Fig. 2: Shape relaxation of amyloid fibril liquid crystalline tactoids.

a–c Evaluation of \({{{{{\mathcal{R}}}}}}\) (defined as \(\tfrac{R\left(t\right)\,-\,{R}_{{{{{{\rm{equil}}}}}}.}}{{R}_{{{{{{\rm{init}}}}}}.}\,-\,{R}_{{{{{{\rm{equil}}}}}}.}}\) where R_equil., R_init., and R(t) are the half-length of the long axis of tactoid at equilibrium, at the initial time, and at a given time t, respectively) with respect to time for tactoids that relax to homogenous, R_equiv. (V^1/3, with V the volume) = 9.4 µm (a), bipolar, R_equiv. = 19.3 µm (b) and cholesteric, R_equiv. = 27.8 µm (c) configurations at equilibrium. Symbols and black lines denote the experimental and numerical simulation results, respectively; colored lines show the fitting (\({{{{{\mathcal{R}}}}}}={{{{{\rm{exp }}}}}}\left(-\frac{t}{{\tau }_{{{{{{\rm{s}}}}}}}}\right)\)) that is used to obtain the characteristic shape relaxation time, τ_s. d Evaluation of \({{{{{\mathcal{R}}}}}}\) with respect to scaled time t⁄τ_s resulting in a universal curve, \({{{{{\mathcal{R}}}}}}={{{{{\rm{exp }}}}}}\left(-\frac{t}{{\tau }_{{{{{{\rm{s}}}}}}}}\right)\), for shape relaxation of the different classes of tactoids with various volumes and initial elongation values. e Circle, triangle, and square symbols denote homogenous, bipolar, and cholesteric tactoids, respectively. The error bars represent standard deviation. The developed theory, solid line, predicts the τ_s for different classes of BLG and SCNC liquid crystalline tactoids, confirming the generality of our approach to predict the bio-colloidal liquid crystalline tactoids relaxation behavior. Here, ω is the anchoring strength, ck_B T is the thermal energy per unit volume of dispersion, K and K₂ are the Frank elastic constants, ξ is the coherence length, M_∅ is the mass mobility, M_Q is the rotational mobility, γ is the interfacial tension, βμ_I is the effective viscosity, and b is a single constant pre-factor.