Shape is a crucial geometric property of surfaces, interfaces, and membranes in biology, colloidal and interface science, and many areas of physics. This paper presents theory, simulation and scaling of local shape and curvedness changes in moving surfaces and interfaces, under uniform normal motion, as in phase ordering transitions in liquid crystals. Previously presented measures of shape and curvedness are introduced in quantities and equations used in colloidal science and interfacial transport phenomena to separate shape effects from those of curvedness. Considering in parallel the new shape formalism with the classical curvature formalism, this paper sheds new light on what effects originate only from shape. The new shape evolution equations are solved under uniform normal surface flow. It is found that the solutions obey the so-called "astigmatism equation" fixing the linear relation between the radii of curvature. Astigmatic trajectories in the shape-curvedness phase plane, can be clearly classified into two modes: (i) constant shape evolution, and (ii) variable shape-variable curvedness. Shapes between spheres and cylinders follow the former mode for large curvedness and transition at smaller curvedness into the latter. Shapes' transitions between cylinder and saddles only follow the second mode. Under geometry-driven stagnation (i.e. zero normal velocity) shapes can be frozen. Evolving spheres and cylinders freeze into the same original shape, but perturbed cylinders can freeze into a variety of shapes including saddles. The results provide a useful complementary view on how to describe and control shape evolution in surfaces and interfaces, of wide interest in soft matter materials.

The shear rate dependence of material functions such as shear viscosity (g) and the first normal stress difference (N 1 ) were given and interpreted earlier by Kiss and Porter. Their widely accepted work revealed the possibility of having a negative minimum of N 1 for polymeric liquid crystals. In this work, we disclose for the first time the evidence of two negative N 1 minima on a sheared cellulosic lyotropic system. The lower shear rate minimum is ascribed to the uncoiling of the cholesteric helix, as theoretically predicted earlier. Our findings contribute also to the understanding of the other minimum already reported in the literature and attributed to the nematic director tumbling mode. Moreover, the elastic change that the LC-HPC sample undergoes during the helix unwinding of the cholesteric structure is also by means of oscillatory measurements. This study is a contribution for the understanding of the structure-properties relationship linked with the complex rheological behavior of chiral nematic cellulose-based systems and may help to improve their further processing.This evidence served as motivation for theoretical studies where the dynamics of the cholesteric structure of PLC under shear flow conditions were studied using continuum mechanics approach, as in the work by Rey. 15,16 These Additional Supporting Information may be found in the online version of this article.

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Dilute isotropic collagen solutions are usually flow processed into monodomain chiral nematic thin films for obtaining highly ordered materials by a multistep process that starts with complex inhomogeneous flow kinematics. Here we present rigorous theory and simulation of the initial precursors during flow steps in cholesteric collagen film formation. We first extract the molecular shape parameter and rotational diffusivity from previously reported simple shear data of dilute collagen solutions, where the former leads the reactive parameter (tumbling function) which determines the net effect of vorticity and strain rate on the average orientation and where the latter establishes the intensity of strain required for flow-birefringence, both crucial quantities for controlled film formation flow. We find that the tumbling function is similar to those of rod-like lyotropic liquid crystalline polymers and hence it is predicted that they would tumble in the ordered high concentration state leading to flow-induced texturing. The previously reported experimental data is well fitted with rotational diffusivities whose order of magnitude is consistent to those of other biomacromolecules. We then investigate the response of the tensor order parameter to complex flow kinematics, ranging from pure vorticity, through simple shear, to extensional flow, as may arise in typical flow casting and film flows. The chosen control variable to produce precursor cholesteric films is the director or average orientation, since the nematic order is set close to typical values found in concentrated cholesteric type I collagen solutions. Using the efficient four-roll mill kinematics, we summarize the para-nematic structure-flow process diagram in terms of the director orientation and flow type. Using analysis and computation, we provide a parametric envelope that is necessary to eventually produce well-aligned cholesteric films. We conclude that extensional flow is an essential ingredient of well-ordered film precursors with required Deborah numbers on the order of unity.

In vitro non-equilibrium chiral phase ordering processes of biomacromolecular solutions offer a systematic and reproducible way of generating material architectures found in Nature, such as biological plywoods. Accelerated progress in biomimetic engineering of mesoscopic plywoods and other fibrous structures requires a fundamental understanding of processing and transport principles. In this work we focus on collagen I based materials and structures to find processing conditions that lead to defect-free collagen films displaying the helicoidal plywood architecture. Here we report experimentally-guided theory and simulations of the chiral phase ordering of collagen molecules through water solvent evaporation of pre-aligned dilute collagen solutions. We develop, implement and a posteriori validate an integrated liquid crystal chiral phase ordering-water transport model that captures the essential features of spatio-temporal chiral structure formation in shrinking film domains due to directed water loss. Three microstructural (texture) modes are identified depending on the particular value of the time-scale ratio defined by collagen rotational diffusion to water translational diffusion. The magnitude of the time scale ratio provides the conditions for the synchronization of the helical axis morphogenesis with the increase in the mesogen concentration due to water loss. Slower than critical water removal rates leads to internal multiaxial cellular patterns, reminiscent of the classical columnar-equiaxed metallurgical casting structures. Excessive water removal rates lead to destabilization of the chiral axis and multidomain defected films. The predictions of the integrated model are in qualitative agreement with experimental results and can potentially guide solution processing of other bio-related mesogenic solutions that seek to mimic the architecture of biological fibrous composites.