Slide 1

Entropy and free energy of a polymer chain from dynamic Monte Carlo simulations on the lattice. The extension of the statistical counting method. W. Nowicki, G. Nowicka and A. Maka Faculty of Chemistry, A. Mickiewicz University, Umultowska 89b, PL-61714, Poland E-mail: gwnow@amu.edu.pl Slide 2 INTRODUCTION: ASSUMPTIONS OF THE SC METHOD SOME APPLICATIONS OF THE SC METHOD AND ITS LIMITATIONS THE MICRO-MODIFICATION PROBABILITIES (MMP) METHOD EXEMPLARY APPLICATIONS OF THE MMP METHOD Slide 3 INTRODUCTION: ASSUMPTIONS OF THE SC METHOD the total number of conformations of the chain Effective coordination number of the lattice = the number of empty lattice nodes Zhao, D.; Huang, Y.; He, Z.; Qian, R. J. Chem. Phys., 1996, 104, 1672. (001) (112) Slide 4 Zhao, D.; Huang, Y.; He, Z.; Qian, R. J. Chem. Phys., 1996, 104, 1672. INTRODUCTION: ASSUMPTIONS OF THE SC METHOD the conforrmational entropy of the chain of N segments the Rosenbluths weighting factors weighted average Slide 5 Zhao, D.; Huang, Y.; He, Z.; Qian, R. J. Chem. Phys., 1996, 104, 1672. INTRODUCTION: ASSUMPTIONS OF THE SC METHOD the effective coordination number = f( probability of a chain micromodification) (112) Slide 6 VERIFICATION OF THE SC METHOD Zhao, D.; Huang, Y.; He, Z.; Qian, R. J. Chem. Phys., 1996, 104, 1672. Slide 7 SOME APPLICATIONS OF THE SC METHOD W Nowicki, G Nowicka, J. Narkiewicz-Michaek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J, 46, 112-122 (2010) W Nowicki, G Nowicka, J. Narkiewicz-Michaek "Influence of confinement on conformational entropy of a polymer chain and structure of polymer- nanoparticles complexes", Polymer, 50, 2161- 2171 (2009) (112) Chain anchored to the convex surface Chain anchored to the concave surface Chain with the forced position of the terminal segmet Slide 8 Chain near the interface Chain anchored to the interface Chain translocationt through the narrow hole SOME APPLICATIONS OF THE SC METHOD W Nowicki, G Nowicka, J. Narkiewicz-Michaek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J, 46, 112-122 (2010) Slide 9 Translocation coordinate TRANSLOCATION THROUGH THE HOLE W Nowicki, G Nowicka, J. Narkiewicz-Michaek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J, 46, 112-122 (2010) Slide 10 TRANSLOCATION THROUGH THE HOLE The whole landscape of the conformational entropy the approach of the terminal segment to the interface the translocation of the chain through the hole W Nowicki, G Nowicka, J. Narkiewicz-Michaek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J, 46, 112-122 (2010) approach translocation walk away Slide 11 The whole landscape of the conformational entropy (escape from cavity) TRANSLOCATION THROUGH THE HOLE W Nowicki, G Nowicka, J. Narkiewicz-Michaek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J, 46, 112-122 (2010) Slide 12 TRANSLOCATION THROUGH THE HOLE W Nowicki, G Nowicka, J. Narkiewicz-Michaek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J, 46, 112-122 (2010) Translocation coordinate Slide 13 MACROMOLECULE NEAR THE INTERACE W Nowicki, G Nowicka, J. Narkiewicz-Michaek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J, 46, 112-122 (2010) Slide 14 Conformational pressure of the chain vs. cavity volume Equation of state : Equation of state : MACROMOLECULE IN THE CAVITY W Nowicki, G Nowicka, J. Narkiewicz-Michaek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J, 46, 112-122 (2010) Slide 15 CHAIN ANCHORED TO A ROUGH SURFACE Samples of Bm (a c) and fBm (d f) surfaces generated by the RMD method W. Nowicki, G. Nowicka, M. Dokowicz, A. Maka "Conformational entropy of a polymer chain grafted to rough surfaces", J. Mol. Model., 18(9), 337-348 (2013) Slide 16 The probability distributions of finding a segment in the unperturbed coil ( F ), in the chain terminally attached ( A ) and the cumulative distribution of free sites near the surface ( S ). The difference = F A is also indicated. The vertical line denotes the mean elevation of the surface (zmean), N=100, /a=10. CHAIN ANCHORED TO A ROUGH SURFACE W. Nowicki, G. Nowicka, M. Dokowicz, A. Maka "Conformational entropy of a polymer chain grafted to rough surfaces", J. Mol. Model., 18(9), 337-348 (2013) Slide 17 ADVANATAGES AND DISADVANTAGES OF THE SC METHOD Although the parameter values obtained from MC simulation are correct because of the employment of the Rosenbluths weighting factors, the generated chain conformations remain biased from the true population, which makes them unsuitable for detailed and realistic analysis of the polymer coil structure Metropolis MC method not biased conformations the intermolecular interactions can be incorporated Metropolis MC method not biased conformations the intermolecular interactions can be incorporated Slide 18 DISADVANTAGES OF THE SC METHOD Coil deformation: - effective coordination number increases to 1 - conformational entropy decreases Slide 19 BASIS OF MMP METHOD: THE METROPOLIS MC D. P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, New York, 2000). The Metopolis MC method assumes that the appriopriate equilibration has take place before commencement of the averaging process. The average is simple (not weighted). Transition acceptance based on Boltzmann distribution Transition acceptance based on Boltzmann distribution Metropolis importance sampling Monte Carlo scheme 0. Create a certain conformatin 1. Modify the conformation and calculate its energy U 2. If U0 then generate a random number 4. If numer Xp then reject the new conformation 6. Go to 1 Metropolis importance sampling Monte Carlo scheme 0. Create a certain conformatin 1. Modify the conformation and calculate its energy U 2. If U0 then generate a random number 4. If numer Xp then reject the new conformation 6. Go to 1 Slide 20 C R E K I F CHAIN MODIFICATIONS IN THE METROPOLIS MC METHOD Verdier-Stockmayer local/bilocal micromodifications (non-ergodic) Cut-and-paste non-local modifications: inversion and reflection (ergodic) P. H. Verdier, W. H. Stockmayer, J. Chem. Phys. 36, 227 (1962). P.H. Verdier, J. Comput. Phys. 4, 204 (1969). Slide 21 The kink-jump motion trials for two different locations of a monomer in the chain a) the forbidden move, b) the permitted move PROBABILITIES OF MICROMODIFICATIONS (KINK-JUMP) P S (K) P E (K) The skeleton effect The excluded volume effect A. Maka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013). Kink-jump Slide 22 The dependencies of P E (K) and P E (C) on L H. The horizontal lines indicate the corresponding probabilities determined for the free chain P E (K) 0 and P E (C) 0, respectively (SAW model, N=100). The influence of L H value on probabilities P S (K) and P S (C). The horizontal lines indicate corresponding probabilities determined for the free chain (SAW, N=100). PROBABILITIES OF MICROMODIFICATIONS (DEFORMED COIL) A. Maka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013). Slide 23 Plots of: a) P S (K)/P S (K) 0 against L (inset: enlarged part of the dependence around L=0, coordinates are not marked) and b) (1P S (K)/P S (K) 0 ) against L in log-log scale for positive values of variables; N=100. Tested for 2D RW, 3D RW, 2D SAW and 3D SAW The values of the expression (( 1) eff )1/2 plotted against 1P E (K), obtained for a chain of N=100 in theta and poor solvents. The regression coefficient of the straight line indicated in the figure is equal to 2. P S (K) > L H P E (K) > eff A. Maka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013). standardization Slide 24 CONFORMATIONAL ENTROPY VERSUS L AND eff P. Atkins, Pchysical Chemistry, Oxfeord University Press, 2D RW eff =4 3D RW eff =6 2D NRRW eff =3 3D NRRW eff =5 2D SAW eff =2.638 3D SAW eff =4.684 Slide 25 System eff N=1000N=10000 100 2D RW 3D RW 2D NRRW 3D NRRW 3D SAW 4646646466 4 6 3 5 4.6839 1377 1776 1092 1597 1533 1385 1790 1098 1608 1544 0.58 0.78 0.54 0.68 0.71 13850 17896 10977 16077 15426 13863 17918 10986 16094 15441 0.09 0.12 0.08 0.11 0.10 CONFORMATIONAL ENTROPY VS. P S (K) AND P E (K) = MMP METHOD A. Maka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013). Slide 26 VERIFICATION OF THE MMP METHOD The relationships of a) S vs L H and b) F vs L H ; SAW chain of N=50. For comparison, the same dependencies calculated by means of expanded ensemble MC method (squares) are shown. Vorontsov-Velyaminov, P.N.; Ivanom, D.A.; Ivanom, S.D.; Broukhno, A.V. Colloids Surfaces A: Physicochemical and Engineering Aspects 1999, 148, 171-177. Dependencies of conformational entropies vs. fixed end-to-end distance calculated for 2D RW, 3D RW and 3D SAW (N=100). Perpendicular lines indicate rms end-to-end distances of RW and SAW chains equal to 10b and 15.8b, respectively. The rms end-to-end distance for RW model is independent on the dimensionality of the lattice. A. Maka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013). rms end-to-end distance for free chain rms end-to-end distance for free chain Slide 27 The comparison of relationships of a) S vs L H, b) U vs L H and c) A vs LH, which were determined for three different solvent conditions: =0 (athermal), =1/2 (theta) and 1/2< 2 (poor), N=50. The entropy of SAW chain (N=100) with ends attached to the opposite parallel surfaces versus the surface separation (circles). For comparison, the corresponding results for the chain with fixed ends but in the open space are indicated (squares). Different solvents Geometical constraints (chain between two parallel plates) VERIFICATION OF THE MMP METHOD A. Maka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013). rms end-to-end distance for free chain rms end-to-end distance for free chain Slide 28 Free energy contributions: non-electrostatic (London) energy energy of electrostatic interactions between ions of electrolyte, polyion and charged nanoparticles conformational entropy of polyion entropy of the DLVO layer APPLICATIONS OF THE MMP METHOD +//+ The same dependencies as in previous slide for the case when nanoparticles have the electric charge of the opposite sign to that of the chain (Q/e=20, qP/e=+1, NC=20, I=5.9 10-6 (squares) and I=5.9 10-5 (triangles)). W. Nowicki, G. Nowicka, Conformation and elasticity of a charged polymer chain bridging two nanoparticles, J. Chem. Phys., 139, 214903 (2013) Slide 29 Entropy contributions : - chain conformational entropy reduction caused by chain deformation by the approach of nanoparicles - chain cinformational entropy reduction caused by charged segments adsorption - configurational entropy of the electrical double layer APPLICATIONS OF THE MMP METHOD W. Nowicki, G. Nowicka, Conformation and elasticity of a charged polymer chain bridging two nanoparticles, J. Chem. Phys., 139, 214903 (2013) +//+ Slide 30 APPLICATIONS OF THE MMP METHOD NON-EQUILIBRIUM MC Umbrella sampling Metropolis algorithm The standard Boltzmann weighting for Monte Carlo sampling is replaced by a potential chosen to cancel the influence of the energy barrier present. Ph. Attard, Non-Equilibrium Computer Simulation Algorithms, Oxfrord University Press, 2012 Transition acceptance based on entropy of conformation Slide 31 Translocation through the long channel of an assumed width Slide 32 Thank you for your attention