Boundary effects in directional freezing

Directional freezing applies a temperature gradient on a fluid or a solid and moves this gradient with respect to the sample. We can either move the sample in a fixed gradient (our case), or the opposite, which occurs in industry (zone melting) and in geology (freeze/thaw cycles in soils, pingos). Both cases are physically identical.

In the case where the solidified medium is a suspension (biphasic), particles will enter the ice in the steady state ; their location and potential structuration will depend on the shape of the liquid/solid interface. For high solidification velocities $V$, this interface forms characteristic needles, whose shape can be modified using anti-freezing agents or an imposed liquid flow. This ice-templating, concept provides useful tools to structure materials after sublimation of the solid (ice) matrix.

Below the critical solidification velocity, the solid/liquid (hereafter water/ice) front is plane. In this case, the particle arrangement in the solid phase is not well-known. To address this question, our experiment relies on this setup:

The motor (left) pushes a sample holder (two glass sheets separated by spacers) in a large temperature gradient. A digital camera observes the sample from above the 1 cm window between the hot and cold ovens, providing frames where the 0°C isotherm is always visible. The camera setup works in (simple) light scattering mode.

Volume models

We have observed a strong particle rejection when ice forms, generating a dense particle strip. This strip can exist beyond the 0°C line. In this situation, and in the reference frame of the glass sheets, fluid muse be continuously provided to the solid phase at a rate proportional to $V$ for a steady state to exist (otherwise, the ice phase will recede). This water muse cross the strip, modelled as a porous medium, in which a Darcy flow sets in. The strip stops growing when the pressure difference induced by this flow equals the chemical potential difference linked to the temperature gradient.

With this simple model, the strip is much larger than what is reported. Another way to tell this is that the Darcy micro-channels are too large (experiments used 300 nm particles). A refined model suggested that these micro-channels present at temperatures below 0°C may be partially frozen, reducing the effective size of the micro-channels and limiting the size of the strip. An adjustable-parameter model then recovers the experimentally measured sub-cooling $T_f - T_0$ where $T_0 = 0^\circ {\rm C}$.

We use larger particles ($d =$ 3μm) that leave very large micro-channels which should eventually be almost frozen. Since this model is only one-dimensional, it does not include any dependence of the strip width with the sample height, $e$. Our experimental observations showed no structural or aspect modification of the strip during freeze/thaw cycles, and a dependence of the strip width with $e$. Something in the model is therefore lacking.

Boundary effects

The pressure gradient applied on the strip may be transmitted to the glass plates through an isotropisation of the particle (solid) stresses (this is the Janssen model). The stress balance of a slice of strip then shows a fast (exponential) increase of the friction at the system boundary. A common life example of this happens when too much coffee grains are put in a French press machine. Dimensional analysis shows that the exponential law characteristic length is

$\mu$ being the Coulomb solid friction ratio and, $K$ is the stress isotropisation constant (of order 1 in our case), $e$ being the height of the sample. We should therefore observe a significant change in the stripe width with $e$, which is observed experimentally.

We are now working on a microscopic description of the particle admission process in the solid phase along with models to explain the patterns observed in the stripe..

Brice