3.2.1. Short-time self-diffusion in spherically confined bidisperse suspensions

A confined test particle experiences, at all positions, a hydrodynamic coupling with the enclosure, making the short-time self-diffusion tensor anisotropic (Aponte-Rivera et al. Reference Aponte-Rivera, Su and Zia2018) (an overview of diffusive regimes relevant to this work is given in Appendix A). The short-time self-diffusion tensor of a confined particle can be orthogonally decomposed into diffusion parallel and perpendicular to the line of centres connecting a test particle with the cavity, the same decomposition applied to an unconfined pair of particles interacting with each other. In the present work, we computed the self-diffusion coefficients along the radial direction from the cavity centre to the wall, $D_{0}^{s_\parallel }(\phi ,\phi _2/\phi ,\lambda _{c_2},\lambda _{p_2})$ and transverse to that, the angular diffusion, $D_{0}^{s_\perp }(\phi ,\phi _2/\phi ,\lambda _{c_2},\lambda _{p_2})$. We computed these for a range of total volume fraction, size composition, cavity size and particle-size ratio, for bidisperse confined suspensions following the procedure we described in section § 2.4.

We start with the semi-dilute regime. The self-diffusion along the radius of the cavity, $D_{0}^{s_\parallel }$, is shown in figure 5 as a function of position in the cavity $y/R$ for various combinations of large and small particles, $0.0 \leq \phi _2/\phi \leq 1.0$, at fixed total volume fraction $\phi = 10\,\%$, particle-size ratio $\lambda _{p_2} = 2$ and confinement $\lambda _{c_2}=0.2$. In the limit of all-small or all-large particles (uppermost and lowermost curves), radial diffusion is fastest near the centre of the cavity and decreases radially outward until it approaches zero near the wall. The expected monodisperse case is recovered: small particles diffuse faster than large ones. Starting with all large particles, the gradual replacement of volume fraction with small particles drives diffusivity up everywhere in the cavity (except at the wall where it must decay to zero). Thus, at the dilute condition shown in figure 5, polydispersity plays only a quantitative role. A common discontinuity at $y/R=0.8$ corresponds to the closest a large-particle centre can get to the wall. Beyond that position, only small particles directly contribute to average calculation of the radial diffusion coefficient. In fact, all the curves overlap onto the small-particle monodisperse curve, showing that, in the region near the wall, bidispersity does not affect the diffusivity of small particles, at least in this semi-dilute regime.

Figure 5.

The radial component of the short-time self-diffusivity obtained here utilizing our new confined mobility functions (see supplementary material, equation (S3)), implemented in polydisperse confined Stokesian dynamics simulations. Plotted as a function of the normalized particle positions in the cavity, for volume compositions of the large particles (symbols) $0.0 \leq \phi _2/\phi \leq 1.0$ with particle-size ratio $\lambda _{p_2} = 2$, volume fraction $\phi =0.10$ and large particle-to-cavity relative size $\lambda _{c_2} = 0.2$. Dashed lines are the proposed ‘colloidal Dalton's law’ where a simple number average of the two monodisperse curves is used to predict the bidisperse $D_{0}^{s_\parallel }/D_{0,1}$.

We captured the quantitative effect of semi-dilute bidispersity at $\phi =0.1$ using a simple mixing rule – a ‘colloidal Dalton's law’ – based on the number of particles, indicated by the dashed lines in figure 5. The linear additivity of the monodisperse curves reflects the rarity of particle–particle interactions in semi-dilute suspensions, even rarer for encounters between different species. The same behaviours are observed for the perpendicular $D_{0}^{s_\perp }$ self-diffusivity (Appendix B, figure 16), and whether the dynamics monitored is that of the small $D_{0,1}^{s_\parallel }$ or large particles $D_{0,2}^{s_\parallel }$ (Appendix B, figure 17).

Crowding changes this picture substantially, as shown in figure 6 for $\phi =0.40$, where we have plotted the diffusivity averaged over all particles at various positions in the cavity for various size compositions. In the two monodisperse limits, peaks and troughs in the diffusivity curves correlate with liquid-like microstructure, i.e. high (low) density of particle centres corresponds to zones of low (high) mobility, as expected (Aponte-Rivera et al. Reference Aponte-Rivera, Su and Zia2018). But polydispersity gives a new result: a non-monotonic change in radial diffusivity with volume composition at some positions in the cavity. This non-monotonic behaviour indicates that at high volume fraction, pair- and higher-order interactions change substantially with size composition. A direct consequence of such interactions is that replacement of even one large particle with small particles drives average diffusivity up everywhere in the cavity; that is, bidispersity increases mobility from the viewpoint of an all-large particle confined suspension. The origin of this behaviour is the disruption of ordered structure: replacing even 5 % volume with small particles destroys the structure (see Appendix B, figure 19), which increases the mobility of the suspension-averaged diffusivity (figure 6).

Figure 6.

The radial component of the short-time self-diffusivity obtained here utilizing polydisperse confined Stokesian dynamics simulations, plotted as a function of the normalized particle positions in the cavity. Volume compositions of the large particles are $0.0 \leq \phi _2/\phi \leq 1.0$; particle-size ratio is $\lambda _{p_2} = 2$, total volume fraction is $\phi =0.40$ and large particle-to-cavity size ratio is $\lambda _{c_2} = 0.2$. Transverse short-time self-diffusivity data can be found in Appendix B, figure 18.

Surprisingly, the average radial diffusivity in bidisperse suspensions can exceed that of the seemingly ‘fastest’ scenario of all-small particles, where individual particles have the highest mobility owing to small size. This increment is seen in figure 6 at several positions in the cavity and is most pronounced near the cavity wall, $y/R> 0.8$, where only small particle centres reside. Although mobility vanishes at contact, small-particle diffusivity first increases markedly close to the wall. To understand this counterintuitive result, we recognize that the annular volume is semi-dilute in small-particle volume fraction for $\phi _2/\phi =0.95$, and becomes more concentrated as $\phi _2/\phi$ decreases, which allows small particles to increasingly populate it. Even though the centres of large particles do not reside within the wall annulus, the volume of large particles do occupy it; that is, the annulus is never dilute in total volume fraction. Thus, the increased mobility of small particles near the wall (as the total volume composition of large particles increases) arises not from the emergence of a depleted layer, but from a decrease in the number of hydrodynamic couplings. More specifically, the near-field two-body couplings underlie this near-wall behaviour. This can be understood as follows. Although many-body far-field and two-body near-field interactions are present everywhere in the cavity, the former dominate qualitative dynamical behaviour throughout the bulk, and the latter dominate qualitative dynamical behaviour near the wall. To illustrate, we extract the far-field contributions to the short-time self-diffusion ($D_{0}^{s,ff,_\parallel }$) from the total diffusion coefficient $D_{0}^{s_\parallel }$ and plot both everywhere in the cavity (figure 20 in Appendix B), which confirms that many-body interactions are sufficient to describe the qualitative behaviour in the bulk but not near the wall, and that two-body interactions are necessary to capture the qualitative behaviour near the wall (and are also required to produce quantitative accuracy in the bulk). Having established that near-field pair interactions set the particle dynamics near the wall, we return to the observation that small-particle mobility increases as the total volume composition of large particles increases. Recall from § 3.1 that the number of neighbours near a small particle in the annulus near the wall decreases with increased large-particle fraction (cf. figure 12a in Appendix B). Obviously, the pair interactions with the wall do not change with volume composition, so we conclude that changes in the number of nearest neighbours of a tracer near the wall drives the change in the near-wall particle dynamics. Overall, for a small particle near the wall, as the annulus becomes bidisperse, the number of lubrication neighbours decreases, and this decrease in the number of near-field interactions drives up self-mobility and reliably predicts the dynamics near the wall (but this simple predictive connection does not hold in the bulk of the cavity where many-body interactions dominate). To see if the small-particle mobility is faster also elsewhere in the cavity, we compute the average diffusivity within each population: diffusion of either small or large particles in a bidisperse suspension. This analysis is plotted in figure 7(a), which shows the radial self-diffusion averaged over only the small particles $D_{0,1}^{s_\parallel }$ in bidisperse suspensions, made dimensionless on the Stokes–Einstein diffusivity of small particles, $D_{0,1}=kT/6{\rm \pi} \eta a_1$. It is evident that small particles in a confined bidisperse suspension are more mobile everywhere in the cavity than small particles in a monodisperse confined suspension. This speedup becomes more pronounced as volume composition shifts to higher and higher fraction of large particles. Small-particle diffusion is faster for almost all positions in the cavity – except at $y/R=0.5$: small particles get faster monotonically as the composition shifts to more large particles, but around $95\,\%$ large particles, there is a high probability that small particles have a large particle nearby, which is close to contact with the cavity and thus slow. Counterintuitively, the more a suspension comprises large (less mobile) particles (at fixed total volume fraction), the faster its smaller particles diffuse. This surprising result can be understood as emerging from a decrease in the number of which, as mentioned above, comprise both far-field many-body and near-field two-body couplings. The former scales as the number of particles in the suspension. Thus, a bidisperse suspension will always have fewer many-body couplings than a monodisperse small-particle suspension; having fewer many-body hydrodynamic couplings increases the mobility of small particles everywhere in the cavity. Overall, small-particle dynamics is found to be faster everywhere in the cavity in a bidisperse suspension, owing to a decrease in the number of many-body couplings. Near the wall, however, the effect of reduction in many-body couplings adds to the effect of the decrease in the number of lubrication interactions with bidispersity to produce a dynamical zone near the wall in which small-particle diffusivity is fastest in bidisperse suspensions.

Figure 7.

Species-specific contributions to the radial component of the short-time self-diffusivity, plotted as a function of distance from the cavity centre, for the same conditions as figure 6. (a) Diffusivity of small particles in a suspension at fixed total volume fraction $\phi =0.40$, as large particles become more numerous. (b) Diffusivity of large particles in a suspension at fixed total volume fraction $\phi =0.40$, as small particles become more numerous.

A similar examination of the radial self-diffusivity of large particles, $D_{0,2}^{s_\parallel }$, given in figure 7(b), reveals similarities and differences to the small-particle behaviour. The monodisperse case exhibits the expected vanishing mobility at the wall, and a trough half-way between that point and the centre, where there is a high density of particle centres. Accordingly, there are two prominent peaks of high mobility ($y/R\approx 0.6$ and $y/R\approx 0.2$) in the regions where there is a low density of particle centres. As large particles are replaced by an equal volume of small particles, the structure is disrupted (see Appendix B, figure 13a). Now the interstitial space is populated by small particles, displacing large particles from the previously packed regions. This structural change smooths the dynamical heterogeneity: troughs in the large-particle diffusivity in the radial direction are shallower, and the peaks in mobility are lower. There are new peaks in $D_{0,2}$, because large particles now sample a greater range of positions. Although structural changes explain the smoothing, hydrodynamics drives the pronounced decrease at $y/R\approx 0.6$ and $y/R\approx 0.2$: bidisperse suspensions have more particles than a monodisperse large-particle suspension, which increases the number of many-body hydrodynamic couplings experienced by large particles, in turn decreasing their mobility. Overall, large-particle short-time self-diffusion is made more spatially uniform by bidispersity.

Finally, the separate contributions of the small and large particles now explain the highly non-monotonic behaviour in figure 6 – the inhomogeneities (peaks and troughs) change as a function of volume composition reflecting the structural changes of the suspension – which is a consequence of the change in structure induced by polydispersity (see Appendix B, microstructure plots $g_1(y)$ and $g_2(y)$ in figure 13a,b) and the change in hydrodynamic couplings (far-field many body and near-field two body). At short times, the self-diffusion of a particle is always influenced by many-body couplings no matter its location in the cavity, and even though it describes motion within a nearest-neighbour cage and reflects the coupling with those nearest neighbours, those nearest neighbours are hydrodynamically coupled to every other particle in the suspension and thus so is the caged particle. At longer times, further spatial exploration driven by Brownian motion will lead to not only hydrodynamic but also entropic interactions with other suspended particles and with the cavity. In the next section, we study this self-motion as the particles move throughout the entire cavity.

3.2.2. Long-time behaviour: mean-square displacement

Here, we measure the MSD as a function of time, as described in § 2.4. The slope of these data indicates whether motion is diffusive, giving confirmation of the short-time self-diffusivity calculation from hydrodynamic theory in § 3.2.1, and also exposes the long-time dynamics. We will examine vertical shifts in MSD curves that arise from changes in volume composition and lag time.

The radial MSD, $\langle r^{2}_{\|} \rangle$, for monodisperse and bidisperse suspensions, is plotted in figure 8(a) for a moderately concentrated, confined suspension at total volume fraction $\phi = 0.25$ and several volume compositions $\phi _2/\phi$. The two monodisperse limits recover behaviour observed previously: a linear (diffusive) growth at short times, followed by transition regions of sublinear (subdiffusive) growth at intermediate times and lastly a long-time plateau. Unsurprisingly, the smaller particles reach the plateau earlier and have a slightly higher terminal displacement plateau, both owing to their smaller size. Bidisperse suspensions display a smooth quantitative transition from one monodisperse limit to the other, as smaller particles replace larger ones; in fact, this progressive growth at moderate concentration is a simple linear combination of the confined monodisperse MSD data. This is made obvious by extracting the average over small particles at each of the six compositions and plotting them in figure 8(b), where the curves overlay the monodisperse average, showing that the absolute value and the temporal evolution of MSD of small particles is the same regardless of their relative abundance. The same procedure is followed for the large-particle population, where the average MSD of large particles within the suspension at the six volume compositions all fall together on the monodisperse curve for large particles (figure 8c). This linear-superposition effect reflects the structural observations that polydispersity does not significantly change structure in the semi-dilute regime (Appendix B, figure 15). We conclude that, in semi-dilute to moderately concentrated regimes, particles that sample the entire confined suspension behave as though they are in a monodisperse suspension at the same volume fraction. One can thus know the average MSD at any volume composition by knowing only the total volume fraction – provided that $\phi \le 0.25$. Therefore, at low and moderate volume fractions, hydrodynamic and entropic interactions between particles produce an effective or mean-field viscosity. However, recalling figure 7, we expect more qualitative effects at higher concentrations, which we examine next.

Figure 8.

The MSD $\langle r^{2}_{\parallel } \rangle /a_1^{2}$ as a function of time at moderate volume fraction ($\phi =0.25$) for particle-size ratio $\lambda _{p_2} = 4/3$ and confinement $\lambda _{c_2} = 0.2$. Volume composition $\phi _2/\phi$ varies from one monodisperse limit to the other as shown in legend. (a) The MSD is averaged over all particles, giving a measure of how polydispersity affects the entire suspension-averaged dynamics. The average dynamics of (a) small and (b) large particles in a surrounding suspension, as they change in the presence of more (a) large or (b) small particles, at fixed total volume fraction. (a) Averaged over all particles. (b) Small-particle contribution. (c) Large-particle contribution.

For more concentrated suspensions, figure 9(a), radial packing leads to a more pronounced caging plateau at intermediate times, followed by a superdiffusive regime not seen at lower volume fractions but observed previously for confined monodisperse suspensions (Aponte-Rivera et al. Reference Aponte-Rivera, Su and Zia2018). More importantly, at higher volume fractions, volume composition plays a qualitative role in the radial MSD. At short times, the transition from all-small to all-large particles is monotonic, but some of the MSD curves shift toward the large-particle limit and the others toward the small-particle limit – the dynamics is dominated by the most numerous species, $N_i/N$, as shown in the legend. At intermediate times, bidispersity lengthens and prolongs the correlation plateaus beyond those in the monodisperse limits. While the all-large-particle system has a pronounced plateau and ostensibly the slowest particles, we find the surprising result that having even slight bidispersity produces a flatter and more prolonged caging plateau. Similarly, replacing small particles with some large ones also produces flatter and more prolonged caging plateaus. In fact, the changes in caging behaviour induced by polydispersity is present in the radial MSD curves for each particle size in bidisperse suspensions, as shown in figure 9(b,c) for small $\langle r^{2}_{\|} \rangle _1$ and large $\langle r^{2}_{\|} \rangle _2$ particles, respectively. Also, in both (b) and (c), the strength of caging follows the same non-monotonic behaviour with volume composition as observed in figure 9(a), where the radial MSD of either small or large particles at $\phi _2/\phi = 90\,\%$ exhibits the strongest caging. This leads to a more complex influence of volume composition on the long-time dynamics. This behaviour is analysed next.

Figure 9.

The MSD $\langle r^{2}_{\parallel } \rangle /a_1^{2}$ for (a) monodisperse and bidisperse suspensions at total volume fraction $\phi =0.40$, with particle size ratio $\lambda _{p_2} = 4/3$, confinement $\lambda _{c_2} = 0.2$, for several volume compositions of large particles $\phi _2/\phi$. The MSD of either (b) small $\langle r^{2}_{\parallel } \rangle _1/a_1^{2}$ or (c) large $\langle r^{2}_{\parallel } \rangle _2/a_1^{2}$ particles at the same suspension compositions. (a) Averaged over all particles. (b) Small-particle contribution. (c) Large-particle contribution.

At long times, the volume-composition effect in figure 9(a) is not monotonic: the polydisperse curves move downward from all-small particles until $\phi _2/\phi =70\,\%$, but suddenly jump below the all-large particles at $\phi _2/\phi =90\,\%$, then move upward. In figure 9(b), we observe that small particles in bidisperse suspensions shift from a more to a less mobile behaviour than those in monodisperse suspensions around $t/(a^{2}_1/D_{0,1}) \sim O(1)$. This behaviour arises because a small particle can explore its immediate (polydisperse) surroundings faster than it could in a monodisperse suspension, owing to a decrease in the number of many-body hydrodynamic couplings. An inset is provided, figure 9(b), to show this speedup more clearly. However, at longer times its motion is suppressed – it is more difficult for small particles to explore longer distances when it must exchange places with particles larger than itself. Physically, small particles get radially trapped between less mobile large particles, which ultimately decides their long-time fate. This behaviour is reversed for large particles: figure 9(c) shows that large particles in bidisperse suspensions are less mobile than they are in monodisperse suspensions for short to intermediate times $t/(a^{2}_1/D_{0,1}) \lesssim O(10)$, while at longer times $t/(a^{2}_1/D_{0,1}) \sim O(100)$, more mobile behaviour than in monodisperse suspensions is only possible for large particles in suspensions with a high relative number of small particles $N_2/N=0.30$ and $0.15$. Physically, large particles in bidisperse suspensions are slower to explore their immediate surroundings (in-cage diffusion) than in monodisperse suspensions (inset figure 9c), owing to an increase in the number of many-body hydrodynamic couplings. As large particles try to break out of their initial cages, they encounter small, very mobile particles, but these are densely packed around them, overall creating more durable cages than an all-large-particle cage. Overall, parsing the MSD into the separate contributions of the two species reveals that polydispersity increases cage durability; this reverses the early-time polydispersity trend of small particles but reinforces the early-time trend of large particles; and size polydispersity slows down long-time motion, regardless of size: both small and large particles diffuse slower when placed in a polydisperse confined suspension (cf. figure 9), indicating a mechanism for polydisperse diffusion that is independent of particle size. The next step is to understand the extent to which this polydisperse cage dynamics is influenced by confinement (figure 9a).

The result that bidispersity suppresses the long-time radial MSD of each species relative to the monodisperse limit can be understood by connecting cavity-length-scale structural fluctuations to the dynamics. The plots of the radial distribution function $g(y)$ in figure 3 in § 3.1 and figure 21 in Appendix B showed that polydispersity to some extent homogenizes cavity-induced layering, but not to the extent that polydispersity smears out structure in unbound suspensions. Evidently, this structural smoothing slows MSD, starting with either all-small or all-large particles. The suppression of MSD arising from smoother structural variations can be understood by recognizing that structural homogeneity suppresses velocity fluctuations, where we recall that MSD is the autocorrelation of velocity fluctuations

(3.2)\begin{equation} \langle r^{2} (t) \rangle = 2t\int^{\infty}_0 \langle \boldsymbol{u}(\tau)\boldsymbol{\cdot}\boldsymbol{u}(0) \rangle\,\mathrm{d}\tau. \end{equation}

The angle brackets signify an inner product or average with weighting set by the particle distribution. A mobile particle moving through a more heterogeneous structure – encountering densely packed regions alternating with solvent pores, or large particles alternating with small particles – will experience pronounced fluctuations in its velocity (Einstein Reference Einstein1906; Batchelor Reference Batchelor1970; Batchelor & Green Reference Batchelor and Green1972; Batchelor Reference Batchelor1976; Ermak & McCammon Reference Ermak and McCammon1978; Russel & Glendinning Reference Russel and Glendinning1981; Batchelor Reference Batchelor1983; Rallison & Hinch Reference Rallison and Hinch1986; Davis & Hill Reference Davis and Hill1992). Conversely, a more homogeneous structure will induce fewer such fluctuations and hence reduce Brownian MSD. The plots for the structure $g({y})$ confirm this view and produce the same trends with added polydispersity: as one moves from all-small or all-large particles (from one curve to another) to a more balanced mixture, the structure becomes more homogeneous. As a probe travels radially outward (left to right along one $g(y)$ curve), it encounters the heterogeneity that produces velocity fluctuations, which becomes less and less pronounced with added polydispersity, weakening fluctuations. At low volume fractions, the structure is nearly uniform, explaining the absence of this effect in figure 8. This analysis of $g({y})$ explains the reduction in the magnitude of the MSD, including the seemingly non-monotonic dependence on volume composition. The correlations induced by the cavity also explain the long-time dynamics, as follows. As a particle traverses the cavity, it will get caged between zones of high particle density. Thus, the width between peaks in $g({y})$ (see figure 21 in Appendix B) establishes cavity-length-scale caging, and such width increases for bidisperse suspensions, setting the onset and length of the caging plateau. Parsing the two contributions to $g(y)$ from the two particle sizes gives two curves with separate but very closely spaced correlation peaks – because the peaks coincide, this generates wider zones between the double peaks, producing stronger caging, as shown in figure 3 in § 3.1 and figure 21 in Appendix B.

Now we turn our attention to transverse MSD in in the concentrated regime. In contrast to the rich radial dynamics discussed above, the angular motion in figure 10(a) shows a simple monotonic and progressive growth of MSD between the two monodisperse limits as the relative abundance of small particles is increased. This behaviour can be understood by examining the MSD computed for only large or only small particles (figure 10b,c), which shows almost no volume-composition dependence, leading to a simple weighted-average behaviour. This behaviour is quite similar to both the radial and angular MSD in semi-dilute suspensions in figure 8, and has a similar explanation here for concentrated transverse MSD: there is very high likelihood that a particle will encounter only particles of its same size when exploring in the transverse direction.

Figure 10.

The MSD $\langle r^{2}_{\perp } \rangle /a_1^{2}$ for (a) monodisperse and bidisperse suspensions at total volume fraction $\phi =0.40$, particle-size ratio $\lambda _{p_2} = 4/3$ and confinement $\lambda _{c_2} = 0.2$, for several volume compositions of large particles $\phi _2/\phi$. The MSD of either (b) small $\langle r^{2}_{\perp } \rangle _1/a_1^{2}$ or (c) large $\langle r^{2}_{\perp } \rangle _2/a_1^{2}$ particles at the same suspension compositions. (a) Averaged over all particles. (b) Small-particle contribution. (c) Large-particle contribution.

To further illustrate this idea, we constructed contour plots of the average distribution of particle types (small or large) in our dynamic simulations at high concentration (figure 11). The cavity-induced layering is clear, as is the impact of polydispersity and volume composition: together they lead to distinct radial domains that vary with the degree of polydispersity. Small and large particles are effectively segregated from one another; as a result, a particle moving transversely through the cavity at fixed radius will encounter mostly other particles of its own size and thus experience much weaker fluctuations than one moving radially across the layered microstructure. The particle ordering shown in figure 10 explains why, in contrast to the angular MSD, the radial MSD of small and large particles in bidisperse suspensions is noticeably distinct from their respective monodisperse radial MSD, i.e. the volume-composition-dependent de-mixing explains the non-monotonicity in figure 9. Suspensions with a high volume composition of large particles have wider radial domains, which correlate with stronger caging; as large particles are replaced with small ones, the radial domains shrink, producing weaker caging. While this in turn leads to radial long-time MSD that changes qualitatively with volume composition, the caging effect leaves the angular dynamics insensitive to polydispersity.

Figure 11.

Confinement-induced layering convolves with polydispersity to induce de-mixing into size-segregated radial layers that change with volume composition. Contour plots of the average distribution of particle types, calculated by monitoring the occupants in each of the radial bins as particles diffuse over a time $1200 \ a^{2}_1/D_{0,1}$, the average time required for particles to explore the length of the cavity. Occupants were identified only based on size, with a value 1 assigned to small particles and a value 2 assigned to large particles. Seven plots are shown, corresponding to different volume compositions. Colour scheme shown in legend.

Overall, in the concentrated regime, many-body interactions produce a new effect: polydispersity and confinement combine to induce radial de-mixing into size-segregated populations. The cavity becomes the most influential ‘nearest neighbour’, setting the length scale of and dynamics within intermediate-length-scale radial domains or cages.