Diffusion of a chain : concentration effects

2014 It is shown that the cross-over concentration between dilute and semi-dilute solutions is the same for dynamical properties as for static properties of flexible polymers in a good solvent. For semi-dilute solutions, the exponent 03B2 in the concentration dependence of the diffusion coefficient of a blob Dc ~ C03B2 is expected to depend on both temperature and concentration. The variation of this effective exponent is non monotonous. A combined neutron elastic and hight quasielastic scattering experiment is proposed to observe this effect. J. Physique-LETTRES 41 ( 1980) L217-L-220 1l’I’MAI 1980, Classification Physics Abstracts 05.20 36.20 61.40 64.90 Recently, Weill and des Cloizeaux [1], Akcasu and Han [2] (A.H.W.C.) have proposed a model for the dynamics of a flexible polymer chain in a good solvent [3]. This approach, based on the blob model [10], shows that the cross-over effects between theta and good solvents are much more important for the dynamical properties, where one has to calculate mean values of inverse distances than for the radius of gyration for instance, where r 2 > is calculated. As a result, the experimental studies [4] measure an effective exponent, as was suggested by Stockmayer and Albrecht [5] some time ago. In this note, we wish to clarify the cross-over between dilute and semi-dilute solutions in a good solvent, and extend the A.H.W.C. approach to semidilute solutions. The cross-over concentration C* is usually defined via the (static) radius of gyration RG as [10] where N is the number of statistical units in the chain, C the monomer concentration, and v the excluded volume exponent [3], [6]. If we consider now the diffusion coefficient Do of a chain, or the viscosity of a dilute solution, we have where RH is called the hydrodynamic radius [7] and where the effective exponent [1 ], [2] v’ is less than v [4] (usually v’ N 0.54). where ~S is the viscosity of the solvent, M the intrinsic viscosity [7] and KH the Huggins coefficient, which is a constant [3], [8]. It can be shown that [9] A straight forward (but wrong) dimensional analysis would then lead to different laws for the cross-over between dilute and semi-dilute solutions, namely Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:01980004109021700 L-218 JOURNAL DE PHYSIQUE LETTRES The flaw in the analysis lies in the fact that there is not only one, but two lengths [10] in the problem (none of them being RH). RH cannot be considered as a characteristic length, except in the asymptotic region of very high N, which according to refs. [1] and [2] seems impossible to reach in the present state of the art. In order to calculate RH, [1], [2] one has to separate short from large distances [10] namely We may call the substructure associated to Nt a flof. This analysis [1], [2], valid for dilute solutions, can be extended to semi-dilute solutions, as shown, I below. So all the following concerns semi-dilute solutions. Let us consider the diffusion coefficient, which is more appropriate, as relation (3) does not hold for high concentrations [11]. In semi-dilute solutions, another substructure, called the blob [10], appears. It is associated to a number of monomers g > Nt, and to an average squared length ~2 between successive contacts. We know that inside a blob (i.e. for short distances) the segment has a single chain behaviour [12]. Thus the blob may be considered as a single chain, with a flof substructure identical to that of a single chain [15] (see Fig. 1). This substructure has not been directly checked experimentally, but we and we may define an effective exponent by Notice that the last term in (8) becomes dominant if ~ ~> N~, which is realized if the temperature is very high or the concentration very low. (Usually the limit VH = v is not achieved.) Of course this analysis is valid above C* (relation (1)). When the monomer concentration goes to C*, g goes to N and D crosses over smoothly from relation (7) valid in semi-dilute solutions to relation (2) valid in dilute solutions. This can be seen very clearly also in relation (8). So this shows that the cross-over concentration is C* and is the Fig. 1. a) The local behaviour of a chain is theta-like. (See notes [10] and [14].) For larger distances, the excluded volume interaction is present. b) In a semi-dilute solution, inside the blob, there is the same structure as for the single chain. consider that the discrepancy between the static and dynamic measurements [13] of the radius of the blob is an evidence for this substructure. The same analysis as above can be done for the diffusion of the blob : In order to calculate the diffusion coefficient Dc one has to separate the short distance behaviour (inside the flof) from the larger distance behaviour The only difference between (7) and (5) is that one has to replace N by the number g of statistical units in the blob. same for the diffusion coefficient as for the radius of gyration. The preceding analysis can be generalized to any other property. One has just to take care of performing the correct dimensional analysis. An interesting effect is exhibited by this analysis : g depends on both the temperature and the monomer concentration C [15], [16] where we know the asymptotic behaviour of the function f (x) L-219 DIFFUSION OF A CHAIN : CONCENTRATION EFFECTS On the other hand NT depends on temperature only (relation (6)). So by varying either the temperature at a concentration, or the reverse, one varies the ratio g/NT, and therefore the value of the exponent VH. We can define effective exponents for the concentration and temperature dependence of ÇH.