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# MACROMOLEC SYS I CHEM 4010

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This 14 page Class Notes was uploaded by Javonte Swift on Tuesday October 13, 2015. The Class Notes belongs to CHEM 4010 at Louisiana State University taught by Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/223116/chem-4010-louisiana-state-university in Chemistry at Louisiana State University.

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Date Created: 10/13/15

A CHEM 4010 Computer simulation of polymeric systems An overview Francisco R Hung Cain Department of Chemical Engineering Louisiana State University httpwwwchelsuedufacultyhung Disclaimer Modeling polymeric systems is not my main area of expertise however I am trying to get into that area Outline UPDATE Today s material adapted from Keith Gubbins lecture notes on Advanced Chemical Engineering Thermodynamics North Carolina State University David Kofke s lectures on Molecular Simulation SUNY Buffalo httn39 wwwenn buffalo quot 39 quot 39 html Sharon Glotzer s lectures on Computational Nanoscience of Soft Materials University of Michigan Doros paper de Pablo et al paper Simulation Methods General Features References A R Leach Molecular Modeling 2 d ed Ch 6 8 Prentice Hall 2001 D Frenkel and B Smit Understanding Molecular Simulation 2nd ed Academic Press 2002 M P Allen and D J Tildesley Computer Simulation of Liquids Clarendon Press 1987 T Schlick Molecular Modeling and Simulation SpringerVerlag 2002 D C Rapaport TheArt of Molecular Dynamics Simulation 2 d ed Cambridge University Press 2004 Cnlnplmsr 51 nullnlmn www amazon com 4 Simulation Methods General Features Role of computer simulations Assist in interpretation of experimental results 9 g capture subtle details of molecu ar motion structure other CONSTRUCT APPROXIMATE THEDRI ES THEORETICAL PREDICTIONS p Predict explore effect of variables I eg explore conditions for which experiments would be impossible expensive Give insights complement experimental work influence experimental efforts Test validity of theories which are computationally less expensive than simulations From Allen and Tildesley Compute Simuati on ofLqui ds l 987 Simulation methods Co nti n uu m TIME Methods ls 10 Mesoscale methods quot15 10391 Lattice Monte Carlo Atomistic Simulation Brownian dynamics us 105 methods Dissipative particle dynamics quot5 we Semiem pirical methOds Monte Carlo Molecular Dynamics quot5 1039 Ab initio methOdS tightbinding MNDO INDOS 395 10 5 1 10quot 10399 10 3 10397 10quot mu m LENGTHmeters Molecular Dynamics Introduction Molecular Dynamics MD simulates the real dynamics of a collection of atoms molecules particles or other extended objects MD is one of the most commonly used methods for materials simulations Positions and velocities of each molecule are followed in time by solving Newton s equations of motion F dzri 3U imiaimi dtz i123N U interintra molecular potential ie interactions between atoms r position vector of atom i F force acting over atom i m mass of atom i Thus MD is a deterministic method the state of the system at any future time can be predicted from its current state in principle at least MD How does it work Newton s equations of motion for the N particle system where E ml 39 1 EU F 7 Force acting on pamcle z m 2 Mass of pamclel dzr 1 dtz Acceleranon of part1clel In MD these equations are integrated numerically to obtain the time evolution of the system under the given potential There are several approximate methods to numerically solve this system of equations Each method has tradeoffs Accuracy Stability Time reversibility Memory requirements Complexity Verlet Alg orithm 2 Flow diagram OneMD Cycle One force evaluation per time step Total pair energy breaks into a sum of terms UrNUm UW U U U tors disp Upol elec Um stretch Ubend A Utors 39 tOI Sion Udisp dispersion van der Waals U eectrostatic elec Upol polarization ATypical Force Field Bond stretch Valence angle bend k1 kt 52 V Intra 2 quotlcosnp y Torsional mglec Iarsiom N N 6 Intro 44 0quot U 2 39 1 43 39 an il rij x j rij rij Infquot molecular Electrostatic Coulomb van der39 Wools Lennard Jones Atomistic MD and polymeric systems The time step At for numerical integration of Newton s equations of motion in atomistic MD is determined by the fastest modes At 1fs 1015 s Bond vibrations However for polymer melts the longest relaxation times are on the order of milliseconds seconds for typical molecular weights under processing conditions W 4810 fs Avery optimized MD package Gromacs v 4 dealing with a system of an ion channel placed in a model membrane solvated with water and ions totaling 121 449 atoms running on 128 cores 32 nodes in parallel 66 nsday 66 x 10399 sday Hess el al J Chem Theoy Compur 2003 4 435 Aballpark estimate to simulate 1 s of real time would take Computing time 15151515 days 41483 years 1s 66X1039g sday With current computers we can model a few microseconds at most Atomistic MD is unable to equilibrate a longchain polymer melt Modeling polymeric systems First introduce simplifications to atomistic methods to remove the faster degrees of freedom andor treat groups of atoms blobs of matter as individual entities interacting through effective potentials gt coarsegraining the system Although it might appear simple properly coarsegraining a system is difficult Application of coarsegraining to polymers has been reviewed J quot 39 39et al Advances in Polymer Science 2000 152 41 F MullerPlathe ChemPhysChem 2002 3 754 D Reith et al J Comp Chem 2003 24 1624 13 Modeling polymeric systems Coarsegraining polymeric systems Three levels of representation of atactic polystyrene melt specimens at 500K and 1 bar a Detailed united atom model formed from four 350 dyad long parent chains molar mass 36500 g mol 1 Segments coming from one of the parent chains are traced in red for clarity b Coarse grained model formed from four 350 dyad long parent chains wherein each dyad of monomers is represented as an interaction site superatom The two types of dyads m r are shown in different colors D N Theodorou Chem Eng Sci 2007 62 5697 14 Modeling polymeric systems Monte Carlo method generate configurations of a system by making random changes to positionsorientationsconformations of the atomsmolecules over and over again Differences between MD and MC MD generate successive configurations by calculating conservative forces derived from gradients in the potential energy and then solve deterministic equations of motion MC generate random configurations with a probability that depends on the potential energy of a new configuration compared with the previous one At each iteration of a MC simulation a new configuration is generated This is usually done by making a random change to the coordinates of a randomly chosen particle using a random number generator Moves are accepted with a probability exp Um UM kT A great variety of random moves can be attempted Trial Moves in MC simulation Significant increase in efficiency of algorithm can be achieved by the introduction of clever trial moves Example simulating chain molecules eg polymers Relaxation times are large exploration of phase space is very slow need concerted moves to disentangle chains whim Reptation crankshaft Example of moves for polymers 2 Ji 0 Chain regrowth and many more Using these physically unrealistic moves we can explore phase space faster than in MD Trial Moves in MG simulation a xmuumzcuuk BRIDGI ue gm Dasha embrl gm gt w quotW 39539 e mm mm panning wnnmhviiymzlnaum rm pi e mm allquot m irime s and chum l r39 W quot391 quotW i V anesmmnw cimmmm M 3 C A m n 1 mum 5 mm 1 D 1 39 m undying marl MW mums i macho DC An em nllhu chain is hm I I in an Imsmsi WW vsmq a gt f m j g v Ilimeiin woman m f 39 aquot mu In a linear an lniarnal and magma mu me many mmalms IV mm in in mm Duirii gt r39m f 739 n W a way M w lt Wequot mummy geomeuy dues not change m w pmioln Mm of me nmnm cam V 23 Mm db b mi v e 39 i quot 8 D quot399 am creams an em bum Figure 1 39 mums L D Perisleras er Macromolecules 2005 38 386 Connectivityaltering moves provide vigorous sampling of the longrange conformational features of chains endtoend distance radius of gyration and are thus extremely efficient in inducing equilibration in longchain polymer systemsquot D N Theodorou Chem Eng Sci 2007 62 5697 17 MC simulations coarsegrained polymeric systems mu m Speci cVoiumeicmSg S N Fig l Experimenlnl circles and simulated isqunras speci c vulu um i m for sens of monadlspelse polyemyienr mehs at 450K and aim Kamyiunnis ZOOZL el al Sk 1 0 200 Mm sou Chain Length 800 mm 0 2 limo epenmen am his mblidgmg min mum bun 39 mm m in PE The imulmlan u 2 450K um Inn mi Are A 4301 and lulrn 5 WW g mmpmu to am L xpcrlnnlnm udurunzlc 1 Smu ummm lliuuvr mi lnr monotqume lintm cm polyullyr D N Theodorou Chem Eng Sci 2007 62 5697 Rheology of coarsegrained polymeric systems Rheological properties important for polymer processing Ato istic MD not suitable long relaxation times MC not suited to measure timedependent properties only equilibrium properties A hierarchical strategy can be used 1 un MC simulations of model polymer systems Properly equilibrate systems 2 Run short MD simulations of systems previously equilibrated via MC simulations 3 Map timedependent results time correlation functions meansquare displacements from short MD simulations onto me oscopictheories of polymer dynamics e Rouse reptation model 4 Use the mesoscopictheories with parameters from MD simulations to estimate rheological properties T M J This mapping process can also a e Fliclion ratlei per urban mom chain selledll39l39uslvlu and me shear 1 g39Ve quot15399th ab e or mail awa chains as m e bv in unitedralom MD ranges of applicability of the onto me Rouse mudCI from Harm lal 998 mesoscoplctheoriesY 2 cm pleaded Miamim Pinpoint inadequacies in the 7 7 r the mesoscoplc a 00 311quot f g3 33 theories which will help in the B lO cmls H lA iOJ 14 development of better theories nu lcp 36 i 005 9 1 D N Theodorou C iemEig Scizuuzszsew m CHEM 4010 Computer simulation of polymeric systems An overview Francisco R Hung Cain Department of Chemical Engineering Louisiana State University httpwwwchesuedufacultyhung Ab Initio Methods including DF T Calculate properties from first principles solving the Schrodinger or Dirac equation numerically Pros Can handle processes that involve bond breakingformation or electronic rearrangement eg chemical reactions Methods offer ways to systematically improve on the results making it easy to assess their quality Can in principle obtain essentially exact properties without any input but the atoms conforming the system Cons 2 Electron locallmtlon function 39 Can handle only small systems about 010 atoms for a an isolated ammonium ion and b an ammonium ion Can only study fast processes usually 010 ps whims rst 801mm she Approximations are usually necessary to solve the from 1 W molefu39ar dynamics From Y Lin ME eqns Tuckerman J Phys Chem B 105 6598 2001 Semiempirical Methods Use simplified versions of equations from ab initio methods eg only treat valence electrons explicitly include parameters fitted to experimental data Pros 0 Can also handle processes that involve bond breakingformation or electronic rearrangement 0 Can handle larger and more complex systems than ab initio methods often of 0103 atoms 0 Can be used to study processes on longer timescales than can be studied with ab initio methods of about 010 ns Cons o Difficult to assess the quality of the results 0 Need experimental input and large parameter sets Structure of an oligomer of polyphenylene sul de phenyleneamine obtained with the PM3 semiempirical method From R Giro DS Galvz39lo Int J Quant Chem 95 252 2003 Atomistic Simulation Methods Use empirical or ab initio derived force fields together with semiclassical statistical mechanics SM to determine thermodynamic MC MD and transport MD properties of systems SM solved exactly Pros Can be used to determine the microscopic structure of more complex systems 000 atoms 0 Can study dynamical processes on longer timescales up to 01 us Cons 0 Results depend on the quality of the force field used to represent the system 0 Many physical processes happen on length and timescales inaccessible by these methods eg diffusion in solids many chemical reactions protein folding micellization Adsorption of Ar molecules in a model MCM41 silica pore From B Coasne F R Hung R JM Pellenq F R Siperstein and K E Gubbins Langmuir 22 194 2006 Mesoscale Methods Introduce simplifications to atomistic methods to remove the faster degrees of freedom andor treat groups of atoms blobs of matter as individual entities interacting through effective potentials Pros Can be used to study structural features of complex systems with 000 atoms Can study dynamical processes on timescales inaccessible to classical methods even up to 01 s Cons Can often describe only qualitative tendencies the quality of quantitative results may be difficult to ascertain In many cases the approximations 39 the ability to physically interpret the results limit Phase equilibrium between a lamellar surfactantrich phase phasein supercritical 00 from a lattice MC s39mu1ation From Lr Scanu CK Han KE Gubbins Langmuir 20 514 2004 Continuum Methods Assume that matter is continuous and treat the properties of the system as field quantities Numerically solve balance equations coupled with phenomenological equations to predict the properties of the systems Pros Can in principle handle systems of any macroscopic size and dynamic processes on long timescales Cons Require input viscosities diffusion coeffs eqn of state etc from experiment or from a lowerscale method that can be difficult to obtain Cannot explain results that depend on the electronic or molecular level of detail 7 Temperature pro le on a laserheated slrface obtained with the nite element m Rajadhyaksha P Michaleris nt J Numer Meth Eng 47 13072000 The Monte Carlo Method procedure At each iteration of the simulation a new configuration is generated bi OO O O In a NVT simulation this is usually done by making a 0 random change to the coordinates of a randomly chosen 0 particle using a random number generator xnew xold 2amp1 1Vmax Where 5 9 2 9 3 are random numbers in the range 01 ynew 310161 2amp2 1 rmax rmaxis the maximum allowed Znew 2 gold 2453 1rmax displacement in any direction exm tW also calculate the difference in potential energy between the new and the old configuration 6U UrN UrN new old Always Calculate the Boltzmann factor exp gUkT WP Accept lf 5U lt O the new configuration is accepted 0 64V 61 lf 5U 2 0 the Boltzmann factor is compared to a random From Allen and TIldesley number 6 In the range 01 If f s ex10 5UkT the new configuration is accepted Formore details abounhe If f gt ex10 UkT the new configuration is rejected Theo ybehmd 7 5 a gor mm please consult any ofthe The old configuration IS retained for the next Iteration mammarsimwa on textbooks 27

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