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Date Created: 10/15/15
CHE596M MultiScale Modeling of Matter Instructor Keith E Gubbins Lecture 18 The Meso scale Lattice Monte Carlo Simulations 5 NC STATE UNIVERSITY Outline Introduction Interaction potentials Moves Discretization Applications NC STATE UNIVERSITY Introduction Why use Lattice Monte Carlo Can be used to simulate large systems up to about 500 nm eg surfactants polymers proteins etc Off lattice simulations cannot go much beyond 30 nm Can be used to probe equilibrium properties for processes that require long time scales eg micelle formation coilglobule transitions for polymers protein folding Off lattice MD simulations with large systems cannot go much beyond 10 ns Can be discretized to improve accuracy is NC STATE UNIVERSITY Introduction Limitations Difficult to implement directional contribution to intermolecular potentials such as hydrogen bonding In the usual implementation of lattice MC cannot include long range forces Has restrictions on bond angles and lengths Only discrete values of bond angles amp lengths are allowed Quantitative comparison with experimental results is difficult is NC STATE UNIVERSITY Introduction Single Polymer chain Molecules occupy discrete positions inside a lattice A molecule can occupy a single lattice site or it can occupy several lattice sites as in a polymer chain Each molecule attracts or repels a neighboring molecule depending on the potential energy between them We usually use a square well potential extending to the nearest and next nearest neighbor but long range potentials can be used also g NC STATE UNIVERSITY Introduction Interaction potentials There are five different types of molecules in a this lattice A B o D and E 39eEC The potentials between E and the neighboring atoms are eEA eEBt eECt eED 9 7 3 NC STATE UNIVERSITY Introduction Coordination number Z 2D Lattice gt 3DLattice gt 26 226 In a 2D lattice E can interact with a maximum of 8 molecules Therefore the coordination number is 8 In a 3D lattice the maximum coordination number is 26 NC STATE UNIVERSITY Introduction Lattice simulation Discontinuous potential 1 0 U1CU1 U1l Reduced Potential 0 15 2 0 25 3 05 1 15 2 Reduced distance 2 1 a 45 Square well 1 6 05 D 39C U D g 05 Long range potential E 1 used by 15 Panag39OtOPOUIOS 2o 05 1 15 2 25 3 Reduced distance Off lattice simulation Continuous potential Reduced Potential 3 J 1 quot1208 1 12 14 16 18 2 22 24 Reduced distance lt3 7 LennardJones potential NC STATE UNIVERSITY Introduction Effective interaction parameter A lattice system containing several types of molecules with different interaction potentials can be represented by a single effective interaction parameter Imagine a system in the NVT ensemble containing two different types of particles A and B The number ofA particles NA number of B particles NB Therefore NA NB N 1 Let the number of AA contacts be NAA number of 88 contacts NBB and number of A B contacts NAB 7 NC STATE UNIVERSITY Introduction Effective interaction parameter E is the total energy of the system ENAAeAANBBeBBNABeAB 2 The system changes to a new configuration and the change in energy is AE AE ANAAeAA ANBBeBB ANABeAB 3 Let the coordination number be 2 so that zNA 2NAA NAB and ZNB 2NBB NAB 4 1 1 Therefore ANAA ANAB and ANBB EANAB 5 7 NC STATE UNIVERSITY 7 Introduction Effective interaction parameter Substituting 5 in 3 we get 1 1 AEZANAB6ABEeAAEeBB 6 WAB WAB is called the effective interaction parameter When the system assumes a new configuration the change in energy AE and hence the probability of occurrence of the new configuration is dependent on WAB not the individual potentials 7 NC STATE UNIVERSITY 7 Simulation scheme 1 A molecule chain solvent is chosen randomly 2 The molecule is moved to a new position 3 Energy of the new configuration is calculated 4 The move is accepted according to the canonical acceptance probability Page min exp 1 CT AE 2 EW E01d T temperature k Boltzmann factor 7 NC STATE UNIVERSITY Moves Moves are employed to change the positions of the molecules during simulation Single molecules can be moved using 1 Switch Polymer chains can be moved using 1 Reptation 2 Twist 3 Regrowth using Configurational Bias NC STATE UNIVERSITY 7 Switch Two solvent molecules exchange positions NC STATE UNIVERSITY Reptation Each segment of a chain is moved through one lattice unit NC STATE UNIVERSITY Twist A single segment is moved through one lattice unit keeping the rest of the chain fixed NC STATE UNIVERSITY Chain regrowth Configurational bias The entire chain or a part of the chain is grown in a different part of the box 2 NC STATE UNIVERSITY Chain regrowth Configurational bias algorithm Ref Frenkel amp Smit Understanding Molecular Simulation Ch 13 Leach Sec 811 1 Insert the first segment randomly 2 Compute w1n zxexp u1nkT z coordination no u1 n energy of the first segment here n new config 3 In turn insert the second segment in all 2 positions adjacent to the first atom Calculate W201 exp u2 energy of second segment in 1 position i NC STATE UNIVERSITY Chain regrowth Configurational bias 4 Insert the second atom in one of the 2 positions according to prob P2n eXp u2 n kT WM 5 Repeat for all the atoms 6 Calculate Rosenbluth Factor Wn of the new configuration I W H mm I chain length no of chain segments P201 2 NC STATE UNIVERSITY Chain regrowth Configurational bias 7 Calculate Rosenbluth Factor Wo of the old configuration 8 Accept the trial move with a probability PM min M I W0 7 NC STATE UNIVERSITY 7 Chain regrowth Configurational bias Acceptance Criteria Derivation Detailed balance POgtltP0 gtngtltP PngtltPn gt0 7 P0 Pni probability of old new configuration P eXp U0 kT eXp Uch0 kTgtlt exp Um kT8 0 Q Q UoBox energyUchoChainUestRest of box H NC STATE UNIVERSITY Chain regrowth Configurational bias Probability of change from old to new P0 gtnPlngtltPZngtlt Pln z X eXp u1 n kT X W W2 gtlt 9 w ZX eXp Uchn kT Wn Similarly Pngt0zzxexp UchokT 10 W0 and P eXp UchnkTgtlteXp Umt kT 11 Q 7 NC STATE UNIVERSITY 7 Chain regrowth Configurational bias Substituting 8 9 10 and 11 in 7 we get PM min W01 1 mm i7 NC STATE UNIVERSITY 7 Lattice Discretization Norma attice Discretized lattice Qualitative comparisons Quantitative comparisons possible possnble A Z Panagiotopoulos J Chem Phys 11216 7132 2000 7 Discretization Lattice Discretization parameter Q Particle diameter Lattice size C NC STATE UNIVERSITY Discretization Comparison of discretized lattice with offlattice simulationsi Phase behavior of C LennardJones particles 3921 f X C 2 18 T f Increasing 0 C 3 ii nc5 Offlattice ne simulation 1 A Z Panagiotopoulos J Chem Phys 11216 7132 2000 NC STATE UNIVERSITY Applications Lattice MC has various applications in simulations of Surfactant solutions gt Surfactant water gt SurfactantWateroil Supercritical fluids MCM MCF materials Polymers Proteins NC STATE UNIVERSITY Applications Larson s model Simple cubic lattice Fully occupied Coordination number Z 26 Site interacts equally with all its neighbors I Single energy parameter Wm 812 811 822 1 NC STATE UNIVERSITY 7 Applications Larson s model Binary system WaterSurfactant Simulation1 Experiment2 M bl L 343 quot 30 a LAMEle 75 f FLUID 2 s 323 Lanme L u msonnsnsn l 39 31 H SWEHES 5395 lLll l 39 OFDERED 303 5 so 5 msonusHED l 5 55 711 quot39 P119155 WORM5 ID 20 30 40 50 El 70 EU 90 IX If n M 6 UFFO to 20 30 40 5 ID a so so VOLUME 9t HT HAT4 surfactant in Water Lithium pen luorooctanoate in water Simulation and experiment are qualitatively similar Ref 1 R GLarson J Chem Phys 9611 7904 1992 2 P Kekicheff G J T Tiddy J Phys Chem 93 2520 1989 3 NC STATE UNIVERSITY Applications Larson s model Ternary system WaterOilSun actant Simulation1 Experiment2 Sodium nTclradccnnuzhle A nunurn nrllnimnl W lcr H4T4 surfaCtant 0 water Sodium1tetradecanoate 1decanol water We notice signi cant discrepancies between simulation and experiment Need to extend the simple lattice model Ref 1 R GLarson J Phys II 6 1441 1996 2 P K Kilpatrick J C Blackburn and T A Walter 8 219211992 NC STATE UNIVERSITY Applications Surfactant Solutions Some interesting results in surfactant solutions using Lattice MC A drop in free monomer concentration with increasing total surfactant concentration above CMC in contrast to equilibrium theories1 Characterization of size and shape distributions of micelles2 Thermodynamic properties of micelles34 Spontaneous vesicle formation5 gt 1 AD Mackie and AZ Panagiotopoulos Langmuir13 5022 1997 2 PH Nelson GC Rutledge and TA Hatton J Chem Phys 107 10777 1997 3 MA Floriano and E Caponetti Langmuir15 3143 1999 4 CM Care and T Dalby Europhys Lett 45 38 1999 5 AT Bernardes Langmuir 1 2 5763 1996 2 NC STATE UNIVERSITY Applications Supercritical Fluids Supercritical Fluids 39 Surfactant Surfactant in supercritical CO2 LT Vacancies to account for 002 compressibility Surfactants Micelles Conc T gt CMC 2 NC STATE UNIVERSITY Applications Supercritical Fluids Qualitative comparison with experiment Simulation1 Surfactants in supercritical CO2 Experiment2 00025 quot1 a b 00020 l A mfg spherical micelles c 15 pucxpltale 1 g 001 00015 I g 3 0 5 X3 390 5 00010 39 5 i g t 30001 f l a re topo ylners L p 00005 000001 t i quot40039quot 39 12 13 14 15 p 16 17 10 19 975 039 034 m 036 um H5T4 surfactant in scCO2 088 ms PVACbPTAN surfactant in scCO2 Ref 1 Lauriane F Scanu Keith E Gubbins and Carol K Hall Langmuir 20 2 514523 2004 2 E Buhler A V Dobrynin J M Desimone and M Rubinstein Macromolec ules 31 7347 1998 NC STATE UNIVERSITY Applications Mesoporous Materials Synthesis of MGM41 Surfactant Silica 9600 surfactant chains 17400 silica units Reduced temperature 65 0 100000 200000 300000 400000 500000 cycles Ref 1 F R Siperstein K E Gubbins Langmuir 19 20492003 NC STATE UNIVERSITY Applications Mesoporous Materials Synthesis of Mesostructured Cellular Foam Structure obtained by Lattice MC Pore size distribution it Transmissio S Bhattacharya and KE Gubbins Modeling Triblock SurfactantTemplated Mesostructured Cellular Foams J Chem Phys 123 134907 2005 NC STATE UNNERSH39Y Applications Surfactant Adsorption Simulation1 Experiment2 1 pmol m 2 Adsorption of HTZ on Hpreferred surface1 C8E4 in controlled pore glass CPG10 24 nm2 Opposite temperature dependency in simulations compared to experiments Shows the inadequacy of simple interaction potentials and the need to include hydrogen bonding in the model Ref 1 U Reimer M Wahab P Schiller HJ MOgel Langmuir 17 8444 2001 2 R Dabiri and GH Findenegg personal communication 2002 3 H Bock and KB Gubbins Phys Rev Letters 92 135701 2004 NC STATE UNIVERSITY Applications Confined polymers Diblock Copolymer confined between heterogeneous substrates Simulation1 Experiment2 Con ned PSbPMMA copolymer Similar structures found in experiments and simulations Simulation also predicts other structures depending on Pattern spacing Ls y Patterned Surface Con ned H12T1zcopolymer x y 39 Ref 1 Q Wang Q Yan P F Nealey and J J de Pablo Macromolecules 33 4512 2000 2 L Rockford Y Liu P Mansky T P Russell M Yoon and S G J Mochrie Phys Rev Len 82 26021999 NC STATE UNIVERSITY