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CHE596M MultiScale Modeling of Matter Instructor Keith E Gubbins Lecture 9 Uniqueness Theorem Reverse Monte Carlo Grand Canonical amp Microcanonical Ensembles Outline Uniqueness theorem for correlation functions Reverse Monte Carlo method Grand Canonical ensemble Microcanonical ensemble 7 NC STATE UNIVERSITY 7 Uniqueness Theorem for Correlation Functions References R L Henderson Phys Lett 49A 197 1974 Gray and Gubbins Theory of Molecular Fluids Clarendon Press pp 178180 1984 R Evans Mol Sim 4 409 1990 G Toth and A Baranyai J Chem Phys 114 2027 2001 Uniqueness Theorem For substances in which the potential is painvise additive we can show that different pair potentials lead to different pair correlation functions g 12601602 Different means different by more than just a constant Conversely for a given g12 there is one and only one unique u12 This can be generalized to nonadditive potentials 7 NC STATE UNIVERSITY Uniqueness Theorem for Correlation Functions Proof we consider 2 different materials with potentials u and uo N We introduce a potential 1 Z CON SUCh that u1 u01u u0 91 and ulzozuo uqu I The average of ltu uogtA is defined as J a zNa afVe ulkT u 10 201 where 201 J dZNdwNeTMlkT ltu u0gtl 92 1 NC STATE UNIVERSITY 7 Uniqueness Theorem for Correlation Functions Differentiate 92 a u uo L dgNdwNeulkT u uo 2 61 67 Tszd 6201 1 51 Z dW LQA From 91 u uo 81 and aZ d iJ dZNdCON euAkT iZclltu u0gtz 81 kT 81 kT so that Kim um ltu u02gti ltu u0gtj ltu uO ltu u0gti2gtl 93 1 NC STATE UNIVERSITY 7 Uniqueness Theorem for Correlation Functions k Tare positive so a Eat um 0 94 Integrate this from 2L 0 to 21 altuugtkdlltuuogt ltuugtmgo 1 la quit X1 x0 This is the GibbsBoqoliubov inequally 95 In 95 the equality holds only if u u0 u u01for all configurations see 93 ie only if u uo is a constant or zero Thus if u is different from uo the inequality holds So far we didn t assume additivity 7 NC STATE UNIVERSITY 7 Pair Additive Case If we assume pair additivity it is easy to show from 91 92 and 95 that 95 becomes j dl12dwldw2 u12 uo 12f12 s j dzlzdwldwz u12 uo 12f0 12 or j d1d2u12 u012f12 f012s 0 96 where u12 uz12a1a2 f12 fl12601602 etc When u and uO differ by more than a constant the inequality holds in 96 The inequality can only hold if 75 0 Hence if u and uO differfandf0 must differ QEQ mportance We can measure the pair correlation function for simple fluids by radiation scattering If we know that manybody forces are negligible a unique pair potential corresponds to the pair correlation function In principle we may be able to find u from g using Reverse Monte Carlo for example 7 NC STATE UNIVERSITY SiteSite Potentials For this case the theorem states that for a given set of gag there is a unique set of ua and vice versa When manvbodv potentials are not neqligible the theorem is generalized to If 2 and 3body forces are present the set f12f123 must differ in some way from the set 12f0 123 if u012 differs from 12 and u0123 differs from u123 Questions 1 Givenfor g how does one finds u 2 How sensitive isfto changes in u and vice versa 3 How sensitive is f12 to 3body forces 7 NC STATE UNIVERSITY Reverse Monte Carlo The Reverse Monte Carlo RMC method R L McGreevy and LPusztai Mol Sim 1 359 1988 was originally developed to produce threedimensional particle configurations consistent with experimentally measured structure factors For liquids if pair additivity is a valid approximation one could in principle determine the pair potential by carrying out atomistic simulations and adjusting u to obtain a simulated structure factor that matches the experimental one RMC has been used to get realistic atomic models of disordered materials eg porous carbons The structure of the models produced by RMC agree quantitatively with results from experimental techniques eg diffraction data No interatomic potential is required and data from different sources neutrons xrays EXAFS can be combined In practice three body forces are nearly always significant Need to include some constraints to take into account three body forces 7 NC STATE UNIVERSITY Some Possible Applications of RMC Diffraction Experiment D39ff t39 E 39 t Structure Factor Sq 39rac39on Xperlmen Structure Factor Sq Radial distribution function gr compare Radial distribution function gr add constraints if any Reverse Monte Carlo l 139 NC STATE UNIVERSITY Conventional MC or MD V 3D particle configurations lnteratomic Potential 7 An Application Activated Carbons Strong adsorbatecarbon forces Carbonaceous porous materials SID2 hybridized C atoms o High surface area i up to 1600 mZg Different than graphite 3 g 1 football field 3 P Most wider used of all J A generalpurpose adsorbents 090 1b Organic material Carbonization 0 Activation t l 3 NC STATE UNIVERSITY Activated Carbons Reconstruction Method We want the model to reproduce experimental structure Reverse Monte Carlo Idea Produce an atomic configuration that is consistent with a set of experimental data data of real carbons l Simplest Approach Changing the atomic positions through a stochastic procedure 2 2 minimize 12 SsimqiSexpqil 0r 2 gmngm 0 il i 1 Calculate 263M 2 Randomly select an atom 3 Random displacement 4 Calculate If 5 Accept the move with probability Page min 1 exp TLZVTZW 1101 1 O O OO 00 Multiple solutions 1 Need constraints J Pikunic et al Langmuir 19 8565 2003 NC STATE UNIVERSITY Activated Carbons Reconstruction Method Uniqueness of RMC How many con gurations can be obtained om one structure factor Do these con gurations have the same higherorder correlation functions Uniqueness theorem of statistical mechanics Pairwise potential 2body correlation function Disordered carbons fall in this class i We need to describe the manybody nature of the interactions 2body potential 3body potential 2body correlation function 3body correlation function NC STATE UNWERSITY Z 2 ThreeBody Constraints We assume that all carbon atoms have sp2 hybridization as in graphite 0 Most carbon atoms have 2 or 3 C neighbors Based on experimental HC and OC ratios we calculate the fraction of atoms with 3 C neighbors The bond angle distribution is centered at 120 equilibrium angle for sp2 hybridized C atoms Minimize gm rdgem all n 5 2 N3 N Jim N target NC STATE UNIVERSITY I Activated Carbons Sexp gsim gexmgt J Pikunic et al Langmuir 19 8565 2003 Monte Carlo gr Sq 21p47rr2gr l dr Truncation errors unphysical gr An alternative MCGR AK Soper 1989 1 Assume gr 2 Calculate Sq 3 Change gr 4 Recalculate Sq 5 Accept the move if there is a better match with Swag Fast No truncation errors NC STATE UNIVERSITY NC STATE UNIVERSHV Simulated Annealing Z Bree min 1 exp Ti anew 1111 iltw3ew lld 6021d 11 a a A gt Start With random configuration of carbon atoms no bias and high artificial temperature T Ty T5 gt Fix the parameters T1 and TI gt Gradually decrease the artificial temperature T 5 NC STATE UNIVERSITY Activated Carbons Reconstruction Method n 2 2 A 5 mmme ufRMC maves 2 A 5 2 m mxlhms afRMC maves J Pikunic at a Langmuir 19 8565 2003 i NC TTE UNIVERSITY z 4 6 8 mllmns nfRMC mnves Activated Carbons Monte Carlo gr OUTLINE XRD and SAXS to obtain Sexpq 1 Experiment gt Density Hg porosimetry He picnometry Chemical composition HC OC ratios 2 M CGR gt Obtain gemr om Sexpq No atomic con guration is obtained from this part 3 Reconstruction gt Obtain an atomic con guration that matches gexp quot method Results Two carbons prepared by pyrolysis of saccharose at 400 C C8400 and 1000 C CSlOOO HC OC Hg p gml CS400 053 0123 1275 CSlOOO 015 0041 1584 J Pikunic et al Langmuir19 8565 2003 NC STATE UNIVERSITY 7 71 Activated Carbons Monte Carlo gr Structure factor 4 4 MCGR ex an en expen39ment E 2 g 2 A I v M K A VVVVJ Iaff IaA C8400 CSlOOO J Pikunic et al Langmuir19 8565 2003 7 NC STATE UNIVERSITY Activated Carbons Reconstruction Method Radial distribution function gexp and gsim g0 g0 ref rd C8400 CSlOOO tar et model J Pikunic et al Langmuir19 8565 2003 i if 3 NC STATE UNVERSTY CS400 J Pwkumc ea Lemma1198565 2003 Activated Carbons Pore Size Distribution Accessible to nitrogen 06 7 cs1ooo 05 7 cs4oo J Pikunic et al Langmuir19 8565 2003 7 7 3 NC STATE UNVERSTY u 4 A Surfaces and Collmds Reallstin Molecular Models lnr Porous Oarhuns m 1 Bl n I 4 TEM IM LATI t n ATV CS4 0z Results md Campa son with experiments expe ment simulations Dotlike segments L7TEM IM LATI N CS4 0z Results nd Cempaiison with eiiments expeIiment simulations ShOIt segments sometimes branched NC STATE JMIVERS T vquot AR GON ABSORPTION Isosteric heats of adsorption at 77 K CSlOOO experiment I CSlOOOsimulati0ns CS400 experiment qst kJmol 0 CS400 simulations 4 l l I 00 02 04 06 08 10 fro Experiment performed by Phillip Llewellyn CNRS Marseille 3 NC STATE UNIVERSITY NC STATE Ji39l x EPS T v ARGON DIFFUSION Diffusion modes Fickian OX0 9 1nltzrgt zlto2gt2pt Single le W 112ltzr z02gt2F D is the selfdiffusivity F is the single file mobility NC STATE UNIVERSITY AR GON DIFFUSION Mean square displacement 300 K 001 01 1 10 100 1000 001 01 1 10 100 1000 t ps t vs C8400 CSIOOO 5 NC STATE UNIVERSITY Grand Canonical Ensemble In the grand canonical ensemble the variables are u V T for mixtures uA 13 um V T Thus the number of molecules Ncan fluctuate open system This is the natural ensemble to use for inhomogeneous systems such as micellar solutions adsorbents composites In adsorption problems the adsorbed fluid is in equilibrium with the fluid in the bulk reservoir The equilibrium conditions are that the temperature and chemical potential of the fluid inside and outside the adsorbent must be equal Fixing u Tfor the reservoir determines the equilibrium concentration inside the adsorbent M Adsorbent in contact with a reservoir that imposes constant chemical potential and temperature by exchanging particles and energy From D Frenkel and B Smit Understanding Molecular Simulation 2nd Edition Academic Press San Diego 2002 NC STATE UNIVERSITY 7 Some Nanoporous Materials Material Surface Pore Shape Pore Width nm A Crystalline Aluminosilicate 0 Al Si Cylinder cage 0310 B Regular Carbon nanotube C Cylinder 210 Templated silica 0 Al Si Cylinder 215 C Disordered Porous glasses 0 Si Cylinder 5104 Pillared clays O Si Al Slitpillars gt05 Activated carbon fiber C Slit 0615 7 NC STATE UNIVERSITY 7 Grand Canonical Ensemble Some examples Grand Canonical Monte Carlo simulations of adsorption of LJ nitrogen colored molecules at 77 K in a model porous glass of 47 porosity and 323 nm mean pore size figure taken from Dr Lev Gelb s website at Washington University in St Louis httpwmANchemistrywustledugel bcpgvishtml Reference L D Gelb and K E Gubbins Langmuir14 2097 1998 1 f 9 he 39 mitt 39r 4 V 3 NC STATE UNIVERSITY Grand Canonical Ensemble Some examples Snapshot of GCMC simulations of LJ nitrogen at 77 K in a model activated carbon 081000 at a bulk pressure of PP01 The rods and the spheres represent CC bonds and nitrogen molecules respectively The model activated carbon was obtained from constrained reverse Monte Carlo simulations figure taken from J Pikunic et a Langmuir 19 8565 2003 NC STATE UNIVERSITY Grand Canonical Ensemble Some examples T290 K T252 K T228 K T192 K on l Reduced local density p 0 1 2 3 4 5 Reduced radial position r a Freezing of CCI4 in a multiwalled carbon nanotube with D 5 nm a Density profile b snapshots of typical configurations ofthe adsorbed phase The carbon nanotube walls are not shown for clarity taken from F R Hung et a Mol Phys 102 223 2004 5 NC STATE UNIVERSITY Grand Canonical Ensemble Classical Thermo GibbsDuhem eguation Prausnitz et alMolec Thermo of FluidPhase Equilibria p 18 For an open phase dU 2 MS PdV yidnl 97 l Integrate this equation from a state of zero mass to a state of finite mass USVrz1 at constant T P xi i12m Thus T P and 1 remain constant and UTS PVZ1nj 98 I Since U is a state function this equation is independent of the integration path Differentiating this equation dU TdS Sa T PdV VdP Ziaan andJz 99 7 NC STATE UNIVERSITY 7 Grand Canonical Ensemble From 97 and 99 Sa T VdPZnidyi 0 GibbsDuhem equation 910 Grand Potential or Grand Free Enerqy For the grand canonical ensemble the thermodynamic potential is QZA ZNa aZA GZ PV bulksystem 911 dPV PdV MP 2 PdV SdT ZltNa gta ua For change at fixed T V ua d9 5 O Q is minimum at equilibrium NC STATE UNIVERSITY 7 1 Grand Canonical Ensemble Thus P8LV j 912 6V Thu T3 Swj 2 913 6T m M m ltNgt 61 m 69 914 8 VT 8 VT i7 NC STATE UNIVERSITY 7 Grand Canonical Ensemble Statistical Mechanics Probabilit distribution law 3Ek N e PNk 11 probability system is in quantum 915 state k and has N molecules where lkT 6G u 2 chemical potential per molecule Ek energy of system in qs k TP E 2 grand partition function 2 EuVT therefore I N Ek MN 06 Zkle Ze QN 916 N2 N20 71 NC STATE UNIVERSITY 7 Grand Canonical Ensemble where QN canonical partition function for a system with N molecules We also have MN 6 PN 2PM QN probability system contains N k molecules regardless of quantum state I 917 4 NC STATE UNIVERSITY 7 Grand Canonical Ensemble Thermo Properties Thermodynamic properties in terms of E 1 Averaqe number of molecules N ZZPNkN ze NeMw N k 5 or Nk 1 alnE N lt gt 6 5V 2 The function PV A thermodynamic identity is 914 81 1 918 919 NC STATE UNIVERSITY 7 Grand Canonical EnsembleThermo Properties 2 The function PV From 918 and 919 PVziIdyalndzllnEC 3 art3 Where C is independent of u It can be shown that C must vanish in order that 920 yield the known virial equation of state derived from the virial theorem of mechanics Thus 920 PVkT1nE 33313132322 2223 921 3 The eguation of state From 912 P w gt P 4 922 W T 6V T NC STATE UNIVERSITY 7 i7 Grand Canonical Ensemble Thermo Properties 4 Entropy From GibbsDuhem SMj k1nEkTalndj 923 GT V 6T V y Example ideal gas equation of state N For ideal gas QN q where q is the molecular partition function N From 916 y 6 c 1 lnazlnz m zl e qlq8PV 924 N20 where we used 921 and A E 63 1 NC STATE UNIVERSITY 7 Grand Canonical Ensemble Ideal Gas EOS Also from 918 ltNgt iFMLEJW kTMW kT3e q 3 81 81 81 e l q 2 Ag 2 BPV from 824 Thus 1 N BPV EPV PV kTltNgt 2 RT I i NC STATE UNIVERSITY 7 Distribution Functions in the GCE fzhah probability density for a set of h molecules to be in dildwld12da2 dzhdah simultaneously regardless of N ZPNfN Zhawh N N N 1 e u Z Nf ngN N IdZN hda N he M a N2 Nlh At Ar wherefN Eh wk 2 canonical distribution function for N molecules UN 5NaN potential energy foerolecules 7 NC STATE UNIVERSITY 7 Distribution Functions in the GCE and E V e Nq d Nd N WN 4m Z a 6 Also Alhawhfzlw1fzzaw2fz3aw3flhawzzglhawh h g1hah for an isotropic homogeneous fluid h h h h N W a Wquot ltltN hgtgt NC STATE UNIVERSITY 7 THE MICRCOCANONICAL ENSEMBLE Consider a system at constant number of molecules N volume Vand energy E A collection of such systesm all at the same N E is a microcanonical ensemble The system can still exist in an enormous number of quantum states but they are all of the same energy Suppose the number of possible states is Q ie Q is the degeneracy Our basic postulate says that the probability of observing a particular quantum state depends only on its energy Thus all states are equally probably in this ensemble is NC STATE UNIVERSITY 7 THE MICRCOCANONICAL ENSEMBLE g 925 1 Q We already showed that the entropy is given by S 421P lnP Thus i9 1 1 S 42 51115 kln Q 926 In the micrcocanonical ensemble TS plays the role of thermodynamic potential or free energy and is minimized at equilibrium 1 NC STATE UNIVERSITY CHE 596M Multi Scale Modeling of Matter Instructor Keith E Gubbins Lectures 20 Brownian Dynamics and Dissipative Particle Dynamics OUTLINE Coarse Graining Brownian Dynamics BD BD implementation the algorithm BD pros and cons BD examples Dissipative Particle Dynamics DPD DPD implementation DPD examples Coarsegraining By matching gR Meihmc Finc a ueff sum that gRatemitic g Rkoarsegrained 1 Is there such a unique ueff Coarsegraining By matching gR Metho Find a um such that gR3a tomis t c g Rkoarsmgrainsd Uniqueness theorem If only pair potentials exist there is a unique uR that gives rise to a particular gR Unique gR 4 uR R L Henderson Phys Lett 1974 C G Gray and K E Gubbins Theory of Molecular fluids Vol 1 p 178 1984 Coarsegraining By matching gR Method Find a self such that gRatomistic gRcoarsegrained Uniqueness theorem If only pair potentials exist there is a unique uR that gives rise to a particular gR Unique gR 4 uR But even if the atomistic system gt only pair interactions the coarsegrained system gt all Nbody interactions Therefore this uniqueness is strictly valid only if 3body and higher interactions in the coarsegrained system are negligible How to find ueff quotJ i eff L 2 approaches to find uefffrom gR atomistic 1 Potential of mean force 2 Iterative Boltzmann inversion 1 Potential of Mean Force PMF uef len gRatomistic Ethanol H20 o 8 wt 46 wt 70 Wto 175 1 1 a 175 1 1 1 1 b 175 C 15 O Atomistic 7 15 7 1 O Atomistic lt 15 7 o atomistlcl 1 25 Mesoscale 125 1 Mesoscae 125 7 J 680808 e 10 7 10 7 10 075 075 075 0395 A90 0 5 A9R 0 5 02 7 1 01 AgR 7 5 7 01 025 1f 39L v 025 39 inkhf 0000 25 510 75 100 125 150 0000 25 5 0 7 5 160 1239s 150 00 W 1 1 1 1 RA 39 39 1 00 25 50 75 100 125 15 RA RA Conclusion Exact at low concentration Good gR match Deviation at higher concentrations 2 Iterative Boltzmann lnversion ueff is iteratively corrected R i if lie7 In mesoscale gRatomistic 1457 2 Potential of mean force kT In gRatomistiC Iteration is repeated until max AgR lt 002 A K Soper Chem Phys 1996 2 Iterative Boltzmann Inversion EthanolH20 46 Wt 70 Wt 175 7 1 1 a 175 7 b a 15 7 15 r o Atomistic O atomlsncl Mesoscale 125 7 esoscae 7 125 7 10 7 g 10 m 075 7 075 05 7 AgR 05 01 AQR 025 00 025 7 01 00 1 1 1 1 0 0 4 I 1 1 00 25 50 75 100 125 150 00 25 50 75 100 125 15 RA RA Conclusion Good gr match for all concentrations studied oJR Silbermann SHL Klapp M Schoen N Chennamsetty H Book and KE Gubbins Mesoscale Modeling of Complex Binary Fluid Mixtures Towards an Atomistic Foundation of Effective Potentials Journal of ChemisePhysics 124 074105 2006 Coarse graining by matching a specific property Coarsegraining by matching gR Dimyristoylphosphatidylcholine DMPC Lipid lt3ng 5 F CH gen 7w 4 r o 3 c 2 l y 0 Airman nioch Causegrained model 8 arms 105m Fig I Mmemcm and mmgmmed nu 1mm models d 2 The inter and intra molecular correlation functions found from a dilute aqeuous atomistic MD simulation Intramolecular RDF 3 4 5 05 The corresponding CG potentials are found using the iterative Boltzmann inversion u I method 5 7 r M BeadBead Bond Length Distributions Lyubartsev AP Eur Biophys J 35 5361 2005 Coarse graining by matching a specific property Coarsegraining by Matching gR pg M l39mm unnrdLrcd sinirtum Non Dimyristoylphosphatidylcholine DMPC Lipid simulation is shown in m The potentials found are state dependentll However if the same potentials are used for a high concentration system self assembly of the lipids into bicells occurs Lyubarisev AP Eur Biophys J 35 5361 2005 Langevin equation In the following we concentrate on one say the Xcomponent of the vectorial velocity Although the motion of a Brownian particle is quite F t random it must follow Newton s second law m v total Our earlier observations justify the gz Gaussian introduction of a random force This mV 7V 0g random gives the LANGEVIN EQUATION variable 0 noise amplitude 02 The FluctuationDissipation Theorem 7 2 says 2kT Ref Robert Zwanzig Nonequilibrium Statistical Mechanics Oxford University Press New York 2001 Brownian Dynamics BD BD amp DPD Starting point Langevin equation mvi Fit lvz39 0 ilt0 jii C Here F17 1s the conservatlve force between the large part1cles i and Method ED is just the solution of the Langevin equation with the conservative forces included Integrating the Langevin equation generates the system s trajectory in phase space From this trajectory all properties of interest can be calculated as time averages The equilibrium solution of this eq is the canonical distribution That means that the equilibrium properties of a Brownian Dynamics simulation are those of the canonical ensemble Introduction of the BD method DL Ermak J Chem Phys 6210 4189 1975 Brownian Dynamics BD BD amp DPD Like MD and MC Brownian dynamics is particle based What is a BD particle complete molecule or colloid part of a molecule polymer gt chain of BDbeads BD Technical Details BD amp DPD What is E conservative force acting on particle 139 generated by particle j pair interaction How do I get it effective interaction given by the appropriate coarse graining BD Technical Details BD amp DPD C Can be approx1mated Assuming pairwise additiVity for the determination of the potential of mean force for nonbonded interactions is usually the only feasible way Empirical potentials are often used rather than carrying out coarse graining over atomistic potentials Combinations of both empirical and coarse graining can be used e g empirical bond length bond angle torsion potential of mean force all other nonbonded interactions inter and intramolecular BD Technical Details BD amp DPD What is 7v What is 030 The friction and the random force arise from the dynamics of the degrees of freedom that have been coarse grained out The noise is Gaussian White noise See discussion of the Langevin eq The noise amplitude 039 and friction coefficient 7 are not independent but connected by a uctuationdissipation theorem BD Technical Details BD amp DPD How to choose 7 The determination of the friction coefficient is nontrivial Remember the equilibrium properties don t depend on the friction coefficient If we are considering a solution say a colloidal solution the friction coefficient can be derived approximated from bulk properties FD drag force 39 Stokes law FD 7V 2 67r77av n Viscosity of the solvent a diameter of the spherical Brownian particle kT m7 D diffusion coef cient of a Brownian particle in solvent Einstein s relation D BD Conclusions BD amp DPD For algorithm to solve BD see Appendix For systems with large differences in the time scales of the molecular motion BD has advantages compared to MD because Removal of many particles gt less computation per time step dissipation stabilizes the equation gt increase of the time step by a factor 23 time step is now determined by the slow modes gt increases time step by a factor of several hundreds The available observation time still depends on system size but is 0104 fold longer at the cost of atomistic detail however BD Conclusions BD amp DPD Problems 1 Because the random force is random it does not re ect any system structure 2 Length scales must be well separated solvent much smaller than solute 3 Impact of the solvent on the structure must be included in the effective interactions excluded volume 4 Doesn t give correct hydrodynamics no momentum conservation because of the random forces solvent mediated velocity coupling due only to the conservative forces between the Brownian particles BD Example 1 S Glotzer BD amp DPD HOW to assemble nanoparticles Thanks to Prof Sharon Glotzer for providing the material The application of nanotechnology to photonics molecular electronics chemical and biological sensors energy storage and catalysis requires manipulation of nanoobjects into functional arrays and structures a A A Challenge How do we organize thousands or billions of these building blocks into predictable ordered structures for materials or devices with useful properties and behavior Taken from lecture by Sharon Glotzer How to assemble nanoparticles How do we organize thousands or billions of these building blocks into predictable ordered structures for materials or devices with useful properties and behavior Add tethers to the nanoparticles short chains the nanoparticle chain becomes a surfactant and this enable us to control selfassembly through suitable use of solvents BD Example 1 S Glotzer BD amp DPD El Use a coarse grained representation whereby a N 0000 group of atoms in the nanoparticle or tether is replaced by a single atom RedLlceS force calculations 6 Retains nanoscale roughness Empirical pair potentials between atoms van der Waals interactions Excluded volume interactions Minimal model Thermodynamic immiscibility z Geometrical constraints El 3 Our goal discover trends and construct general design strategies Taken from lecture by Sharon Glotzer BD Example 1 S Glotzer BD amp DPD Expect in surfactants Here panice can be a gold her 3 herical nanoparticle Solvent poor for nanoparticles good for tethers Taken from lecture by Sharon Glotzer BD Example 1 S Glotzer BD amp DPD Again expected in surfactants ere we have in mind spherical pam39cles with single tethers Reverse selectivlt 27L Zhang MA Hersch MH Lawn and sce Solvent cm for NanaLzHers In press P tethers good for nanoparticles Taken from lecture by Sharon Glotzer BD Example 1 S Glotzer BD amp DPD Solvent poor for white tether and nanopanicle good for purple tether Taken from lecture by Sharon Glotzer BD Example 1 S Glotzer BD amp DPD Increase overall volume fraction Hexagonal cylinder phase Z L thmg MA Horscl39L MHV Lawn and 566 NanoLeHers m press Taken from lecture by Sharon Glotzer BD Example 1 S Glotzer BD amp DPD Similar to oligome icallyistabilized discofic LC s Percec et a Science 2002 24 Zhang M A Hursch M H Lawn and see prepz im Taken from lecture by Sharon Glotzer BD Example 1 S Glotzer Taken from lecture by Sharon Glotzer BD amp DPD Solvent poor for nanoparticle good for tethers Hexagonal packing w thm lamellae sheets BD Example 2 BD amp DPD Permeation of Ions Across the Potassium Channel 0 sodium inside outside potas sium selectivity lter The potential of the channel depends on the positions of all ions in the channel The channel selectively permiw passage of K r but not Na ShiniHo Chung Biophysics Group Department ofPhysics The Australian National Universi Toby Allen Departments ofPhysiology and Biochemistry Weill Medical College ofComell University Stuart Ram den Supercomputer Facility The Australian National University very interesting Web site httplangevinanueduau BD Example 3 BD amp DPD Counterion Condensation Shulan Liu from research group of M Muthukumar Department of Polymer Science and Engineering University of Massachusetts httpkalipseumassedu Nshulan ED time 5273 set 547 Dissipative Particle Dynamics DPD BD amp DPD In DPD the force between particles 239 and j is given by the sum C FDPD RC FD FR E conservatwe force I I I 1 FD I dlss1patlve force R E random force Instantaneous force acting on particle i EltrgtZlEf 15 1le jii We assume pairwise additivity for all three types of force Dissipative Particle Dynamics DPD BD amp DPD The equation of motion is given by the Langevin equation 12 Fm ZlF 12p Efl jii Since all forces are pairinteractions Newton s third law is obeyed This is not the case in BD As a result DPD looks like MD D Integration of the equation of motion as in MD possible but some methods are more ef cient than others gt Implementation DPD The particle BD amp DPD What is a DPD particle 0 molecule part of a molecule uid element many molecules DPD particles don t necessarily have a hard core can be soft The colored area symbolizes the interaction range DPD Differences to BD BD amp DPD Without specifying the force contributions some differences to BD are apparent In BD the dissipative force describes only friction with the small particles but in DPD the friction generates a drag on neighboring DPD particles mediated by the small particles Since all interactions are pairinteractions momentum is conserved Conservation of mass and momentum D Macroscopic behavior is NavierStokes hydrodynamics ED is diffusive and does not give correct hydrodynamic behavior Couplings through dissipative force causes long range velocity correlations D hydrodynamics with much less particles than in MD DPD The conservative force BD amp DPD People have often used E39jc fmanCrj ij r force on i engendered by j l39 1 fmaX force parameter maX force I 2 r I 1 EC 79 weight function A 1 unit vector e U from 39 to 139 describes how E decays With 1 7339 J FlC becomes more complicated for chains of DPD particles due to bond length angle and torsion DPD The conservative force BD amp DPD For very large molecules traditionally C A 1 2 r lt7 fmanCrjezj l c Iquot ZUCOi j 0 C else 3 fmaX 39 39 39 39 fmach purely repulsive very soft no hard core 0 I I 20 10 0 10 20 Last two pomts imply a large 73 time step if consistent with re 10 fmaX 25 dissi ative and random force p MeX1can hat function DPD The conservative force BD amp DPD Even in the traditional case it remains to determine fmax This can be done by a t of experimental or other simulation results If the description of the system is obtained from coarse graining than EJC is de ned by the coarse graining procedure The smaller the structural atomistic unit assigned to a DPD particle is the harder the potential becomes DPD The dissipative frictional force BD amp DPD friction coef cient D A A E 7wD 73eij vijeij J 3039 D 73 weight functlon positive de nite component of the relative speed I vi relative Speed of z w1th respect to 1n the d1rectlon of 1 J positive move apart D force towards j l 1 negative move closer D force away from j The dissipative force always damps relative movement along the con nection vector This generates a direct drag thus cooperative motion 7 and ZUDUU are yet to be de ned DPD The random force BD amp DPD noise amplitude A 0 U g1 R 1 U HR 13 weight function QU Gaussian noise variable Symmetry g 1 g necessary to obey Newton s third law EJR FJ thus momentum conservation Stochastic properties of the noiseglj em 0 ltgijtgijt39gt 511451 5115J5z 139 gt ltgyt2gt 1 time autocorrelation function variance The time autocorrelation function of the noise states that the noise is entirely uncorrelated The Fourier transform of the time autocorrelation function is a constant 1 gt white noise aand ZURUU are yet to be de ned DPD The FluctuationDissipation Theorem BD amp DPD Results The canonical ensemble NVT is the stationary solution of the conservative part of the DPD equation gt If we run a simulation without the dissipative and the random forces y 5 O we generate a canonical distribution If we wish the canonical distribution to be the stationary equilibrium solution with dissipative and random forces present we must satisfy 2 0 7 2kBT and 01303 ta1102 The first equation is the uctuationdissipation theorem identical to the one in BD See Espanol and Warren Europhys Left 304 191 1995 DPD BD amp DPD Consequences Reduction of the number of unknown parameters If we choose 013031zaR1jZ and satisfy the uctuationdissipation theorem then the equilibrium solution of our simulation is the canonical ensemble at temperature T independent on the specific choice of 7 039 and zaR7j EDGE The dynamics does depend on the specific form of the dissipative and the random force DPD The random and dissipative forces BD amp DPD How to chose the parameters for the dissipative and the random force As long as only equilibrium properties are concerned the parameters and weight functions can be chosen arbitrarily Using a Verlettype algorithm the simulation is found to be stable and efficient if 2 2 1 if r1ltl c 3lt0lt8 ZUDUZURIlj c J 0 otherwise Groot and Warren J Chem Phys 10711 4423 1997 A more realistic choice of the parameters which is identical to relating the dynamics of the coarse grained system to that of the full system is nontrivial DPD The problems BD amp DPD A more realistic choice of the parameters Which is identical to relating the dynamics of the coarse grained system to that of the full system is nontrivial and an unsolved problem Attempts have been made but With little practical use The second problem is the physical time scale of DPD Since the random and the dissipative force do not have a well de ned physical origin the physical time scale is ambiguous In some cases the dynamics of the simulation can be related to the dynamics of an experimental system Which gives a physical time scale DPD The advantages BD amp DPD If equilibrium properties in the canonical ensemble are desired than DPD is much easier to program than MC no complex moves for long chains can be applied to systems of arbitrary complexity with little or no modi cation is faster than MC for soft potentials DPD shows hydrodynamic behavior velocity correlations with far less particles than lD As BD DPD is orders of magnitude faster than MD if only mesoscopic properties are of interest This is because the number of interacting particles is signi cantly reduced and the potentials are soft and we can use a much larger time step DPD Implementation BD amp DPD Since the dissipative force depends on the velocities the equilibrium temperature has been found to depend on the time step More stable algorithms 1 modi ed velocityVerlet algorithm Groot and Warren J Chem Phys 107 l l 4423 1997 2 iterative LeapFrog algorithm Frenkel and Smit Understanding Molecular Simulation 2nd edition Academic Press San Diego 2002 DPD Example 1 BD amp DPD Polymer solution fm 28 1111 Palm 1 e pa1rs 1 if 1 lt rC 0DjwRjz wcz l0 else 7230 r5 1 Polymer chains 0 o 4 0 0 39 sp ng O Fij C b o 0 C 20 RD Groot and PB Warren 1 Chem Phys 10711 4423 1997 SM Willemsen TJH Vlugt HCJ Hoefsloot and B Smit J Comp Phys 147 507 1998 DPD Example 1 DPD was done in an explicit interface simulation gt much bigger box than Gibbs ensemble 070 060 D D In lt12 E D 050 D a El El 397 a as X 040 DPD N6000 ltgt Gibbs ensemble N1296 a D 030 r B Gibbsrensemble N384 I 1 2 Pl chain length 10 particles BD amp DPD Polymer solution Gibbs ensemble DPD 384 particles 6000 particles 30 min 270 min For the given problem Gibbs ensemble MC is more appropriate than DPD 47 sparticle 27 sparticle SM Willemsen TJH Vlugt HCJ Hoefsloot and B Smit J Comp Phys 147 507 1998 DPD Example 2 BD amp DPD Time evolution of A3B7 diblock copolymer melt 2000 time units Gyroid like structure mean eld predicts this structure to be unstable 6000 time units rods are completely aligned with some side Ward connections http www lee hut researchpolymer so simuZUUZgroot shtml RD Great and TJ Madden Chem Phys 10820 8713 1998 4000 time units less symmetric rods start to reorient 8000 time units hexagonal phase DPD Example 3 BD amp DPD Time evolution of A3B7 diblock copolymer melt DPD or BD DPD BD I 1500 7500 15000 tlme umts BD nds hexagonal ordering In DPD haxagonal ordering is quickly reached locally but hexagonal domains Whereas BD does not nd this state don t grow over the domain of the box RD Groot T J Madden and DJ Tildesley J Chem Phys 110 19 9739 1999 DPD Example 3 BD amp DPD Time evolution of A3B7 diblock copolymer melt DPD or BD BD time 103T hexagonal DPD a fast BD 9 never lamellar DPD amp BD identical lamellar 4 spinodal decomposition mixture is unstable hexagonal 4 nucleation and growth mixture is metastable initial growth of the nucleus is di asive second growth regime is visc ous hydrodynamic RD Groot T J Madden andDJ Tildesley J Chem Phys 110 19 9739 1999 DPD Example 4 BD amp DPD Formation of a liposome 86000 DPD pal cles from the group of Mikko Karttunen Biophysics amp Statistical Physics Group Laboratory of Computational Engineering Helsinki University of Technology full object N80000 Water particles are removed for clarity httpWWWlcehut researchpolyrnerdpdshtml APPENDIX BD Implementation A good starting point is MP Allen and DJ Tildesley Computer Simulations of Liquids Oxford University Press New York 1987 The goal is to find an efficient method to integrate the Langevin equation nevir EU 2F r mt actr jii We replace the integral by a discrete sum leading to a finite time step A I As in atomistic MD we integrate the Langevin equation over one time step This is however possible only if we make some assumptions about the time dependence of the conservative force e g we take EJC to be constant over At BD Implementation BD amp DPD Simplest method assume that the conservative force is constant during At The integration of the velocity part is analogous to our earlier calculations for the Langevin equation For one component 1 l l M wrgt agltr Ff Ff ECU 212 dt m m m 1 E multiplying by e and using the product rule gives 0 E 1 C K emvl 0 ZE 6 dt m BD Implementation BD amp DPD now we integrate from t to 1 At ytAt tAt e m vtAt evt j dr39iagt391 e t m rearranging terms and executing the integration over the timeindependent force gives amp 1 amp IAt 1 yz z Az vtAte mvt EC 1 6 m Idl39 0gl39e m r m This equation tells us how to update the velocities in our algorithm BD Implementation BD amp DPD We define a new stochastic variable 92 tAt a dr39 Var I M The reason for introducing this variable is that l O39gl39e m the inverse of the collision frequency is much m smaller than the time step It can be shown that if QC is Gaussian then 92 is Gaussian too Since we now know how to update the velocities it remains to derive an equation for the updated positions tAt xl At w Idl39vU39 BD Implementation BD amp DPD We have already derived an equation for the velocities A variable transformation is necessary The integration is then straight forward Let tAtt39 The algorithm is then vt At covt chic Q A 1 m xl At w clmvo Ff gx c1 L r BD Implementation BD amp DPD It is obvious that xand 95v are not independent since they are integrals involving the same random process They are sampled from a bivariate Gaussian distribution For details see MP Allen and DJ Tildesley Computer Simulations of Liquids Oxford University Press New York 1987 The algorithm we have developed here is a simple predictor algorithm If we would have assumed the force to be linear in time then we would have obtained a Verlet type of algorithm which is more efficient
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