3.1. General observations

Microstructural evolution of ice and the complementary brine manifests unique spatio-temporal features during directional freezing. The brine specimen is under supercooling before nucleation, and the initial temperature profile is linear along the vertical direction between the warm and cold ends, which were maintained at constant temperatures throughout the freezing process. When the nucleation barrier was overcome, numerous ice crystals emerged in several seconds. These nascent crystals, shown as dark areas and coexisting with the bright brine, are disordered, as shown in the CT image taken soon after the nucleation (figure 1). The brine trapped in the porous ice gradually gets pushed out of the porous ice afterwards and accumulates at the warm end. Meanwhile, the boundary between porous ice and pure brine, or the freezing front, advances at first, but retreats later. Within the porous ice, the initially disordered pattern is swiped out towards the warm end by vertically striped patterns. Later, a cleaner layer of ice emerges within the porous ice. In addition, brine localises in the centre to form brine columns during downward freezing, but in the periphery to form brine skirts during upward freezing. Overall, the evolution of trapped brine emulates waterfalls in downward freezing and flames in upward freezing (see SI movies 1 and 2, which are time-lapse animations). The final state of upward freezing at ∼24 h resembles that of downward freezing except for the centre column, although the evolution path is obviously different.

Figure 1.

Typical raw images capturing microstructural evolution of brine (bright) and porous ice (dark). Panels (a) and (d) are time sequences of vertical slice images in downward and upward freezings. Panels (b), (c), (e), and (f) are respectively the horizontal slices from bottom to top corresponding to vertical slices at 8.22, 5.07, 5.07 and 2.06 mm out of the 10 mm specimen height. Full time sequences of horizontal and vertical slices are provided in figure S4.

The horizontal sections (figure 1 b,c,e and f) reveal complementary structural evolution in addition to vertical features. Several groups of porous ice with striped patterns emerge soon after nucleation. These groups resemble the distribution of single crystals within polycrystals, which has also been reported in previous research (Kamm Reference Kamm2023). Localisation of unfrozen brine is clearer in downward freezing, and later facilitates the formation of brine columns. We speculated that the location of these brine columns could be associated with group boundaries, but careful investigation of raw images does not support this. Although the vertex of the brine columns shows strong correlation with the group boundaries, the rest of the columns can cut through one group of crystals. The brine in the porous ice structure forms a swirl distribution in figures 1(b) and 1(c) for the 0.6 mol l⁻¹ case, while swirls are absent in both the 0.3 and 1.2 mol l⁻¹ cases. All striped and localised brine patterns are swiped by polygonal patterns during the afterward evolution. This transformation from striped to polygonal patterns coincides with the vertical alignment observed in the vertical sections.

The observed initial patterning is unlikely to be caused by the applied mechanical disturbance introduced to facilitate the nucleation. For the cases with 0.3 mol l⁻¹, mechanical disturbance was not required to initiate the nucleation, and these patterns still emerge. The brine stripes and disordered ice crystals are not localised near the tapped site but spread throughout the specimen. We speculate that the first nucleation occurs randomly, while the following propagation spreads from the first nucleation sites. The pattern is governed by the complex interaction between heat and mass transfer, ice-brine interface instability, as well as preferential growth direction associated with symmetry broken in crystallisation.

3.2. Evolution stages determined by spatial-temporal scales

The overall directional solidification can be categorised into three transitional stages, separated by four critical states (S1–S4), as detailed in figure 2. Initially (S1), a linear temperature profile is established between the imposed cold and warm boundaries prior to nucleation (grey line in figure 2 a), where the brine concentration is uniform throughout the specimen. Rapid (within seconds) crystal growth occurs after nucleation, and the system then transitions to the S2 state. During this short period, long-range energy and mass transfer can be neglected; and therefore both local energy and mass conservation can be assumed. Latent heat released during this period raises the local temperature (pink line). The critical state S3 (purple line) is defined considering that heat transfer is faster than mass migration limited by diffusion, where we consider a steady state of heat transfer is reached while mass migration is yet to take effect. Local mass conservation is still assumed, while one-dimensional heat conduction is used to calculate the local temperature. Since the thermal conductivity of ice is ∼4 times that of water, the range of colder regions enlarges and the freezing front advances. The last transition spanning S3–S4 (yellow line) occurs over a lengthy time compared with earlier stages. Here, brine migration by diffusion plays a dominant role. As ions get purged out of the porous ice, the salt concentration in the pure brine region increases, and the freezing front retreats to a lower temperature as determined by the phase boundary (see the detailed calculation process in § S4).

Figure 2.

Vertical profiles of (a) temperature $T$ , (b) concentration $C$ , (c) ice volume fraction $\phi$ of downward freezing and (d) upward freezing predicted for four critical states (solid lines) and experimental measurements (broken lines). Note that $t$ indicates freezing duration and, as t increases, the ice volume fraction profiles evolve following the directions of the black arrows.

Figure 2(c) illustrates the predictions with solid lines and the volume fraction profiles of ice with broken lines extracted from CT images. The stage from S1 to S2 state (first 20 min) images are omitted because they are blurry, and the process occurs too rapidly for clear visualisation. In general, the broken lines evolve from S3 at one hour (blue broken line) towards S4 at approximately 22 h (red broken line). Note that the actual cases deviate from the ideal predictions because the assumptions are not strictly met. For example, the difference between the blue broken line and purple solid line at Z = 0–2 mm is caused by actual mass migration. The ice volume fraction one hour after nucleation (blue broken line) during upward freezing (figure 2 d) matches well with the predicted value (purple solid line), while the mismatch shown in the grey shadow during downward freezing indicates a quick brine transport mechanism, likely gravity-driven convection, soon after nucleation. Because brine convection accelerates the rejection of condensed brine out of the porous ice, the average salt content within the porous ice decreases, which then allows the formation of more ice at the same temperature. The shorter distribution of ice, only significant in downward freezing, reflects an elevated brine concentration at the ice-brine interface, which is also consistent with the mismatch in the shadow.

Both convection in single brine pockets and larger-scale convection in the porous ice through connected brine pockets is considered. Convection in single brine pockets is possible when the Rayleigh number is larger than 1708 (Wanschura, Kuhlmann & Rath Reference Wanschura, Kuhlmann and Rath1996; Shishkina Reference Shishkina2021). Meanwhile, between rigid and highly conducting boundaries, convection is verified in inclined systems without a threshold in Rayleigh number (Nield Reference Nield1994; Daniels, Plapp & Bodenschatz Reference Daniels, Plapp and Bodenschatz2000; Ortiz-Pérez & Dávalos-Orozco Reference Ortiz-Pérez and Dávalos-Orozco2011). However, we should also consider that the thermal expansion coefficient is negative for water when the temperature is below 4 °C, at which the water density reaches a maximum. Considering the temperature within the porous ice layer is mostly below −1 °C, which is below the maximum density temperature even considering the effect of dissolved salt, the direct effect of the temperature gradient should stabilise the fluid. We therefore consider that the density variation due to the salt concentration gradient, which is also associated with the temperature gradient, should be the driving force for convection in the first hour after nucleation. We define an equivalent expansion coefficient $\beta '$ due to the salt concentration gradient determined by the temperature gradient according to the phase diagram within the porous ice, to obtain a calculated $\beta '$ ≈ 3.28 × 10⁻² K⁻¹, which is two orders of magnitude higher than for pure thermal expansion, for which $\beta$ ≈ −1 × 10⁻⁴ K⁻¹ at −1 °C. This highlights the dominant role of the salt concentration gradient over thermal expansion. Correspondingly, the equivalent Rayleigh number within the 8 mm-thick porous ice, which considers the fluid density change due to both temperature and salt concentration gradients, is approximately 3.2 × 10⁶ (detailed derivation and calculation in § S4). Alternatively, if we enforce the equivalent Rayleigh number to be 1708, then the corresponding critical thickness $L_{{c}}$ = 1.2 mm, which means that convection is possible in any brine pockets taller than 1.2 mm. Similarly, we can obtain the Darcy–Rayleigh number within the porous ice to evaluate large-scale convection through connected brine pockets. The Darcy–Rayleigh number ranges from 285, when we consider only the fine brine stripes, to 7120, when we look into the thick brine columns. Therefore, convection can only occur within the thick brine columns, in which the analysis with Rayleigh number is more appropriate, but not in porous ice with fine brine stripes. Alternatively, this points to the possibility that the brine columns are the main contributor to the convection. It has been reported that there is a critical thickness, typically several centimetres, for the mushy ice layer to produce convection (Wettlaufer, Worster & Huppert Reference Wettlaufer, Worster and Huppert1997). However, several millimetres of mushy layer is shown to yield convection, which can be related to the applied higher temperature gradient. Furthermore, double diffusion due to the different diffusion coefficients of mass and heat can occur. In addition, brine expulsion caused by volume expansion also contributes to brine exclusion, which can potentially explain the mild difference between the blue broken line and the purple solid line at Z = 3–8.5 mm, as shown in figure 2(d).

3.3. Brine columns and skirts

Brine columns emerge at or near the centre of the specimens in all downward freezing cases. (figure 3 a, also see SI movie 3). Smaller brine stripes are filtered out in the visualisation to highlight the evolution of major brine columns. These brine columns continuously fall towards the warm end with time. Brine skirts (figure 3 b, also see SI movie 3) around the central porous ice develop during upward freezing while brine columns are absent (see details in § S5). Note that the downward freezing also forms a reverse skirt in the end, shorter than that formed during upward freezing.

Figure 3.

Microstructural evolution of brine structures. (a) Three-dimensional visualisation of large brine columns and their evolution during downward freezing. (b) Three-dimensional morphological evolution of brine skirts during upward freezing. More details of brine structural evolution are presented in figure S4.

We infer that both the brine columns and skirts are induced by the coupling effect of gravity and the imperfect thermal boundary. As the side thermal insulation does not fully stop heat transfer from the environment to the specimen, the temperature around the periphery is always higher than that at the centre. During downward freezing, the salt concentration of brine coexisting with ice is higher in the centre according to the phase diagram, inducing a higher density. Gravity then favours the local downward brine flow. As the brine with higher concentration migrates to warmer areas, it melts the contacted ice and widens the flow path, which then facilitates additional flow. This positive feedback loop continues to form these thick brine columns. These brine columns likely resemble the brine channels seen in natural sea ice (Worster et al. Reference Worster and Wettlaufer1997). We have also demonstrated that brine columns can develop within porous media (Jia et al. Reference Jia, Gao, Wu and Lei2025), following a similar mechanism. More than one column can be formed. Therefore, we suggest that random localisation rather than the imperfect thermal boundary is dominant. The thickening of the brine columns feeds on the salt collected from adjacent areas. When brine in adjacent areas depletes or gets disconnected, the column thickening stops. During upward freezing, salt concentration and brine density are also higher in the centre, which can drive the saltier and denser brine flow towards the periphery. This transverse salt migration then impedes the ice growth at the periphery, which facilitates the formation of brine skirts.

3.4. Pattern falling or rising

Earlier formed striped patterns near the porous ice-brine interface, with regularly spaced brine and adjacent ice crystals, either fall in downward freezing or rise in upward freezing. These patterns form within the first hour after nucleation and gradually get pushed out of the porous ice within 4–6 h after nucleation. Migration towards the warm end can be quantified by tracking representative points A (see one example in figure 8(d) and more examples in § S6, similarly, points B, C and D are selected to track other features). Figure 4 displays positions of points A versus time. The pattern migration velocity is higher with lower initial salt concentrations, regardless of downward or upward freezing. This velocity decreases with time. The difference between upward and downward freezing is not clear because the absolute position is dependent on point selection and the migration velocity demonstrates significant variations. Note that the falling or rising of patterns, involving extensive ice formation and melting, is complicated. These migrations include not only motion and stretching, but also merging and separation of brine pockets, and constant evolution of brine pocket outlines. Ions within one brine pocket are unlikely to stay within the original brine pocket.

Figure 4.

Movement of falling or rising brine stripes versus time obtained by tracking typical features.

3.5. Vertical aligning

Initially formed irregular patterns, with random inclinations, gradually transform into nearly vertical patterns. The final orientation of these patterns matches well with the thermal gradient (see details in § S7). We track the positions of points B at the interface between irregular and vertical patterns to investigate the alignment velocity (see details in § S6). Like the migration of points A, lower initial salt concentrations lead to faster transformation (figure 5). If we normalise the advancing velocity of points B by the case of 0.3 mol l⁻¹ in either downward or upward freezing, the ratios among the three cases of 0.3, 0.6 and 1.2 mol l⁻¹ are 1 : 0.89 : 0.80, showing little dependence on the influence of gravity.

Figure 5.

Movement of vertically aligned patterns versus time tracked by raw CT images.

3.6. Brine pockets migration

A great number of brine pockets within the aligned ice layer slowly migrate towards the freezing front. As individual brine pockets are difficult to identify, we filtered out just large brine pockets via image processing to track a group of smaller brine pockets (points C, see details in § S6). Since these brine pockets are nearly parallel to the temperature gradient, we infer that ion diffusion driven by the brine concentration gradient $\mathrm{d}c/\mathrm{d}z$ , induced by the temperature gradient $\mathrm{d}T/\mathrm{d}z$ and phase boundary $\mathrm{d}c/\mathrm{d}T$ , is the dominant mechanism. As ions diffuse towards the warm end, ice forms at the cold end of the brine pocket and melts at the warm end. According to Fick’s law and mass conservation, this mechanism is described by a migration rate of (Harrison Reference Harrison1965; Hoekstra et al. Reference Hoekstra, Osterkamp and Weeks1965)

(3.1) \begin{equation} \left| v\right| =-\frac{D}{c}\frac{\mathrm{d}c}{\mathrm{d}T}\frac{\mathrm{d}T}{\mathrm{d}z} ,\end{equation}

where the $v$ is the velocity of brine pockets, $c$ is the local salt concentration and $D$ is the mass diffusion coefficient. Figure 6 demonstrates both the experimental results (black diamonds) and theoretical calculations (dashed line). The good match, regardless of freezing direction, suggests that temperature gradient-induced diffusion rather than gravity is the dominant factor.

Figure 6.

Vertical positions of brine pockets over time for both downward and upward freezings in 0.6 mol l⁻¹ cases (the moment nucleation occurs is regarded as the origin). The scattered points represent individual brine pockets, while black diamonds indicate their mean positions.

3.7. Cleaner ice layer thickening

A layer of cleaner ice becomes obvious at around 3 h after the nucleation. Note that ice in this layer is not 100 % pure, but contains minute brine pockets, as discussed in the previous section. This layer thickens with time as the trapped brine stripes move towards the warm end. We assess the velocities of thickening by manually tracking positions of representative points D at the boundary of the cleaner ice layer in the raw tomograms (see details in § S6). Figure 7(a) shows the positions of these points versus time under various conditions, and the relation between them looks linear. In three downward freezing cases, a lower initial salt concentration produces a thicker layer and higher thickening rate. If we consider salt diffusion in the adjacent striped brine as the governing mechanism, i.e. the transfer of salt in the porous ice determines the thickening rate, the thickening rate should be only dependent on the local salt concentration at points D and the temperature gradient, as seen in (3.1). The initial thought is that initial salt concentration should be irrelevant. However, the initial salt concentration affects the boundary between porous ice and brine, which then affects the local temperature gradient. The temperature gradient in low initial salt concentration cases is higher in the porous ice due to a larger portion of porous ice in the whole specimen (see the calculation in § S8). The ratio among thickening rates for three initial salt concentrations match well with the ratio among the corresponding temperature gradients (figure 7 b). We therefore suggest that diffusion of adjacent striped brine is the dominant factor for cleaner ice layer thickening.

Figure 7.

(a) Thickening of cleaner ice layers as a function of time. (b) Initial concentration effects on dimensionless temperature gradients $\partial T/\partial z$ . The initial concentration varies from 0.3, 0.6 to 1.2 mol l⁻¹.

Similar trends can be found in upward freezing cases, i.e. thicker layers and higher thickening rates correlate with lower initial salt concentrations. However, the ratio among thickening rates does not match with the ratio among temperature gradients (figure 7 b). If we look at the original CT images (figure 1 b), the formation of a cleaner ice layer is obviously more difficult in upward freezing, especially with higher initial salt concentrations. Some brine stripes persist in the porous ice until the end of experiments in the 0.6 and 1.2 mol l⁻¹ cases. Ideally, upward salt diffusion driven by the concentration gradient should not be influenced by gravity. However, gravity seems to play an important role in wider brine stripes and hinders the upward salt migration. Although at a smaller scale, gravity-driven salt migration towards lower elevations, similar to the formation mechanism of a brine skirt, can potentially explain this trend.

Note that residual brine pockets persist throughout the freezing process within the relatively clean ice layer. Although the overall volume fraction of trapped brine is lower near the cold end and gradually decreases with time, there are tiny brine pockets (mostly thinner than 35 µm) even at the cold end by the end of the experiments. This behaviour cannot be explained by ion diffusion. If the governing equations based on diffusion are universally applicable to all brine pockets, complete expulsion from the porous ice matrix would be expected over time. Therefore, we suggest that there is a mechanism to trap some brine pockets or at least slow down the diffusion and impede the formation of a pure ice layer. Higher free energy level at the boundaries between single crystals can also potentially play a role.

Figure 8.

Evolution of representative points within typical groups during freezing. Panels (a), (b) and (c) display cases with 0.3, 0.6 and 1.2 M of initial salt concentration. (d) Separation of different zones and examples of representative points A pattern falling or rising, points B vertical alignment, points C brine pockets migration and points D cleaner ice layer. Multiple points have been selected for points A and B, therefore there are several lines. We use dotted lines to denote the position of points at the front, broken for that in the middle and solid for that at the tail. Grey lines represent the predicted position of pockets following ideal diffusion.

3.8. Comparison of migration rates

The vertical sections of the specimens can be separated into three zones, as shown in figure 8(d). Zone I features fast falling or rising inclined brine stripes (purple lines). The border between zones I and II is the shift from random inclined stripes to aligned vertical stripes, which are not perfectly horizontal but zigzagged (blue broken lines). Brine stripes in zone II fall slower than in zone I (red lines). Behind the tale of falling brines in zone II, a relative but not 100 % pure ice layer is leftover and classified as zone III. The boundary between zones II and III is smooth (pink broken lines). Brine columns mainly occur in zone I.

The overall migration rate follows the order of (A) brine stripe falling in zone I ≈ brine column sinking > (B) vertical aligning > (C) brine stripe falling in zone II > (D) cleaner ice layer thickening. The grey solid lines are the predicted migration of ideal brine pockets, initiated at different vertical positions, driven solely by diffusion (see the derivation in § S9). Red and blue lines, which represent ice crystals aligning and brine pocket migration in vertical porous ice, are roughly parallel to the theoretical derivations, indicating diffusion-dominant processes. The migration rate of brine stripes in zone I (slope angle of purple lines) is obviously faster in the first hour after nucleation but gradually converges to the prediction by diffusion in the following hour. This reduction in migration rate occurs in both downward and upward freezings, which indicates that gravity does not contribute. We suspect it to be related to heat transfer coupled with phase transformation because the duration of roughly one hour agrees well with the time required for the system to reach thermal equilibrium (corresponding to state S3).

Note that the tale of thicker brine columns does not advance faster than the thinner brine stripes, which indicates that the contribution of convection in the larger channels is not distinguishable from the convection in the smaller channels, even when convection plays a role. The expansion rate of the cleaner ice layer (zone III), as displayed by the pink lines, is slower than the prediction by solely diffusion. This means that there is a limiting factor that impedes diffusion in the porous ice, which could be associated with the complex brine network geometry and potentially interfacial interactions such as adsorption. This slower migration than diffusion is also a new mechanism, which requires further theoretical and experimental investigation.

3.9. Concentration of trapped salt in the porous ice

The average salt concentration in the porous ice is a critical parameter for seawater desalination and geophysical studies on sea ice. We calculated the salt mass according to the brine volume and local concentration as determined from the phase boundary and local temperature. The vertical salt concentration profile and its evolution during either downward or upward freezing for the 0.6 mol l⁻¹ cases are plotted in figures 9(a) and 9(b), with the cold end at Z = 0 mm and Z = 10 mm, respectively. As the trapped brine migrates towards the warmer end, the salt concentration in the porous ice gradually decreases. If we consider only one-dimensional diffusion, the salt concentration profile should follow a vertical translational motion and mild stretching. However, the salt concentration at each point decreases with time, which proves the brine pockets’ horizontal thinning in addition to the vertical motion. This indicates there is a driving mechanism for ice to grow horizontally to push the brine pockets out. Furthermore, the violation of mass conservation in seemingly individual brine pockets, as identified in CT images, also implies smaller paths in the ice to transport the excluded ions. The size of these paths exceeds the resolution of CT scanning in this study. It therefore could be at the nanoscale. We speculate the brine veins among adjacent single crystals (Vetráková et al. Reference Vetráková, Neděla, Runštuk and Heger2019) serve as these paths.

Figure 9.

(a), (b) Mass profiles of solute for each slice along the vertical height. (c) Desalination efficiency as a function of time at different initial salt concentrations.

The overall desalination efficiency, defined as the mass of salt rejected from the porous ice to the liquid reservoir divided by the initial mass of salt, increases with time (figure 9 c). Downward freezing (blue lines) achieves higher desalination efficiency than upward freezing, highlighting the facilitation of gravity. Interestingly, the desalination efficiency in the high molarity brine case is higher during downward freezing, while the trend is the opposite during upward freezing. In all cases except for the upward freezing of the 1.2 mol l⁻¹ case, the desalination efficiency exceeds 90 % after 22 h, producing 5–7 mm thick cleaner ice.