1. gwnow@amu.edu.pl Complexation of nanoparticles with a long polymer chain Department of Physical Chemistry Faculty of Chemistry UAM, Poznań Waldemar Nowicki

2. Problem Determination of the structure of the multiplet composed of a single very long and flexible polymer chain and a number of nanoparticles (particles are small as compared to the linear dimension of the polymer chain)

3. Recent approaches 1. The mean field model treating polymer-micelle aggregate as a one-dimensional fluid [1] (cmc of polymer bound surfactant < cmc of free surfactant) 2. The mean field model of polymer brush with attached nanoparticles [2] (there is an optimum particle number at which the free energy minimizes) 3. Off-lattice simulations of particles-polymer aggregate [3] (direct analysis of chain conformation) [1] Sear, R.P., J. Phys.: Condens. Matter, 10, 1677-1686 (1998) [2] Curie, E.P.K., van der Gucht, J., Borisov, O.V., Cohen Stuart M.A., Pure Appl.Chem, 71, 1227-1241 (1999) [3] Chodanowski, P., Stoll, S., Colloid Polym.Sci., 278, 406-417 (2000)

4. Model assumptions: a very low concentration of polymer (the multiplets are mutually independent) athermal contitions a very long, flexible polymer chain nanoparticles small as compared with the polymer coil size weak adsorption – the polymer coil conformation is very slightly perturbed strong (irreversible) adsorption

5. Methods the SAW chain on the 3D cubic lattice simulations performed with – the Rosenbluth-MC algorithm [1] (the method of SAW improvement to approach the athermal condition) – the statistical counting method (SCM) [2] (calculation of the conformational entropy of the coil) – the standard Metropolis-MC method [3] [1] Rosenbluth, M.N., Rosenbluth, A.W., J. Chem. Phys., 23, 356, 1955 [2] Zhao, D., Huang, Y., He, Z., Qian, R., J. Chem Phys., 104, 1672, 1996 [3] Metropolis, N.A., Rosebluth, A.W., Rosenbluth, M.N, Teller, A., Teller, E., J. Chem. Phys., 21, 1087, 1953

6. Segment distributions around the mass centre of the polymer coil The reduction of the chain conformational entropy caused by the insertion of a particle The reduction of the chain conformational entropy caused by the insertion of a particle The effect of the particle shape on the entropy penalty The effect of the particle shape on the entropy penalty Free particle in a polymer coil – the effect on the coil entropy

7. The effect of the chain length on the entropy loss caused by the terminal chain attachment to particles of different volumes (DCM method) Relative coordination number vs. chain length dependence obtained for different particle volumes The effect of the chain length on the entropy loss caused by the terminal chain attachment to particles of different volumes Polymer chain terminally attached to a particle

8. The mean position of terminally attached particle in the polymer coil (dashed line) The grey area ­ the projection of the ellipsoid of the polymer coil: the light region determined by the average separation of the most distant segments the dark region ­ by the radius of gyration

9. The minimum adsorption energy required for the attachment of a nanoparticle to the polymer coil

10. Cavity distribution in the polymer coil The density of cavities versus the relative distance from the coil centre (solid lines – free chain, dashed lines – adsorbed chain, N=1000; cavities of volumes equal to 1, 8, 27 and 64 are considered) Adsorption layer thickness  particle radius

11. The probability of particle motion in the polymer coil - penetration depth - escape probability

12. The density of adsorption sites versus the relative distance from the coil centre (N=1000; particles of volumes equal to 1, 8, 27, 64, 125 and 216 are considered).

13. The loss of conformational entropy of the chain due to the insertion of a non- adsorbing nanoparticle into the coil domain depends on the nanoparticle size and shape, its position in the coil and the length of the polymer chain. The conformational entropy reduction effect pushes the terminally attached particle towards the coil peripheries. The conformational entropy penalty increases with increasing chain length except for systems containing extremely small particles that induce only local perturbations in the coil. A minimum adsorption energy is required to outweigh the penalty of conformational entropy. There is the critical particle radius, below which the complexation between the polymer and the nanoparticles does not occur. The diffusion of particles from the inner part of the coil without any collision with the polymer strands is very unlikely, even in an athermal solvent in which the coil structure is quite open. The adsorption of large polymers on nanoparticles is the “pseudo-irreversible” process ­ the particle detachment from the polymer strands can proceed quite readily, but the probability of the particle escape from the coil is low.