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H andan Arkın 1,2 , W olfhard Janke 1

Thermodynamics of a p olymer c hain in an a ttractive spherical cage. H andan Arkın 1,2 , W olfhard Janke 1. 1 Institut f ü r Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany 2 Department of Physics Engineering, Faculty of Engineering,

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H andan Arkın 1,2 , W olfhard Janke 1

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  1. Thermodynamics of a polymerchain in anattractive spherical cage Handan Arkın1,2, Wolfhard Janke1 1Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany 2Department of Physics Engineering, Faculty of Engineering, Ankara University Tandogan, 06100 Ankara, Turkey

  2. Outline ■Motivation ■ Model: Off-lattice Polymer inside an Attractive Sphere ■Multicanonical Monte Carlo Method ■ Some Simulation results

  3. Motivation • Investigating basis structure formation mechanisms of biomolecules at different interfaces is one of the major challenges of modern interdisciplinary research and possible application in nanotechnology • Applications: • Polymer adhesion to metals, semiconductors • biomedical implants and biosensors, smart • drugs etc. • Due to the complexity introduced by the huge amount of possible substrate structures and sequence variations this problem is not trivial. • Therefore the theoretical treatment of the adsorption of macromolecules within the framework of minimalistic coarse-grained polymer models in statistical mechanics has been a longstanding problem.

  4. The polymer Model Interaction energy: ■ Coarse-grained off-lattice ■ Semi-flexible ■ Three-dimensional Bending energy Lennard-Jones potential Adjacent monomers are connected by rigid covalent bonds and the distance between them is fixed and set to unity The position vector of the ith monomer is

  5. Polymer Chain inside an Attractice Sphere Potential ■The interaction of polymer monomers and the attractive sphere is of van der Walls type, modeled by LJ 12-6 We integrate this potential over the entire sphere inner surface Where the surface element in spherical coordinates is

  6. Polymer Chain inside an Attractice Sphere Potential The radius of the sphere The distance of a monomer to the origin Set to unity Varied during simulation

  7. Polymer Chain inside an Attractice Sphere Potential

  8. Idea: choose ensemble that allows better sampling Example: Multicanonical Ensemble General Procedure: Determine the weight factors Large scale simulation Calculate expectation values for desired temperatures Problem: Find good estimators for the weights Advantages: Any energy barrier can be crossed. The probability of finding the global minimum is enhanced. Thermodynamic quantities for a range of temperatures Simulations in Generalized Ensembles

  9. Multicanonical Monte Carlo Method • The canonical ensemble samples with the Boltzmann probability density • where x labels the configuration. The probability of the energy E is • where n(E) is the density of states. • The Muca ensemble is based on a probability function in which the different energies are equally probable: • where w(E) are multicanonical weight factors.

  10. Observables Specific Heat: Radius of gyration: Mean number of adsorbed monomers to the inner wall of the sphere: We define a monomer i is being adsorbed if and this can be expressed as .

  11. Observables ■Gyration Tensor Transformation to principal axis system diagonalizes S Where the eigenvalues are sorted in descending order

  12. Observables ■The first invariant of Gyration Tensor The second invariant shape descriptor is Shape Anisotropy (reflects both symmetry and dimensionality) Where The last shape desciptor is the asphericity parameter Derivative of all these quantities

  13. Results: Pseudo Phase Diagram D : Desorbed A: Adsorbed E: Extended G: Globular C:Compact

  14. Results: Specific -Heat

  15. Results: Radius of Gyration

  16. Results: The mean number of adsorbed monomers

  17. Results: Relative Shape Anisotropy

  18. Results: The eigenvalues of the Gyration tensor

  19. Results: Fluctuations of the Observables

  20. Results: Low Energy Conformations

  21. Acknowlegments Thank you for your attention. Thanks to ... CQT Team, Institute of Theoretical Physics, Leipzig

  22. Results: The ratio of the greatest eigenvalue to the smallest

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