Sedimentation of colloidal plate-sphere mixtures and inference of particle characteristics from stacking sequences

We investigate theoretically the effect of gravity on a plate-sphere colloidal mixture by means of an Onsager-like density functional to describe the bulk, and sedimentation path theory to incorporate gravity. We calculate the stacking diagram of the mixture for two sets of buoyant masses and different values of the sample height. Several stacking sequences appear due to the intricate interplay between gravity, the sample height, and bulk phase separation. These include the experimentally observed floating nematic sequence, which consists of a nematic layer sandwiched between two isotropic layers. The values of the thicknesses of the layers in a complex stacking sequence can be used to obtain microscopic information of the mixture. Using the thicknesses of the layers in the floating nematic sequence we are able to infer the values of the buoyant masses from the colloidal concentrations and vice versa. We also predict new phenomena that can be experimentally tested, such as a nontrivial evolution of the stacking sequence by increasing the sample height in which new layers appear either at the top or at the bottom of the sample.

Figure 1

Dimensions of the hard spheres (green) and hard cylinders (orange) used to model the colloidal plate-sphere mixture, together with a sketch of a cuvette of height $h$ under a gravitational field $\mathit{g}$. The stacking sequence is a floating nematic phase $INI$, i.e., top isotropic, middle nematic, and bottom isotropic. Both the stacking sequence and the thickness of the layers are consistent with one of the experimental samples reported in Ref. [11]. Schematics of the particles in the isotropic (no orientational order $S_{\mathrm{p}}=0$) and in the uniaxial nematic (orientational order $S_{\mathrm{p}}>0$) phases are also shown.

Figure 2

(a) Model bulk phase diagram in the plane of chemical potentials ${\mu }_1$ and ${\mu }_2$. The phases $A$ and $B$ coexist along a binodal (black-solid line) that ends at a critical point (empty circle). The line segments are finite sedimentation paths. The gray path crosses the binodal and corresponds to a stacking sequence $AB$ (from top to bottom). The grey arrow indicates the direction of all paths from top to bottom. Illustrative examples of paths that form boundaries between different stacking sequences are depicted: (i) paths that start (red) or end (yellow) at the binodal, (ii) paths tangent (blue) to the binodal, and (iii) paths that cross (orange) the critical point. A displacement of any of such paths can alter the stacking sequence. The dotted-gray line is a sedimentation path in the limit of infinite height ($a$ and $b$ are the intersects of the path with the ${\mu }_2$ and the ${\mu }_1$ axes, respectively). (b) Stacking diagram, plane of average chemical potentials ${\overline{\mu }}_1$ and ${\overline{\mu }}_2$, of the bulk phase diagram depicted in (a). Each region is a different stacking sequence, as indicated. The boundary lines between sequences are sedimentation binodals of type I (solid lines) or type II (dashed-blue line), and a terminal line (dotted-orange line). A sedimentation path in (a) is a point in (b) given by the coordinates of the average chemical potentials along the path. See, e.g., the grey circle in (b) that corresponds to the gray sedimentation path (finite height) in (a).

Figure 3

Bulk phase diagram in the plane of chemical potential of plates ${\mu }_{\mathrm{p}}$ and spheres ${\mu }_{\mathrm{s}}$ (a), and also in the plane of packing fractions of plates ${\eta }_{\mathrm{p}}$ and spheres ${\eta }_{\mathrm{s}}$ (b). The solid-black line in (a) is the binodal at which the isotropic $I$ and the nematic $N$ phases coexist. Dotted lines in (b) are tie lines connecting coexisting points along the binodal (solid-black line). The grey area is the two-phase region. The pink line in (a) is a sedimentation path corresponding to the $INI$ stacking sequence with layer thicknesses of $9.3\mathrm{mm}, 16.5\mathrm{mm}$, and $9.6\mathrm{mm}$ as found experimentally in Ref. [11]. The pink curve in (b) is the same sedimentation path in the plane of packing fractions. The pink arrows indicate the direction of the path from the top to the bottom of the sample. Stacking diagram in the plane of average chemical potential of plates ${\overline{\mu }}_{\mathrm{p}}$ and spheres ${\overline{\mu }}_{\mathrm{s}}$ [(c)–(e)], and in the plane of average packing fractions of plates ${\overline{\eta }}_{\mathrm{p}}$ and spheres ${\overline{\eta }}_{\mathrm{s}}$ [(f)–(h)] for three different sample heights: $h=5\mathrm{mm}$ [(c),(f)], $10\mathrm{mm}$ [(d),(g)], and $30\mathrm{mm}$ [(e),(h)]. Each colored region correspond to a different stacking sequence (except the pure sequences $I$ and $N$ depicted in white). The sequences are labeled from the top to the bottom of the sample. The black crosses in panels (e) and (h) indicate the position of the sedimentation path plotted in panels (a) and (b). In the stacking diagram, sedimentation binodals of type I (type II) are represented with solid (dashed)-black lines.

Figure 4

Average packing fraction of plates ${\overline{\eta }}_{\mathrm{p}}$ (a) and spheres ${\overline{\eta }}_{\mathrm{s}}$ (b) indicated by the color map as a function of the gravitational lengths of plates ${\xi }_{\mathrm{p}}$ and spheres ${\xi }_{\mathrm{s}}$. Each point in the diagrams represents a sedimentation path that produces an isotropic-nematic-isotropic $\left(INI\right)$ stacking sequence with layer thicknesses of $9.3\mathrm{mm}, 16.5\mathrm{mm}$, and $9.6\mathrm{mm}$, from top to bottom. Illustrative paths together with a sketch of the sample are shown in the inset of panel (a). No solution exists above the red-dotted line in panels (a) and (b) due to the path being too flat to cross the binodal twice. The inverse relations, i.e., gravitational lengths as a function of average packing fractions, are depicted in panels (c) and (d). The samples labeled 1 – 3 correspond to the paths shown in the inset of panel (a). The samples labeled 4 – 7 are close to the boundaries of the diagrams shown in panels (a) and (b). The black crosses in (a) and (b) indicate the value of the gravitational lengths used here.

Figure 5

Stacking diagram in the plane of average chemical potential of plates ${\overline{\mu }}_{\mathrm{p}}$ and spheres ${\overline{\mu }}_{\mathrm{s}}$ [(a)–(c)] and also in the plane of average packing fractions of plates ${\overline{\eta }}_{\mathrm{p}}$ and spheres ${\overline{\eta }}_{\mathrm{s}}$ [(d)–(f)] for sample heights of $h=20\mathrm{mm}$ [(a), (d)], $32\mathrm{mm}$ [(b), (e)], and $40\mathrm{mm}$ [(c), (f)]. The buoyant mass ratio is $s=2$ in all cases. The crosses indicate at each height the location of the samples with average packing fractions $({\overline{\eta }}_{\mathrm{p}},{\overline{\eta }}_{\mathrm{s}})=(0.063,0.14)$ labeled with white squares as samples 1, 2, and 3. The stacking sequences are labeled from top to bottom of the sample. The inset in (b) is a close view of a small region of the stacking diagram. Sedimentation binodals of type I (type II) are represented with solid (dashed)-black lines.

Figure 6

Stacking diagram at fixed colloidal concentrations. Stable layer at elevation $z$ as a function of the total sample height $h$ for samples with fixed concentrations $({\overline{\eta }}_{\mathrm{p}},{\overline{\eta }}_{\mathrm{s}})=(0.063,0.14)$. The buoyant mass ratio is $s=2$. The vertical-dashed lines mark the samples 1, 2, and 3 with heights $20\mathrm{mm}, 32\mathrm{mm}$, and $40\mathrm{mm}$. The same samples are also labeled and marked with crosses in Fig. 5. The solid-black line indicates the sample-air interface.

Figure 7

Stacking diagram for samples in the limit of infinite height in the plane of inverse slope $1/s$ and root $b$ of the sedimentation path. The auxiliary-horizontal axis indicates the value of the slope $s$. Each region corresponds to a different stacking sequence in the limit of infinite height. Sequences are labeled from top to bottom of the sample if $m_{\mathrm{p}}>0$ and from bottom to top if $m_{\mathrm{p}}<0$. Black-solid lines are sedimentation binodals formed by the paths that are tangent to the bulk binodal. Vertical-dashed lines are asymptotic terminal lines formed by the two sets of paths that are parallel to the bulk binodal in the limits ${\mu }_{\mathrm{s}}\to -\infty$ ($1/s=0$) and ${\mu }_{\mathrm{s}}\to +\infty$ ($1/s\approx 0.39$).