Statistics of polymers in random media S. V. Panyukov P. N Lebedev Physics Institute, Moscow (Submitted 11 November 1992) Zh. Eksp. Teor. Fiz. 103,1287-1304 (April 1993) A theory of localized states of linear and branched polymer macromolecules in a medium with "quenched" disorder is constructed. The conditions under which short chains are localized in one potential well are found, and the size distribution of such wells is obtained. It is shown that it is more energetically favorable for long chains to be distributed over several wells. The free energy and the sizes of macromolecules situated in a random medium in the presence of two-particle interaction between their links are calculated, as are the same quantities in the case of three- particle and long-range interactions. The reasons why these results differ from the results obtained previously in the framework of a variation approach are discussed. Characteristics of the disorder of polymer networks are calculated, and the dependence of the anisotropic deformation of macromolecules placed in them on the degree of stretching of the network is studied. The correlation functions of such macromolecules of a polymer solvent are also calculated, and it is shown that the results obtained are in agreement with the experimental data. 1. INTRODUCTION The theory of polymers situated in random media1-' has numerous applications. Polymer chains adsorbed on a rough surface constitute a typical two-dimensional situa- tion. Polymer chains or porous media with chains inserted in them can serve as a physical realization of a three-dimen- sional system. Recently, various biological systems consist- ing of molecules interacting with impenetrable particles have also been studied intensively. The simplest model of a polymer in a random medium is a Gaussian chain situated in a random lattice of impenetra- ble obstacles. Such a system was first studied in computer experiment^,^ in which it was demonstrated that the size of a sufficiently long chain is asymptotically independent of its length. These results provided the impetus for a subsequent anaytical examination of the influence of quenched impuri- ties with a given concentration v on the statistics of the poly- mer chain." It was shown that in space of dimensionality d = 2 or d = 3 a sufficiently long chain is localized over a scale R,,, - v - - d, . In Refs. 11 and 12 the influence of the interaction of the monomer links of the chain on the possibility of its localiza- tion in a medium with quenched impurities was studied. By means of variational estimates it was shown that impurities screen the two-particle interaction, leading to Gaussian sta- tistics of the chains. With further increase of the concentra- tion of impurities localization of the chains on the scale R,,, is predicted. In the case of three-particle interaction a com- pact state of the chain was obtained, with a density of mon- omer links that is independent of its length.'' It was also stated that, in the presence of long-range interaction, an in- termediate, "native" state of the chain should be formed. '' An approach substantially different from that of Refs. 10-12 was used in Ref. 13 to construct a theory of localized states of polymer chains. A more detailed examination of localized states of linear and branched polymer molecules is given in Secs. 2 and 3 of this article. On the basis of this approach, in Sec. 4 we find the average size of a chain in a random medium. The interest in this quantity is due to the fact that this is the quantity that was calculated in Refs. 11 and 12. The results that we obtain do not agree with the results of these papers. Therefore, first and foremost, the question of the applicability of the corresponding ap- proaches arises. In contrast to the variational estimates of Refs. 10-12, the method that we use is a regular expansion in a small parameter, which we find in Sec. 3. The reason for the differ- ences under discussion is buried in the specific physical fea- tures of the problem under consideration. A large number of potential wells with a broad distribution of sizes cannot be simulated by a single-well potential in the framework of a variational approach. From the experimental point of view, the most interest- ing realization of the systems under consideration here is provided by chains immersed in a polymer network. The theory of small-angle neutron scattering by these chains in such a polymer system was constructed in Refs. 5 and 6, and below we shall give a brief review of the corresponding ex- perimental data. In Refs. 14 and 15 a study was made of the influence of the degree of crosslinking X of the network (as judged from the degree of equilibrium swelling of the network) on the intensity of small-angle neutron scattering by free deuterat- ed chains situated in the network. For small values of X the intensity turns out to be the same as in the case of a mixture of deuterated chains with uncrosslinked network chains of the same density. Under uniaxial deformation of the network the signal increases strongly in the direction paral- lel to the stretching, but remains unchanged in the perpen- dicular direction (see Fig. la). In the case of intermediate values of X the scattering in the swollen state of the network is greater than that by a mixture of uncrosslinked chains. Upon stretching, the signal increases in the direction parallel to the stretching and de- creases in the perpendicular direction, returning, with in- crease of the stretching, to its value for the mixture (Fig. lb). In the case of large values of X the scattering in the swollen state is considerably greater than the scattering by a mixture of uncrosslinked chains. In the direction parallel to the stretching the signal does not change with increase of the 631 JETP 76 (4), April 1993 1063-7761 I931040631 -09$10.00 @ 1993 American Institute of Physics 631