The extension of the statistical counting method. 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. Mańka Faculty of Chemistry, A. Mickiewicz University, Umultowska 89b , PL-61714 , Poland E-mail: gwnow@amu.edu.pl

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

INTRODUCTION: ASSUMPTIONS OF THE SC METHOD (001) (112) Effective coordination number of the lattice = the number of empty lattice nodes the total number of conformations of the chain Zhao, D.; Huang, Y.; He, Z.; Qian, R. J. Chem. Phys., 1996, 104, 1672.

INTRODUCTION: ASSUMPTIONS OF THE SC METHOD weighted average the Rosenbluths’ weighting factors the conforrmational entropy of the chain of N segments 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) Zhao, D.; Huang, Y.; He, Z.; Qian, R. J. Chem. Phys., 1996, 104, 1672.

VERIFICATION OF THE SC METHOD Zhao, D.; Huang, Y.; He, Z.; Qian, R. J. Chem. Phys., 1996, 104, 1672.

SOME APPLICATIONS OF THE SC METHOD (112) Chain anchored to the convex surface Chain anchored to the concave surface Chain with the forced position of the terminal segmet W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Influence of confinement on conformational entropy of a polymer chain and structure of polymer-nanoparticles complexes", Polymer, 50, 2161- 2171 (2009) W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J,  46, 112-122 (2010)

SOME APPLICATIONS OF THE SC METHOD Chain near the interface Chain translocationt through the narrow hole Chain anchored to the interface W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J,  46, 112-122 (2010)

TRANSLOCATION THROUGH THE HOLE coordinate W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J,  46, 112-122 (2010)

TRANSLOCATION THROUGH THE HOLE The whole landscape of the conformational entropy walk away approach translocation the approach of the terminal segment to the interface the translocation of the chain through the hole W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J,  46, 112-122 (2010)

TRANSLOCATION THROUGH THE HOLE The whole landscape of the conformational entropy (escape from cavity) W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J,  46, 112-122 (2010)

TRANSLOCATION THROUGH THE HOLE W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J,  46, 112-122 (2010)

MACROMOLECULE NEAR THE INTERACE W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J,  46, 112-122 (2010)

MACROMOLECULE IN THE CAVITY Equation of state: Conformational pressure of the chain vs. cavity volume W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J,  46, 112-122 (2010)

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. Mańka "Conformational entropy of a polymer chain grafted to rough surfaces", J. Mol. Model., 18(9), 337-348 (2013)

CHAIN ANCHORED TO A ROUGH SURFACE 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. W. Nowicki, G. Nowicka, M. Dokowicz, A. Mańka "Conformational entropy of a polymer chain grafted to rough surfaces", J. Mol. Model., 18(9), 337-348 (2013)

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

DISADVANTAGES OF THE SC METHOD Coil deformation: - effective coordination number increases to  – 1 - conformational entropy decreases

THE METROPOLIS MC METHOD 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 U<0 then accept the new conformation and go to 1 3. If U>0 then generate a random number 4. If numer X<p then accept the new conformation 5. If numer X>p then reject the new conformation 6. Go to 1 The Metopolis MC method assumes that the appriopriate equilibration has take place before commencement of the averaging process. The average is simple (not weighted). D. P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, New York, 2000).

CHAIN MODIFICATIONS IN THE METROPOLIS MC METHOD Verdier-Stockmayer local/bilocal micromodifications (non-ergodic) C R E K Cut-and-paste non-local modifications: inversion and reflection (ergodic) F I P. H. Verdier, W. H. Stockmayer, J. Chem. Phys. 36, 227 (1962). P.H. Verdier, J. Comput. Phys. 4, 204 (1969).

PROBABILITIES OF MICROMODIFICATIONS (KINK-JUMP) PS(K) The skeleton effect PE(K) The excluded volume effect The kink-jump motion trials for two different locations of a monomer in the chain a) the forbidden move, b) the permitted move A. Mańka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013).

PROBABILITIES OF MICROMODIFICATIONS (DEFORMED COIL) The influence of LH value on probabilities PS(K) and PS(C). The horizontal lines indicate corresponding probabilities determined for the free chain (SAW, N=100). The dependencies of PE(K) and PE(C) on LH. The horizontal lines indicate the corresponding probabilities determined for the free chain – PE(K)0 and PE(C)0, respectively (SAW model, N=100). A. Mańka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013).

PS(K) – > LH PE(K) –> eff Plots of: a) PS(K)/PS(K)0 against L (inset: enlarged part of the dependence around L=0, coordinates are not marked) and b) (1–PS(K)/PS(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 1–PE(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. A. Mańka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013).

CONFORMATIONAL ENTROPY VERSUS L AND eff 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 P. Atkins, Pchysical Chemistry, Oxfeord University Press,

CONFORMATIONAL ENTROPY VS. PS(K) AND PE(K) = MMP METHOD System  eff N=1000 N=10000 100 2D RW 3D RW 2D NRRW 3D NRRW 3D SAW 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 A. Mańka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013).

VERIFICATION OF THE MMP METHOD The relationships of a) S vs LH and b) F vs LH; 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. rms end-to-end distance for free chain 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. Mańka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013).

VERIFICATION OF THE MMP METHOD Different solvents The comparison of relationships of a) S vs LH, b) U vs LH 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. rms end-to-end distance for free chain Geometical constraints (chain between two parallel plates) 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). A. Mańka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013).

APPLICATIONS OF THE MMP METHOD 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 double layer +/+/+ The dependence of a) the free energy A and b) the elastic force F on the separation distance between nanoparticles’ surfaces bearing the electric charge of the same sign as that of beads (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)

APPLICATIONS OF THE MMP METHOD 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 +/–/+ 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)

APPLICATIONS OF THE MMP METHOD 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 W. Nowicki, G. Nowicka, “Conformation and elasticity of a charged polymer chain bridging two nanoparticles”, J. Chem. Phys., 139, 214903 (2013)

Metropolis algorithm Umbrella sampling APPLICATIONS OF THE MMP METHOD – NON-EQUILIBRIUM MC 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. Umbrella sampling Transition acceptance based on entropy of conformation Ph. Attard, Non-Equilibrium Computer Simulation Algorithms, Oxfrord University Press, 2012

Translocation through the long channel of an assumed width

Translocation through the long channel of an assumed width

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