DCDT

DCDT processes sedimentation velocity data using the time derviative to eliminate systematic noise and produces a plot of the concentration gradient with respect to the radial axis expressed in svedbergs. The DCDT plot [g(s) vs s] represents a snapshot of the sedimentation process at a particular time. It preserves diffusion information allowing accurate estimation of diffusion coefficients and, therefore, calculation of molar masses by fitting the g(s) vs s function to a gaussian and extracting the diffusion coefficient from the variance of the gaussian.

Advantages of g(s*) from dc/dt

  • The DCDT plot [g(s) vs s] represents a snapshot of the boundary at a particular time.

  • Uses a narrow time interval.

  • Boundary shape is preserved.

  • Diffusional information is preserved allowing accurate estimates of diffusion coefficients - even for multiple overlapping boundaries - and, therefore, reliable calculation of molar masses.

  • Boundary spreading characteristics of interacting systems are also preserved.

Derivation

\[ \left(\frac{\partial c}{\partial t} \right)_{t} = A~\text{exp}\left[- \frac{\left(\frac{\bar{r}}{r_{m}} - \frac{r^{*}}{r_{m}}\right)^{2}}{\epsilon(\text{exp}(2\omega^{2}st)-1)} \right] \]
\[ \epsilon \equiv \frac{2D}{s\omega^{2}r_{m}^{2}} \]
\[ \frac{\bar{r}}{r_{m}} = ~\text{exp}(\omega^{2}st) \approx 1 + \omega^{2}st. \]
\[ \left(\frac{\partial c}{\partial t} \right)_{t} = A~\text{exp}\left[-\frac{(s-s^{*})^{2}}{2\sigma^{2}} \right] \]
\[ \sigma^{2} = \frac{2Dt}{(\omega^{2}tr_{m})^{2}} \]
\[ D = \frac{(\sigma\omega^{2}tr_{m})^{2}}{2t} = 0.5t(\sigma\omega^{2}r_{m})^{2} \]
\[ g(s^{*}) = \left(\frac{\partial c}{\partial t} \right)_{t} \omega^{2}r_{m}t \]
\[ g(s^{*}) = A'~\text{exp}\left[-\frac{(s-s^{*})^{2}}{2\sigma^{2}}\right] \]