1. 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

2. 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.

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

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

5. N  64, B  25 System I – Compressible Homopolymer Melts

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

8. System II – Compressible Symmetric Diblock Copolymers

10. 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.

11. System II – Compressible Symmetric Diblock Copolymers

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

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

14. 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

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

16. System III – Compressible Homopolymer Melts in 1D x s2s2s  1,3s  4,6s5s5 Density of States g(E) Wang-Landau – Transition-Matrix MC 

17. FLMC  LSCF @ finite C En0En0 fcfc  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

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

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

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

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

22. 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)

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

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

25. N  40, n  0  1  5 System V – Mushroom in an Explicit Solvent (S)

26. N  40, n  0  1  5 System V – Mushroom in an Explicit Solvent (S)  * LSCF ≈1.412

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

28. 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)

29. 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)

30. 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.

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