internal modes of relaxation, hydrodynamic interactions and interactions between chains such as entanglements. Other types of diffusion will effect dynamic scattering depending on the complexity of the material, i.e. At infinite dilution (d p /dc) T, m ' = N AkT, so at infinite dilution (no interactions) and K = 0 (size scales much larger than the particles of interest), D m = kT/f m, as previously discussed.Ĭenter of mass diffusion is the simplest case to describe. Where f m is the mutual friction factor, (d p /dc) T, m ' is the osmotic susceptibility. large size scales, the mutual diffusion coefficient, D m, is defined by the Stokes-Einstein relationship, The mutual diffusion coefficient is dependent on the wavevector (scattering angle) since at different size scales, r = 1 /K, the mechanisms of diffusion differ. Normalizing by yields the electric field correlation function, g (1), Where D m is the mutual diffusion coefficient and D C is the concentration change as a function of time and wavevector, K. Chu's book cited above.Īpplication of Fick's second law for diffusion to scattering in K-space results in the following expression for the molecular correlation function for center of mass motion, The DLS instrumentation is well described in B. The correlator usually calculates the intensity correlation function directly. The dynamic light scattering instrument will require a high power laser, typically an Argon gas laser, a temperature controlled sample cell, a sensitive detector such as a photomultiplier tube, and a time correlator capable of recording intensity (or current from the photomultipler tube) on an extremely short time scale (nanoseconds). Where g (2)(t) is the square of the normalized autocorrelation function for electric field, g (2)(t) = |g (1)(t)| 2.ĭynamic light scattering offers a direct measure of C v(t). If the intensity correlation function is normalized by the autocorrelation function results, Since the electric field vector relies on the presence of scattering matter at position R at time t', it is expected that there is a relationship between C(t) and the intensity correlation function, q is related to size r by r = 2 p /q = 1/K. The angle q is usually converted to wavevector, K = 2 sin( q /2)/ l, or momentum transfer vector q = 2 p k. "*" indicates the complex conjugate and "T" the transpose (this is how you square a complex vector). Where t' is the time of irradiation and emission and t is the time of observation by the PMT (there is an incidental time lag in measurement). The observed intensity is proportional to the square of the resulting electric field associated with the combination of light emanating from the irradiated volume.Ī photomultiplier tube of quantum efficiency Q e records the scattered intensity associated with a separation distance R at a fixed angle q as a function of time t, Constructive interference between the emitted light from two molecules or parts of a polymer separated by a vector, r, results in the scattering pattern. This oscillating electric field produces light of the same wavelength as the incident light that is irradiated from the molecules essentially in a uniform manner in space. This results in an electric field E(t) associated with the position of the molecules at a given time. When monochromatic, collimated visible light irradiates matter the state of polarization of the molecules oscillates at the frequency of the irradiating light. Because it is an analytic technique unfamiliar to the majority of polymer scientists a brief overview is given here.Īs mentioned above, the autocorrelation function C(t), or correlation function for position is given by: Bee, "Quasielastic Neutron Scattering: Principles and Applications in Solid State Chemistry, Biology and Materials Science", 1988.ĭynamic light scattering, as well as dynamic neutron and x-ray scattering (recently), is a main tool to understand and verify models pertaining to the dynamics of polymers in dilute solution. Chu, "Laser Light Scattering: Basic Principles and Practice, Second Edition", Academic Press, 1991. "An Introduction to Dynamic Light Scattering by Macromolecules" p.31ī. Doi, "Introduction to Polymer Physics" p.
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