Diffusion coefficient scattering function

Water vapor diffusion coefficients as functions of time are shown in Fig, 8. Values obtained from (2), (3), and (4) are all shown for comparison. Due to the scatter obtained, coefficients resulting from (2) are shown as a band of values rather than a single curve. Such a coefficient should be a function of the partial pressure gradient across the diffusional boundary layer. However, under the ambient conditions existing during this study, this quantity was essentially constant, and no partial pressure relationship was determined. However, the effect of boundary layer condensation without subsequent adherence to the container surface is illustrated by the data shown in Fig. 8. Equations (2), (3) and (4) pro-... 

The temporal evolution of P(r,t 0,0) is determined by the diffusion coefficient D. Owing to the movement of the particles the phase of the scattered light shifts and this leads to intensity fluctuations by interference of the scattered light on the detector, as illustrated in Figure 9. Depending on the size of the polymers and the viscosity of the solvent the polymer molecules diffuse more or less rapidly. From the intensity fluctuations the intensity autocorrelation function... 

In Ref. [107] it has been demonstrated how, based on the scaling law for the diffusion coefficient, molar mass distributions can be calculated from time correlation functions obtained from scattering experiments. 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

Stillinger and Rahman have also considered the diffusion coefficient, velocity autocorrelation function and scattering function for simulated water. For discussion of these interesting calculations the reader is referred to their papers 3>. 

The cooperative diffusion coefficient in the salt-free limit is thus strongly 7 -dependent. In this limit the equilibrium scattering function g k) exhibits a peak atk =. Approximating g k) in Eqs. (282) and (285) by its value at the peak... 

In contrast, in dynamic light scattering (DLS) the temporal variation of the intensity is measured and is represented usually through what is known as the intensity autocorrelation function. The diffusion coefficients of the particles, particle size, and size distribution can be deduced from such measurements. There are many variations of dynamic light scattering, and... 

Fig. 53. Computed decay of the dynamic scattering function for a rather stiff dumbbell according to Ref.215). The rotatory diffusion coefficient is 0, qL = 2, a = L/50, r = 0/2500. L = length of the dumbbell, a2 = mean square amplitude of the bond stretching, r = a2/4 D stretching relaxation time321 ...

In dynamic light scattering (DLS), or photon correlation spectroscopy, temporal fluctuations of the intensity of scattered light are measured and this is related to the dynamics of the solution. In dilute micellar solutions, DLS provides the z-average of the translational diffusion coefficient. The hydrodynamic radius, Rh, of the scattering particles can then be obtained from the Stokes-Einstein equation (eqn 1.2).The intensity fraction as a function of apparent hydrodynamic radius is shown for a triblock solution in Fig. 3.4. The peak with the smaller value of apparent hydrodynamic radius, RH.aPP corresponds to molecules and that at large / Hs,Pp to micelles. 

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

Figure 2. A pictorial representation of the mode coupling theory scheme for the calculation of the time-dependent friction (f) on a tagged molecule at time t. The rest of the notation is as follows Fs(q,t), self-scattering function F(q,t), intermediate scattering function D, self-diffusion coefficient t]s(t), time-dependnet shear viscosity Cu(q,t), longitudinal current correlation function C q,t), longitudinal current correlation functioa...

K represents the following constant parameters n is the index of refraction of the liquid, X is the laser wavelength in air, and 0 is the angle at which the scattering intensity is measured. For polydisperse samples, the autocorrelation function plot is the sum of exponentials for each size range. Once the average translational diffusion coefficient of the sample is determined, the equivalent spherical diameter can be determined by using the Stokes-Einstein... 

Fig. 3 Mass and thermal diffusion coefficients D and Z>r as functions of reduced temperature . Literature PCS data for D taken from Meier [8] and Sato [92] (scattering angle 60° (open diamond) and 130° (open square)). See text for a discussion of the fit functions. Also shown Dj (upper curve, right y-axis) for the same temperature range together with fit function containing only thermal activation (dotted line). Open diamonds data with unclear error bars due to very long equilibration times. Note the different units of the two y-axes...

Many excellent introductions to quasi-elastic light scattering can be found in the literature describing the theory and experimental technique (e.g. 3-6). The use of QELS to determine particle size is based on the measurement, via the autocorrelation of the time dependence of the scattered light, of the diffusion coefficients of suspended particles undergoing Brownian motion. The measured autocorrelation function, G<2>(t), is given by... 

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