen algorithm for modeling olloidal systems. d Atually, sine the Euler routine is simpler than the Van Gunsteren/Berendsen algorithm, eah loop through the Euler routine will be slightly faster than eah loop through the Van Gunsteren/Berendsen algorithm. However, the omputation effiieny gains due to simpliity of the algorithm are not enough to make up for the additional omputation time required due to the smaller time step.

39 Effets of Gravity on Colloidal Partiles After the Van Gunsteren/Berendsen algorithm was suessfully implemented, one additional modifiation was made before using the program to simulate deposition of hematite nanopartiles onto a substrate. Namely, the fore of gravity was added to the program. Gravity was implemented as a system fore F ating in only the z diretion. The fore was added to the Lennard-Jones fore in the z diretion and was alulated using F G = mg, ( 48 ) where g = 9.81 m/s 2 is the aeleration due to gravity. The systems studied in this hapter onsisted of magnetite nanopartiles dispersed in blood, while the systems studied in hapter 4 onsisted of hematite nanopartiles in hexane. The gravitational fore ating on magnetite nanopartiles was 2.13 x N, while that for hematite nanopartiles was 2.16 x N. e The standard deviation of the random fore for magnetite in blood was 3.81 x N, while that for hematite in hexane was 8.31 x N. The mean of the random fore is zero, and we an expet the fore to flutuate quite a bit, but the standard deviation gives us an idea of a typial order of magnitude for the random fore. At times, the magnitude of the random fore will be muh greater than that of gravity. At times it will at in onert with gravity, and at times in opposition. Sine the gravitational fore ats only in the negative z diretion, we an expet the partiles to gradually settle onto the substrate. However, we an also expet that the partiles will be buffeted about by the random flutuating fore, so that their desent to the substrate will not follow a perfetly smooth urve. To verify that the gravitational fore works as expeted, the Van Gunsteren/Berendsen routine was exeuted for a system of 100 nanopartiles over a time period of 1 million ns, or 1 ms (10 million yles), with data snapshots taken every 1,000 yles. The program was set up with periodi boundary onditions eliminated in the z diretion to simulate a substrate, and physial e These alulated fores assume a partile radius of 10 nm.

40 31 parameters used inluded density of hematite (to alulate partile mass) and visosity of hexane (to alulate visous/flutuating fores). The Lennard-Jones fore did not at between partiles. As the program was initiated, partiles were imparted with random positions and veloities. Partiles were onsidered to have reahed the substrate at the value z = 15 nm, equal to the hydrodynami radius of the partile f. Figure 10 depits the average height of all partiles in this system versus time. As expeted, the partiles gradually approah the substrate due to the ation of gravity, but average height above the substrate flutuates due to the random fore. Output data show that all partiles had reahed the substrate after 467 μs. The Mathematia FindFit funtion was used to exponentially urve fit the data. Note that the oeffiient in front of t in the exponent is in units of μs -1 and is the inverse of the time onstant. Figure 10. Derease in z average for all partiles over time for a system of 100 partiles ated upon by gravity and visous and flutuating fores. f Note that z refers to the position of the enter of the partile, so the bottom edge of the partile is on the substrate when z = 15 nm.

41 CHAPTER 4 APPLICATION OF MOLECULAR DYNAMICS SIMULATION TO HEMATITE NANOPARTICLE DEPOSITION ONTO A SUBSTRATE The moleular dynamis algorithm desribed in the previous hapter was applied to hematite nanopartile deposition onto a silion dioxide substrate. Deposition times and substrate deposition patterns for systems with various numbers of partiles were reorded and analyzed. This hapter desribes the proedures, analysis, results, and onlusions of this appliation. The simulation algorithms used in this study an be found in Appendix Introdution Prior to onduting the hematite nanopartile simulations, the moleular dynamis algorithm was adjusted slightly for the speifi system under study. As with the investigation of gravitational fore in the previous hapter, periodi boundary onditions in the z diretion were modified to simulate a solid substrate at z = 0. In addition to gravity, a Van der Waals fore between eah partile and the substrate, given by equation (2), was added ating in the z diretion. The Lennard-Jones potential was adjusted to math as losely as possible the Van der Waals and steri interpartile fores desribed in hapter 1. A variable L was added to the program to represent the ligand shell thikness. The hydrodynami radius in the program was represented by the sum of the radius of the hematite nanopartile plus the ligand shell thikness, as shown in Table 3. Hydrodynami radius was used both to alulate the frition with Stoke s formula, and in determining when the partiles had reahed the substrate. After a partile s height above the substrate reahed a value less than or equal to 15 nm, the hydrodynami radius, the partile was

42 33 onsidered to have hit the substrate, and the height of the partile above the substrate was held fixed at a value of 15 nm. Finally, the density of hematite and visosity of hexane, as reported in Table 3, were used to alulate variables in the program suh as mass and frition oeffiient. Table 3. Physial properties of nanopartile system. Physial Property Value Density of Hematite kg/m 3 Visosity of Hexane x 10-4 Pa-s Radius of Hematite Nanopartile 10 nm Hydrodynami Radius 15 nm Temperature 298K kt x J Lennard-Jones Fore ε x J Parameters σ nm 1 (Hematite) 2 (Dynami Visosity of Some Common Liquids) 3 Boltzmann s onstant times temperature Below we will present results of deposition times, deposition rates, and pattern formation Deposition Time One goal of this study was to determine whether the number of partiles in a system influenes partile deposition time or rate onto a substrate. To minimize the influene of initial positions on deposition rate, partiles were initially plaed within an ordered array as depited in Figure 11 for 80 partiles. g The x and y values for the initial positions were spaed 50 nm apart at a height above the substrate of 150, 200, or 250 nm. No matter how many partiles were in the system, approximately one-third of all partiles were plaed at eah of the three heights above the substrate. The reason for this plaement is that partiles loser to the substrate should theoretially hit the substrate before partiles further away, both beause they have less distane to travel and beause the substrate-to-partile Van der Waals fore is stronger. If initial positions had been g Figures depiting initial positions for systems with 20 and 50 partiles are provided in Appendix 6.

43 34 randomly assigned instead, a system that had a larger perentage of partiles loser to the surfae ould have been predisposed to produing an unrealistially short partile deposition rate. More importantly, it would have been diffiult to ompare different simulations with the same number of partiles if eah had started with a different distribution of partiles above the substrate surfae. A partile spaing of approximately 50 nm between enters in all three dimensions was hosen to ensure that partiles were far enough apart that Lennard-Jones fores between partiles were not exessive. Thus, at high partile densities, partiles had to be plaed in three vertial layers. Although systems with fewer partiles ould have allowed plaement in fewer layers, the Figure 11. Initial positions for 80 partiles. The partiles are shown by balls and the olors are a guide to the eye to demonstrate the various heights in the z diretion. best way to ensure that partile height did not bias the results was to ensure that an approximately equal perentage of partiles was plaed in eah of the three vertial layers, no matter how many partiles omprised the system. Partile deposition times and rates were analyzed for systems with 20, 50, and 80 partiles. For eah system, 30 simulations were onduted, where deposition time for all partiles was reorded. After the partiles reahed the substrate, the z position was held fixed, but the x and y positions were allowed to vary due to random flutuations and interpartile fores. In other words, partiles were stuk to the surfae after striking but were able to diffuse in two dimensions on the substrate. For eah simulation run, average settling time for all partiles, standard deviation of the settling times, and times for the first and last partiles to hit the substrate were tabulated. Next, the

44 35 mean deposition time of all data runs at eah partile density (the mean of the means), along with the standard error of the means, were alulated. From this information, the onfidene intervals for mean deposition time in systems with 20, 50, and 80 partiles ould be determined. The results are shown in Figure 12, where the mean of the mean deposition or settling time is plotted as a funtion of the number of partiles h. The trend is that higher numbers of partiles in the system result in higher average times for all partiles to hit the substrate. We an Figure 12. Differene in partile mean deposition time for systems with 20, 50, and 80 partiles. Mean deposition times are written in text on the figure. Error bars denote standard errors. also see that the standard error is larger for systems with fewer partiles. Some analytial alulations will point to the major fators at work in produing these results. First of all, we an use Newton s seond law to alulate the time required for a single partile at zero initial veloity to fall to the substrate when ated upon only by gravity. By alulating this value for heights above the substrate of 250 nm, 200 nm, and 150 nm, and then averaging the results, we find a fall time of 193 μs. We an use the leapfrog numerial integration sheme to alulate the fall time for a partile ated upon by both gravity and the Van der Waals substrate fore. Averaging the results for all three heights above the substrate, we find a fall time of 47.2 μs. Sine the mean fall times shown in Figure 12 are all less than the fall time of gravity ating alone, we an dedue that both gravity and the Van der Waals substrate fore play signifiant roles in partile deposition. The fat that the mean times are greater than that found when both h Some of the raw data used to generate the Figures in this setion an be found in Appendix 7.

45 36 gravity and the Van der Waals fore at on the partiles indiates the interpartile fores affet deposition time. To understand the effets of interpartile fores on deposition time, we must onsider the spae available on the substrate and how many partiles will fit on the substrate surfae. If we try to ram too many partiles on the substrate, they will not all fit, and mean deposition time for all partiles goes to infinity! The diameter of the hematite nanopartile plus the ligand shell oating is 30 nm. Thus, we an fit 10 partiles aross one row that spans the entire length of the 300 nm long substrate. If we stagger the rows, we find that partiles fit on the 300 nm square substrate, as depited in Figure 13. Figure 13. Maximum number of partiles that fit on square substrate when all ligand shells are touhing. Cirles represent diameter of partile with ligand shell. When fewer partiles are on the surfae, we an alulate the perent overage of the substrate by looking at the ratio of the ross setional area of all the partiles on the substrate to the surfae area of the substrate itself, as depited in Figure 14 for 20 partiles i. We an surmise that as the number of partiles on the substrate inreases, the opportunities for interpartile interations to affet substrate deposition will inrease. In fat, in systems with large numbers of partiles, we an dedue that these interpartile interations on the substrate will influene the average time for all partiles to reah the substrate. As progressively more partiles i Similar figures for 50 and 80 partiles appear in Appendix 6.

46 37 are attahed to the substrate, interpartile interations will affet deposition of additional partiles more and more. We an imagine that falling partiles will be attrated to partiles already on the substrate via the Lennard- Jones fore when the partile enter-enter distane is beyond 2 1/6 σ ( 34.6 nm), but that these same partiles will beome repelled at distanes Figure 14. Calulation of perent overage for a substrate ontaining 20 partiles. Cirles represent diameter of partile with ligand shell. less than 2 1/6 σ. Thus, whether or not a given partile is deposited onto the surfae is determined by a vetor summation of several ompeting fores: interpartile Lennard-Jones fore; Van der Waals substrate fore; flutuating fores, and gravity. The interpartile Lennard-Jones fores will gradually ome to play a more important role in deposition rate as the number of partiles in a system inreases. Whereas with 20 partiles the perent overage is 15.7%, with 50 partiles it is 39.3%, and with 80 partiles, it rises to 62.8%, taking up over half of the area on the substrate surfae! If we further onsider the fat that the partiles experiene a repulsive fore within a enter-enter distane of 34.6 nm, we an understand that in systems with very high partile numbers, most available onfigurations will be suh that partiles are repelled by the other partiles that are losest to them. We an visualize this most easily by looking at Figure (15), a diagram where irles are sized at 34.6 nm, the distane where the Lennard-Jones fore of the system transitions between being attrative and repulsive. In other words, the red irles represent areas that, if overlapping, would result in repulsive fores between

47 38 partiles. The maximum number of partiles that an fit onto the substrate surfae without feeling any repulsive fore at all is 86.8 and the partiles must be plaed in the staggered depited. onfiguration Thus, for systems with partile numbers greater than 86.8, we surmise that the only way partiles an be deposited onto the substrate is for the sum of the Van der Waals substrate fore, Figure 15. Depition of maximum number of partiles that an fit on the substrate without any repulsive fores ating. Diameter of irles represents distane where Lennard-Jones fore will beome repulsive. If any red shells overlap, a repulsive fore results. flutuating fores, gravity, and attrative fores with nearby partiles to overome the repulsive fore between the losest partiles Deposition Rate Another way to analyze the vertial fall of the partiles is to use an exponential fall rate, rather than average time to hit the substrate. This may be espeially useful as the average time to hit has a great deal of variation due to the last partile taking a very large time to find a spae on the substrate in whih to sit. The average height of all partiles an be plotted over time and fit to an exponential deay urve to determine the time onstant. Figure 16 displays the output data for one of the 50 partile simulations, with the exponential deay urve plotted on top of the data. Mathematia s Findfit

48 39 funtion was used to determine the deay rate for the urve. The exponential fit looks adequate to desribe the data, with a alulated R- squared value of Note that the fator in front of t in the exponential is the inverse of the time onstant in units of μs -1. Reall that Figure 16. Derease in z average for all partiles over time for a system of 50 partiles ated upon by Van der Waals substrate fore, Lennard-Jones interpartile fore, gravity and visous and flutuating fores. the exponential deays to a height of 15 nm (not zero!) sine the partile enters are 15 nm above the substrate when the partiles have settled. Repeating this fitting proedure for all data runs, the average time onstants were found and are plotted in Figure 17 for the three partile numbers onsidered. The results reveal a slightly different piture from that of Figure 12. Here, the mean time onstant, or mean fall time, is very lose for systems with all numbers of partiles. The value does inrease slightly for systems with higher numbers of partiles (from 136 μs for 20 partiles to 140 μs for 80 partiles), but not to the extent of the inrease in mean deposition time. Like the graph for mean deposition time, the graph for mean time onstant shows wider variation in the system with the fewest number of partiles. The error band is learly tighter for systems with larger numbers of partiles. Figure 17. Differene in mean time onstant for systems with 20, 50, and 80 partiles. Error bars denote standard errors.

49 40 The reason that Figure 12 shows suh a wide differene in deposition times between systems with different numbers of partiles is most likely due to the effet of outliers in the system. For some simulation runs in systems with higher numbers of partiles, the last ouple of partiles took signifiantly longer to deposit onto the substrate. Thus, during a relatively long period of time, the average height above the substrate flutuated just above 15 nm. Suh flutuations would have little effet on the exponential urve fit. However, they ould have a pronouned effet on the average deposition time. As a onrete example, during data run 11 for a system with 80 partiles, 78 partiles had reahed the substrate after about 0.72 ms. The 79 th partile was deposited at 1.2 ms, and the last partile at 1.9 ms. These last 2 partiles inreased the average deposition time by 34 μs, but inlusion of the last two partiles did not greatly influene the alulated value of time onstant. j Thus, Figure 12 shows us that there are signifiant differenes in the amount of time required for all partiles to reah the substrate. However, Figure 17 shows us that the time required for most partiles to reah the substrate does not depend signifiantly on the number of partiles in the system. In applying this to a system where we wish to deposit hematite nanopartiles onto a silion dioxide substrate for some purpose, however, we must onsider that more partiles will be deposited in a given amount of time if the number density of partiles in the fluid is higher. For example, from Figure 17, we see that in a system with 20 partiles, the mean fall time is 20 partiles/136 μs partiles/μs, while that for a system with 80 partiles is 80 partiles/140 μs partiles/μs. Thus, we an onlude that as long as the partiles will all fit onto the substrate, it is best to use a higher onentration of partiles in our solvent when onduting some experiment to deposit hematite nanopartiles onto a silion dioxide substrate. j The average deposition time for this data run hanged from 162 μs to 128 μs when the last two partiles were exluded from the alulation. Calulated value of time onstant was 131 μs for this data run.

50 Pattern Formation We now turn to onsider whether the interpartile interations result in the partiles forming ordered patterns when they deposit onto the substrate. We have seen that the size of the substrate limits the number of partiles that may be deposited, and that larger numbers of partiles will derease the distane between partiles on the substrate and therefore inrease the fores. Thus, we hypothesize Figure 18. Final positions of partiles on substrate for an 80 partile system ated upon by Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles able to move after reahing substrate. Size of partile equal to hematite nanopartile. that partile positions will be better ordered or more orrelated for larger numbers of partiles. We first look at representative output from a system with 80 partiles and examine both qualitatively and quantitatively whether interpartile fores have aused patterns to form. To determine the influene of interpartile fores, we ompare the results to those of an 80 partile system where Lennard-Jones Figure 19. Final positions of partiles on substrate for an 80 partile system ated upon by visous and flutuating fores. Partiles fell to substrate due to gravity only, and were able to move after reahing substrate. Size of partile equal to hematite nanopartile.

51 42 interpartile fores have been eliminated. Figure 18 shows the partile positions for a system where the Lennard- Jones fore is ative, while Figure 19 shows the partiles for the system where the interpartile fore is absent. (a) Lennard-Jones fores ative between partiles Visually, the partiles in Figure 18 appear to be more ordered. Conversely, in Figure 19, some of the partiles go so far as to overlap, whih is unphysial, but results by treating the partiles as point objets in the simulation. Figures 20 and 21 provide a more quantitative omparison of the two (b) No Lennard-Jones interpartile fores Figure 20. Comparison of histograms showing distane to nearest partile for two different 80-partile systems. (a) In the first system, Lennard- Jones fores prevent the partiles from getting too lose or very far away. (b) When no Lennard-Jones fores are present, as in the seond system, the distane to nearest partile shows a wider spread. systems. The final positions on the substrate were analyzed to develop the two figures. For eah partile, the distane to every other partile on the substrate was alulated and tabulated, then plaed in rank order from losest partile to furthest partile. Figure 20 shows a histogram of the distane to the losest partile for all 80 partiles in eah system. Note that when Lennard-Jones fores are ative, the hematite nanopartiles never overlap (i.e. lowest interpartile distane is 20

52 43 (a) Lennard-Jones fores ative between partiles (b) No Lennard-Jones interpartile fores Figure 21. Comparison of histograms showing distanes to five nearest partiles for two different 80-partile systems. (a) In the first system, Lennard-Jones fores at between partile pairs. (b) When no Lennard- Jones fores are present, as in the seond system, the distanes over a wider spread.

53 44 nm), but when Lennard-Jones fores are absent, the visous and flutuating fores ating on the partile points have put the partile enters as lose as 3 nm. In addition, the spread in losest distanes is muh wider for the system where no interpartile fores are ative. There, losest distanes range between 3 and 49 nm, whereas in the system with Lennard-Jones fores present, the spread is only 15 nm. Figure 21 depits a histogram for the five losest partiles in eah system and shows a similar trend. Again, when the Lennard-Jones fore is ative, we do not see distanes loser than the diameter of the hematite nanopartiles. In addition, this time we an see that the distanes to losest partiles, seond losest, and so on, appear in tighter groupings for the hart where the Lennard-Jones fore is ative. With no interpartile fores, the distanes again show a wider spread. For the system with Lennard-Jones fores, although the hematite nanopartiles do not overlap, the ligand shells do, as depited visually in Figure 22. This is the same system as that shown in Figure 18, exept the partile diameters are sized to represent the hydrodynami diameter. The data in Figures 20 and 21 show that many of the nearest partiles are at distanes suh that repulsive fores will be felt between partiles. In fat, we would expet most of the partiles to be present in the bin ranging from 34 to 36 nm, the loation of the minimum potential in the L e n n a r d - J o n e s u r v e. Figure 22. Final positions of partiles on substrate for an 80 partile system ated upon by Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles able to move after reahing substrate. Dark blue irles represent hematite nanopartiles, while light blue irles represent the ligand shell diameter.

54 45 The reason why the partiles are so lose an be understood by onsidering many fators ating in onert. First of all, although Figure 20 indiates that a given partile is likely to be repelled by the partile losest to it, Figure 21 indiates that the same partile may be attrated to other nearby partiles. In some ases, other nearby partiles will attrat a given partile toward another partile that repels it. In addition, we must onsider that the partiles are onstantly moving due to the ation of the visous and flutuating fores. Thus, Figure 22 represents only a snapshot in time. Also, with 80 partiles on the substrate, we approah the limiting value of 86.8 partiles the maximum number of partiles that will fit on the substrate without any repulsive fores at all, but only if the partiles are not moving. With attrative fores ating on the partiles when distanes are beyond 34.6 nm, and with visous and flutuating fores ontinuously ating upon the partiles, we an intuitively understand why the onfiguration is likely to have at least some repulsive fores ating between partiles. Additional insight an be gained by onsidering representative magnitudes of the fores ating between partiles. Reall that the standard deviation of the random fore is 8.3 x N. For a partile enter-enter distane of 21.4 nm, the Lennard-Jones repulsive fore between partiles is 8.5 x N. Sine this value is less than the standard deviation of the random fore, we an imagine that the random fores ould push the partiles to this distane. In addition, we should onsider that the sum of the Van der Waals and steri interpartile fores at this same distane is 5.2 x N, whih exeeds the standard deviation of the flutuating fore. Thus, more aurate insights ould be gained by using the models of the Van der Waals and steri fores individually, rather than estimating their influene using the Lennard-Jones model. In fat, the Lennard-Jones model underestimates the atual value of the repulsive fore for distanes less than about 29 nm. However, even the sum of the Van der Waals and steri fores is less than that of the standard deviation of the random fores for interpartile distanes greater than about 23 nm. Thus,

55 46 we should be able to attain greater partile separations on the substrate for systems with larger steri fores. Suh systems should be studied in future simulation projets. Finally, we ompare the results for an 80-partile system, as depited in Figure 18, to those for a 50-partile system. Figure 23 depits representative positions on the substrate for a 50-partile system. Visually, we an see there is less order to the Figure 23. Final positions of partiles on substrate for a 50 partile system ated upon by Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles able to move after reahing substrate. Size of partile equal to hematite nanopartile. partile patterns, and more variation in distanes between partiles as ompared to the 80- partile system. These intuitive insights are supported by looking at a histogram of nearest partile distanes, shown in Figure 24. As with the 80-partile system represented by Figure 20a, interpartile repulsive fores Figure 24. Histogram showing distane to nearest partile for 50- partile system where Lennard-Jones fores at between partiles. As with the 80-partile system, interpartile repulsive fores prevent the partiles from getting too lose. However, there is more spread in the distanes for the 50-partile system. prevent the partiles from approahing loser than 20 nm. However, in the 50-partile system some

56 47 of the nearest partiles are over 40 nm away, at distanes where Lennard-Jones attrative fores are relatively low on the order of N. Thus, beause the substrate is less rowded, we an understand that some partiles will be more dispersed due to the relatively low attrative fores at greater distanes.

57 CHAPTER 5 SUMMARY, FUTURE WORK, AND CONCLUDING REMARKS 5.1. Summary A Fortran simulation program was developed to model the behavior of hematite nanopartiles in hexane solution as they deposited onto a silion dioxide substrate. The simulation employed periodi boundary onditions in the x and y diretions, with the substrate simulated in the z (vertial) diretion. The ative box was a ube with 300 nm side lengths. Systems with 20, 50, and 80 partiles were modeled, and simulation results were analyzed to determine whether the number of partiles that omprised a system affeted partile deposition time and rate, as well as partile deposition pattern on the substrate. The mean time for all partiles to settle onto the substrate was learly longer for systems with greater numbers of partiles. However, in systems with greater numbers of partiles, the last few partiles sometimes took signifiantly longer to deposit onto the substrate, thus inreasing the overall average settling time. Fall time was also analyzed using deposition rate, whih was determined for a given partile number by using the mean time onstant for all data runs. The time onstant for eah simulation run was found by fitting an exponential urve to data for the average height above the substrate over time. The mean time onstant was slightly longer for systems with greater numbers of partiles. The dependene of mean time onstant on system partile number was not as pronouned as that for mean time for all partiles to reah the substrate.

58 49 Deposition pattern analysis showed that interpartile fores did ause patterns to form on the substrate surfae, and patterns were more ordered or orrelated for systems with greater numbers of partiles. The results indiate that when depositing hematite nanopartiles onto a substrate, a solution with a greater number of partiles should be hosen to attain the best partile distribution on the substrate in the optimum amount of time. The Fortran simulation program employed to model deposition of the nanopartiles onto a substrate was shown to adequately model the visous and flutuating fores of the system via tests suh as omparing output kineti energy to the equipartition theorem, and omparing partile behavior to random walk motion. In addition, the Lennard-Jones interpartile fores behaved as expeted in output results. However, the Lennard-Jones fore model was not as aurate as the individual Van der Waals and steri fore models for partiles at extremely lose distanes. The differene in the two models ould affet the output partile deposition patterns Future Work Many extensions to this work should be ompleted to expand our knowledge of partile deposition rate and pattern formation onto a substrate. The following five important areas are disussed in the remainder of this setion: improve the fore model; develop a more quantitative pattern analysis tehnique; simulate a more realisti substrate, suh as MCM-41; vary parameters suh as ligand shell thikness to determine the effet on deposition pattern formation; and optimize the simulation algorithm Improve the Fore Model As disussed in Chapter 4, when partiles were at extremely lose distanes the omputed Lennard-Jones fores were orders of magnitude lower than the sum of the Van der Waals and steri fores. The Lennard-Jones fore was hosen due to its omputational effiieny. However, the

59 50 differenes in fore magnitudes at lose distanes ould affet formation of patterns on the substrate surfae. Therefore, a future version of the algorithm should inlude separate Van der Waals and steri interpartile fores in plae of the Lennard-Jones fore. Output results should be ompared with the results attained using the Lennard-Jones model and differenes should be noted. If differenes are signifiant, then the more speifi Van der Waals and steri fores should be inorporated in the simulation in plae of the simpler Lennard-Jones fore. In addition, this projet modeled only Van der Waals and steri interpartile fores, but other fores ould affet partile deposition time and pattern formation. In some ases, eletrostati interations an play a role in nonpolar solvents suh as hexane. (Hsu, Dufresne and Weitz) Thus, the role of the surfatants as harge arriers and their ontribution to eletrostati interations between partiles should be examined. A sreened Coulomb fore should be added to the model as appropriate. Also, magneti interations play a strong role if the ligands are short. Therefore, inorporation of magneti interations into the model should also be onsidered. Finally, when alulating the fore due to gravity, the effet of buoyany on the partiles was not onsidered. In the future, buoyany due to the hexane solution should be inorporated into the simulation algorithm Develop a More Quantitative Pattern Analysis Tehnique This projet analyzed pattern formation qualitatively by looking at output pattern results, and quantitatively by looking at histograms of distanes between partiles. As a baseline, results were ompared to simulations where no interpartile fores were present. Additional quantitative analysis methods should be explored. First of all, the hard sphere model ould be hosen as the baseline for omparison using the histogram data. Furthermore, twodimensional Fourier transforms or Wavelet transforms of partile deposition patterns may reveal mathematially signifiant patterns.

60 Simulate a More Realisti Substrate A flat silion dioxide substrate was used for this study. In appliations, mesoporous siliate substrates suh as MCM-41 are often employed. (Hondow and Fuller) Thus, future studies should simulate these more ompliated porous strutures Vary Parameters to Determine Effet on Deposition Pattern This study modeled partiles of only one size, with a single ligand shell thikness and one distane between ligand headgroups. The simulation program shows great promise for modeling a variety of systems without having to expel the resoures required to experiment with suh systems in the laboratory. For example, as mentioned in Chapter 4, a higher steri fore ould result in patterns where partiles are more widely and evenly spaed. As desribed in Chapter 2, steri fore an be adjusted by either inreasing the ligand shell thikness L or dereasing the distane between headgroups h. A series of simulations an be onduted for various values of L and h to determine the ombination of fators that produes the most desirable substrate pattern. Simulations an be onduted to vary other parameters as well. For example, as mentioned in Chapter 1, all three magneti iron oxide nanopartiles (hematite, magnetite, and maghemite) have interesting properties suited to a variety of appliations. The simulation program ould easily be modified to simulate the properties of one of the other iron oxide nanopartiles. The omposition of the fluid and substrate ould also be easily hanged Optimize the Fortran algorithm Simulations with large numbers of partiles took a great deal of time. For example, trial simulations of systems with 100 partiles took nearly a day to run on a laptop omputer. Simulations for systems with 80 partiles sometimes lasted over 12 hours. For suh large simulations, optimization of the Fortran algorithm would pay huge dividends by reduing the time required to run a given simulation, thus allowing more simulations to be run in a given amount of

61 52 time. Two speifi improvements that ould be easily inorporated are optimizing the fore alulation loop and establishing a ut-off distane for alulation of the interpartile fores. The fore between two partiles aused by the Lennard-Jones potential is the same for eah partile, and thus need only be alulated one. When the Fortran routine was initially established, however, the fore was alulated twie one for eah partile of a two partile pair. The additional alulations were of little onsequene for small numbers of partiles. However, utting the number of alulations in half for large partile numbers an save a great deal of time. Therefore, steps should be taken to alter the fore alulation algorithm to make it more effiient. Figure 3, whih depits the Lennard-Jones potential, indiates that fores between partiles should approah zero at large distanes. In fat, at a distane of 2.5σ, the pair potential is only 1.6 perent of the minimum potential energy. (Allen and Tildesley p. 29) Thus, we an save further time in the fore alulation loop by eliminating fore alulations if the distane between two partiles is beyond a given distane. The optimum ut-off distane to employ in this simulation should be determined, and then the algorithm modified to inlude this distane in future simulations Conluding Remarks A stohasti dynamis Fortran simulation program was developed that suessfully modeled hematite nanopartile deposition onto a silion dioxide substrate. The method shows great promise in answering important questions related to deposition time and pattern formation on a substrate surfae, and an be used to ondut omputational experiments at less ost in time and resoures than suh experiments would ost in the laboratory. For example, ligand shell parameters an be adjusted over a wide range of values to modify the steri fore and determine the effet on pattern formation. With slight modifiations, the method an also be applied to nanopartiles or substrates made from different substanes. This would allow the program to be used to answer

62 53 questions regarding the differene in deposition rates between partiles omposed of different substanes. (Hondow and Fuller)

63 APPENDIX 1 EQUATIONS FOR VAN DER WAALS ENERGIES AND FORCES This appendix shows the steps used to modify Hamaker s equation for Van der Waals energies and fores into the forms that were used in this study. Hamaker s original paper presents his formula in terms of dimensionless variables. This appendix onverts the equation into a form dependent on the radius of the partiles, height above the substrate (for partile-substrate energies and fores), and distane between enters of the partiles (for partile-partile energies and fores). First, we develop the equations used between eah partile and the substrate. Next, the equations used to model the energies and fores between partiles are developed. We begin with Hamaker s formula for the energy between two spheres of diameters D 1 and D 2, where the surfaes are loated a distane d apart: V = A 1 12 { y + x 2 +xy+x y x2 +xy+x + 2 ln }. ( 49 ) x 2 +xy+x+y x 2 +xy+x+y Here, x and y are defined by x = d D 1, and y = D 2 D 1, ( 50 ) and A is known as Hamaker s onstant, determined in Hamaker s original paper using the density of atoms, ionization potential, and polarizability of the atoms involved. For the ase of a sphere approahing a flat substrate, we allow D 1 to represent the diameter of the sphere, and D 2 that of the substrate. In order for D 2 to represent the substrate, we allow the value of D 2 and therefore the dimensionless variable y to approah infinity. We look at eah term in equation (49) in turn, taking the limit as y approahes infinity, to develop an expression for

64 55 the energy between the sphere and flat substrate. We apply L Hopital s Rule to any indeterminate forms in deriving the solution. For the first term, we have lim = lim = 1. ( 51 ) x 2 +xy+x x x y y The seond term is y lim = lim x 2 +xy+x+y y y and the third term is 1 1 = 1 y x+1 x+1, ( 52 ) lim 2 ln x2 +xy+x = lim x x 2 ln = 2 ln. ( 53 ) y x 2 +xy+x+y y x+1 x+1 Substituting these last three expressions into equation (49), we find the expression developed by Hamaker for the energy between a sphere, or spherial partile, and a flat surfae: V Vps = A 1 12 {1 + 1 x + 2 ln }. ( 54 ) x x+1 x+1 Substituting x = d/d 1, we find V Vps = A 1 12 {D d d D1 +D 1 D1 + 2 ln d D1 d D1 +D 1 D1 } = A 1 12 {D 1 d + D ln }. ( 55 ) d+ D 1 d+ D 1 In the algorithm used in this projet, the Van der Waals partile-substrate fore ats in the vertial diretion only. The vertial distane between the enter of the partile and the substrate is denoted by z, while the radius of the partile is denoted by r. Realling that d represents the distane between surfaes, we an substitute D 1 = 2r and d = z r into equation (55) to find V Vps = A 1 6 { r z r + r z+r z r + ln }, ( 56 ) z+r whih is the desired expression for energy in terms of the radius r and height above the substrate z. We alulate the fore ating between partile and substrate by finding the gradient using F = V Vps = x V Vps = 1 V Vps. ( 57 ) d x D 1 x We take the derivative of equation (54), where energy is expressed in terms of x, and find d

65 56 F(x) = A { D 1 x 2 + 2(x+1) (x+1) 2 x [ 1 (x+1) Substituting x = d/d 1 into equation (58), we find x (x+1) 2]} = A { }. ( 58 ) 12D 1 x 2 (x+1) 2 x (x+1) F(d) = A 12 { D 1 d 2 D 1 d 2 +2dD 1 + D d 2 (d+d 1 ) }. ( 59 ) Finally, we substitute D 1 = 2r to obtain F(d) = A 6 {1 d r d 2 1 (d+2r) r (d+2r) 2}, ( 60 ) whih is an expression for fore in terms of the radius r and distane between the partile and substrate surfaes d. Substituting d = z r into equation (60), we modify the equation to F = A 6 { 1 r 1 z r (z r) 2 (z + r) r (z + r) 2}, ( 61 ) whih is the desired expression for fore in terms of radius r and height above the substrate z. To derive the form of Hamaker s equation used for the energy between partiles, we begin again with Hamaker s formula, equation (49). Sine the partiles used in this program were all of equal size, then D 1 equals D 2, and therefore y = 1. Thus, Hamaker s formula an be written more simply as V Vpp = A 1 12 { 1 x 2 +2x + 1 x 2 +2x ln x2 +2x x 2 +2x+1 }. ( 62 ) We substitute x = d/d 1 and find V Vpp = A 1 12 { D 1 2 d 2 +2dD 1 + As before, we next substitute D 1 = 2r to obtain V Vpp = A 1 12 { 4r2 + 4r 2 d 2 +4dr D 1 2 d 2 +2dD 1 + D ln d2 +2dD 1 d 2 +2dD 1 +D 1 2}. ( 63 ) d 2 +4dr + 4r ln d2 +4dr d 2 +4dr+4r 2}, ( 64 ) where, as before, d represents the distane between the surfaes of the partiles. We substitute d = D 2r, where D is the distane between partile enters, and find V Vpp = A 1 6 { 2r 2 D 2 4r 2 + 2r2 D 2 + ln D2 4r 2 D 2 }, ( 65 ) whih is the desired expression for Van der Waals energy between partiles in terms of radius r and distane between partile enters D.

66 57 The fore between two spheres an be found by differentiating equation (62) for the energy. The equation for fore between two equally sized spheres is also provided by Hamaker and is repeated as F Vpp = A 1 2(x+ 1) { 6D 1 x 2 +2x (x+ 1) (x 2 +2x) 2 2 (x+1) 1 (x+1) 3}. ( 66 ) As before, we substitute x = d/d 1 to find an expression in terms of d, the distane between surfaes of the partiles. After some simplifiation, we have the expression F Vpp = A 1 6 {2(d + D 1 ) d 2 +2dD 1 Next, we substitute D 1 = 2r to find F Vpp = A 1 6 (d + D 1 )D 1 2 (d 4 +4d 3 D 1 +4d 2 D 1 2 ) 2 D 2 1 (d + D 1 ) (d 3 +3d 2 D 1 +3dD D 1 3 ) }.( 67 ) + 2r) {2(d 4r2 (d + 2r) 2 4r 2 d 2 +4dr (d 2 +4dr) 2 (d + 2r) (d+2r) 3}. ( 68 ) Finally, we substitute d = D 2r to modify the expression to the form F = A 1 6 { 2D D 2 +4r 2 4r 2 D 2 4r2 (D 2 +4r 2 ) 2 D D3 }, ( 69 ) whih is the desired expression for the fore between two partiles in terms of the radius r and distane between enters D.

67 APPENDIX 2 REPRESENTATIVE MAGNITUDES OF FORCES USED IN THIS STUDY This appendix shows the magnitudes of some of the fores used in this study. Table 4. Representative Magnitudes of Fores Used in this Study Fore Magnitude (N) Comments Van der Waals to Substrate x Calulated for z = 250 nm, r = 10 nm Van der Waals to Substrate x Calulated for z = 50 nm, r = 10 nm Van der Waals to Substrate x Calulated for z = 15 nm, r = 10 nm (On the Substrate) Standard Deviation of Random Fore x For alulation, assume T = 298K, visosity of hexane, hematite nanopartile of 10 nm radius, time step 0.1 ns, and hydrodynami radius of 15 nm (to inlude ligand shell) Gravitational Fore x For alulation, assume mass of hematite nanopartile of 10 nm radius Table 5. Comparison of Van der Waals, Steri, and Lennard-Jones Fores for Various Interpartile Distanes. Distane Between Centers Steri Fore Magnitude (N) Van der Waals Fore Magnitude Differene in Magnitudes of Lennard-Jones Fore 1, 2 (N) D (nm) (N) Van der Waals and Steri Fores 1 (N) x x x x x x x x x x x x x x x x A negative fore indiates repulsion, while a positive fore indiates attration. 2 Lennard-Jones fore alulated using ε = x Joules, and σ = nm.

68 APPENDIX 3 NUMERICAL INTEGRATION SCHEME DESCRIPTIONS This appendix gives detailed desriptions or derivations of numerial integration shemes disussed in the main body of this report. A3.1. Euler Method The simplest numerial integration sheme to derive is the Euler method. We begin with Newton s seond law x (t) = 1 F [x (t), x (t), t], ( 70 ) m whih an be separated into six oupled first-order differential equations for position x and veloity v, given by x (t) = v (t), and ( 71 ) v (t) = 1 F [x (t), v (t), t], ( 72 ) m where t represents time, F is the fore, and m is the mass. The values of x (t) and v (t), whih represent the time derivatives of x and v, an be found by using a Taylor series expansion of x and v around time t, given by (Giordano and Nakanishi p. 456) d 2 x x (t + t) = x (t) + dx t + 1 dt 2 dt 2 ( t)2 +, and ( 73 ) d 2 v v (t + t) = v (t) + dv t + 1 dt 2 dt 2 ( t)2 +. ( 74 )

69 60 If we ignore the terms in the Taylor series expansion of order ( t) 2, solve the expansions for dx /dt and dv /dt, respetively, and then substitute into equations (71) and (72), we obtain x n = v n = 1 t (x n+1 x n ), and ( 75 ) v n = 1 F [x m n, v n, t] = 1 (v t n+1 v n ). ( 76 ) In equations (75) and (76), note that we have replaed x (t + Δt) by x n+1 and v (t + Δt) by v n+1 to represent values of position and veloity at the next time step, as well as x (t) by x n and v (t) by v n to represent values of position and veloity at the urrent time step. Time derivatives of position and veloity at the urrent time step were similarly replaed. Solving equations (75) and (76) for x n+1 and v n+1, we are left with the equations used in the Euler numerial integration method for two oupled first-order differential equations, namely, x n+1 = x n + v n t, and ( 77 ) v n+1 = v n + 1 F [x m n, v n, t] t = v n + a n t, ( 78 ) where the aeleration a n = F [x n,v n,t]/m. In deriving the Euler method, we ignored terms of order (Δt) 2 in the Taylor series expansion, and thus errors of this same order aumulate with eah time step. However, the overall error globally is of order Δt, sine the total error is proportional to the number of time steps N multiplied by the error of eah time step, or (Giordano and Nakanishi p. 457) Overall Error = NΔt 2 = t total Δt 2 t. ( 79 ) Δt A3.2. Verlet Algorithm and is given as The Verlet algorithm (Verlet) provides a method for updating the position at eah time step x n+1 = 2x n x n 1 + F n m t2 + O[( t) 4 ]. ( 80 )

70 61 The method may be derived by expanding both x (t + Δt) = x n+1 and x (t Δt) = x n-1 in Taylor series, adding the expansions, and then solving for x n+1. (Van Gunsteren and Berendsen, A Leap-Frog Algorithm for Stohasti Dynamis) The Taylor series are expressed as x n+1 = x n + v n t a n t x n 1 = x n v n t a n t Adding the two Taylor expansions, we find da n dt t3 + O[( t) 4 ], and ( 81 ) da n dt t3 + O[( t) 4 ]. ( 82 ) x n+1 = 2x n x n 1 + a n t 2 + O[( t) 4 ], ( 83 ) whih is another form of the Verlet algorithm, in terms of aeleration rather than Fore and mass. A3.3. Leapfrog Algorithm The leapfrog algorithm provides a method to update the position at eah time step and the veloity at eah half time step. It is presented as (Van Gunsteren and Berendsen, A Leap-Frog Algorithm for Stohasti Dynamis) x n+1 = x n + v 1 n+ t + O[( t) 3 ], and ( 84 ) 2 v 1 n+ = v 1 n F n Δt + m O[( t)3 ]. ( 85 ) As with the previous methods, the leapfrog algorithm an be derived by starting with a Taylor series expansion for x n+1 = x (t n + Δt), or x n+1 = x (t n + t) = x n + v n t a n( t) 2 + O[( t) 3 ]. ( 86 ) Next, by using the half-time-step Δt/2, we note that the Taylor series expansion for the half step veloity is v (t + t t ) = v (t) + a (t) O[( t)2 ]. ( 87 ) Rewriting in terms of our subsript notation, we see that t v 1 n+ = v n + a n O[( t)2 ], ( 88 )

71 62 We an solve this expression for v n and then substitute bak into equation (86). The aeleration terms anel, and we are left with equation (84), the expression to update position in the leapfrog algorithm. The equation to update veloity in the leapfrog algorithm is derived by using equation (88) for the half time step veloity, along with the equation for half time step veloity at the previous half step v n-1/2. To find the previous half step veloity, we start with the Taylor series expansion v (t t t ) = v (t) a (t) O[( t)2 ]. ( 89 ) Rewriting in terms of our subsript notation, we find the equivalent expression t v 1 n = v n a n O[( t)2 ]. ( 90 ) Note that the right hand sides of equations (89) and (90) are equivalent to those of equations (87) and (88), exept that terms in odd powers of Δt are negative. If we subtrat equation (90) from equation (88) and rearrange terms, the v n terms and terms of order (Δt) 2 anel, and we are left with v 1 n+ = v 1 n + a n Δt + O[( t) 3 ], ( 91 ) 2 2 whih is an alternative form of the equation to update veloity in the leapfrog algorithm, expressed in terms of aeleration rather than fore and mass. In the leapfrog algorithm, aeleration is alulated from the fores ating due to positions at integer inrements of the time step. Aeleration is then used to propagate the veloity forward from one half-time-step to the next. The half-time-step veloities are then used to move the position forward in integer inrements of the time step. Note that errors at eah time step of the leap-frog algorithm are of the order of (Δt) 3. A3.4. Veloity Verlet Algorithm The veloity verlet algorithm is a method that allows alulation of both the veloity and position at integer time steps. To derive the equation for position, begin by adding x n+1 to both

72 63 sides of equation (83) in the Verlet algorithm to find (Sandvik) 2x n+1 = x n+1 + 2x n x n 1 + a n t 2 + O[( t) 4 ]. ( 92 ) Define veloity at the urrent time step by using (Van Gunsteren and Berendsen) v n = x n+1 x n 1 2 t + O[( t) 2 ]. ( 93 ) Substitute equation (93) into equation (92) to derive the expression used to update position in the veloity Verlet algorithm, whih is x n+1 = x n + v n t a n t 2. ( 94 ) To find the expression for veloity, begin by writing the Verlet position algorithm, equation (83), for the urrent time step instead of the next time step as x n = 2x n 1 x n 2 + a n 1 t 2. ( 95 ) Now, add equations (83) and (95) and rearrange terms to obtain (Sandvik) x n+1 x n 1 = x n x n 2 + (a n 1 + a n ) t 2. ( 96 ) We an rearrange equation (93), and also modify the subsripts, to find x n+1 x n 1 = 2v n t, and ( 97 ) x n x n 2 = 2v n 1 t. ( 98 ) Substituting equations (97) and (98) into equation (96) and rearranging terms, we obtain v n = v n (a n 1 + a n ) t, ( 99 ) an expression for veloity at the urrent time step. We onvert this equation to one of veloity for the next time step and find v n+1 = v n (a n + a n+1 ) t, ( 100 ) whih is the expression used to update veloity in the veloity Verlet algorithm.

73 64 A3.5. Veloity Verlet/Leapfrog Algorithm One undesirable aspet of the veloity Verlet algorithm is that both the old and new aelerations are required in order to obtain the new value of veloity. The veloity Verlet/Leapfrog method is a modified form of the veloity Verlet algorithm that uses less memory and does not require the old value of aeleration. (Strobl and Bannerman) The position values are updated using equation (94), just as in the veloity Verlet algorithm. Then, the half time step value of veloity is alulated using equation (88). Now, if we solve equation (88) for the urrent time step veloity, we find t v n = v 1 n+ a n. ( 101 ) 2 2 If we substitute equation (101) into equation (100), we obtain v n+1 = v n a n+1 t, ( 102 ) 2 whih is the Veloity Verlet/Leapfrog algorithm expression to update the veloity. A3.6. Leap-Frog Algorithm for Stohasti Dynamis Van Gunsteren and Berendsen (Van Gunsteren and Berendsen, A Leap-Frog Algorithm for Stohasti Dynamis) developed a stohasti dynamis algorithm that redues to the leapfrog algorithm in the limit of small frition oeffiient. The algorithm was designed to solve the pair of first-order differential equations x i (t) = v i (t), and ( 103 ) m i v i(t) = F i [x i (t)] m i γ i v i + R i (t), ( 104 ) where m is mass, γ is the frition oeffiient, R(t) represents the random, flutuating fore, and the subsript i refers to eah of the three oordinate diretions for eah of the N partiles present in a stohasti system. Thus, i ranges in value from 1 to 3N. Note that for a system with partiles of equal size and mass, we an drop the subsript i from the mass m and frition oeffiient.

74 65 The equations for the Van Gunsteren/Berendsen stohasti dynamis algorithm were verified mathematially as part of this projet. In this appendix, we merely present the equations that were used in the Fortran program developed, and do not present their detailed derivation or explanation. Additional equations and further explanation of their derivation an be found in the paper by Van Gunsteren and Berendsen. The omputational sheme of the method developed is summarized in the following paragraphs. using When using the Van Gunsteren and Berendsen algorithm, position and veloity are updated x n+1 = x n + v n+ 1 Δt 2 (γδt) γδt γδt [exp (+ ) exp ( 2 2 )] + X n+ 1 ( Δt 2 2 ) X n+ 1 2 ( Δt 2 ) + O[(Δt) 3 ], and ( 105 ) v n+ 1 2 = v 1 n exp( γδt) + 2 F n Δt [1 exp( γδt)] + V m(γδt) n ( Δt ) 2 exp( γδt) V n ( Δt 2 ) + O[(Δt)3 ], ( 106 ) whih are the stohasti dynamis equivalent of equations (84) and (85). Like the leapfrog algorithm presented previously, note that the errors in position and veloity for eah time step are of order (Δt) 3. In addition, note that alulation of the next values of position and veloity depends upon the variables X 1 n+ ( Δt ), X n+ ( Δt ), V 2 2 n ( Δt ), and V 2 n ( Δt ). Eah of these expressions 2 represents an integral of the random fore R (t) multiplied by various quantities evaluated over a ertain integral of time. Both X 1 n ( Δt ) and V 2 2 n ( Δt ) are evaluated over the time integral (tn-1/2, 2 t n), and thus are orrelated. They will be related in the method below through the random Gaussian variable Y v. Similarly, X 1 n+ ( Δt ) and V 2 2 n ( Δt ) are orrelated sine they are both different 2 integrals of R (t) over the same time period (t n, t n+1/2). The random Gaussian variable Y x will be used to orrelate the latter two variables.

75 66 To initiate the stohasti dynamis leapfrog algorithm, initial positions and veloities must be given to all of the partiles in the olloidal system. The values used for initial onditions in this study were presented in the main body of this report. The method also requires an initial value for the variable X n-1/2 (Δt/2) k. An initial value for this variable an be found by sampling from a Gaussian distribution with zero mean and variane where σ 2 1 = X 1 2 n ( Δt ) = kt 2 2 mγ C ( γδt 2 2 C (γδt 2 ), ( 107 ) γδt ) = γδt exp ( ) exp( γδt), ( 108 ) 2 and where k is Boltzmann s onstant and T is the temperature of the system. After the initial onditions are determined, the systemati fore F i [x i (t)] ating on eah partile in eah of the three oordinate diretions is evaluated based upon the initial positions. Note that these are the fores that depend upon position (and thus a onservative potential field) only. and variane where C ( γδt 2 Next, the variable Y v is determined by sampling from a Gaussian distribution of zero mean σ Yv 2 = kt m B (γδt 2 ) /C (γδt), ( 109 ) 2 ) is given by equation (108), and B (γδt) is given by 2 B ( γδt γδt ) = γδt[exp(+γδt) 1] 4[exp (+ ) 1] 2. ( 110 ) 2 2 Reall that Y v is used to orrelate X n-1/2(δt/2) and V n(-δt/2). The values of Y v and X n-1/2(δt/2) are then used to determine V n(-δt/2) using V n ( Δt ) = X 1 2 n 2 ( Δt 2 ) γd (γδt) / C (γδt ) + Y 2 2 v, ( 111 ) k This is the random variable denoted in equation (105) by X n+1/2 (Δt/2), but at the previous half-time-step.

76 67 where C ( γδt ) is alulated using equation (108), and D (γδt ) using 2 2 D ( γδt 2 γδt γδt ) = 2 exp (+ ) exp ( ). ( 112 ) 2 2 The random Gaussian variable V n(δt/2) is determined by sampling from a Gaussian distribution of zero mean and variane ρ 1 2 = V n 2 ( Δt 2 ) = kt m [exp( γδt) 1]. ( 113 ) After this variable is determined, we an alulate the next half-time-step of veloity using equation (106). The random Gaussian variable Y x, used to orrelate X n+1/2(-δt/2) and V n(δt/2), is determined by sampling from a Gaussian distribution of zero mean and variane σ Yx 2 = kt mγ 2 γδt B( 2 ) [exp( γδt) 1]. ( 114 ) Using Y x, we an now find X n+1/2(-δt/2) using X 1 n+ ( Δt ) = V 2 2 n ( Δt 2 ) 1 γ D( γδt 2 ) [exp( γδt) 1] + Y x. ( 115 ) X n+1/2(δt/2) is then determined by sampling from a Gaussian distribution of zero mean and variane given by equation (107), the same equation that was used to determine the variane of X n-1/2(δt/2). Finally, the positions at the next time step are determined using equation (105). In a three-dimensional moleular dynamis routine, eah of these steps is performed for eah partile in all three dimensions. The routine then repeats step-by-step for a speified time period, or until some predetermined ondition is reahed.

77 APPENDIX 4 RANDOM NUMBER GENERATION As disussed in the main body of this report, proper generation of random numbers is vital to produing an aurate moleular dynamis algorithm. Two types of random numbers are of interest in moleular dynamis: uniform random numbers falling on the interval from zero to one, and random numbers falling on a Gaussian distribution with zero mean and some speified standard deviation. A4.1. Uniform Random Number Generators Uniform random number generators supplied by most software routines produe pseudo random numbers using linear ongruential generators (LCG) (Press, Flannery and Teukolsky p. 267), whih use an equation of the form (Giordano and Nakanishi p. 512) x n+1 = (ax n + b)mod m, ( 116 ) where x n, a, b, and m are integers, and mod refers to the modulus funtion. The equation is initialized with a random seed x 0. The reason the numbers generated using LCGs are alled pseudo random is that any sequene started with the same random seed will produe the same sequene of random numbers. The value of the onstants a and b in equation (116) has been a subjet of extensive researh. Values should be hosen arefully to ensure high quality random numbers. For example, if a is even, then x n+1 is always even if b is even, and always odd if b is odd, effetively utting the number of random numbers able to be produed in half. (Giordano and Nakanishi pp )

78 69 Important fators to onsider when hoosing an appropriate pseudo random number generator inlude its speed, period, auray, lak of orrelation, and uniformity. (Giordano and Nakanishi p. 513) In any pseudo random number generator, the random numbers will eventually repeat. The period tells us how many random numbers are produed before the sequene repeats itself. Auray gives us the resolution of the random values produed between 0 and 1. Correlation refers to similarities between random numbers produed in sequene. For example, if a random number generator is bad, then suessive random pairs of numbers produed will appear to fall on lines when plotted, while suessive random triplets of numbers produed will fall on planes in three-dimensional plots. (Katzgraber) Uniformity tells us whether the numbers produed tend to fill the interval of spae where they are generated, or whether they tend to luster in various regions. A variety of statistial tests are available to test the quality of random number generation shemes. (Katzgraber) Press, et. al., have disussed and analyzed four alternative random number generators that are all superior to the standard LCG random number generator. (Press, Flannery and Teukolsky pp ) The ran0( ) and ran1( ) methods disussed by Press, et. al. were inappropriate for this study sine they do not pass all statistial tests and sine the periods are not long enough. (Katzgraber) The ran2( ) method presented by Press, et. al. was hosen sine it passes all statistial tests and has a period on the order of (Katzgraber) The period for this projet had to exeed 100 billion to be aeptable. To understand why, for the Fortran program implemented to analyze hematite nanopartiles in Chapter 4, the number of random numbers required for a 100-partile system analyzed for 20 million yles simulating 2 mse of real time was on the order of 100 billion. Random statistial tests for orrelation and for uniformity were onduted using the ran2( ) random number generator. To test for orrelation, 1000 random triplets were suessively produed and then plotted in three dimensions using Mathematia. If the random number generator

79 70 was bad, then the random triplets produed would appear to fall on planes within the three-dimensional ube. However, the ran2( ) random number generator produed points that tended to fill the spae of the ube, as shown in Figure 25. To test for uniformity, Figure 25. Correlation test for Ran2( ) random number generator. One thousand suessive triplets of random numbers were produed and plotted. Sine the suessive triplets tend to fill the spae, and do not fall into visible planes, suessive random numbers are not orrelated. 1,000 random numbers were generated and plaed in equally-sized bins from 0 to 1, then plotted on a histogram as depited in Figure 26. If the distribution was perfetly uniform, we would expet that 100 random numbers would fall into eah bin. Clearly, the numbers are not perfetly uniform. However, we an Figure 26. Uniformity test for Ran2( ) random number generator. One thousand random numbers were generated and then plaed into 10 equally-sized bins between 0 and 1. Approximately 100 numbers should fall within eah bin. Note that tik marks are loated at the enter of eah bin. determine a onfidene level in the uniformity of the distribution by applying the hi-square test to the data

80 71 depited in Figure 26. (Giordano and Nakanishi pp ) (Jain) We begin by alulating (Giordano and Nakanishi p. 520) χ 2 = (N i n ideal ) 2 i, ( 117 ) n ideal where N i represents the number of random numbers falling into bin i, and n ideal is the theoretial number of random numbers that should fall within a bin. The hi-square test is a hypothesis test. In this ase, we hypothesize that our distribution is uniform. Tabulated values of hi-square for various onfidene levels exist. If the value of hi-square alulated using equation (117) is less than the tabulated value, then we an aept the hypothesis that our distribution is uniform. For the data shown in Figure 27, the alulated value of hi-square is The tabulated value of hisquare for 10 bins at a 90% onfidene level is (Jain) Sine the alulated value of hisquare is less than the tabulated value, we may aept our hypothesis that the ran2( ) routine has produed a uniform distribution with 90% onfidene. A4.2. Gaussian Random Number Generators The stohasti dynamis algorithm required seletion of random numbers from a Gaussian distribution of zero mean and speified standard deviation. Allen (pp ) disusses three methods that may be used to produe random numbers on a Gaussian distribution. For this study, all three methods were verified to produe random variables on a Gaussian distribution, but the third method was implemented in the stohasti dynamis algorithm. To initiate the method, we first alulate 12 R = ( i=1 ξ i 6) /4, ( 118 ) where ξ i represent uniform random numbers generated on the interval from 0 to 1. Next, we alulate a random number on a Gaussian distribution with zero mean and unit standard deviation using the polynomial

81 72 ζ = ((((a 9 R 2 + a 7 )R 2 + a 5 )R 2 + a 3 ) R 2 + a 1 ) R, ( 119 ) where the polynomial oeffiients are defined by a 1 = , a 3 = , a 5 = , a 7 = , and a 9 = ( 120 ) After using equation (119) to find a random number on a Gaussian distribution of zero mean and unit width, the number may be onverted to a random Gaussian number of standard deviation σ by ζ = σζ, ( 121 ) where ζ is the random number sampled from the speified distribution. To test the random Gaussian number generator, the routine was used to generate 100,000 random numbers hosen from a Gaussian distribution of zero mean and unit width. The numbers generated were then plaed in bins of width 0.1 and plotted in Figure 27. The Gaussian funtion was plotted on the same figure for omparison. The alulated mean of all 100,000 values was , while the alulated variane was Based upon these results, the performane of the random Gaussian number generator was onsidered to be suffiient for use in this projet.

82 Figure 27. Verifiation of Gaussian random number generator. The 100,000 numbers generated fell on a normal distribution with approximately zero mean and unit width. 73

83 APPENDIX 5 SIMULATION ALGORITHMS This appendix presents the Fortran omputer ode used for two of the simulations performed as part of this projet. The first ode presented is the Van Gunsteren/Berendsen stohasti dynamis algorithm implemented in three dimensions to simulate hematite nanopartile depositions onto a substrate. The seond algorithm is a two dimensional version of the program that was implemented in order to allow partiles that had already deposited to move on the substrate under the ation of the Lennard-Jones, visous, and flutuating fores for an input period of time. A5.1. Three-Dimensional Stohasti Dynamis Algorithm to Model Hematite Nanopartile Deposition onto a Substrate This program uses a Moleular Dynamis simulation to model the interations between N dipoles in 3 dimensions. All dipoles are able to move within a ubi grid of size 300 nm. The initial positions of the dipoles are assigned suh that the partiles are in an ordered pattern near the top of the box, while the veloities are assigned randomly. Fore between the dipoles is modeled using the Lennard-Jones potential. After aeleration is alulated using the Lennard-Jones fore, a Leapfrog algorithm developed by van Gunsteren and Berendsen is used to alulate the next steps of position and veloity, taking into aount fritional and flutuating fores. The speifi Lennard-Jones fore used in this program was designed to model the Van der Waals partile-partile fore between hematite partiles in hexane solution with an SiO2 substrate, along with a Steri fore between partiles that have a 5 nm thikness of ligand shell and 5 nm distane between head groups. This version of the program also inorporates the fore of gravity in the z diretion. Thus, periodi boundary onditions on the bottom surfae of the box are eliminated to simulate a substrate surfae.

84 75 This version of the program also adds a Van der Waals fore ating between the partiles and the substrate in the z diretion. When the partiles reah a value of z equal to the radius of the partile plus the thikness of the ligand shell, the partile is assumed to have "hit bottom." This version of the program uses a random number generator reommended by the text "Numerial Methods in Fortran 77: The Art of Sientifi Computing." The user is asked to input a negative integer seed for the random number generator. This version also only prints out snapshots of the data. The user is asked to input the frequeny of snapshots taken. The user is also prompted to input the number of nse for whih the simulation should be run. Variables are defined as follows: r = radius of nanopartile (in meters) L = thikness of ligand shell mag = magnetization mu = magneti dipole moment mass = mass of dipoles pi = the onstant pi mu0 = the onstant mu0 T = temperature T (Kelvin) k = Boltzmann's onstant k ktimet = k*t sigma, eps = variables used to ompute Lennard Jones fore/potential DeltaT = time step size = size of 3D box where partiles are moving vis = visosity of fluid we are working with gamma = oeffiient used to alulate visous fore N = number of nanopartiles (input by user) real*8 r, L, mag, mu, mass, pi, mu0, T, k, ktimet real*8 sigma, eps, DeltaT, size, vis, gamma integer N Set values for r, L, mag, T, sigma, eps, vis, DeltaT, and size: r = 10.0D-9 L = 5.0D-9 mag = D0 T = 298.0D0!298.0K Room temperature sigma = 30.85D-9 eps = 4.976D-23 vis = 2.97D-4!2.97D-4 Visosity of hexane DeltaT = 1.0D-10 size = 300D-9 Define values for onstants pi, mu0, and k:

85 76 pi = D0 mu0 = 4.0D0*pi*1D-7 k = D-23 Request user input for number of partiles: print*, 'Enter Number of Partiles, N: ' read*, N Begin alulating various quantities, and then printing them to the sreen: ktimet = k*t print*, 'k * T = ', ktimet Calulate Magneti Dipole Moment mu: mu = mag*(4.0d0/3.0d0)*pi*r**3.0d0 print*, 'mu = ', mu Calulate mass of nanopartile. Assume partile is made of hematite, whih aording to the internet has a density of 5255 kg/m**3: mass = D0*(4.0D0/3.0D0)*pi*r**3.0D0 Calulate frition oeffiient for visous fore. gamma = 6.0D0*pi*vis*(r + L)/mass!This is for 3D print*, 'frition oeffiient = ', gamma print*, 'mass = ', mass pause!pause to preview values printed to sreen thus far. all posvel(sigma, eps, mass, DeltaT, r, N, size, gamma, ktimet) pause stop end ****************************************************************************** The following subroutine exeutes a "do loop" that repeatedly alulates the new positions and veloities resulting from the Lennard-Jones fore ating between two magneti dipoles. subroutine posvel(sigma, eps, mass, DeltaT, r, N, size, gamma, &ktimet) integer N, k1, k2, k3, k4, j1, j2, l1, l2, l3, idum, nsnaps integer ttime, ltime real*8 ran2, FluF, gamma, ktimet real*8 sigma, eps, LJFore, LJnrg, mass, time, DeltaT, r, size real*8 L, h real*8 rxn(n), rxnp1(n), vxnmhalf(n), vxnphalf(n) real*8 ryn(n), rynp1(n), vynmhalf(n), vynphalf(n) real*8 rzn(n), rznp1(n), vznmhalf(n), vznphalf(n) real*8 Fzn(N), Fxn(N), Fyn(N), Fn(N) real*8 distx, disty, distz, dist, T(N), V(N), Enrgy(N) real*8 gdt, gdthalf, Bgdthalf, Cgdthalf, Dgdthalf real*8 BMgdthalf, DMgdthalf, eq318, SDeq318

86 77 real*8 eq321, SDeq321, eq329, SDeq329, eq332, SDeq332 real*8 xxnmhalf(n), yxnmhalf(n), zxnmhalf(n) real*8 xyv(n), yyv(n), zyv(n), xvnmhalf(n), yvnmhalf(n) real*8 zvnmhalf(n), xvnhalf(n), yvnhalf(n), zvnhalf(n) real*8 xyx(n), yyx(n), zyx(n), xxneghalf(n), yxneghalf(n) real*8 zxneghalf(n), xxhalf(n), yxhalf(n), zxhalf(n) real*8 TEnrgy, TKineti, TPoten real*8 TFx, TFy, TFz, Tvx, Tvy, Tvz, TFore, Tvel real*8 vxum, vyum, vzum real*8 Gfore, bottomtime, avex, avey, avez real*8 pbttmtime(n), pbttmx(n), pbttmy(n) logial hitbottom(n), allbottom Calulate Fore of gravity ating on partile in z diretion: Gfore = 9.81D0*mass Ask user to input seed for ran2 random number generator funtion. print*, 'input idum (negative integer): ' read*, idum Ask user to input total amount of time (tenth of nse) simulation will be run: print*, 'input time to run simulation (tenth of nse): ' read*, ttime Ask user to input a number indiating how often data "snapshots" are taken. An inner do-loop runs from 1 to "nsnaps," and then data is output to a fort.10 file. print*, 'input how many loops to skip before snapshot: ' read*, nsnaps Calulate number of iterations of outer loop, based upon total time of simulation and number of snapshots: ltime = ttime/nsnaps Initialize time and bottomtime at 0.0: time = 0.0D0 bottomtime = 0.0D0 Initialize logial variable allbottom to.false. allbottom =.false. Input values for thikness of ligand shell L and distane between heads h: L = 5.0D-9 h = 5.0D-9 Set initial x, y, and z positions suh that partiles are in an ordered array; initial positions are assigned suh that, no matter what the partile density, approximately one-third of all partiles are in eah of three vertial layers. (NOTE: Previous routine for assigning initial positions randomly now appears in omment lines); assign initial veloities randomly; these are initial onditions for time t = 0. When we enter the do-loop initially, we'll be alulating the positions at time t = 1. Also initialize logial array hitbottom(n) to.false. for all partiles. Initialize real array pbttmtime(n) to 0.0D0 for all partiles. Initialize pbttmx(n) and pbttmy(n) to -1.0D0 for all partiles. do 33 k2 = 1, N, 1 hitbottom(k2) =.false.

87 78 pbttmtime(k2) = 0.0D0 pbttmx(k2) = -1.0D0 pbttmy(k2) = -1.0D0 rxn(k2) = (MOD(k2,6))*50.0D-9 ryn(k2) = (MOD(k2,36)/6)*50.0D-9 rzn(k2) = 250.0D-9 - (MOD(k2 + ((k2-1)/36),3))*50.0d-9 76 rxn(k2) = (size/3.0d0)*(1.0d D0*ran2(idum)) ryn(k2) = (size/3.0d0)*(1.0d D0*ran2(idum)) rzn(k2) = (size/3.0d0)*(1.0d D0*ran2(idum)) too_lose = 0.0 do 79 k3 = 1, k2-1 dist = dsqrt((rxn(k3) - rxn(k2))**2.0d0 + & (ryn(k3) - ryn(k2))**2.0d0 + (rzn(k3) - rzn(k2))**2.0d0) if (dist.lt.sigma) then too_lose = too_lose print*, 'partile ',k2,' is too lose to partile ', k3 endif 79 ontinue if (too_lose.gt.0.1) goto 76 vxnmhalf(k2) = 2.0D0*(ran2(idum) - 0.5D0) vynmhalf(k2) = 2.0D0*(ran2(idum) - 0.5D0) vznmhalf(k2) = 2.0D0*(ran2(idum) - 0.5D0) 33 ontinue The veloities above may have a enter of mass motion. We don't want this beause the temperature (found using kineti energy) should not depend on enter of mass veloity. Therefore, the next setion alulates the average veloity in the x, y, and z diretions and sets it to zero. vxum = 0.0D0 vyum = 0.0D0 vzum = 0.0D0 do 50 k4 = 1, N vxum = vxum + vxnmhalf(k4) vyum = vyum + vynmhalf(k4) vzum = vzum + vznmhalf(k4) 50 ontinue vxum = vxum/n vyum = vyum/n vzum = vzum/n do 60 k4 = 1, N Print initial positions and veloities to sreen; Before printing veloities, subtrat net motion in x, y, z diretions; also print out initial positions and veloities to fort.16 file: print*, 'rxn(', k4,') = ', rxn(k4) print*, 'ryn(', k4,') = ', ryn(k4) print*, 'rzn(', k4,') = ', rzn(k4) vxnmhalf(k4) = vxnmhalf(k4) - vxum vynmhalf(k4) = vynmhalf(k4) - vyum vznmhalf(k4) = vznmhalf(k4) - vzum print*, 'vxnmhalf(', k4,') = ', vxnmhalf(k4) print*, 'vynmhalf(', k4,') = ', vynmhalf(k4)

88 79 print*, 'vznmhalf(', k4,') = ', vznmhalf(k4) write(16,78) k4, rxn(k4)*1.0d9, ryn(k4)*1.0d9, rzn(k4)*1.0d9, &vxnmhalf(k4), vynmhalf(k4), vznmhalf(k4) 60 ontinue Add pause statement to preview initial onditions on sreen: pause Calulate some of the quantities that will be used in the van Gunsteren and Berendsen algorithm. gdt = gamma*deltat gdthalf = gdt/2.0d0 Cgdthalf = gdt - 3.0D D0*(dexp(-gdthalf)) - dexp(-gdt) Dgdthalf = 2.0D0 - dexp(gdthalf) - dexp(-gdthalf) Bgdthalf = gdt*(dexp(gdt) - 1.0D0) - 4.0D0*(dexp(gdthalf) - &1.0D0)**2.0D0 BMgdthalf = gdt*(-dexp(gdt) - 1.0D0) - 4.0D0*(dexp(-gdthalf) - &1.0D0)**2.0D0 DMgdthalf = 2.0D0 - dexp(-gdthalf) - dexp(gdthalf) eq318 = ktimet*cgdthalf/(mass*gamma**2.0d0) SDeq318 = dsqrt(eq318) eq321 = ktimet*bgdthalf/(mass*cgdthalf) SDeq321 = dsqrt(eq321) eq329 = ktimet*(1.0d0 - dexp(-gdt))/mass SDeq329 = dsqrt(eq329) eq332 = ktimet*bmgdthalf/(mass*(dexp(-gdt) - 1.0D0)* &gamma**2.0d0) SDeq332 = dsqrt(eq332) Print quantities obtained above and pause to hek results: print*, 'gdt = ', gdt print*, 'gdthalf = ', gdthalf print*, 'Bgdthalf = ', Bgdthalf print*, 'Cgdthalf = ', Cgdthalf print*, 'Dgdthalf = ', Dgdthalf print*, 'BMgdthalf = ', BMgdthalf print*, 'DMgdthalf = ', DMgdthalf print*, 'eq318 = ', eq318 print*, 'SDeq318 = ', SDeq318 print*, 'eq321 = ', eq321 print*, 'SDeq321 = ', SDeq321 print*, 'eq329 = ', eq329 print*, 'SDeq329 = ', SDeq329 print*, 'eq332 = ', eq332 print*, 'SDeq332 = ', SDeq332 pause Calulate initial values for xxnmhalf(n), yxnmhalf(n), zxnmhalf(n): do 11 k3 = 1,N xxnmhalf(k3) = FluF(SDeq318, idum) yxnmhalf(k3) = FluF(SDeq318, idum) zxnmhalf(k3) = FluF(SDeq318, idum)

89 80 11 ontinue Write initial positions and veloities to sreen and to fort.10 file: write(*,77) time, (rxn(k1), ryn(k1), rzn(k1), vxnmhalf(k1), &vynmhalf(k1), vznmhalf(k2), k1=1,n) 77 format (' ', 350(1p1e15.6)) write(10,*) 'time ','avex ','avey ','avez ','Tvx ','Tvy ', &'Tvz ', 'TFx ', 'TFy ', 'TFz ', 'TKineti ', &'TPoten ','TFore ', 'Tvel ', 'TEnrgy ' write(10,77) time*1.0d9, (rxn(k1)*1.0d9, ryn(k1)*1.0d9, &rzn(k1)*1.0d9, k1 = 1, N),(vxnMhalf(k1),vynMhalf(k1), vznmhalf(k1) &, k1 = 1, N) Purpose of outer do-loop is to print out the results to the sreen after "nsnaps" nse have elapsed. do 41 l2 = 1, ltime, 1 Inner do-loop alulates positions and veloities for "nsnaps" nse. do 42 l1 = 1, nsnaps, 1 Enter do loop to update fores for all partiles. Reset Total Energy, Total/Net Fore (all partiles), Total/Net veloity (all partiles), Total Kineti/Potential Energies, Total (sum of) Fores, and Total (sum of) veloities at 0.0 prior to entering this do loop. TEnrgy = 0.0D0 TFore = 0.0D0 Tvel = 0.0D0 TKineti = 0.0D0 TPoten = 0.0D0 TFx = 0.0D0 TFy = 0.0D0 TFz = 0.0D0 Tvx = 0.0D0 Tvy = 0.0D0 Tvz = 0.0D0 do 3 j1 = 1, N, 1 Initialize values of fores, potential energy, and kineti energy at 0.0 before entering inner do loop: Fn(j1) = 0.0D0 Fxn(j1) = 0.0D0 Fyn(j1) = 0.0D0 Fzn(j1) = 0.0D0 T(j1) = 0.0D0 V(j1) = 0.0D0 do 8 j2 = 1, N, 1

90 81 if (j1.eq.j2) goto 8 distx = rxn(j2) - rxn(j1) disty = ryn(j2) - ryn(j1) distz = rzn(j2) - rzn(j1) all sep(distx, disty, distz, size) dist = dsqrt(distx**2.0d0 + disty**2.0d0 + &distz**2.0d0) Fn(j1) = LJFore(dist, sigma, eps) Fn(j1) = 0.0D0 Fxn(j1) = Fxn(j1) + Fn(j1)*(distx/dist) Fyn(j1) = Fyn(j1) + Fn(j1)*(disty/dist) Fzn(j1) = Fzn(j1) + Fn(j1)*(distz/dist) V(j1) = V(j1) + LJnrg(dist,sigma,eps) V(j1) = V(j1) + 0.0D0 8 ontinue Subtrat gravity and substrate/partile fore from the Fore in the z diretion: print*, 'Fzn(',j1,') = ', Fzn(j1),' before gravity and &substrate fores added.' print*, 'V(',j1,') = ', V(j1),' before gravity and &substrate fores added.' Fzn(j1) = Fzn(j1) - Gfore - FSubstrate(r, rzn(j1)) print*, 'Fzn(',j1,') = ', Fzn(j1),' after gravity and &substrate fores added.' Modify potential energy to inlude potential energy with substrate: V(j1) = V(j1) + VSubstrate(r, rzn(j1)) print*, 'V(',j1,') = ', V(j1),' after gravity and &substrate fores added.' pause print*, 'Fn(',j1,') = ',Fn(j1) print*, 'Fxn(',j1,') = ',Fxn(j1) print*, 'Fyn(',j1,') = ',Fyn(j1) print*, 'Fzn(',j1,') = ',Fzn(j1) 3 ontinue Pause to review fores on sreen: pause Calulate half step veloities and new value of position. Comments in the following do-loop desribe several interim steps. do 15 k3 = 1,N Sample values of xyv(n), yyv(n), and zyv(n) from a Gaussian distribution, and then alulate xvnmhalf(n), yvnmhalf(n), and zvnmhalf(n), per step 3, page 180, of paper. xyv(k3) = FluF(SDeq321, idum) xvnmhalf(k3) = xxnmhalf(k3)*gamma*dgdthalf/cgdthalf + xyv(k3) yyv(k3) = FluF(SDeq321, idum) yvnmhalf(k3) = yxnmhalf(k3)*gamma*dgdthalf/cgdthalf + yyv(k3) zyv(k3) = FluF(SDeq321, idum) zvnmhalf(k3) = zxnmhalf(k3)*gamma*dgdthalf/cgdthalf + zyv(k3)

91 82 Sample values of xvnhalf(n), yvnhalf(n), and zvnhalf(n) from a Gaussian distribution, and then alulate the half-step veloities from equation 3.6: xvnhalf(k3) = FluF(SDEq329, idum) vxnphalf(k3) = vxnmhalf(k3)*dexp(-gdt) + Fxn(k3)*DeltaT* &(1.0D0 - dexp(-gdt))/(mass*gdt) + xvnhalf(k3) - dexp(-gdt)* &xvnmhalf(k3) yvnhalf(k3) = FluF(SDEq329, idum) vynphalf(k3) = vynmhalf(k3)*dexp(-gdt) + Fyn(k3)*DeltaT* &(1.0D0 - dexp(-gdt))/(mass*gdt) + yvnhalf(k3) - dexp(-gdt)* &yvnmhalf(k3) If partile has hit substrate at z = r+l, then we want to set future values of veloity and position in z diretion equal to zero. Before alulating z values, hek to see if partile has hit substrate: if (abs(rzn(k3)).le.r+l) then!partile has hit substrate vznphalf(k3) = 0.0D0 else!partile has not hit substrate zvnhalf(k3) = FluF(SDEq329, idum) vznphalf(k3) = vznmhalf(k3)*dexp(-gdt) + Fzn(k3)*DeltaT* &(1.0D0 - dexp(-gdt))/(mass*gdt) + zvnhalf(k3) - dexp(-gdt)* &zvnmhalf(k3) endif Finally, xyx(n), yyx(n), and zyx(n) are sampled from Gaussian distributions. Then the random variables xxneghalf(n), yxneghalf(n), and zxneghalf(n) are alulated. xxhalf(n), yxhalf(n), and zxhalf(n) are sampled from Gaussian distributions. From all of these variables, the positions at the next step are alulated. xyx(k3) = FluF(SDeq332, idum) xxneghalf(k3) = xvnhalf(k3)*dmgdthalf/(gamma*(dexp(-gdt) - &1.0D0)) + xyx(k3) xxhalf(k3) = FluF(SDeq318, idum) rxnp1(k3) = rxn(k3) + vxnphalf(k3)*deltat*(dexp(gdthalf) - &dexp(-gdthalf))/gdt + xxhalf(k3) & - xxneghalf(k3) yyx(k3) = FluF(SDeq332, idum) yxneghalf(k3) = yvnhalf(k3)*dmgdthalf/(gamma*(dexp(-gdt) - &1.0D0)) + yyx(k3) yxhalf(k3) = FluF(SDeq318, idum) rynp1(k3) = ryn(k3) + vynphalf(k3)*deltat*(dexp(gdthalf) - &dexp(-gdthalf))/gdt + yxhalf(k3) & - yxneghalf(k3) If partile has hit substrate at z = r+l, then we want to set future values of veloity and position in z diretion equal to zero. Before alulating z values, hek to see if partile has hit substrate:

92 83 if (abs(rzn(k3)).le.r+l) then!partile has hit substrate rznp1(k3) = r+l else!partile has not hit substrate zyx(k3) = FluF(SDeq332, idum) zxneghalf(k3) = zvnhalf(k3)*dmgdthalf/(gamma*(dexp(-gdt) - &1.0D0)) + zyx(k3) zxhalf(k3) = FluF(SDeq318, idum) rznp1(k3) = rzn(k3) + vznphalf(k3)*deltat*(dexp(gdthalf) - &dexp(-gdthalf))/gdt + zxhalf(k3) & - zxneghalf(k3) endif Print interim values to sreen as desired for troubleshooting: print*, 'xyv(',k3,') = ',xyv(k3) print*, 'xvnmhalf(',k3,') = ',xvnmhalf(k3) print*, 'xvnhalf(',k3,') = ',xvnhalf(k3) print*, 'vxnphalf(',k3,') = ',vxnphalf(k3) print*, 'xyx(',k3,') = ',xyx(k3) print*, 'xxneghalf(',k3,') = ',xxneghalf(k3) print*, 'xxhalf(',k3,') = ',xxhalf(k3) print*, 'rxnp1(',k3,') = ',rxnp1(k3) print*, ' ' print*, 'yyv(',k3,') = ',yyv(k3) print*, 'yvnmhalf(',k3,') = ',yvnmhalf(k3) print*, 'yvnhalf(',k3,') = ',yvnhalf(k3) print*, 'vynphalf(',k3,') = ',vynphalf(k3) print*, 'yyx(',k3,') = ',yyx(k3) print*, 'yxneghalf(',k3,') = ',yxneghalf(k3) print*, 'yxhalf(',k3,') = ',yxhalf(k3) print*, 'rynp1(',k3,') = ',rynp1(k3) print*, ' ' print*, 'zyv(',k3,') = ',zyv(k3) print*, 'zvnmhalf(',k3,') = ',zvnmhalf(k3) print*, 'zvnhalf(',k3,') = ',zvnhalf(k3) print*, 'vznphalf(',k3,') = ',vznphalf(k3) print*, 'zyx(',k3,') = ',zyx(k3) print*, 'zxneghalf(',k3,') = ',zxneghalf(k3) print*, 'zxhalf(',k3,') = ',zxhalf(k3) print*, 'rznp1(',k3,') = ',rznp1(k3) print*, ' ' 15 ontinue Pause to preview interim values: pause do 19 k1 = 1,N Calulate Final Kineti energy for eah partile, Total Kineti energy of all partiles, Total potential energy of all partiles, Total fore (sum of all fores) in x, y, and z diretions, Total veloity (sum of all veloities) in x, y, and z diretions, and Total Energy of all partiles. T(k1) = (0.5D0)*mass*(vxnPhalf(k1)**2.0D0 +

93 84 &vynphalf(k1)**2.0d0 +vznphalf(k1)**2.0d0) TKineti = TKineti + T(k1) TPoten = TPoten + V(k1) TFx = TFx + Fxn(k1) TFy = TFy + Fyn(k1) TFz = TFz + Fzn(k1) Tvx = Tvx + vxnphalf(k1) Tvy = Tvy + vynphalf(k1) Tvz = Tvz + vznphalf(k1) For now, the following two lines are omment lines. If added bak in, the lines provide a method to alulate the total energy of eah partile, and also a different method to alulate total energy. Enrgy(k1) = T(k1) + V(k1) TEnrgy = TEnrgy + Enrgy(k1) Chek to make sure the newly alulated positions are not outside of the box. If they are, use the modulus funtion to bring the partile in at the other side of the box: if (rxnp1(k1).gt.size) then rxnp1(k1) = dmod(rxnp1(k1),size) elseif (rxnp1(k1).lt.0.0d0) then rxnp1(k1) = rxnp1(k1) + size endif if (rynp1(k1).gt.size) then rynp1(k1) = dmod(rynp1(k1), size) elseif (rynp1(k1).lt.0.0d0) then rynp1(k1) = rynp1(k1)+ size endif For the top of the box, we don't want to use periodi boundary onditions. Rather, if the partile adjusts to a height higher than the box, we will readjust the partile height so that it is *at* the top of the box. if (rznp1(k1).gt.size) then rznp1(k1) = size To simulate a substrate, eliminate the periodi boundary onditions in the negative z diretion. Note that it is still possible for the partile to "hop" bak up in the z diretion due to random thermal flutuations. The logial array hitbottom(n) keeps trak of whih partiles have hit the substrate. The partile is onsidered to have "hit bottom" when its enter is loated at a distane of z = r+l elseif (rznp1(k1).le.(r+l)) then if (hitbottom(k1).eqv..true.) then goto 4!Already set values below. endif rznp1(k1) = r+l hitbottom(k1) =.true. pbttmtime(k1) = time pbttmx(k1) = rxnp1(k1) pbttmy(k1) = rynp1(k1) vznphalf(k1) = 0.0D0 endif Now, set "final" values equal to "initial" values to be used

94 85 during next iteration: 4 rxn(k1) = rxnp1(k1) ryn(k1) = rynp1(k1) rzn(k1) = rznp1(k1) xxnmhalf(k1) = xxhalf(k1) yxnmhalf(k1) = yxhalf(k1) zxnmhalf(k1) = zxhalf(k1) vxnmhalf(k1) = vxnphalf(k1) vynmhalf(k1) = vynphalf(k1) vznmhalf(k1) = vznphalf(k1) print*, 'T(',k1,') = ', T(k1) 19 ontinue Pause to look at partile kineti energies on sreen: pause Chek to see if all partiles have hit bottom yet. If all have hit bottom, then reord the bottom time. if (allbottom.eqv..false.) then do 88 l3 = 1,N if (hitbottom(l3).eqv..false.) then goto 89 endif 88 ontinue allbottom =.true. bottomtime = time 89 endif time = time + DeltaT 42 ontinue Calulate total fore, net veloity of all partiles, and total energy for those ases where data is output to sreen and fort.10: TFore = (TFx**2.0D0 + TFy**2.0D0 + TFz**2.0D0) Tvel = (Tvx**2.0D0 + Tvy**2.0D0 + Tvz**2.0D0) TEnrgy = TKineti + TPoten Calulate average values of x, y, and z for all partiles. avex = 0.0D0 avey = 0.0D0 avez = 0.0D0 do 1 j1 = 1, N avex = avex + rxn(j1) avey = avey + ryn(j1) avez = avez + rzn(j1) 1 ontinue avex = avex/n avey = avey/n avez = avez/n Write results to sreen: write(*,77) time, avex, avey, avez, Tvx, Tvy, Tvz, TFx, TFy,

95 86 &TFz, TKineti, TPoten, TFore, Tvel, TEnrgy Write same quantities to a fort.10 file. Time in nanoseonds and distane in nanometers. write(10,77) time*1.0d9, avex*1.0d9, avey*1.0d9, &avez*1.0d9, Tvx, Tvy, Tvz, TFx, TFy, TFz, TKineti, TPoten, &TFore, Tvel, TEnrgy if(allbottom.eqv..true.) goto 90!stop exeution of program 41 ontinue Output bottom time for all partiles to fort.11 file. Output x and y oordinates of partiles when they hit bottom, alongside final x, y, and z oordinates, to fort.12 file: 90 do 2 j1 = 1, N write(11,*) j1, pbttmtime(j1)*1.0d9 write(12,78) j1, pbttmx(j1)*1.0d9, pbttmy(j1)*1.0d9, &rxn(j1)*1.0d9,ryn(j1)*1.0d9, rzn(j1)*1.0d9, vxnmhalf(j1), &vynmhalf(j1), vznmhalf(j1) 2 ontinue 78 format (' ', i3, 8(1p1e15.6)) print*, 'bottomtime = ',bottomtime,' seonds' write(11,*) bottomtime*1.0d9 return end ****************************************************************************** The following funtion is used to alulate the Lennard-Jones Fore. funtion LJFore(dist, sigma, eps) real*8 LJFore, dist, sigma, eps LJFore = dist*((24.0d0*eps/(dist**2.0d0))* & ((sigma/dist)**6.0d0)*(1.0d0-2.0d0*((sigma/dist)**6.0d0))) return end ****************************************************************************** ****************************************************************************** The following funtion alulates the Lennard-Jones energy of the dipole with respet to another dipole. Typial Lennard-Jones energy is divided in half beause only half of the energy "belongs" to the partile under onsideration. The other half "belongs" to the partile it is interating with. funtion LJnrg(dist, sigma, eps) real*8 LJnrg, dist, sigma, eps LJnrg = 2.0D0*eps*(((sigma/dist)**12.0D0) -((sigma/dist)**6.0d0)) return end ******************************************************************************

96 87 ****************************************************************************** The following funtion alulates the attrative fore between the substrate and the partile. For purposes of determining the Hamaker onstant, it is assumed that the substrate is SiO2, the solution is hexane, and the partile is made of hematite. funtion FSubstrate(r, z) real*8 FSubstrate, r, z FSubstrate = ((-4.0D-21)/6.0D0)*(1.0D0/(z-r) - r/((z-r)**2.0d0) & - 1.0D0/(z+r) - r/((z+r)**2.0d0)) return end ****************************************************************************** ****************************************************************************** The following funtion alulates the Van der Waals potential energy between a partile and substrate. For purposes of determining the Hamaker onstant, it is assumed that the substrate is SiO2, the solution is hexane, and the partile is made of hematite. funtion VSubstrate(r,z) real*8 VSubstrate, r, z VSubstrate = ((-4.0D-21)/6.0D0)*(r/(z-r) + r/(z+r) & + dlog((z-r)/(z+r))) return end ****************************************************************************** ****************************************************************************** The following subroutine ensures that the partiles are interating with the partiles that are losest to them. Note that sign(a,b) returns the value of A with the sign of B. subroutine sep(distx, disty, distz, size) real*8 distx, disty, distz, size if (abs(distx).gt. 0.5D0*size) distx = distx - sign(size,distx) if (abs(disty).gt. 0.5D0*size) disty = disty - sign(size,disty) Routine eliminated in z diretion sine we only look at z values between z = 0 and z =300 if (abs(distz).gt. 0.5D0*size) distz = distz - sign(size,distz) return end ******************************************************************************

97 88 ****************************************************************************** The following funtion alulates a random number on a Gaussian distribution with zero mean and with width speified by the standard deviation (input to the funtion). funtion FluF(StdDev, idum) real*8 ranu(12), rang, R, a1, a3, a5, a7, a9, StdDev, FluF real*8 ran2 integer l2, idum Set values for polynomial oeffiients: a1 = D0 a3 = D0 a5 = D0 a7 = D0 a9 = D0 Initialize rang: rang = 0.0D0 do 43 l2 = 1, 12, 1 ranu(l2) = ran2(idum) rang = rang + ranu(l2) 43 ontinue R = (rang - 6.0D0)/4.0D0 rang = ((((a9*r**2.0d0+a7)*r**2.0d0+a5)*r**2.0d0+a3)* &R**2.0D0+a1)*R FluF = StdDev*ranG return end ****************************************************************************** ****************************************************************************** This program used a slightly modified version of a random number generation routine alled ran2 that was found in NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN X). Only the first line of the funtion is provided here. Please onsult this referene for full details of random number generation algorithms. FUNCTION ran2(idum)

98 89 A5.2. Two-Dimensional Stohasti Dynamis Algorithm to Model Partile Motion on the Substrate This program uses a Moleular Dynamis simulation to model the interations between N dipoles in 2 dimensions. It is a modified version of the three dimensional program designed to examine motion of the partiles on the substrate after all partiles have settled. All dipoles are able to move within a square of size 300 nm. The initial positions of the dipoles are read from a data file, while veloities are assigned randomly. Fore between the dipoles is modeled using the Lennard-Jones potential. After aeleration is alulated using the Lennard-Jones fore, a Leapfrog algorithm developed by van Gunsteren and Berendsen is used to alulate the next steps of position and veloity, taking into aount fritional and flutuating fores. The speifi Lennard-Jones fore used in this program was designed to model the Van der Waals partile-partile fore between hematite partiles in hexane solution, along with a steri fore between partiles that have a 5 nm thikness of ligand shell and 5 nm distane between head groups. This version of the program uses a random number generator reommended by the text "Numerial Methods in Fortran 77: The Art of Sientifi Computing." The user is asked to input a negative integer seed for the random number generator. This version also only prints out snapshots of the data. The user is asked to input the frequeny of snapshots taken. The user is also prompted to input the number of nse for whih the simulation should be run. Variables are defined as follows: r = radius of nanopartile (in meters) L = thikness of ligand shell mag = magnetization mu = magneti dipole moment mass = mass of dipoles pi = the onstant pi mu0 = the onstant mu0 T = temperature T (Kelvin) k = Boltzmann's onstant k ktimet = k*t sigma, eps = variables used to ompute Lennard Jones fore/potential DeltaT = time step size = size of 2D square where partiles are moving vis = visosity of fluid we are working with gamma = oeffiient used to alulate visous fore N = number of nanopartiles (input by user) real*8 r, L, mag, mu, mass, pi, mu0, T, k, ktimet

99 90 real*8 sigma, eps, DeltaT, size, vis, gamma integer N Set values for r, L, mag, T, sigma, eps, vis, DeltaT, and size: r = 10.0D-9 L = 5.0D-9 mag = D0 T = 298.0D0!298.0K Room temperature sigma = 30.85D-9 eps = 4.976D-23 vis = 2.97D-4!2.97D-4 Visosity of hexane DeltaT = 1.0D-10 size = 300D-9 Define values for onstants pi, mu0, and k: pi = D0 mu0 = 4.0D0*pi*1D-7 k = D-23 Request user input for number of partiles: print*, 'Enter Number of Partiles, N: ' read*, N Begin alulating various quantities, and then printing them to the sreen: ktimet = k*t print*, 'k * T = ', ktimet Calulate Magneti Dipole Moment mu: mu = mag*(4.0d0/3.0d0)*pi*r**3.0d0 print*, 'mu = ', mu Calulate mass of nanopartile. Assume partile is made of hematite, whih aording to the internet has a density of 5255 kg/m**3: mass = D0*(4.0D0/3.0D0)*pi*r**3.0D0 Calulate frition oeffiient for visous fore. gamma = 6.0D0*pi*vis*(r + L)/mass print*, 'frition oeffiient = ', gamma print*, 'mass = ', mass pause!pause to preview values printed to sreen thus far. all posvel(sigma, eps, mass, DeltaT, r, N, size, gamma, ktimet) pause stop end ****************************************************************************** The following subroutine exeutes a "do loop" that repeatedly alulates the new positions and veloities resulting from the Lennard-Jones fore ating between

100 91 two magneti dipoles. subroutine posvel(sigma, eps, mass, DeltaT, r, N, size, gamma, &ktimet) integer N, k1, k2, k3, k4, j1, j2, l1, l2, l3, idum, nsnaps integer ttime, ltime real*8 ran2, FluF, gamma, ktimet real*8 sigma, eps, LJFore, LJnrg, mass, time, DeltaT, r, size real*8 L, h real*8 rxn(n), rxnp1(n), vxnmhalf(n), vxnphalf(n) real*8 ryn(n), rynp1(n), vynmhalf(n), vynphalf(n) real*8 rzn(n), Fxn(N), Fyn(N), Fn(N) real*8 distx, disty, dist, T(N), V(N), Enrgy(N) real*8 gdt, gdthalf, Bgdthalf, Cgdthalf, Dgdthalf real*8 BMgdthalf, DMgdthalf, eq318, SDeq318 real*8 eq321, SDeq321, eq329, SDeq329, eq332, SDeq332 real*8 xxnmhalf(n), yxnmhalf(n) real*8 xyv(n), yyv(n), xvnmhalf(n), yvnmhalf(n) real*8 xvnhalf(n), yvnhalf(n), zvnhalf(n) real*8 xyx(n), yyx(n), xxneghalf(n), yxneghalf(n) real*8 xxhalf(n), yxhalf(n) real*8 TEnrgy, TKineti, TPoten real*8 TFx, TFy, Tvx, Tvy, TFore, Tvel real*8 vxum, vyum real*8 avex, avey, avez Open data file ontaining initial positions of partiles. Be sure to hange the name as required, depending upon where data is stored: open(20, FILE = 'part100.sv', STATUS = 'OLD') Ask user to input seed for ran2 random number generator funtion. print*, 'input idum (negative integer): ' read*, idum Ask user to input total amount of time (tenth of nse) simulation will be run: print*, 'input time to run simulation (tenth of nse): ' read*, ttime Ask user to input a number indiating how often data "snapshots" are taken. An inner do-loop runs from 1 to "nsnaps," and then data is output to a data file. print*, 'input how many loops to skip before snapshot: ' read*, nsnaps Calulate number of iterations of outer loop, based upon total time of simulation and number of snapshots: ltime = ttime/nsnaps Initialize time at 0.0: time = 0.0D0 Input values for thikness of ligand shell L and distane between heads h: L = 5.0D-9 h = 5.0D-9 Initial positions are read from a data file, while initial veloities are assigned randomly.

101 92 do 33 k2 = 1, N, 1 read(20,*) rxn(k2), ryn(k2), rzn(k2) Convert values to meters (from nanometers): rxn(k2) = rxn(k2)*1.0d-9 ryn(k2) = ryn(k2)*1.0d-9 rzn(k2) = rzn(k2)*1.0d-9 vxnmhalf(k2) = 2.0D0*(ran2(idum) - 0.5D0) vynmhalf(k2) = 2.0D0*(ran2(idum) - 0.5D0) 33 ontinue The veloities above may have a enter of mass motion. We don't want this beause the temperature (found using kineti energy) should not depend on enter of mass veloity. Therefore, the next setion alulates the average veloity in the x and y diretions and sets it to zero. vxum = 0.0D0 vyum = 0.0D0 do 50 k4 = 1, N vxum = vxum + vxnmhalf(k4) vyum = vyum + vynmhalf(k4) 50 ontinue vxum = vxum/n vyum = vyum/n do 60 k4 = 1, N Print initial positions and veloities to sreen; Before printing veloities, subtrat net motion in x and y diretions; also print out initial positions and veloities to fort.17 file: print*, 'rxn(', k4,') = ', rxn(k4) print*, 'ryn(', k4,') = ', ryn(k4) print*, 'rzn(', k4,') = ', rzn(k4) vxnmhalf(k4) = vxnmhalf(k4) - vxum vynmhalf(k4) = vynmhalf(k4) - vyum print*, 'vxnmhalf(', k4,') = ', vxnmhalf(k4) print*, 'vynmhalf(', k4,') = ', vynmhalf(k4) write(17,78) k4, rxn(k4)*1.0d9, ryn(k4)*1.0d9, rzn(k4)*1.0d9, &vxnmhalf(k4), vynmhalf(k4) 60 ontinue Add pause statement to preview initial onditions on sreen: pause Calulate some of the quantities that will be used in the van Gunsteren and Berendsen algorithm. gdt = gamma*deltat gdthalf = gdt/2.0d0 Cgdthalf = gdt - 3.0D D0*(dexp(-gdthalf)) - dexp(-gdt) Dgdthalf = 2.0D0 - dexp(gdthalf) - dexp(-gdthalf) Bgdthalf = gdt*(dexp(gdt) - 1.0D0) - 4.0D0*(dexp(gdthalf) - &1.0D0)**2.0D0 BMgdthalf = gdt*(-dexp(gdt) - 1.0D0) - 4.0D0*(dexp(-gdthalf) - &1.0D0)**2.0D0 DMgdthalf = 2.0D0 - dexp(-gdthalf) - dexp(gdthalf) eq318 = ktimet*cgdthalf/(mass*gamma**2.0d0)

102 93 SDeq318 = dsqrt(eq318) eq321 = ktimet*bgdthalf/(mass*cgdthalf) SDeq321 = dsqrt(eq321) eq329 = ktimet*(1.0d0 - dexp(-gdt))/mass SDeq329 = dsqrt(eq329) eq332 = ktimet*bmgdthalf/(mass*(dexp(-gdt) - 1.0D0)* &gamma**2.0d0) SDeq332 = dsqrt(eq332) Print quantities obtained above and pause to hek results: print*, 'gdt = ', gdt print*, 'gdthalf = ', gdthalf print*, 'Bgdthalf = ', Bgdthalf print*, 'Cgdthalf = ', Cgdthalf print*, 'Dgdthalf = ', Dgdthalf print*, 'BMgdthalf = ', BMgdthalf print*, 'DMgdthalf = ', DMgdthalf print*, 'eq318 = ', eq318 print*, 'SDeq318 = ', SDeq318 print*, 'eq321 = ', eq321 print*, 'SDeq321 = ', SDeq321 print*, 'eq329 = ', eq329 print*, 'SDeq329 = ', SDeq329 print*, 'eq332 = ', eq332 print*, 'SDeq332 = ', SDeq332 pause Calulate initial values for xxnmhalf(n), yxnmhalf(n): do 11 k3 = 1,N xxnmhalf(k3) = FluF(SDeq318, idum) yxnmhalf(k3) = FluF(SDeq318, idum) 11 ontinue Write initial positions and veloities to sreen and to fort.18 file: write(*,77) time, (rxn(k1), ryn(k1), rzn(k1), vxnmhalf(k1), &vynmhalf(k1), k1=1,n) 77 format (' ', 350(1p1e15.6)) write(18,*) 'time ','avex ','avey ','avez ','Tvx ','Tvy ', &'TFx ', 'TFy ', 'TKineti ', &'TPoten ','TFore ', 'Tvel ', 'TEnrgy ' Purpose of outer do-loop is to print out the results to the sreen after "nsnaps" nse have elapsed. do 41 l2 = 1, ltime, 1 Inner do-loop alulates positions and veloities for "nsnaps" nse. do 42 l1 = 1, nsnaps, 1

103 94 Enter do loop to update fores for all partiles. Reset Total Energy, Total/Net Fore (all partiles), Total/Net veloity (all partiles), Total Kineti/Potential Energies, Total (sum of) Fores, and Total (sum of) veloities at 0.0 prior to entering this do loop. TEnrgy = 0.0D0 TFore = 0.0D0 Tvel = 0.0D0 TKineti = 0.0D0 TPoten = 0.0D0 TFx = 0.0D0 TFy = 0.0D0 Tvx = 0.0D0 Tvy = 0.0D0 do 3 j1 = 1, N, 1 Initialize values of fores, potential energy, and kineti energy at 0.0 before entering inner do loop: Fn(j1) = 0.0D0 Fxn(j1) = 0.0D0 Fyn(j1) = 0.0D0 T(j1) = 0.0D0 V(j1) = 0.0D0 do 8 j2 = 1, N, 1 if (j1.eq.j2) goto 8 distx = rxn(j2) - rxn(j1) disty = ryn(j2) - ryn(j1) all sep(distx, disty, size) dist = dsqrt(distx**2.0d0 + disty**2.0d0) Fn(j1) = LJFore(dist, sigma, eps) Fn(j1) = 0.0D0 Fxn(j1) = Fxn(j1) + Fn(j1)*(distx/dist) Fyn(j1) = Fyn(j1) + Fn(j1)*(disty/dist) V(j1) = V(j1) + LJnrg(dist,sigma,eps) V(j1) = V(j1) + 0.0D0 8 ontinue 3 ontinue Calulate half step veloities and new value of position. Comments in the following do-loop desribe several interim steps. do 15 k3 = 1,N Sample values of xyv(n) and yyv(n) from a Gaussian distribution, and then alulate xvnmhalf(n) and yvnmhalf(n) per step 3, page 180, of paper. xyv(k3) = FluF(SDeq321, idum) xvnmhalf(k3) = xxnmhalf(k3)*gamma*dgdthalf/cgdthalf + xyv(k3) yyv(k3) = FluF(SDeq321, idum) yvnmhalf(k3) = yxnmhalf(k3)*gamma*dgdthalf/cgdthalf + yyv(k3)

104 95 Sample values of xvnhalf(n) and yvnhalf(n) from a Gaussian distribution, and then alulate the half-step veloities from equation 3.6: xvnhalf(k3) = FluF(SDEq329, idum) vxnphalf(k3) = vxnmhalf(k3)*dexp(-gdt) + Fxn(k3)*DeltaT* &(1.0D0 - dexp(-gdt))/(mass*gdt) + xvnhalf(k3) - dexp(-gdt)* &xvnmhalf(k3) yvnhalf(k3) = FluF(SDEq329, idum) vynphalf(k3) = vynmhalf(k3)*dexp(-gdt) + Fyn(k3)*DeltaT* &(1.0D0 - dexp(-gdt))/(mass*gdt) + yvnhalf(k3) - dexp(-gdt)* &yvnmhalf(k3) Finally, xyx(n) and yyx(n) are sampled from Gaussian distributions. Then the random variables xxneghalf(n) and yxneghalf(n) are alulated. xxhalf(n) and yxhalf(n) are sampled from Gaussian distributions. From all of these variables, the positions at the next step are alulated. xyx(k3) = FluF(SDeq332, idum) xxneghalf(k3) = xvnhalf(k3)*dmgdthalf/(gamma*(dexp(-gdt) - &1.0D0)) + xyx(k3) xxhalf(k3) = FluF(SDeq318, idum) rxnp1(k3) = rxn(k3) + vxnphalf(k3)*deltat*(dexp(gdthalf) - &dexp(-gdthalf))/gdt + xxhalf(k3) & - xxneghalf(k3) yyx(k3) = FluF(SDeq332, idum) yxneghalf(k3) = yvnhalf(k3)*dmgdthalf/(gamma*(dexp(-gdt) - &1.0D0)) + yyx(k3) yxhalf(k3) = FluF(SDeq318, idum) rynp1(k3) = ryn(k3) + vynphalf(k3)*deltat*(dexp(gdthalf) - &dexp(-gdthalf))/gdt + yxhalf(k3) & - yxneghalf(k3) Print interim values to sreen as desired for troubleshooting: print*, 'xyv(',k3,') = ',xyv(k3) print*, 'xvnmhalf(',k3,') = ',xvnmhalf(k3) print*, 'xvnhalf(',k3,') = ',xvnhalf(k3) print*, 'vxnphalf(',k3,') = ',vxnphalf(k3) print*, 'xyx(',k3,') = ',xyx(k3) print*, 'xxneghalf(',k3,') = ',xxneghalf(k3) print*, 'xxhalf(',k3,') = ',xxhalf(k3) print*, 'rxnp1(',k3,') = ',rxnp1(k3) print*, ' ' print*, 'yyv(',k3,') = ',yyv(k3) print*, 'yvnmhalf(',k3,') = ',yvnmhalf(k3) print*, 'yvnhalf(',k3,') = ',yvnhalf(k3) print*, 'vynphalf(',k3,') = ',vynphalf(k3) print*, 'yyx(',k3,') = ',yyx(k3) print*, 'yxneghalf(',k3,') = ',yxneghalf(k3)

105 96 print*, 'yxhalf(',k3,') = ',yxhalf(k3) print*, 'rynp1(',k3,') = ',rynp1(k3) print*, ' ' print*, 'zyv(',k3,') = ',zyv(k3) print*, 'zvnmhalf(',k3,') = ',zvnmhalf(k3) print*, 'zvnhalf(',k3,') = ',zvnhalf(k3) print*, 'vznphalf(',k3,') = ',vznphalf(k3) print*, 'zyx(',k3,') = ',zyx(k3) print*, 'zxneghalf(',k3,') = ',zxneghalf(k3) print*, 'zxhalf(',k3,') = ',zxhalf(k3) print*, 'rznp1(',k3,') = ',rznp1(k3) print*, ' ' 15 ontinue Pause to preview interim values: pause do 19 k1 = 1,N Calulate Final Kineti energy for eah partile, Total Kineti energy of all partiles, Total potential energy of all partiles, Total fore (sum of all fores) in x and y diretions, Total veloity (sum of all veloities) in x and y diretions, and Total Energy of all partiles. T(k1) = (0.5D0)*mass*(vxnPhalf(k1)**2.0D0 + &vynphalf(k1)**2.0d0) TKineti = TKineti + T(k1) TPoten = TPoten + V(k1) TFx = TFx + Fxn(k1) TFy = TFy + Fyn(k1) Tvx = Tvx + vxnphalf(k1) Tvy = Tvy + vynphalf(k1) For now, the following two lines are omment lines. If added bak in, the lines provide a method to alulate the total energy of eah partile, and also a different method to alulate total energy. Enrgy(k1) = T(k1) + V(k1) TEnrgy = TEnrgy + Enrgy(k1) Chek to make sure the newly alulated positions are not outside of the box. If they are, use the modulus funtion to bring the partile in at the other side of the box: if (rxnp1(k1).gt.size) then rxnp1(k1) = dmod(rxnp1(k1),size) elseif (rxnp1(k1).lt.0.0d0) then rxnp1(k1) = rxnp1(k1) + size endif if (rynp1(k1).gt.size) then rynp1(k1) = dmod(rynp1(k1), size) elseif (rynp1(k1).lt.0.0d0) then rynp1(k1) = rynp1(k1)+ size endif Now, set "final" values equal to "initial" values to be used

106 97 during next iteration: 4 rxn(k1) = rxnp1(k1) ryn(k1) = rynp1(k1) xxnmhalf(k1) = xxhalf(k1) yxnmhalf(k1) = yxhalf(k1) vxnmhalf(k1) = vxnphalf(k1) vynmhalf(k1) = vynphalf(k1) print*, 'T(',k1,') = ', T(k1) 19 ontinue time = time + DeltaT 42 ontinue Calulate total fore, net veloity of all partiles, and total energy for those ases where data is output to sreen and fort.10: TFore = (TFx**2.0D0 + TFy**2.0D0) Tvel = (Tvx**2.0D0 + Tvy**2.0D0) TEnrgy = TKineti + TPoten Calulate average values of x, y, and z for all partiles. avex = 0.0D0 avey = 0.0D0 avez = 0.0D0 do 1 j1 = 1, N avex = avex + rxn(j1) avey = avey + ryn(j1) avez = avez + rzn(j1) 1 ontinue avex = avex/n avey = avey/n avez = avez/n Write results to sreen: write(*,77) time, avex, avey, avez, Tvx, Tvy, TFx, TFy, &TKineti, TPoten, TFore, Tvel, TEnrgy Write same quantities to a fort.18 file. Time in nanoseonds and distane in nanometers. write(18,77) time*1.0d9, avex*1.0d9, avey*1.0d9, &avez*1.0d9, Tvx, Tvy, TFx, TFy, TKineti, TPoten, &TFore, Tvel, TEnrgy 41 ontinue Output final x, y, and z oordinates to fort.19 file: 90 do 2 j1 = 1, N write(19,78) j1, rxn(j1)*1.0d9,ryn(j1)*1.0d9, rzn(j1)*1.0d9, &vxnmhalf(j1), vynmhalf(j1) 2 ontinue 78 format (' ', i3, 5(1p1e15.6)) return end ******************************************************************************

107 98 The following funtion is used to alulate the Lennard-Jones Fore. funtion LJFore(dist, sigma, eps) real*8 LJFore, dist, sigma, eps LJFore = dist*((24.0d0*eps/(dist**2.0d0))* & ((sigma/dist)**6.0d0)*(1.0d0-2.0d0*((sigma/dist)**6.0d0))) return end ****************************************************************************** ****************************************************************************** The following funtion alulates the Lennard-Jones energy of the dipole with respet to another dipole. Typial Lennard-Jones energy is divided in half beause only half of the energy "belongs" to the partile under onsideration. The other half "belongs" to the partile it is interating with. funtion LJnrg(dist, sigma, eps) real*8 LJnrg, dist, sigma, eps LJnrg = 2.0D0*eps*(((sigma/dist)**12.0D0) -((sigma/dist)**6.0d0)) return end ****************************************************************************** ****************************************************************************** The following subroutine ensures that the partiles are interating with the partiles that are losest to them. Note that sign(a,b) returns the value of A with the sign of B. subroutine sep(distx, disty, size) real*8 distx, disty, size if (abs(distx).gt. 0.5D0*size) distx = distx - sign(size,distx) if (abs(disty).gt. 0.5D0*size) disty = disty - sign(size,disty) return end ****************************************************************************** ****************************************************************************** The following funtion alulates a random number on a Gaussian distribution with zero mean and with width speified by the standard deviation (input to the funtion). funtion FluF(StdDev, idum) real*8 ranu(12), rang, R, a1, a3, a5, a7, a9, StdDev, FluF

108 99 real*8 ran2 integer l2, idum Set values for polynomial oeffiients: a1 = D0 a3 = D0 a5 = D0 a7 = D0 a9 = D0 Initialize rang: rang = 0.0D0 do 43 l2 = 1, 12, 1 ranu(l2) = ran2(idum) rang = rang + ranu(l2) 43 ontinue R = (rang - 6.0D0)/4.0D0 rang = ((((a9*r**2.0d0+a7)*r**2.0d0+a5)*r**2.0d0+a3)* &R**2.0D0+a1)*R FluF = StdDev*ranG return end ****************************************************************************** ****************************************************************************** This program used a slightly modified version of a random number generation routine alled ran2 that was found in NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING (ISBN X). Only the first line of the funtion is provided here. Please onsult this referene for full details of random number generation algorithms. FUNCTION ran2(idum)

109 APPENDIX 6 ADDITIONAL FIGURES This appendix provides supplemental bakground figures and diagrams for topis disussed in the main body of this report. A6.1. Initial Partile Positions for Systems with Various Numbers of Partiles Figure 28. Initial positions for 20 partiles. The partiles are shown by balls and the olors are a guide to the eye to demonstrate the various heights in the z diretion.

110 101 Figure 29. Initial positions for 50 partiles. The partiles are shown by balls and the olors are a guide to the eye to demonstrate the various heights in the z diretion. A6.2. Additional Figures Supporting Deposition Time and Rate Analysis Figure 30. Derease in z average for all partiles over time for a system of 20 partiles ated upon by Van der Waals substrate fore, Lennard-Jones interpartile fore, gravity and visous and flutuating fores.

111 102 Figure 31. Derease in z average for all partiles over time for a system of 80 partiles ated upon by Van der Waals substrate fore, Lennard-Jones Interpartile fore, gravity and visous and flutuating fores. Figure 32. Perent overage for a system ontaining 50 partiles.

112 103 Figure 33. Perent overage for a system ontaining 80 partiles. A6.3. Additional Figures Supporting Pattern Formation Analysis Figure 34. Final positions of partiles on substrate for a 20 partile system ated upon by Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles able to move after reahing substrate. Size of partile equal to hematite nanopartile.

113 104 Figure 35. Final positions of partiles on substrate for a 20 partile system ated upon by Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles able to move after reahing substrate. Size of partile equal to hematite nanopartile plus thikness of ligand shell. Figure 36. Final positions of partiles on substrate for a 50 partile system ated upon by Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles able to move after reahing substrate. Size of partile equal to hematite nanopartile plus thikness of ligand shell.

114 105 Figure 37. Final positions of partiles on substrate for a 20 partile system where partiles fell under influene of gravity, Van der Waals substrate fore, Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles were loked in plae upon reahing substrate. Size of partile equal to hematite nanopartile. Figure 38. Final positions of partiles on substrate for a 20 partile system where partiles fell under influene of gravity, Van der Waals substrate fore, Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles were loked in plae upon reahing substrate. Size of partile equal to hematite nanopartile plus thikness of ligand shell.

115 106 Figure 39. Final positions of partiles on substrate for a 50 partile system where partiles fell under influene of gravity, Van der Waals substrate fore, Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles were loked in plae upon reahing substrate. Size of partile equal to hematite nanopartile. Figure 40. Final positions of partiles on substrate for a 50 partile system where partiles fell under influene of gravity, Van der Waals substrate fore, Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles were loked in plae upon reahing substrate. Size of partile equal to hematite nanopartile plus thikness of ligand shell.

116 107 Figure 41. Final positions of partiles on substrate for an 80 partile system where partiles fell under influene of gravity, Van der Waals substrate fore, Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles were loked in plae upon reahing substrate. Size of partile equal to hematite nanopartile. Figure 42. Final positions of partiles on substrate for a 80 partile system where partiles fell under influene of gravity, Van der Waals substrate fore, Lennard-Jones interpartile fores, visous, and flutuating fores. Partiles were loked in plae upon reahing substrate. Size of partile equal to hematite nanopartile plus thikness of ligand shell.