= Capillary electrophoresis heat-dissipation tradeoff
The \b[<capillary electrophoresis heat-dissipation tradeoff>] is the balance between rapid electrically driven separation and the temperature rise caused by <Joule heating>. Narrow capillaries offer a large wall area per liquid volume, while high <electric fields> increase migration speed and electrical heat production. The 1981 primary experiment demonstrated efficient separations in a 75-micrometre-bore, one-metre glass capillary at 30 kV; it does not imply that arbitrarily high fields are thermally harmless.<Free-zone electrophoresis in glass capillaries by J. W. Jorgenson and K. D. Lukacs>[Abstract, capillary dimensions, applied potential and fluorescent separations]{ref}
A simple model makes the competing scalings explicit. For a cylindrical liquid column of radius $a$, uniform <electrical conductivity> $\sigma$ and axial field $E$, current and heat production per unit length are
$$
I=\sigma E\pi a^2,\qquad P'=IE=\sigma E^2\pi a^2.
$$
At fixed conductivity and bore, doubling the field doubles drift speed for constant <electrophoretic mobility> but quadruples generated heat. Halving the bore radius reduces heat per length by four at fixed field.
For steady radial <heat conduction>, let the liquid's <thermal conductivity> be $\kappa$ and impose a fixed wall temperature $T_w$. With uniform volumetric heating $q'''=\sigma E^2$, the cylindrical heat equation gives
$$
\frac1r\frac{d}{dr}\left(r\frac{dT}{dr}\right)=-\frac{\sigma E^2}{\kappa},\qquad
T(r)-T_w=\frac{\sigma E^2}{4\kappa}(a^2-r^2).
$$
Regularity at the axis and the wall condition fix the integration constants. The center-to-wall rise is \b[$\Delta T_{\max}=\sigma E^2a^2/(4\kappa)$]. The symbol $\kappa$ here denotes thermal conductivity, not an optical decay constant.
For stipulated $\sigma=0.1\,\mathrm{S/m}$, $E=30\,\mathrm{kV/m}$, $a=25\,\mathrm{\mu m}$ and $\kappa=0.6\,\mathrm{W/(m\,K)}$, the model gives $P'=0.1767$ W/m and center-to-wall rise 0.02344 K. Doubling the radius or field multiplies that rise by four. This ideal calculation holds the wall temperature fixed; it does not include heating through the capillary wall, instrument housing or cooling system.
Temperature gradients can change liquid <viscosity>, migration and band dispersion. Conversely, reducing bore size also reduces sample volume and optical path through the sample. A useful instrument experiment therefore records current and separation behavior while controlling cooling conditions, rather than judging success only by reduced migration time. The thermal model predicts what to test; it is not a measured temperature profile from the cited experiment.
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