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Quantifying Fluctuation/Correlation Effects in Inhomogeneous Polymers by Fast Monte Carlo Simulations Department of Chemical & Biological Engineering and School of Biomedical Engineering q.wang@colostate.edu David (Qiang) Wang Laboratory of Computational Soft Materials Jing Zong, Delian Yang, Yuhua Yin, and Pengfei Zhang

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Coarse-Grained Simulations of Multi-Chain Systems Conventional Monte Carlo (MC) simulations: Hard-core excluded- volume interactions: u(r→0)→∞ + Model of chain connectivity or SMAW on lattice 1.Orders of magnitude faster (better) sampling of configuration space; 2.All advanced MC techniques can be used; Advantages: Fast MC Simulations:Finite u(r→0) Q. Wang and Y. Yin, JCP, 130, 104903 (2009); Q. Wang, Soft Matter, 5, 4564 (2009); 6, 6206 (2010). 3.Much wider range of N controlling system fluctuations can be studied; 4.No parameter-fitting when compared with polymer field theories.

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Part 1: Fast Off-lattice Monte Carlo (FOMC) Simulations

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System I – Compressible Homopolymer Melts n: number of chains; N: number of segments on each chain.

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N 64, B 25 System I – Compressible Homopolymer Melts

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N 64, B 25 Q. Wang and Y. Yin, JCP, 130, 104903 (2009) System I – Compressible Homopolymer Melts

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System II – Compressible Symmetric Diblock Copolymers

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Canonical-Ensemble Simulations Replica Exchange (RE) Exchange configurations between simulations at different N to greatly improve the sampling efficiency. Multiple Histogram Reweighting (HR) Interpolate at any point within the simulation range; Minimize errors using all the information collected; Accurately locate the order-disorder transition. Trial moves: Hopping, Reptation, Pivot, and Box-length change.

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System II – Compressible Symmetric Diblock Copolymers

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Field Theories vs. Particle-Based MC Simulations of Multi-Chain Systems

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Direct Comparison Based on the SAME Hamiltonian (not vs. ) No parameter-fitting

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Direct Comparison Based on the SAME Hamiltonian (not vs. ) Kronecker -function interactions are isotropic on a lattice (while nearest-neighbor interactions are anisotropic) and straightforward to use; Lattice simulations are in general much faster than off-lattice simulations. FLMC simulation is very fast due to the use of Kronecker -function interactions and multiple occupancy of lattice sites (MOLS). Advantages of FLMC Simulations: No parameter-fitting

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Part 2: Fast Lattice Monte Carlo (FLMC) Simulations and Direct Comparison with Lattice Self- Consistent Field (LSCF) Theory

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System III – Compressible Homopolymer Melts in 1D x s2s2s 1,3s 4,6s5s5 Density of States g(E) Wang-Landau – Transition-Matrix MC

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FLMC LSCF @ finite C En0En0 fcfc s c /k B R 2 e,g N 0 (no correlations) 000 finite N 0 N →∞ (no fluctuations) 0 P. Zhang, X. Zhang, B. Li, and Q. Wang, Soft Matter, 7, 4461 (2011). System III – Compressible Homopolymer Melts in 1D

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System IV – Confined Compressible Homopolymers in 3D L x 10 L

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(C →∞ ) Closest to wall Middle of film L x 10 L System IV – Confined Compressible Homopolymers in 3D

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Q. Wang, Soft Matter, 5, 4564 (2009); 6, 6206 (2010). System IV – Confined Compressible Homopolymers in 3D

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Q. Wang, Soft Matter, 5, 4564 (2009); 6, 6206 (2010). System IV – Confined Compressible Homopolymers in 3D

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A polymer mushroom refers to a group of n chains grafted at the same point onto a flat, impenetrable, and neutral substrate. x A 3D lattice, BFM2, having six bonds of length 1 and twelve bonds of length (in units of the lattice spacing) with equal a priori probability for bonds of different lengths is used, which minimizes lattice anisotropy; see Q. Wang, JCP, 131, 234903 (2009). System V – Mushroom in an Explicit Solvent (S)

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LSCF, N 40 System V – Mushroom in an Explicit Solvent (S)

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LSCF, N 40 System V – Mushroom in an Explicit Solvent (S)

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N 40, n 0 1 5 System V – Mushroom in an Explicit Solvent (S)

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N 40, n 0 1 5 System V – Mushroom in an Explicit Solvent (S) * LSCF ≈1.412

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Asymmetric 1 Asymmetric 2 Symmetric N 40, n 0 1 5 System V – Mushroom in an Explicit Solvent (S)

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n 64, 1.4n 64, 1.45n 64, 1.5 n 64, 1.55 N 40, n 0 1 5 System V – Mushroom in an Explicit Solvent (S)

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N 40, n 0 1 5 n 4, 1.6n 8, 1.6 n 64, 1.6 n 2, 1.6 System V – Mushroom in an Explicit Solvent (S)

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Coarse-Grained Simulations of Multi-Chain Systems Conventional Monte Carlo (MC) simulations: Hard-core excluded- volume interactions: u(r→0)→∞ + Model of chain connectivity or SMAW on lattice Fast MC Simulations:Finite u(r→0) 1.Orders of magnitude faster (better) sampling of configuration space; 2.All advanced MC techniques can be used; Advantages: Q. Wang and Y. Yin, JCP, 130, 104903 (2009); Q. Wang, Soft Matter, 5, 4564 (2009); 6, 6206 (2010). 3.Much wider range of N controlling system fluctuations can be studied; 4.No parameter-fitting when compared with polymer field theories.

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L(n)L(n) j ( x,y,z) n t System II – Compressible Symmetric Diblock Copolymers SCFT, Incompressible, CGC, Dirac .

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