Differential Dynamic Algorithm

All Scattering in the Near Field (SINF) techniques rely on Fourier analysis to extract the sample static power spectrum out of the acquired images thus being able to separate the information at many different wave vectors. If image differences are analyzed (instead of images) the resulting power spectra are depleted by some amount which is a signature of the amount of correlation between the images taken into account. Quite obviously this depletion also depends on the wave vector, since the characteristic times are wave vector-dependent. From this simple consideration the analysis over image differences is performed on couples of images with different time delays and eventually the signal is analyzed for each wave vector as a function of the delay time.

This is the principle of the Differential Dynamic Algorithm which was first applied to two SINF techniques, namely the Shadowgraph and the Schlieren. The same algorithm has later been applied to a variety of optical techniques of the SINF family: Near Field Scattering, X-ray scattering, and microscopy, which generated the Differential Dynamic Microscopy technique.

In Fig.1 the power spectra for many different time delays are shown for a Shadowgraph measurement on concentration non equilibrium fluctuations in a Soret experiment. The bottom line is the electronic background of the experiment B(q), the other lines are the power spectra for increasing (from bottom to top) delay times.

Fig.1 Power spectra for different time delays.

In Fig.2 the values of the power spectrum for a single wave vector are reported as a function of the time delay. This function is known as stucture function of the sample and contains information about the time correlation function.

where Cm is the structure function, S is the sample static power spectrum, T is the technique transfer function, G is the time correlation function and B is the background noise. Hence, from fitting the structure function values for each wave vector by means of the above formula allows getting the values of SxT, G and B, without the need of a separate measurement for B, which is required with other techniques.

In Fig.3 the resulting static analysis is showed for a Shadowgraph measurement on concentration non equilibrium fluctuations in a Soret experiment for 4 different values of the temperature difference over the two sides of the sample cell.

Fig.3 Values of the sample power spectra S multiplied by the Shadowgraph transfer function T.

The time correlation function can be further analyzed by fitting with a known expression. For example in the case of concentration non equilibrium fluctuations it is predicted to be a simple exponential decay, therefore it is possible to obtain the value of the time constant for each wave vector. The result of such analysis is reported in Fig. 4.

Fig.4 Values of the sample time constants as a function of the wave vector.

In the above figure it is possible to see the diffusive decay of concentration non equilibrium fluctuations in a Soret experiment for wave vectors larger than a critical value. For smaller wave vectors the effect of gravity is that of reduce the fluctuation lifetimes. Data for 4 different values of the temperature difference are reported.