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J. Phys. Chem. 1985,89, 404-406
The Size and Shape of Macromolecular Structures: Determlnation of the Radius, the Length, and the Persistence Length of Rodllke Micelles of Dodecyldimethylammonium Chloride and Bromide W. Van De Sandet and A. Persoons* Laboratory for Chemical and Biological Dynamics, University of Leuven, Celestijnenlaan 200 D, B- 3030 Leuven, Belgium (Received: October 15, 1984)
We present a method to determine the size and shape of macromolecular structures. The radius of gyration is plotted vs. the volume of the macromolecular structure and the experimental data are compared to the theoretical predictions for various shapes. The method is uniquely adapted to dynamically nonideal systems and permits a determination of the length, the radius, and the persistence length of flexible rods with high accuracy. The micelles of dodecyldimethylammoniumchloride and bromide in aqueous NaCl and NaBr solutions are shown to behave as flexible rods with a radius of 1.6 and 1.8 nm, respectively, and a persistence length of 45 nm.
Introduction Recently, viscosity measurements,' N M R data,* and light scattering experiments, both static3 and dynamic,4vs have indicated a rodlike shape for micelles of dodecyldimethylammonium chloride (DDAC) in concentrated aqueous NaCl solutions. The same behavior was also found for micelles of dodecyldimethylammonium bromide (DDAB) in concentrated aqueous NaBr solutions.6 For DDAC Flamberg and Pecora4v5suggest that the solution at 4 M NaCl falls within the region discussed in the work of Doi and Edwards:' the entanglement of the rods leads to an overestimation of the apparent hydrodynamic radius because of the dynamically nonideal behavior. Also, flexibility complicates the interpretation of the experimental data. We present a method which still permits the determination of the size and shape of macromolecular structures, even in the case of dynamically nonideal behavior, by only assuming thermodynamically ideal behavior. The flexibility of the long rodlike micelles is quantitatively taken into account.
Theory The expressions for the radius of gyration (&), the hydrodynamic radius ( R H ) ,and the volume (VM)of spheres, ellipsoids, and rods are known as a function of the geometrical parameters of these shapes (Table I). Comparing the theoretical predictions with the experimental data one can assign a most probable shape to a macromolecular structure. The volume VM of a micelle can be calculated from MMP v, = (13) MSNA
where MM is the molecular weight of the micelle, M , is the molecular weight of the surfactant molecule, NA is Avogadro's number, and P is the partial molar volume of the surfactant (expressed as volume per mole of surfactant molecules). The molecular weight of the micelle can be evaluated from static light scattering The partial molar volume is calculated from density measurements.' The radius of gyration of the flexible rod is calculated RG2(L)= L * [ ( / ) / ~-L( 1 ) 2 / ~ z+ 2 ( l ) 4 / L 4 ( L / ( l )- 1 + exp(-L/(l)))] (14) where L is the contour length of the rod and ( I ) is the persistence length of the rod. The persistence length is a measure of the resistance to curvature of the rod. The free energy F required to bend a straight W.V.D.S. is a bursar of the Instituut voor Wetenschappelijk Onderzoek in Nijverheid en Landbouw (Belgium).
rod of length L and persistence length (1) into an arc with a radius of curvature q is8*9 F = (1/2)(l)kBTL/q2
(15)
where k~ is Boltzmann's constant and T is the absolute temperature. The entanglement of rigid rods is discussed by Doi and Edwards? A solution of rodlike structures with length L and diameter d is considered with n the number of rods in a unit volume. As long as n
