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Free Energy MD and Nanoscale Polymers

Free Energy MD and Nanoscale Polymers. Lecture notes for Computational Nanotechnogoly and Molecular Engineering Workshop, Pan American Advanced Study Institutes 1/14/2004 Shiang-Tai Lin, Youyong Li, Seung Soon Jang, Prabal Maiti Tahir Çağın, Mario Blanco, and William A. Goddard III

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Free Energy MD and Nanoscale Polymers

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  1. Free Energy MD and Nanoscale Polymers Lecture notes for Computational Nanotechnogoly and Molecular Engineering Workshop, Pan American Advanced Study Institutes 1/14/2004 Shiang-Tai Lin, Youyong Li, Seung Soon Jang, Prabal Maiti Tahir Çağın, Mario Blanco, and William A. Goddard III Materials and Process Simulation Center, Caltech

  2. Free Energy Calculation in Nanotechnology • Free Energy is a key parameter in Nanofabrication • Equilibrium structure is determined by free energy • formation of self-assembled monolayers (SAM) • nanoscale patterns in liquid crystals and block copolymers • secondary structures of DNA, RNA • However, “Many of the ideas that are crucial to the development of this area--"molecular shape", the interplay between enthalpy and entropy, the nature of non-covalent forces that connect the particles in self-assembled molecular aggregates--are simply not yet under the control of investigators.” • George M. Whitesides(http://www.zyvex.com/nanotech/nano4/whitesidesAbstract.html) NanoStructures Functions Weak interactions (vdW, Coulomb, Hb) Free Energy

  3. Outline • How to obtain Free Energies from MD Simulations • Test Particle method • Perturbation method • Nonequilibrium method • Normal mode analysis • A new 2PT approach for efficient Free Energy Estimation • Basic Ideas (with Blanco and Goddard) • Test of method with LJ fluids (with Blanco and Goddard) • Applications of 2PT methods in the study of Dendrimers • Zimmerman H-bond dendrimer (with Jang, Çağın and Goddard) • PAMAM dendrimer (with Maiti and Goddard) • Percec dendrimer (with Li and Goddard)

  4. Free Energy Calculation from Molecular Simulations • Test Particle Method (insertion or deletion) • Good for low density systems • Available in the Sorption Module of Cerius2 • Perturbation Method (Thermodynamic integration, Thermodynamic perturbation) • Applicable to most problems • Require long simulations to maintain “reversibility” • Nonequilibrium Method (Jarzinski’s equality) • Obtaining differential equilibrium properties from irreversible processes • Require multiple samplings to ensure good statistics • Normal model Method • Good for gas and solids • Fast • Not applicable for liquids Reference: Frenkel, D.; Smit, B. Understanding Molecular Simulation from Algorithms to Applications. Academic press: Ed., New York, 2002. McQuarrie, A. A. Statistical Mechanics. Harper & Row: Ed., New York, 1976. Jarzynski, C. Nonequilibrium Equality for Free Energy Differences. Phys. Rev. Lett.1997, 78, 2690.

  5. From Normal Modes to Free Energy • Total number of normal modes = 3N, N=number of particles • For an isolated molecule with N atoms • 3 translation, 3 rotation (or 2 for linear molecules, 0 for monoatomic molecules), 3N -6 vibration • For a crystal with N particles • 3 translation (acoustic modes), 3N-3 vibrational modes • Each vibrational mode can be treated as a harmonic oscillator • The partition function is the sum of contributions from HOs • All the thermodynamic properties are defined

  6. Determine Normal Modes from Molecular Simulations • The density of states S(u) • S(u)dunumber of modes betweenuandu+du • Determination of S(u) • The eigenvalues of the Hessian matrix (vibrational analysis, phonon spectrum ) • Covariance matrix of atomic position fluctuations • Fourier transform of velocity autocorrelation function

  7. Liquid Gas Solid Debey crystal S(v) ~u2 exponential decay The Density of States Distribution S(u) • Singularity at zero frequency • Strong anharmonicity at low frequency regime The 2PT idea: Liquid  Solid+Gas • Decompose liquid S(v) to a gas and a solid contribution • S(0) attributed to gas phase diffusion • Gas component contains anharmonic effects • Solid component contains quantum effects • Two-Phase Thermodynamics Model (2PT) solid-like gas-like

  8. The 2PT Method • The basic idea • The DoS • Thermodynamic properties • The gas component • VAC for hard sphere gas • DoS for hard sphere gas • Two unknowns (a and Ngas) or (so and f) Lin, S. T.; Blanco, M.; Goddard, W. A. The Two-Phase Model for Calculating Thermodynamic Properties of Liquids from Molecular Dynamics: Validation for the Phase Diagram of Lennard-Jones Fluids. J. Chem. Phys. 2003, 119, 11792. Gas S0 f

  9. Determining so and f from MD Simulation • so (DoS of the gas component at u=0) • completely remove S(0) of the fluid • f (gas component fraction) • T or 0 : f1 (all gas) •  : f0 (all solid) • one unknown sHS

  10. Determining sHS • sHS (hard sphere radius for describing the gas molecules) • gas component diffusivity should agree with statistical mechanical predictions at the same T and r • gas component diffusivity from MD simulation • HS diffusivity from the Enskog theory

  11. f fy At Last… • A universal equation for f • Graphical representation

  12. Comparison of the 1PT and 2PT methods Run a MD simulation (trajectory information saved) Calculate VAC Apply HO approximation To S(u) 1PT thermodynamic predictions Calculate DoS (FFT of VAC) Calculate S(0) and D Solve for f Apply HO statistics To Ssolid(u) Apply HS statistics to Sgas(u) 2PT thermodynamic predictions Determine Sgas(u), Ssolid(u)

  13. VAC DoS An Overview of the 2PT Method To Phase Behaviors From MD Simulations MD simulations

  14. V(r) r = s 0 -e Supercritical Fluid r Gas Liquid Solid Test the 2PT Method Using the LJ System • Intermolecular potential • Phase diagram • stable • metastable • unstable • critical point • triple point (T*=kT/e r*=rs3)

  15. VAC and DoS of LJ Fluids Velocity Autocorrelation Density of States gas liquid solid gas liquid solid

  16. 2PT DoS Decomposition • Examples liquid solid gas solid-like gas-like solid-like gas-like gas-like solid-like

  17. Pressure and Energy Total Energy Pressure Pressures and MD Energies agree with EOS values Quantum Effect (ZPE) most significant for crystals (~2%)

  18. Entropy 2PT model 1PT gas crystal liquid • Overestimate entropy for low density gases • Underestimate entropy for liquids • Accurate for crystals • Accurate for gas, liquid, and crystal • Accurate in metastable regime • Quantum Effects most important for crystals (~1.5%)

  19. Gibbs Free Energy 1PT 2PT model liquid crystal gas • Underestimate free energy for low density gases • overestimate entropy for liquids • Accurate for crystals • Accurate for gas, liquid, and crystal • Accurate in metastable regime

  20. Why does 2PT work? HS fy = 0.036 HS fy = 0.309 gas QHO CHO liquid • 1PT overestimates Wsgas for gas for modes < 5 cm-1 • 1PT underestimates Wsgas for liquid for modes between 5 and 100 cm-1 • 2PT properly corrects these errors

  21. gas (r*=0.05 T*=1.8) liquid (r*=0.85 T*=0.9) Convergence of 2PT • For gas, the entropy converges to within 0.2% with 2500 MD steps (20 ps) • For liquid, the entropy converges to within 1.5% with 2500 MD steps (20 ps).

  22. 2PT for Melting and Solidification • Initial amorphous structure is used in the cooling process • The fluid remains amorphous in simulation even down to T*=0.8 (supercooled) • The predicted entropy for the fluid and supercooled fluid agree well with EOS for LJ fluids Simulation conditions supercritical fluid metatstable unstable supercritical fluid solid solid starting with amorphous liquid • Initial fcc crystal is used in the heating process • The crystal appears stable in simulation even up to T*=1.8 (superheated) • The predicted entropies for the crystal and superheated crystal agree well with EOS for LJ solids Entropy starting with fcc crystal

  23. Extension to Mixtures LJ Mixtures at T*=0.85 Combination Rules: sij= lij ( sii+sjj)/2 eij= bij ( eii+ejj)1/2 s22/s11=0.80 e22/e11=0.85 l12=0.95 b12=0.70 2PTLiterature x1(I) ~0.020.04 x1(II) ~0.790.84

  24. Efficiency of 2PT for Mixtures 100 ps 400 ps 12.5 ps 25 ps

  25. VAC DoS An Overview of the 2PT Method To Phase Behaviors From MD Simulations MD simulations

  26. Summary of 2PT • Thermodynamic and transport properties are determined simultaneously. • Only short simulation times (20 ps) are needed to obtain high accuracy. For a system with N particles, we expect 2PT to be N times faster than methods such as particle insertion and thermodynamic integration. • The efficiency of 2PT does not deteriorate with increasing density (a severe limitation in most other techniques). • The properties are obtained under real equilibrium conditions (no perturbation in the simulation itself). • Zero point energy and corrections for quantum effects are included. • 2PT can be used to determine the properties in metastable and unstable regimes. • 2PT could also be used for nonequilibrium systems to estimate effects of transient effects, reaction, and phase transitions, since it is only necessary to have stabilities over time scales of ~20 ps.

  27. R= g 1 g 2 g 3 g 4 O O H O O O O O O O O O O O O O O O O O O O O H O R O O O O O O O O O O O O O O O dendrons O O O O O O O O O core H O O O O O N O O O O O O O O O O O H O O Application: Zimmerman H-bond Dendrimers Dendrimer study on Zimmerman system Initiative: Science, 271, 1095 (1996) • Experimental Observations • Aggregates found in non-polar solvents such as CH2Cl2 and CHCl3 but not in polar solvents such as THF and DMSO • Circular and ladder coexist in lower generations (gen 1) • Circular type is dominant for generations 2, 3 and 4 • The generation dependent stability behavior is attributed to the subtle interplay between H-bond, vdW and steric repulsions

  28. Application: Zimmerman H-bond Dendrimers • Relative stability determined by the free energy differences is in consistent with experimental observations • Linear and circular forms are energetically similar • Stability of Zimmerman H-bond mediated dendrimers are dominated by entropic effects • Opens up possibility to design thermodynamically stable suprastructures • This method is potentially useful for study other supramolecules such as Protein, DNA, etc Free Energy Energy Entropy

  29. Application: PAMAM dendrimers EDA core repeating (monomer) unit Generation 3 • Applications of PAMAM dendrimer • Catalysts • Environment applications (metal encapsulation) • Medical applications (drug delivery, gene therapy) • Surface active agents • Viscosity modifier • New electrical materials

  30. Free Water Domain Surface Water Domain Inner Water Domain Application: PAMAM dendrimers Water molecules in PAMAM Schematic Atomistic • Number of water molecules

  31. Application: PAMAM dendrimers • Engetically slightly less favored for water at the PAMAM surface • Entropically less favored for water inside PAMAM • Entropically slightly less favored for water at PAMAM surface • Surface and Inner Waters are in a higher free energy state compared to the bulk

  32. Application: Percec Dendrimer Crystals

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