Relaxation dynamics in bio-colloidal cholesteric liquid crystals confined to cylindrical geometry

Para-nematic phases, induced by unwinding chiral helices, spontaneously relax to a chiral ground state through phase ordering dynamics that are of great interest and crucial for applications such as stimuli-responsive and biomimetic engineering. In this work, we characterize the cholesteric phase relaxation behaviors of β-lactoglobulin amyloid fibrils and cellulose nanocrystals confined into cylindrical capillaries, uncovering two different equilibration pathways. The integration of experimental measurements and theoretical predictions reveals the starkly distinct underlying mechanism behind the relaxation dynamics of β-lactoglobulin amyloid fibrils, characterized by slow equilibration achieved through consecutive sigmoidal-like steps, and of cellulose nanocrystals, characterized by fast equilibration obtained through smooth relaxation dynamics. Particularly, the specific relaxation behaviors are shown to emerge from the order parameter of the unwound cholesteric medium, which depends on chirality and elasticity. The experimental findings are supported by direct numerical simulations, allowing to establish hard-to-measure viscoelastic properties without applying magnetic or electric fields.

p   and if the phase is para-nematic (e.g. in our simulation, the initial phase is para-nematic specified by Supplementary

Supplementary Note 5.
In order to reveal the mechanism of front propagation during Phase (II) for the BLG relaxation, we examined the formation of the cholesteric layers and found that the formation of each half-pitch, 2 p  , which is equivalent to one sigmoid-like step, obeys four stages  respectively. Supplementary Figure 9 shows a representative fibers orientation and their order parameter captured by simulation. Note that, throughout our work, the order parameter was obtained by direct numerical simulation. 27

Supplementary Note 7.
The order-disorder phase transformation is considered as a first-order transition because physical quantities undergo a sharp change through the interface 4 . In this study, we employed the LdG and FOM theories; for this approach, the order parameter has been analytically derived The unique role of the pitch on S was previously discussed 5 . The typical behavior of the order parameter thus looks like Supplementary Figure 10. If concentration does not reach the phase transition threshold, the phase is isotropic and S=0. Upon exceeding the concentration threshold, the liquid-crystalline phase emerges, and S achieves a finite value less than unity.
The order parameter at this jump is called the critical order parameter which has been substantially discussed [6][7][8][9][10] . The relatively low critical order parameter in our study is thus In this study, S<Sc takes place while the concentration remains at the cholesteric bulk (see section 'Direct numerical simulation').

Supplementary Note 8.
The homogeneous free energy is a polynomial of uniaxial, S, and biaxial, P, order parameters because of 6,11 ( ) In order to understand the behavior of the homogeneous free energy, h f , with respect to the order parameter, it is reasonable to use the widely-accepted assumption of neglecting the biaxial contribution because the biaxial order parameter is considerably smaller compared to uniaxial order parameter, 0 P  7 . Thus, the homogeneous free energy is expressed as 28 Supplementary Figure 11 shows that homogenous free energy decreases when the order parameter evolves from a low value to the equilibrium point. Hereafter, for the sake of simplicity, we use the order parameter rather than the uniaxial order parameter throughout the article.
Given the formulation of gradient elastic free energy along with the Q-tensor definition, one can see that there is a factor S in all terms. where ζ is the 3D unit dyadic. Reduction in the order parameter leads to decrease of the Q-tensor, Supplementary Equation (5), in turn, all penalty terms in gradient elasticity, Supplementary Equation (6), become smaller. For simplicity to show the impact of order parameter reduction on the gradient elastic free energy contribution, it is reasonable to assume that the order parameter is independent of space. In this case, the long-range elastic free energy is rewritten as More specifically, the elastic-free energy is weighted by the squared order parameter and the squared coherence length. Therefore, a reduction in the magnitude of the order parameter leads to a lower gradient elastic free energy.
Aside from the impact of order parameter and coherence length on the elastic free energy, the pitch length can also have an impact on elastic free energy, especially affecting the early relaxation. The excess elastic free energy is higher for smaller pitch lengths as the deviation from the ground state increases. Supplementary Table 1 summarizes how elastic free energy is affected by the order parameter, coherence length, and pitch length. It should, however, be mentioned that we did not apply the simplifying assumption of the constant order parameter in direct numeric simulations. 29

Supplementary Note 9.
In the given confinement, the relaxation dynamics depend on pitch length and coherence length. In this section, we focus on how a relaxation dynamic gradually switches from a slowfast relaxation dynamic to a smooth relaxation dynamic and vice versa.
The impact of pitch length, coherence length, and order parameter on the elastic free energy are summarized in Supplementary Table 1  y,t f K q y     −    (9) By use of Supplementary Equation (9), the Leslie-Ericksen model reduces to 2 2 2 K ty   =    (10) where  is rotational viscosity of a single helix. Given Supplementary Figure 13, the initial and boundary conditions are expressed by ( ) Therefore, the analytical solution of relaxation dynamics reads ( ) Note that the upper boundary condition comes from the steady solution, ( ) , q y q  = ). Therefore, the relaxation progress, R, is expressed as The Supplementary Figure 14 illustrates that relaxation generally obeys the first-order dynamic in an unconfined planar geometry. Thus, slow-fast relaxation does not emerge for the unconfined planar system.
In conclusion, relaxation dynamics also depend on the curvature. Characterizing the curvature impact on relaxation is beyond the scope of this work. 31

Supplementary Note 11.
As can be seen in Supplementary Figure 15, the best curves fitting the experiment and simulation result in estimation of rotational viscosity coefficient, η, and coherence length, ξ.
Then, L1 and rotational viscosity are estimated by use of Equation (2) and Equation (3), respectively.

Supplementary Note 12.
The capillaries were filled with the birefringent solution which is the bulk cholesteric.
According to the thermodynamics of phase equilibria, the bulk cholesteric is at a constant concentration equal to the upper binodal curve, see Supplementary Figure 16.

Supplementary Note 13.
The finite element (FE) technique with biquadratic basis functions was employed to carry out the simulations. In this regard, the governing equations, Equations (5-9), along with axillary conditions, Equations (10)(11), and parameters tabulated in Supplementary Table 2  32

Supplementary Note 14.
The dimensionless normalized relaxation progress curve, R, is computed in the lateral plane (xy-plane), see Figure 1(o). The ideal director field representing a monodomain along with xaxis reads   = sin( ) cos( ) 0 q z q z  − n (16) Moreover, we have 2 :: S   =   Q Q nn nn -this identity is proven in accordance with Q-tensor definition and tensorial operations, explained as follows.
The Q-tensor is defined as

33
Owing to fact that Supplementary Equation (21) is derived for the ideal director field represented in Supplementary Equation (16), and knowing that ( ) qt is computed by direct numerical simulation, there could be a slight difference in the prediction of q  at equilibrium.
In this regard, we re-define the chiral wavevector as ( , ) : : where t A stand for the total area of the lateral plane (xy-plane).
 plays a scaling role in order to reach q  .
 is quite close to the ideal value that is 0. Now that the normalized relaxation progress, Supplementary Equation (23) or Equation (12), is formulated and our experimental-theoretical approach reveals that the dark zone in the POM images is a para-nematic phase with the order parameter of nearly 0 to 0.4, the conceptual understanding of this quantity (R) deserves more discussion. As explained in the paper, when R is computed via the discretization of the time-series POM images, R is taken to be 0 and 1 in dark and fingerprint partitions, respectively. R=0 is equivalent to q=0 signifying that the phase is para-nematic and R=1 indicates q=q∞ representing that the phase is cholesteric. In the case of BLG relaxation (i.e. Figure 2), the R distribution theoretically becomes as what is shown in Supplementary Figure 18.
In the case of CNC, the para-nematic phase loses its order parameter to an extremely low value, S~10 -2 , see Supplementary Figure 19.
The actual fibers' orientation can be understood in light of two factors. First, the uniaxial director field, n, representing the average fibers' orientation. Second, the uniaxial order parameter, S, describing the strength of fibers alignment around n. Fibers lie perfectly parallel to n if S=1 and the resulting phase becomes more crystal-like. In the case of S≈Sc~0.7, fibers retain both fluidity and crystallinity (orientational order), corresponding to the liquidcrystalline phase. Finally, S<Sc indicates that the actual fibers' orientation can be less aligned around n; hence, the phase possesses more fluid-like characteristics rather than crystalline ones 5,8,10,[14][15][16] . As explained in the Supplementary Note 7, there is no unanimous agreement on the Sc value; however, Sc=0.25 suits for the theory used in our study. Accordingly, wherever S<Sc, the fibers' orientation is randomly visualized in order to emphasize the concept of critical order parameter and the fact that orientational ordering is weak, see Supplementary Figure 20.
Wherever S<Sc, the phase can also be called isotropic due to the fact that the correlation existing among fibers is insignificant. However, the distinguishing point that should be taken into account is that the concentration of the isotropic phase is still at the upper binodal curve, which is unequivocally greater than the critical order-disorder transition, see "Direct numerical simulation" section for discussion on the concentration field in the present study. Additionally, we know that the mechanical bulk total elastic stress tensor T is