Molecular simulation of adsorption

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Content MOLECULAR SIMULATION OF ADSORPTION by Anu Deep Vij A Thesis Presented to the FACULTY OF THE SCHOOL O F ENGINEERING UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree M ASTER OF SCIENCE (Chemical Engineering) August 1994 UMI Number: EP41783 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI EP41783 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 This thesis, written by AMU .£&££ .Via.............. under the guidance of Faculty Committee and approved by all its members, has been presented to and accepted by the School of Engineering in partial fulfillment of the re quirements for the degree of . M a s t e r o£ cc if n >' - { • : x n .... .CH.EB.I..CAL &inMME.£.RJ.N&...-... Date. . . . a u . w i . . . 6 . , . . j M k ....... Faculty Qjommittee hairman Ch ACKNOWLEDGEMENTS I am thankful fo r the guidance provided by ‘ Dr. ‘ Jfatherine Shing in writing tfis thesis. It was pleasure Being associated with her. I am indebted to her for knowledge, wisdom and confidence. I am thankful to her fo r helping me in the computational aspects o f the problem. I am thankful to her for introducing me “ Cifonte CarCo ‘ Technique ” . I acknowledge the financial support o f the Chemical ‘ Engineering Department in compfeting this work: I am thankfuC to my parents, brother and sister for constant encouragement. Do all these above peopfe and many more, my eternal'gratitude. Table Of Contents Acknowlegements ii i) Introduction 1 ii) Summary of Literature Survey 3 iii) Formulation o f Program 9 iv) Result and Discussion 13 v) Conclusion and Recommendation for future work 19 vi) References 20 APPENDIX 22 GRAPHS: 1(a) Number of particles vs Configuration plot for C2 H4/C system 23 1(b) Number of particles vs Configuration plot for Kr/C system 24 1(c) Coverage vs Pressure plot for Kr/C system at T*=2.0 25 1(d) Coverage vs Pressure plot for Kr/C system at T*=2.37 26 1(e) Density profile for Kr/C system 27 1 1(a) Density profile for CH4 -AJ/C sytem 28 1 1(b) Density profile for CHj-Ar/C sytem with change in Ba and Bb 29 11(c) Density profile for C H ^A r/C sytem with change in Ba and Bb 30 H(d) Density profile for CH^-Ar/C sytem with change in T* 31 11(e) Density profile for CH4 -AT/C sytem with change in LJ parameters 32 11(f) Density profile for CH 4 -X/C sytem 33 TABLES 34 iii i) Lennard Jones Parameters for C2 H4 /C system 35 ii) Simulation parameters for C2 H4 (a)/C system 36 iii) Lennard Jones Parameters for Kr/C system 37 iv) Simulation parameters Kr(a)/C system 38 v) Simulation results for Kr(a)/C system at T*=2.0 39 vi) Simulation results for Kr(a)/C system at T*=2.37 40 vii) Lennard Jones parameters for CHj-Ar/C system 41 viii) Simulation parameters for CH4 (a)-Ar(b)/C system 42 ix) Simulation results for CH4 (a)-Ar(b)/C system 43 x) Simulation parameters for CH4 (a)-Ar(b)/C system 44 xi) Simulation results for CH4 (a)-Ar(b)/C system at Ba=2.5, Bb=2.5 45 xii) Simulation results for CH4 (a)-Ar(b)/C system at Ba=2.5. Bb=2.0 46 xiii) Simulation parameters for CH4 (a)-Ar(b)/C system 47 xiv) Simulation results for CH4 (a)-Ar(b)/C system at T*=2.5 48 xv) Lennard Jones Parameters for A-B/C system 49 xvi) Simulation results for CH4 (a)-Ar(b)/C system at New LJ Parameters 50 xvii) Lennard Jones Parameters for CH4 -X/C system 51 xviii) Simulation results for CH4 (a)-Ar(b)/C system at New LJ Parameters 52 FIGURES 53 i) Typical Adsorbate-Adsorbent System 54 ii) Grand Canonical Ensemble 55 iii) Simulation Box 56 iv) Flow Chart 57 v 1.Introduction Adsorption is a physical process. Besides its extensive use in chemical industry, in recent years it has found wide application in the environmental field particularly haz ardous waste management. For example, a waste containing heavy metals like Hg, Pb, Cd, As, Se etc. can’t be discharged without treating due to stringent environmental regulations. Traditional methods like incineration fails to deliver goods as the heavy metals being volatile will go out as gas and thereby creating air-pollution problem. Hence one is forced to use some air emission con trol equipment after incineration process and thereby increasing the cost. Under such situation adsorption is the best alternative. Thermodynamically adsorption frequently takes place in an open system. The adsorbate may be as simple as a single component or as complex as a mixture of components and similarly adsorbent may have homogeneous or heterogeneous surface. All these coupled together to make adsorption a complex system. Progress towards a satisfactory theory of physical adsorption, particularly for multilayer region, has been rather slow. Two reasons can be cited for this.[l] Firstly, a general theory must allow for a spatial variation in the adsorbate density normal to adsorbent surface which can range from solid-like to gas-like over a distance of a few molecular diameters. Although general formulations of the problem in which adsorbate is considered to be a fluid in an external field have been developed [2-5], no explicit solution for these statistical equations have been presented. Other rigorous studies of various aspects of the problem have been undertaken [6 - 8 ] but none have been advanced to a stage where contact with experimental data can be made. Secondly, difficulty arises on the experimental side because adsorption measurements are notoriously subject to artifacts beyond the control of experimenter. Among these are: [1 ] i) Surface heterogeneity ii) The possible existence of microporous and of interparticle condensation iii) Uncertainties concerning the intermolecuiar forces Most of the work done in this field is experimental i.e experiment is per formed for a particular adsorbate adsorbing on a particular adsorbent under certain conditions. But experimental work will solve only the problem in hand i.e in other words it is not easy to extrapolate the experimental data. At the most from experimental data one can formulate some empirical correlations suitable in some range of values. But to have a thorough understanding of the system one has to go to molecular level and carry out the molecular simulation of the system in hand. After a preliminary literature survey it was found that not enough work is done in Molecular simulation of adsorption i.e most of the work done is by carrying out the simu 1 lation of pure adsorbate on homogeneous adsorbent. Hence more work is required in this field in actual cases we may deal with mixture of adsobate adsorbing on heterogeneous adsorbent. 2. Summary Of Literature Survey 2.1 Objective And Conclusions The main objective of carrying out literature survey was to find answer to the following ques- tions:- (i) Which simulation methods are used? (ii) What type of adsorbate is studied? (iii) What type of adsorbent is studied? (iv) How well is competitive adsorption understood? (v) What important conclusions can be drawn? As far as simulation methods are concerned, it was found that following methods are used extensively in most papers:- (i) Molecular Dynamics Simulation (ii) Monte Carlo Simulation Molecular Dynamics simulation is a dynamic approach i.e one considers a classical dynamical models for atoms and molecules and trajectory is formed by integrating Newton’ s equation of motion. The procedure thus provides dynamical information as well as equilibrium statistical properties. Any thermodynamic property ‘G ’ can be written as: Monte Carlo simulation is a statistical approach i.e it does not provide straight forward method of obtaining time-dependent dynamical information. Instead of calculating time average of thermodynamic properties which was the'case in Molecular Dynamics Simulation, we consider ensemble average. An ensemble is the assembly of all possible microstates- all states consistent with the constraint with which we characterize the system macroscopically. It has been suggested that constant particle number Monte Carlo calculations which uses simple particle translations to sample configuration space (e.g. isobaric-isothermal and canonical ensemble methods) are not appropriate for adsorption studies because no single simula tion will adequately sample both high and low density regions. Primarily for this reason Grand Canonical Ensemble Monte Carlo method has been preferred for adsorption studies [9]. It has the advantage of using panicle creation and destructions, in addition to particle transla time 1 time Where <G> is known as the time average of G. 3 tions, to sample configuration space. For ‘N ’ measurements on the system, any thermodynamic property ‘G ’ can be written as: N G o b s = S C a 2 a = 1 The above equation can be written as: Where is the number of times state y is observed in the ‘N ’ observations. If By is the probability of observing state y, which can be approximated by T./N then the above equation can be written as: Where <G> is known as the ensemble average of G. Monte Carlo procedure is more generally applicable than molecular dynamics as it can be used to study quantum systems and lattice models as well as classical assemblies of molecules. 2.2 Molecular Dynamics Simulation The following important conclusions were drawn concerning Molecular Dynamic Simulation after studying the papers. Steele et ai have presented a molecular model and simulation technique for the physical adsorp tion of N2 on graphite carbon black (gcb) and has also carried out a study of thermodynamic and structural properties of N2 adsorbed on gcb. [ 1 0 ] Although the desorption of physisorbed atoms at low temperature is a relatively simple activated process, it is still only poorly understood. It is possible to study the desorption dynamics in detail for an ideal surface and low coverage using molecular dynamics. Jansen has presented the results of classical molecular dynamic simulation at low temperature of the desorption of Xe atoms from Pd or Pt. The effect of corrugation, coverage dependence and the effect of steps are studied.[11] Computer simulations of nitrogen adsorbed on the graphite basal plane at 73.6 K are reported for coverages ranging from ~ 1 monolayer to ~ 2.5 layers by Vemov and Steele. Quantities evaluated include average potential energies and the component molecule-solid and molecule-molecule 4 7 4 parts of these energies, translational and orientational order parameters of N2 molecules at the sur face, and molecular pair correlation functions projected onto planes parallel to the surface. Com parison with available data are good.[1 2 ] Pickett et al have reported that results from a molecular dynamics simulation of xenon in silicate at 298 K and 4 atoms per unit cell (AUacjs = -26.9 kJ/mol, D = 1.86 x 10' 9 m^/sec) are in good agreement with the experimental value of -24.5 kJ/mol and the diffusion coefficient derived from the NMR pulsed field gradient method (4.00 x 10' 9 m~/sec).[13] Hufton has calculated thermodynamics and transport properties of methane adsorbed in the zeo lite silicalite from a NVT molecular dynamics algorithm. Methane was modeled as a sphere, with potential parameters determined by comparison of theoretical and experimental Henry’ s con stants. The average potential energy and self-diffusion coefficient o f the spherical adsorbate agreed with previous MD results of June et al. obtained with a five-point methane model. Agree ment was also obtained with experimental data. [14] The phase behavior of a fluid in mesopores just below the critical temperature has been studied experimentally and by Molecular dynamics simulation by Keizer et al. The adsorption of SFg on Controlled-pore glass(CPG) for reduced temperatures Tr = T/Tc of 0.857,0.920 and 0.985. Molecular dynamics simulation have been performed for a system of Lennard-Jones molecules, where the parameters values chosen simulated the SFg-CPG system. Good agreement is found with the experimental findings.[15] Joshi et al have studied the structural properties of O2 adsorbed on a graphite surface using molec ular dynamic simulation and energy minimization techniques with different sets of interaction parameters. The Van der Waals interaction is modelled using four site-site interactions. The qua- drupole moment is modelled by distributing partial chaiges on the molecular axis. The magnetic interaction is not considered. [16] 2-3 Monte Carlo Simulation The following important conclusions were drawn concerning Monte Carlo Simulation after study ing the papers. Cortes et al have presented good agreement between theoretical models introduced by Hill and later by Rudzinsky and Monte Carlo simulation carried out by the authors for the adsorption of a gas on heterogeneous solids characterized by energy distribution and random topography of the superficial sites. [17] Russier et al have investigated the influence of adsorption potential on the structure of a liquid (described as dipolar hard sphere fluid) in the vicinity of a smooth hard wall. They have also reported Monte Carlo simulation results for density profile, the orientation structure, the polariza tion and the electrical potential profiles and make comparisons with the theoretical results obtained from integral equations theories.[18] 5 Any system consisting of adsorbent pius adsorbate constitutes a mixture and therefore poses par ticularly difficult problems for anyone attempted a detailed explanation o f its structures and dynamics because one must not only provide an accurate representation of the interactions among the adsorbate molecules but also a balanced account of the adsobate-adsorbent interactions. In the present case little is known about the interactions among chlorine molecules, but essentially noth ing is known about the chlorine-graphite interactions. Suh and O ’Shea have made an attempt at explaining the structure and dynamics of chlorine adsorbed on graphite.[19] Lane and Spurting have carried out a Monte Carlo simulation for the study of forces between two adsorbing walls for a system in which adsorbate is in equilibrium with either a bulk gas or a liquid phase. The force is correlated with the adsorption, singlet and radial distribution functions, all of which vary with the distance of separation of the walls. [2 0 ] The complex dependence of the desorption spectra on surface coverage has been attributed to two principal causes, namely lateral interactions among adsorbed molecules and surface heterogene ity. Gupta and Hirtzel have used Monte Carlo simulation to study thermal desorption of gas mole cules from mixed adlayers containing two species. The effects of lateral interactions among adatoms on the desorption spectra are discussed.[2 1 ] Knight and Monson have studied Monte Carlo computer simulation of monolayer adsorption of a gas on a solid surface in which potential distribution method was used to directly calculate the adsorption isotherms. The system used for study was two-dimensional, Lennard-Jones fluid and such a fluid in an external field.[2 2 ] Zeolites are microporous crystalline materials usually consisting of silicon, aluminum, oxygen and sodium. Recent improvements in the design of zeolite catalysts suggest that much higher selectivity may be achieved in near future. It is difficult to study the behavior of molecules inside the pores of a zeolite from direct experiments. Smit and den Ouden have reported Monte Carlo computer simulations on the adsorption of methane in the zeolites faujasite, mordenite and ZSM- 5.[23] Finn and Monson have used Monte Carlo method based on an isobaric-isothermal ensemble to study adsorption at fluid-solid interfaces. After reviewing the Monte Carlo algorithm and its gen eral applications to studies of fluid-solid interfaces, the authors have presented a study of the tran sition from gas adsorption to complete wetting of a surface by a pure fluid. The model system used is a 9-3 argon/solid carbon dioxide potential for the gas-solid interactions. [24] Nelson et al have shown that Poisson-distributed event times are required to correctly simulate diffusion and adsorption in Zeolites. The main microscopic assumptions of the diffusion model are: the zeolite is a periodic array of sites, each o f which may contain only one molecule: and transport is achieved by sorbate molecules randomly jumping from one site to an adjacent site. The simulation technique is also extended to independent Poisson-distributed events of more than one type.[25] 6 Van Damme et al have simulated the adsorption of homologous series of molecules on determin istic fractal curves (fractals generated by a self-similar iteration process). Three series of mole cules were used: (i) circles of increasing diameter, r (ii) rectangles of increasing height, t (iii) rectangles of high aspect ratio and increasing length. 1 . The results apply to particular class of fractals that the authors have considered i.e those for which the fractal character is restricted to their boundary. Fractals which stem from a particular internal morphology leads to a different behavior. [26] Razmus and Hall have studied the adsorption of air in 5 A zeolites using Monte Carlo simulations in the grand canonical ensemble. Site-site potentials were used to model the adsorbate-zeolite and adsorbate-adsorbate interactions. The results for pure component isotherms are in excellent agree ment with experimental data. The results for multicomponent adsorption isotherm are qualita tively correct, however, the simulation was not able to quantitatively predict mixture data. [27] Bladon and Higuchi have first concluded that the group interaction model and lattice adsorption model produces a good simulation of the experimental adsorption data of n-alkyl alcohols adsorb ing out o f hexane and dodecane onto CaF2 when all the segment-to-segment interaction energies are the same at the surface as they are in the bulk, then they have used these two lattice models to simulate the experimental adsorption data of n-alkyl amines and alkyl diamines in variety of sol vents in an effort to explore the consistency of this approach.[28] Van Megen and Snook have used the grand canonical ensemble Monte Carlo Method to simulate the behaviour of Lennard-Jones gas near a graphite surface. The results reproduced the shape of the experimentally observed adsorption isotherm. [29] 2.4 Type of Adsorbate And Adsorbent Adsorbate and adsorbent constitutes a mixture in adsorption process and therefore poses difficult problems for anyone attempted a detailed explanation of its structures and dynam ics because one must not only provide an accurate representation of the interactions among the adsorbate molecules but also a balanced account of the adsorbate-adsorbent interactions. A typi cal diagram is shown in Figure 1. In most of the cases, little is known about the interactions among adsorbate molecules, but essentially nothing is known about adsorbate-adsorbent interac tions. It has been found from literature survey that both liquid and gas adsorbate are consid ered, but in most cases pure adsorbate is considered. Similarly, for adsorbent most of the papers dealt with ideal surfaces i.e homogeneous surface. In most of papers, the following combination of adsorbate-adsorbent system are considered: (i) Single adsorbate and homogeneous surface adsorbent. (ii) Single adsorbate and heterogeneous surface adsorbent (iii) Two component adsorbate and homogeneous surface adsorbent. Case like two component adsorbate and heterogeneous adsorbent is not considered in papers studied so far. 7 TVpical examples considered in papers for adsorbate-adsorbent system are as follows: (i) C h adsorbed on Graphite. (ii) N2 adsorbed on Graphite. (iii) Air adsorbed on Zeolite. (iv) CH4 adsorbed on Zeolite. (v) Xe adsorbed on Silicate. (vi) O2 adsorbed on Graphite. In most of cases, adsorbate is considered as a spherical molecule. 2.5 Competitive Adsorption As far as competitive adsorption is concerned, it has been found that very few studies have been performed and it is not clearly understood for the cases of heterogeneous surface adsor bent and component mixture adsorbate. Papers that have dealt with competative adsorption along- with important conclusions are given below: Murli and Aylmore have presented the computer simulations of the effects of Langmuir and Fredlich type competition in a binary system under equilibrium and dynamic conditions. Theses simulations indicate that during competitive adsorption, the presence of a competing species can have a marked effect in depressing the equilibrium isotherm of an adsorbing species to an extent determined by the relative solution concentrations, distribution and selectivity coefficients etc. [30] Schwarz and Smith have examined the adsorption inhibition due to the unavailability of a specific geometric arrangement of adsorption sites by Monte Carlo simulation. When two species having different site requirements for chemisorption compete for available empty sites, there is reduction in the ratio of their concentrations on the fully covered lattice. This reduction increases as the interaction strength between adsorbate increases. The authors have applied this premise to the competitive adsorption of dual- and single- site species. The example used in understanding this is: H 2(dual) + CO(single) --> CH4.[31] Lemos and Cordoba have presented mutually consistent results obtained from Doublet closure approximation and a Monte Carlo simulation for the adsorption-desorption o f two types of inter acting particles on a linear chain. In this model, multistability of steady states does not exist, inspite of interactions between adatoms. This outcome disagrees with that of a Mean field approx imation (MFA), where multistability can appear.[32] 8 3. Formulation Of Program We want to study the Adsorption process and will be using Grand Canonical Ensemble Monte Carlo Simulation method. More precisely, our aim is to study competitive adsorption which so far is not very clearly understood as is found from literature survey. We will now focus on why we have chosen this method? In microcanonical ensemble, each system is enclosed in a container whose walls are neither heat conducting nor permeable to passage of of molecules. In canonical ensemble, each system is enclosed in a container whose walls are heat conducting but impeimeable to passage of mole cules. Thus, conceptually the canonical ensemble corresponds more closely to physical situations. In experiments we never deal with a completely isolated system nor do we directly measure the total energy of a macroscopic system. We usually deal with systems with given temperature - a parameter that we can control in experiments. By the same token we should not have to specify exactly the number of particles of a macroscopic open system, for that is never precisely known. All we can find out from experiments is the aver age number of particles. This is the motivation for introducing grand canonical ensemble. For many experimental systems conducted under phase equilibrium conditions, it is desirable to use the chemical potential as a controlling variable since it specifies the environmental conditions the sytem is subjected to. For example, in adsorption studies, the adsorption isotherm is determined from the equality of the chemical potentials of the bulk phase and those of the adsorbed phase. In grand canonincal ensemble, each system is enclosed in a container whose walls are both heat conducting and permeable to the passage of molecules. The number of molecules in a system can range over all possible values, that is, each system is open with respect to transport of matter. We construct a grand canonical ensemble by placing a collection of such systems in a large heat bath at temperature T and a large reservior of molecules. Configurations are varied by displacement of the molecule and addition/removal of the molecule from the system (more details are given in for mulation of program). After equilibrium is reached, the entire ensemble is isolated from its sur roundings. Since the entire ensemble is at equilibrium with respect to the transport of heat and matter, each system is specified by V, T and ju, where p. is the chemical potential. If there is more than one component, the chemical potential of each component is same from system to sys-_ tem[33]. The systematic picture of Grand Canonical Ensemble is shown in Figure 2. The interactions between adsorbate molecules were modelled with truncated Lennard-Jones potentials:[34] r o.. 1 2 s.. 6 \ Uij(r) = Ai.j (-j?) - i - y ) ....................(r<Rc) o- ...................(r>Rc) , 9 The interactions between adsorbate and adsorbent surface were modelled using the 10-4-3 poten tial, which is an accurate approximation to the full summed 10-4 potential:[34] 2nq ? — - £ < 5 ~ ■ a wi wi ( C F • 10 ( < 5 . \ - ( ~ ) aii 3A ((2 + 0.61A) ) . For the basal plane of graphite q=2 and as=5.24Angstroms2 (needed for interaction between mol ecules and wall). These two quantities refer to the number of carbon atoms and the area of the lat tice unit cell of the surface lattice. A is the spacing between graphite layers and was set at 3.40A°. The parameters for carbon-carbon interactions are Ccc=3.40 Angstroms and ecc/k=28K. and are interaction parameters between wall(carbon in the present study) and adsorbate molecule i. The Lorentz rule was used for unlike collision diameters i.e ay = 0.5(Gij + Cjp and the Berthelot rule was used for unlike well depths i.e £y = (e^ Ejj)0'5. Our system consists of a cube of volume V containing N particles. There are walls on two oppo site faces of the cube. The systematic diagram of the system is shown in Figure 3.1n order to min imize the surface effects we suppose the faces without the wall to be periodic i.e each face is connected to an identical cube, each cube containing N particles in the same configuration. The energy of the system is defined as: N N E = X uij {r) + X (-') 7 i < j = 1 i = l The equation for virial is given by: N S V J F . ® 8 j = 1 J The chemical potential is defined by B as follows: Where ur is the excess chemical potential and N is the mean number of particles. The equation for pressure is given by: P = p k T - (<D )/(3V 0..........10 An initial number of molecules are placed in random locations and the energy E and virial < I > are calculated. A series of Monte Carlo configurations corresponding to various values of N and posi tions of each molecule are generated using the following algorithm: The algorithm of the program can be divided into two steps: Step I: A trial configuration is generated by choosing one particle at random and giving it a small random displacement. We then calculate the change in energy of the system AE, which is caused by the move. If AE<0, i.e.. if the move would bring the system to a state o f lower energy, we allow the move and put the particle in its new position. If AE>0, we allow the move with probability exp(-AJE/kT), i.e., we take a random number c between 0 and 1. and if c<exp(-AE/kT), we move the particle to its new position. If c>exp(-AE/kT), we return it to its old position.[35] Step II: The second step proceeds regardless of the consequences of the first and attempts to change the number of panicles in the volume. If it is decided to attempt removal of a particle then one is chosen at random and is removed with probability Nexp(-B-AE/kT) where AE includes a change in the long range correction due to the simultaneous removal of all the periodic images. If it is decided to attempt adding another panicle then a point is chosen at random and a particle is placed there with probability exp(-B-AE/kT)/(N+l) where AE includes a change in the long-range correction due to simultaneous addition of all periodic images.[36] The energy, virial etc. produced by the above steps are then added to the various running totals from which ensemble averages were calculated. A flow chart is given in figure 4. The various subroutines used in the program and their functions are as follows: i) test, is used to calculate potential energy of a test (add/remove) molecule with respect to other molecules. Lennard- Jones 12-6 potential is considered between the test molecule and other mole cules. If the separation between the molecules in the x-direction is more than cube/2 where cube is a side of the simulation box, then the new x co-ordinate o f the molecule is — previous x-coordi- nate - cube. If the separation is less than -cube/2, then the new position of the molecule is = previ ous x-coordinate + cube. If the separation is less than cube/2 and greater than -cube/2 then the new co-ordinate will be the same as previous co-ordinate. This process is called Minimum Image. 1 1 ii) inter, is used to calculate potential energy of displacement molecule with respect to other mol ecules. Lennard- Jones 12-6 potential is considered between the displacement molecule and other molecules. Minimum Image is also taken care of. iii) upairs, is used to calculate the total potential energy of the system as a whole. Lennard - Jones 12-6 potential is used to calculate potential energy of the system. Minimum image is also taken care of. iv) fee, is used to generate initial configuration and calculate energy of this configuration. Moule- culescules are randomly placed to generate the initial configuration. Minimum image is also taken care of. v) datadu, is used to calculate cumulative and block averages. vi) corr, is used to calculate density profile. vii) interl, is used to calculate potential energy between displacement molecule and wall. 10-4-3 potential is used to calculate potential energy between displacement molecule and wall. viii) inter2 , is used to calculate potential energy between test(add/remove) molecule and wall. 1 0 - 4-3 potential is used to calculate potential energy between test molecule and wall ix) totinter. is used to calculate potential energy of all molecules with the wall. 10-4-3 potential is used to calculate potential energy of all molecules with the wall The length of a typical simulation run was on the order of 107 configurations, plus an equilibrium period of roughly 2 x 1 0 6 configurations, a configuration was generated by one of the displace ment steps described above. 4. Result And Discussion 4.1 Single Component Adsorption 4.1.1 Comparison With Previous Studies Firstly, we have studied adsorption of C2 H4 on graphite, the system has previously been studied. The reason of studying the system was to check the validity of the program. Interaction parame ters used for Lennard-Jones (interparticle) and approximate 10-4 (wall-particle) potentials [29] are shown in Table 1. The important parameters used for this simulation and their meanings are as follows. Here ‘a ’ refers to C 2 H4 . i) T*; reduced temperature = kT /e^ ii) kbmax; maximum number of configurations iii) V*=V/ca a 3 iv) Ba is reduced Chemical Potential Parameter and is defined by: Ba = )+lnNe 11 Where ua is the Chemical Potential and Na is the mean number o f particles, v) Na; number of molecules of component ‘a’, The state conditions for this simulation are given in Table 2. See the graph 1(a) for number of particles vs. configurations plot for the simulation results. The following important conclusions can be drawn from the graph: - i) The average number of particles till 50,000 configurations is 2 8 .1 have started the simulation with 50 particles. This period is a part of equilibrium period. ii) After 60,000 configurations, the number of particles start increasing gradually till they stabilize around 93. THis region in graph represents a phase transition i.e. from a gas-like to a liquid like phase. This means that for the value of temperature, chemical potential and volume used in this simulation, the equilibrium state corresponds to a liquid-like density. The initial starting density (for 50 particles) corresponds to a gas like density. The algorithm used rapidly increased N until the equilibrium density is attained. iii) After 200,000 configurations, the number of the particles remain constant around 93. The above observations were in compliance with Van Megen. W. and Snook, I. K.[29]; thus estab lishing the validity of the program. 13 4.12 Comparison With Experimental Data Next we compare the simulated adsorption isotherm with experimental results for the Kr/C sys tem. In single adsorbate studies, we have simulated the krypton/graphite system. The Kr/Kr interaction parameters are obtained from the gas properties, the C/C parameters from solid state properties of graphite and the Kr/C parameters from the simple combining rules [37]. Interaction parameters used for Lennard-Jortes (inerparticle) and approximate 10-4 (wail-particle) potentials are as shown in Table 3. The adsorption is defined as the surface excess of gas particles divided by area A of the periodic box (refer to Figure 3). The gas density is very low in all of these calculations so that the number of particles Ngas, in a gas phase of the same volume as the particle box, is very small compared with the ensemble average particle number <N> and is generally very m uch less than fluctuations in <N>; hence adsorption is given to certain accuracy by[37] r = ^ ........ 12 A In experimental adsorption studies, the solid is generally in the form of a fine powder whose sur face area cannot be determined accurately and therefore the absolute adsorption is subject to an uncertainity. In an attempt to avoid this problem, the adsorption is reported in terms of the cover age 6 defined by A T 6 = ................ 13 (AT"1 ) where AT*13 is the adsorption when the solid has been covered completely by a single layer of close packed particles. Monte Carlo simulation studies give T directly but there is some arbitrariness in defining I""5 . In their study of Kr adsorption on C, Rowley et al [38] define F™ as the number density per unit area of triangularly close-packed spheres having a diameter 2 1^c(Kr,Kr). If this defination is used the diameter of spheres is 0.412 nm and F" 1 is 6.80 atom/nm2. The value o f important parameters used in the program are as shown in Table 4. See the graph 1(b) for number of particles vs. configurations plot for the simulation results. The following important conclusions can be drawn(for Ba=1.80)from the graph: - 14 i) The average number of particles till 50.000 configurations is 3 9 .1 have started the simulation with 50 particles. This period is a part of equilibrium period. ii) After 60.000 configurations, the number of particles start increasing gradually till they stabilize around 58. iii) After 200,000 configurations, the number of the particles remain constant around 53. We have carried out a series of simulations at various values of Ba for the same temperature to obtain the adsorption isotherm. The results for the simulated <N> and pressure are summerized in Table 5. Graph 1(c) is a plot of coverage vs. pressure. Also, experimental isotherm of Thorny and Duval [39] is plotted in the same graph. One can easily see surface transition in the plot Close agree ment can be seen between simulation studies and experimental work in the above plot. Some of the disagreement may arise through lack of structure in our model graphite. A second isotherm is calculated at T* = 2.37 and the results are given in Table 6 and in Graph 1(d). Also, experimental isotherm of Thorny and Duval [39] is plotted in the same graph. One can eas ily see surface transition in the plot. The qualitative behaviour is in close agreement with experimental data, however the quantitaive agreement is not as good as for the T* = 2.0 isotherm. The differences can arise from the inade quacy of the potential models. The density profile is shown in Graph 1(e) for Ba=1.80. The simulated pressure was compared with the pressure calculated at the simulation ju a and T from equation of state for the LJ gas as developed by Nicolas et al [40]. This pressure is also tab ulated in Table 5 and 6 alongwith simulated pressure. 4.2 Competitive Adsorption Most processes of industrial interest involves the selective adsorption of certain components from a mixture. Therefore, it is important to understand how physical and chemical characteristics of various types of adsorbents and adsorbates affect the competitive adsorption. In competitive adsorption studies CH4 *Ar mixture on graphite was first studied. The CH4 -Ar mix ture was chosen for several reasons. First, the solutions that these molecules form in the bulk are reasonably ideal. Secondly, methane and argon are molecules that might reasonably be modelled with 1 2 - 6 potentials. Interaction parameters used for Lennard-Jones (interparticle) and approximate 10-4 (wall-parti- cle) potentials [33] are as shown in Table 7. 15 The Lorentz rule was used for unlike collision diameters i.e Cy = 0 .5 (0 ^ + 0 y) and the Berthelot rule was used for unlike well depths i.e £y = (qj Ejj)0" 5. The value of important parameters used in the program are as shown in Table 8 . *a’ molecule is CH4 and ‘b ’ molecule is ‘A r’. The results for the simulated <Na>, <Nb> and Pressure are tabu lated in Table 9. Graph II shows density profile for the above case. The density profiles for CH4 and Ar is shown. Following useful remarks can be concluded from the plot: i) More molecules can be found near the surface (excluding the nearest 8 %) than anywhere in the simulation box. This is because the wall is an attractive one. ii) As we go away from the surface the density decreases. iii) In the nearest 8 % of the surface, the density decreases as we go towards the surface because the wall is impermeable to molecules. iv) CH4 adsorbs more on the surface as compared to Ar. To better understand the competitive adsorption, the following effects were studied on the CH4 and Ar syatem. Fluid Density and Composition of the bulk gas phase can be calculated by using the equation of the state of Nicolas et al [40] and assuming van der waals I mixing rules. ua and ub can be calcu lated from Ba and Bb and at equilibrium the |ia and ub for gas should be equal to simulated ,ua and ub and pressure of the gas should be equal to simulated pressure. The bulk density and composi tion in equilibrium with the adsorbed phase are evaluated iteratively using the equation o f state. i ) E f f e c t o f C h e m i c a l P o t e n t i a l The Chemical potential is related to the gas phase partial pressure. Therefore varying the parame ters ba and bb are equivalent to changing the gas phase partial pressures of a and b. The above simulation was carried out at Ba=2.0 and Bb=2.0. Now, the simulation was carried for Ba=2.5 and Bb=2.5. The value of important parameters used in the program are shown in Table 10. ‘a ’ molecule is CH4 and ‘b’ molecule is ‘ A r’. The results for the simulated <Na>, <Nt,> and Pressure are tabu lated in Table 11. Graph 11(b) shows the density profile for above simulation parameters alongwith density profile for Ba(or Bb)=2.0 i.e previous simulation data (using parameters in Table 8 ). Also, the simulation was carried out at Ba=2.5 and Bb=2.0. Graph 13(c) shows the density profile for these simulation parameters alongwith density profile for Ba(or Bb)=2.0 i.e previous simula tion data (using parameters in Table 8 ). The results for the simulated <Na>, <Nh> and Pressure are tabulated in Table 12. 16 I It can be seen that as Chemical potential increases, the density at a particular location in the sytem increases i.e number of molecules in the system will be more i.e more coverage and hence more adsorption. Thus, a d s o r p t i o n w i l l b e m o r e i f c h e m i c a l p o t e n t i a l i s m o r e . Also, at any position, the density of Argon is less as compared to the density of Methane at the same position except near the wall where it is more. Adsorption of Ar is unchanged when only the chemical potential of CH4 i.e Ba is changed from 2.0 to 2.5. This implies that CHyAr is behaving as an ideal system as we expected. i i ) E f f e c t o f T e m p e r a t u r e The above simulation was carried out at T*=2.0. Now, the simulation was carried for T*=2.5 . The value of important parameters used in the program are shown in Table 12. ‘a’ molecule is CH4 and ‘b ’ molecule is ‘ A r’. The results for the simulated <Na>, <Nb> and Pressure are tabu lated in Table 13- Graph 11(c) shows the density profile for above simulation parameters alongwith density profile for T* =2.0 i.e previous simulation data (using parameters in Table 8 ). It can be seen that as temperature increases, the density at a particular location in the sytem decreases i.e number of molecules in the system will be less i.e less coverage and hence less adsorption. Thus, a d s o r p t i o n w i l l b e l e s s i f t e m p e r a t u r e i s m o r e . Also, at any position, the density of Argon is less as compared to the density of Methane at the same position except near the wall where it is more. Temperature has little effect on adsorption selectivity for these rather ideal mixtures. i i i ) E f f e c t o f L e n n a r d - J o n e s P a r a m e t e r s Case I: Consider a hypothetical mixture of A and B with following Lennard Jones parameters and system was compared with sytem whose Lennard Jones parameters are tabulated in Table 14, rest of the parameters in the simulation are identical as in Table 8 . The results for the simulated <Na>, <N^> and Pressure are tabulated in Table 15. Graph 11(d) shows the density profile for above simulation parameters alongwith density profile for previous simulation data (using parameters in Table 7). It can be seen that as Lennard Jones Parameters increases, the density at a particular location in the sytem increases i.e number of molecules in the system will be more i.e more coverage and hence more adsorption. Thus, a d s o r p t i o n w i l l b e m o r e i f L e n n a r d J o n e s P a r a m e t e r s a r e m o r e . This is as expected since increasing c and £ both result in increased attraction with the wall. 17 Also, at any position, the density o f Argon is less as compared to the density of Methane at the same position except near the wall where it is more. Case II: Consider a hypothetical mixture of CH 4 and X with following Lennard Jones parameters and sys tem was compared with sytem whose Lennard Jones parameters are tabulated in Table 16, rest of the parameters in the simulation are identical as in Table 8 .The results for the simulated <Na>, <!%> and Pressure are tabulated in Table 17. Graph 11(e) shows the density profile for above simulation parameters alongwith density profile for previous simulation data (using parameters in Table 7). It can be seen that as Lennard Jones Parameters increases, the density at a particular location in the sytem increases i.e number of molecules in the system will be more i.e more coverage and hence more adsorption. Thus, a d s o r p t i o n w i l l b e m o r e i f L e n n a r d J o n e s P a r a m e t e r s a r e m o r e . Also, at any position, the density o f Argon is less as compared to the density o f Methane at the same position except near the wall where it is more and density of Methane is less as compared to the density of X at the same position except near the wall where it is more Replacing Ar by the more strongly attractive component X have a mild depressing effect on the adsorption of CH4 as <Na> decreases from 65 to 61. 18 5. Conclusion And Recommendation For Future Work In this work we have shown that Monte Carlo method is a useful technique for theoretical studies of the gas/solid interface. For pure adsorbate case we have been able to reproduce the main fea tures o f the experimental properties of the krypton/graphite interface. In addition to comparison of simulation and experimental results, we can calculate properties which are not accessible to experiments. These include the distribution functions which are important in determining the nature of the surface phases and the surface pressures. The work also demonstrates the feasibility of applying the Grand Canonical Monte Carlo simula tion for competitive adsorption. Although in this work we have considered only the ideal adsor bate, but other complex adsorbates can also be studied as the validity o f the program is established. From this point onwards, the work can be started for competitive adsorption on heterogenous sur faces. 19 6. References 1. Rowley, L. A., Nicholson. D. and Parsonage, N. G., Mol. Phys., 31 (19761 365 2. Hill. T. L. and Saito, N., J. Chem. Phys., 24 (1961) 1543 3. Lebowitz, J. L. and Percus, J. K., J. Math. Phys., 4 (1963) 116 4. Steele, W. A. and Ross, M., J. Chem. Phys., 22 (I960) 464 5. Stillinger, F. H. and Buff, F. P., J. Chem. Phys., 22 (1962) 1 6 . Chapyak, E. J., J. Chem. Phys., 52 (1972) 4512 7. Millard, K., J. Math. Phys., 12 (1972) 222 8 . Garrod, C. and Simmons, C„ J. Math. Phys., 12 (1972) 1168 9. Finn, J. E. and Monson. P.A., Mol. Phys., 65 (1988) 1345 10. Steele, W. A., Vemov. A. V., Tildesley, D. J., Carbon, 25 (1987) 7 11. Jansen. A. P. J., Surf. Sci., 222 (1992) 193 12. Vemov, A. V. and Steele, W. A., Langmuir, 2 (1986) 219 13. Pickett, S. D., Nowak, A. K., Thomas, J. M., Peterson, B. K., Swift, J. F. P., Cheetham, A. K., Den Ouden, C. J. J. and Post, M. F. M., J. Phys. Chem., 94 (1990) 1233 14. Hufton, J. R., J. Phys. Chem., 95 (1991) 8836 15. Keizer, A. D., Michalski. T., Findenegg, G. H., Pure and Appl. Chem., 62 (1991) 1495 16. Joshi, Y. P. and Tildesley, D. J., Surf.'Sci., 166 (1986) 169 17. Cortes, J. and Araya, P., J. Chem. Phys., 95.(1991) 7741 18. Russier, V., Rosinberg, M. L„ Badiali, J. P., Levesque, D. and Weis, J. J., J. Chem. Phys., 82 (1987)5012 19. Suh, S. and O ’Shea, S. F . Can. J. Chem.. 65 (1988) 955 20. Lane, J. E. and Spurling, T. H„ Aust. J. Chem., 22 (1980) 231 21. Gupta, D. and Hirtzel, C. S., Chem. Phys. letters, 149 (19881 22. Knight, J. F. and Monson, P. A„ J. Chem. Phys., 84 (1986) 1909 23. Smit, B. and Den Ouden, C. J. J., J. Phys. Chem., 22 (1988) 7169 24. Finn, J. E. and Monson, P.A., Langmuir, 5 (1989) 639 25. Nelson, P. H., Kaiser, A. B. and Bibbly, D. M., J. of Catalysis, 12Z (1991) 101 26. Van Damme, H., Levitz, P., Bergaya, F„ Alcover, J. F., Gatineau, L. and Fripiat, J. J., J. Chem. Phys., 85.(1986)616 27. Razmus, D. M. and Hall, C. K., AIChE Journal, 21 (1991) 769 28. Bladon, J. J. and Higuchi, W. I., J. of Colloid and Interface Sci., 81 (1981) 445 29. Van Megen, W. and Snook. I. K„ Mol. Phys., 42 (1982) 629 30. Murii, V. and Aylmore, L. A. G., Soil Sci.. 135 (1983) 203 31. Schwarz, J. A. and Smith, R. P., J. of Catalysis, £2 (1980) 176 32. Lemos, M. C. and Cordoba, A., J. Chem. Phys., 25 (1991) 6171 33. McQuarrie, D. A., Statistical Mechanics, Harper & Row Publishers, 51 34. Finn, J. E. and Monson, P. A., Mol. Phys., 22 (1991) 661 35. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N„ Teller, A. H. and Teller, E., J. of Chem. Phys., 21 (1953) 1087 36. Adams, D. J . , Mol. Phys., 29 (1975) 307 37. Lane. J. E. and Sparling, T. H., Aust. J. Chem., 29 (1976) 2103 38. Rowley, L. A., Nicholson, D. and Parsonage, N. G., Mol. Phys., 31 (1976) 365 39. Thorny, A. and Duval, X., J. Chem. Phys., £7 (1970) 1101 40. Nicolas J. J., Gubbins, K. E„ Streett, W. B. and Tildesley, D. J., Mol. Phys., 37 (1979) 1429 21 APPENDIX Result: I (a) 100 90 80 60 r 40 30 20. 0.5 1 1.5 C onfigurations 2.5 x100000 3.5 23 Number ol Particles Result: I (b) 70 6 0 50 40 30 20 0 .5 1.5 C onfigurations 2.5 x100000 24 C overage Resultrl(c) 0.9 0.8 0 .7 0.6 - is e x p erim e n tal iso th erm of Thorny a n d Duval 0.5 is M onte C arlo C alcu latio n s 0.4 0.2 0.1 0.2 0 .4 0.8 10 E3 p /T orr 25 C overage Result: !(d) 0.9 0.8 0 .7 0.6 0 g is ex p erim en tal isotherm of T horny a n d Duvj is M onte C arlo C alculations 0.4 0.3 0.2 0.1 0.2 0.4 C.5 0.8 1 10 E 3 p/T orr 26 Density Resultn (e) 0.65 0.6 0.55 0 .5 0.45 0 .4 - 0 .35 i- 0.3 0 .25 0.2 -10 Z /sig of Kr-Kr 27 Result: II A rgon 15 5 10 -10 •5 0 D Z 'sig of m eth .-m eth . 28 Density^ Density Result: 11(b) le lh a n e .a tb a = 2.5,. bb = 2.5 0 .5 0.4 0.2 M e th an e at b a = 2 .0 , bb = 2.0 -10 2Jsia of m eth .-m eth . R e su lt: ll(b) 0.2 0.1 A rgon a t b a = 2 .0 , b b = 2.0 Z 'sig of m eth .-m eth . 2 9 Result: 11(c) ■ x X M eih an e.atb a = 2.5,. bb = 2 .0 0.5 0.4 c 0 .3 0.2 M eth an e at b a = 2 .0 , bb = 2.0 0.1 -10 10 Z 'sig o f m eth .-m eth . R e su lt: ll(c) 0 .5 .rg o rra t b a = 2 :5 .,b b -.2 .0 . 0 .4 c 0.3 0.2 0.1 A rgon at b a = 2 .0 . bb = 2.0 -10 10 Z 'sig of m eth .-m eth . 3 0 Resuit: 11(d) 0 .5 M e th an e a t T* = 2.0 0 .4 > * I 0.3 © o 0.2 M e th an e at T* = 2.5 0.1 -10 -o Z 'sig of m eth .-m eth . R esu lt: !l(d) 0.5 0.4 0.3 0.2 0.1 A rgon a tT * = 2.5, -10 Z 'sig of m eth .-m eth . 31 Density^ ^ Dansily Result: 11(e) .M eth an e at N ew .Leninard J o n e s P a ra m e te rs . 0.5 0.4 0.2 - M e th an e at Old L ennand J o n e s P a ra m e te rs -10 Z 'sig of m eth .-m e th . R e su lt: 11(e) .rgqrt a t N ew L en n ard J o n e s P a ra m e te rs. 0.2 0.1 A rgon at Old L e n n a rd J o n e s P a ra m e te rs -10 Z /sig of m eth .-m eth . 32 Result: 1 1 (f) M e th an e A rgon 15 10 -10 5 ■ 5 0 Z 'sig of m eth .-m e th . 33 TABLES Leonard Jones Parameters for C2 H 4 /C System Table 1: Param eter j 1 j Value £(C2 H 4,C2 H 4)/k 202 K a (C 2H 4,C2H4) | 0.422 nm e(C ,C )/k 28 K a(C ,C ) 0.340 n m e(C 2H 4,C )/k 75.2 K a (C 2H4,C) 0.384 nm 35 Simulation Parameters for C2H4(a)/C System Table 2: Parameter Value T* 2.0 kbmax 1,000,000 Ba 2.0 Na(initial) 50 V* 125 Lennard Jones Parameters for Kr/C System Table 3: Parameter J Value £(Kr,Kr)/k 168.8 K o(Kr,Kr) | 0.367 nm e(C,C)/k | 27.0 K C T(C ,C ) | 0.345 nm e(Kr,C)/k j 67.5 K o(Kr,C) j 0.356 nm 37 Simulation Parameters for Kr(a)/C System Table 4: Parameter Value ■ p * 2.0 kbmax 1,000,000 Ba varying Na(initial) 50 V* 125 Simulation Results for Kr(a)/C System at T* = 2.0 Table 5: S. No. Ba <N> Coverage 103 p/torr (Simulated) 103 p/torr (Eq.of State) 01 1.75 17 0.20 | 0.25 0.28 02 1.80 53 _________ 0.80 ] 0.28 0.29 03 2.10 59 0.85 [ . . J ' 0 1.1 04 2.33 64 0.90 1.3 1.35 05 2.78 65 0.90 1.35 1.38 39 Simulation Results for Kr(a)/C System at T* = 2.37 Table 6: S. No. Ba <N> Coverage 103 p/torr (Simulated) 103 p/torr (Eq. of State) 01 1.75 16 | 0.19 1.10 | 1.15 02 1.80 44 0.73 1.19 1.24 03 2.10 49 0.76 1.26 1.30 04 2.33 51 0.78 1.39 1.43 05 2.78 53 0.79 1.50 1.55 40 Lennard Jones Parameters for CH4-Ar/C System Table 7: Parameter Value j .......... i e(CH4 ,CH4)/k 148.2K 1 .... .. i 0(CH4,CH4) | 0.3817 nm e(A_r,Ar)/k | 119.8 K c(Ar.Ar) 0.3405 nm £(C,C)/k 28K a(C,C) 0.340 nm c t (CH4,C) 0.361 nm e(CH4,C)/k 64.4K a(Ar,C) 0.34 nm e(Ar,C)/k 57.9 K Simulation Parameters for CH4(a)-Ar(b)/C System Table 8: Parameter Value 1'* 2.0 kbmax 1,000,000 Ba 2.0 Bb 2.0 Na(initial) 50 Nb(initial) 50 V* 250 Simulation Results for CH4(a)-Ar(b)/C System Table 9: Parameters Values <Na> 65 <Nb> 59 103 p/torr 1.01 Bulk Phase Density 0.26 Bulk Phase Compositio n, ya 0.7 Simulation Parameters for CH4(a)-Ar(b)/C System Table 10: Parameter Value 2.0 kbmax 1,000,000 Ba 2.5 Bb 2.5 Na(initial) 50 Nb(initial) 50 V* 250 Simulation Results for CH4(a)-Ar(b)/C System at Ba = 2.5 and Bj, = 2.5 Table 11: Parameters ■ Values <Na> 69 <Nb> 63 103 p/torr 1.04 Bulk Phase Density 0.25 Bulk Phase Compositio n .y a 0.74 45 Simulation Results for CH4(a)-Ar(b)/C System at Ba Table 12: Parameters J Values <Na> 71 <Nb> 59 103 p/torr 1.03 Bulk Phase Density 0.26 Bulk Phase Compositio 1 n .y a 0.77 Simulation Parameters for CH4(a)-Ar(b)/C System Table 13: Parameter Value •p * 2.5 kbmax 1,000,000 Ba 2.0 Bb 2.0 Na(initial) 50 Nb(initial) 50 V* 250 Simulation Results for CH4(a)-Ar(b)/C System at T* = 2.5 Table 14: Parameters Values <Na> 61 <Nb> 54 103 p/torr 1.07 Bulk Phase Density 0.2 Bulk Phase Compositio n ,y a 0.61 48 Lennard Jones Parameters for A-B/C System Table 15: Parameter Value £(AA)A 192.7 K i ct(AA) 0.4962 nm j e(B3)/k 155.7 K < t (B 3) 0.4426 nm e(C,C)/k 28K a(C,C) 0.340 nm a( A,C) 0.4693 nm e(A,C)/k 83.7 K o(B,C) | 0.442 nm e(BrC)/k 75.3 K | 49 Simulation Results for CH4(a)-Ar(b)/C System at New Lennard Jones Parameters Table 16: Parameters Values <Na> 77 <Nb> 70 103 p/torr 0.96 Bulk Phase Density 0.27 Bulk Phase Compositio n ,y a 0.73 Lennard Jones Parameters for CH4-X/C System Table 17: Parameter Value | e(CH4,CH4)/k 148.2K a(C H 4,C H 4) | 0.3817 ran e(X ,X )/k 255.0 K o(X ,X ) 0.59 nm j e(C ,C )/k 28K 0(C ,C ) 0.340 nm i l <7(CH4,C) 0.361 nm || e(CH4,C)/k 64.4K 0 (X ,C ) 0.57 nm s(X .C )/k 102.0 K 51 Simulation Results for CH4(a)-X(b)/C System at New Lennard Jones Parameters Table 18: Parameters Values <Na> 61 <Nb> 82 103 p/torr 0.97 Bulk Phase Density 0.26 Bulk Phase Compositio n, ya 0.4 F I G U R E S Figure: 1 54 Figure 2 OR AND CANONICAL ENSEM BLE v , T , ix V,T.(i V , T . u V , T , |X V , T , u V , T , u V.T.n V,T, h V , T , u mwdo FIGURE 3 WALL f W ALL SIM U LA TIO N BOX 56 Eigucfci FLO W CH ART START FINISH TEST INTER INTER I INTER2 FCC UPAIRS TOTTNTER MAIN ADD/REMOVE MAIN DISPLACEMENT DATADU

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Creator Vij, Anu Deep (author)

Core Title Molecular simulation of adsorption

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Advisor Shing, Katherine S. (committee chair), Chang, Wenji Victor (committee member), Shaffer, John Scott (committee member)

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