Excluded-volume effect on quasi-elastic light scattering by flexible macromolecules

Abstract

First-order perturbation expressions are developed for the first cumulant (initial time derivative) of the dynamic structure factor observable by photon-correlation measurements of the light scattered by flexible chain molecules in solution. A dimensionless coefficient C, which measures the initial departure ofthe first cumulant from proportionality to the square ofthe scattering vector, is found to be only slightly altered by excluded-volume effects.Quasi-elastic light scattering (QELS) offers a powerful method for the characterization ofmacromolecules in solution (1). With modem photon-correlation methods, the dynamic structure factor S(q,t) can be measured with quite high precision, and in most cases an accurate value of the first cumulant (initial time derivative) of the dynamic structure factor can be obtained (2, 3). By invocation of the known properties (4) of the usual polymer diffusion equation, Akcasu and Gurol (5) showed that the first cumulant can be expressed as an equilibrium average:where q is the scattering vector with a magnitude q = (41r/A)sin(0/2), involving the scattering angle 6 and the wavelength A in the solvent medium; t is time. The denominator of Eq. 1 is the wellknown (5) equilibrium particle scattering factor:where Rjk iS the vector-between structural elementsj and k. The numerator of Eq. 1 requires a knowledge ofthe diffusion tensor DJk. In most applications, this is given the form applied to polymer problems by Kirkwood and Riseman (6), which reads Djk/kBT = ajk C11k + (1 -6jk)(81T7oROk) (1 + IRRk Ok). [3] In this formula, kB is the Boltzmann constant, T is absolute temperature, ; is the friction coefficient of a chain element, qO is the solvent viscosity, and PJk = IRikI . The Akcasu-Gurol recipe, Eq. 1, has been applied to a number of different polymer models (5, 7-11), and it has been shown (10, 11) how such calculations can aid in the estimation of branching and polydispersity in polymer samples, provided that excluded volume effects can be neglected.In most polymer solutions, excluded volume effects on chainlength-dependent properties must be taken into account (12).Supplementing previous work on the excluded volume effect in QELS, we offer here a rigorous first-order perturbation treatment, using well-established methods. Our results indicate that the estimation of branching or polydispersity from QELS will not be seriously affected by excluded-volume interactions. General formulationIt is convenient to implement the averaging process of Eq. 1 in two stages: first, an average over orientations of R with respect to q, and then the average over magnitudes ok. After the first stage, we have N N r/q2 = (NkBT/I) 1 + (C/41r%)q N2

Effect of short-range correlation between chain elements on the hydrodynamic radius and the first cumulant in dilute polymer solutions

“…Thus, it cannot be explained well why, as described in Introduction, the experimental value is close to the results, 0.0625, calculated for non-PA nondraining chains. In this connection, it may be interesting to note a study by Tanaka and Stockmayer,19 who developed a first-order perturbation expression of by the same scheme as in the present work and showed that, for non-PA chains, limg_" Q/(q3kBT/q0) = 0.0625 independent of excluded volume. After all, we have the following suspicion that the discrepancies between experiments and theories for lim9^" Q/(q3kBT/q0) could be due to (1) the insufficiency of the Oseen description for the hydrodynamic interaction in the Kirkwood formula and/or (2) the approximate nature (or a lack of accuracy) of the Kirkwood formula.…”

Internal structure and dynamics of dilute linear polymers

“…Although this result was first obtained by preaveraging the Oseen interactions, more sophisticated calculations show that only the constant of proportionality C depends on preaveraging,14 solvent quality,15 and combinations thereof. 16 In fact, this general result apparently holds for all systems with a single length scale and hydrodynamic interactions, including liquid consolute points, etc., so studies of the q-dependence of the mean relaxation rate merely allow one to measure the dimension d of space, in case there is any real uncertainty about that.…”

Influence of draining and excluded volume on the intrinsic viscosity of flexible polymers

“…Combining (18) with the definition of the Flory constant and (14) gives the dimensionless ratio (d = 3) (19) which reduces in the nondraining limit to (£, °°,£)/ 0 = [1 -13£/96(1 + £)]/[! -13£/64(1 + £)][1 + 0.224£/(l + £)] (20) In addition the intrinsic viscosity expansion factor of (17) is equal in this limit to The self-avoiding limit for (" = °» of (20) provides a lower bound ( / )nondraining (1 -13/96)/(l -13/64)(1 + 0.224) = 0.886 (22) to / 0 as a function of f in the nondraining limit.…”