Hierarchical Self-Assembly of Cellulose Nanocrystals in a Confined Geometry

Complex hierarchical architectures are ubiquitous in nature. By designing and controlling the interaction between elementary building blocks, nature is able to optimize a large variety of materials with multiple functionalities. Such control is, however, extremely challenging in man-made materials, due to the difficulties in controlling their interaction at different length scales simultaneously. Here, hierarchical cholesteric architectures are obtained by the self-assembly of cellulose nanocrystals within shrinking, micron-sized aqueous droplets. This confined, spherical geometry drastically affects the colloidal self-assembly process, resulting in concentric ordering within the droplet, as confirmed by simulation. This provides a quantitative tool to study the interactions of cellulose nanocrystals beyond what has been achieved in a planar geometry. Our developed methodology allows us to fabricate truly hierarchical solid-state architectures from the nanometer to the macroscopic scale using a renewable and sustainable biopolymer.


Figure S9
Graph demonstrating that the diameter of the droplet does not affect the trend in measured cholesteric pitch, as exemplified by the evaporation of water from droplets with an initial diameter of 140 µm (blue circles) and 50 µm (orange diamonds). The data point at [CNC] = 100% corresponds to the pitch measured by SEM, as illustrated in Fig. 4 in the manuscript. Figure S10. Graph comparing microdroplets with a higher initial concentration (Ø ~ 140 µm, [CNC] = 10.9 wt%) (green diamonds) against a macroscale capillary measured by both laser diffraction (red circles) and optical microscopy (red triangles). Figure S11. Example SEM of the cross-section of (a) a cast film and (b) dry microparticles of the same suspension of CNC, as used for pitch measurement in Fig. 3 in the manuscript. Figure S12. Phase diagram reporting the ratio of the volume of anisotropic phase to the total sample volume, as observed in capillaries after 4 days (blue circles) and 95 days (red squares). Slight evaporation of the sample was observed after 95 days, as compared to the initial filling levels, which was taken into account when reporting the concentrations in abscissae.

PHASE DIAGRAM IN THE CAPILLARIES
The phase diagram (Figure 1b in the manuscript) reports the proportion of anisotropic phase as a function of the total concentration of CNC in the suspension, calculated as the ratio of the volume of anisotropic phase to the total sample volume, and accounting for the uneven bottom and upper meniscus. From this diagram one can observe that the suspension is fully isotropic below cI = 5.97 wt% and fully anisotropic above cA = 14.59 wt% (values obtained from linear extrapolation). The observation of the capillaries after 4 and 95 days did not show evolution of those critical concentrations ( Figure S12). As partial desulfation of CNC occurs at room temperature over such a timescale, we considered the measurement after 4 days as most relevant to the microdroplet studies.
As previously observed by Dong et al. 1 the coexistence regime follows a higher slope at lower concentrations, with a transition around 9.7 wt% in our case, and related to the change of pH and ionic strength as the sample concentration varies.

PITCH MEASUREMENTS IN THE CAPILLARIES
The cholesteric pitch in the glass capillaries was determined using both polarized optical microscopy and laser diffraction, as described below.
Optical microscopy. Long working distance objectives (TPlan Nikon, 20x (NA=0.30, WD=30mm) and 50x (NA=0.40, WD=22mm) objectives) were used in order to increase the depth of focus (~61µm (20x) and 19µm (50x) as estimated from Berek's formula), which in turn helps with blurring out the fingerprint pattern of misaligned cholesteric domains. As a result, only the domains whose helices lay parallel to the observation plan contribute to visible fingerprints, with a tolerance angle δθ below 10 degrees allowing little uncertainty on the determined pitch values (error [1-1/cos(δθ)] < 2%) for the considered pitch values). Figure S14 was used for the average optical index, ñ, calculated as the average of the effective optical indices given by Bruggeman modeling (described below). In this geometry, the diffracted peaks are observed in transmission, allowing higher diffraction orders to be observed for larger pitches (see Figure   S15) and improve the pitch measurement. The diffracted light is mainly linearly polarized along the helix axis of the diffracting domains. This contrasts with the case of smaller pitches comparable with the wavelength of visible light. In the latter case, the first order diffraction is observed in reflection. At the limit of normal incidence, the diffracted light is bound in wavelengths (photonic band-gap) and is fully circularly polarized, with no existence of higher order diffraction peaks as long as the cholesteric helix profile remains sinusoidal. 2

Pitch variation along the vertical dimension of a capillary
The strong agreement between the two techniques confirms the validity of the observation conditions, in contrast to other pitch measurements available in the literature. Moreover, it allows detecting slight change of the pitch value from the bottom to the top of the capillaries, as reported in Figure S16, where measurements were performed at regular intervals. To our knowledge, this is the first clear observation of vertical pitch gradients in the anisotropic phase. Local size fractionation, salting-out gradients, or building-up of hydrostatic pressure in the lower levels of the anisotropic phase could explain this variation. Given this variation, we attributed to each prepared sample the average pitch of each series, as reported in Figure 3 in the manuscript, and we displayed as error bars the minimum and maximum of the locally measured values.

The change of pitch with concentration in a capillary
The pitch values in the capillaries initially follow a power law of ∝ , as expected from

Comparison of microdroplets of different size with a low starting concentration
Microdroplets of different size (~140 and ~50 µm) both with a low starting concentration of 7.3 wt% were prepared and their change in pitch upon concentration compared in Figure S9.
In both cases the cholesteric domains form a spherical shell and consequentially the trend in the pitch measurements overlaps reasonably well, following a power law from ~ to ~ / . This indicates there is negligible dependence on the size of the droplets on the pitch measurement.

Comparison between microdroplets at a high starting concentration with the capillary.
Microdroplets of large size (~140 µm) were prepared from a higher starting concentration ([CNC] = 10.9 wt%) and the measured pitch is reported together with the capillary series in Figure S10. The concentration dependence of the pitch in both cases overlap, with a change of power law (matching previous reports), [5][6][7] initially closer to ~ and then to ~ / .
Importantly, both these droplets and the capillaries display a polydomain structure, which reduces topological constraint on the pitch relaxation (i.e. the local nematic director on each end of a large cholesteric domain has to rotate fast to accommodate for the creation of more cholesteric bands). This polydomain structure can explain the small discrepancy noted in Figure 3 of the manuscript, where geometrically-confined self-assembly into a large monodomain cholesteric shell (~140 µm, [CNC] = 7.3 wt%) leads to long-range constraints on pitch relaxation. It is interesting to note that the trend in pitch in the measured droplets switches to ~ / at a threshold concentration at a value similar to cg observed in Figure 3 of the manuscript.

THE RATE OF WATER LOSS FROM SHRINKING DROPLETS
The aqueous microdroplets are submerged beneath a thin layer of hexadecane oil, and consequentially water loss from the droplets to the air is dependent upon diffusion through this barrier. The rate of water loss was found to be most dependent on two parameters that

MODELING OF THE OPTICAL BEHAVIOR OF CHOLESTERIC DROPLETS
In order to confirm the formation of a uniform radial chiral nematic phase inside the microdroplets, the polarized microscopy images of our droplets were compared with those obtained from numerical simulations, following a method well established in the literature. [8][9][10] These were produced assuming a director field of spherical components , , = 0, cos + , sin + , 2 where p is the helix pitch, as described in the Frank-Pryce model. 11 The liquid crystal phase is described as an effective birefringent medium with the optical axis parallel to the local nematic axis. The effective medium refractive indices and local birefringence are obtained from those of cellulose nanocrystals (n∥ = 1.6180, n⊥ = 1.5436) 12 and water (nw = 1.33), 13 with the effect of CNC concentration accounted by Bruggeman's theory applied to a dense assembly of aligned and non-conductive, intrinsically birefringent rods (see next section and Figure S19). The resulting birefringence is typically low and, as The mass fraction [CNC] in wt% and its volume fraction ϕ are related using the formula: CNC % = + 1 − using ρ = 1,630 kg.m -3 for cellulose and ρ0 = 1,000 kg.m -3 for water.