1. 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 , Poland

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

3. 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.

4. 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.

5. 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.

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

7. 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, (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, (2010)

8. 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, (2010)

9. 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, (2010)

10. 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, (2010)

11. 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, (2010)

12. 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, (2010)

13. 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, (2010)

14. 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, (2010)

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

16. 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), (2013)

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

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

19. 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).

20. 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).

21. 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).

22. 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).

23. 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).

24. 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,

25. 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).

26. 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,
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).

27. 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).

28. 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, (2013)

29. 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, (2013)

30. 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, (2013)

31. 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

32. Translocation through the long channel of an assumed width

33. Translocation through the long channel of an assumed width

34. Thank you for your attention