Bulk Elasticity Research Articles

Curvatures of smectic liquid crystals and their applications

ABSTRACTCurvature is an important concept for understanding layering structures in soft matters, ranging from complex macromolecular self-assembled structures to simple lipid bilayers. Among the various kinds of soft matters, smectic liquid crystal (LC) phases have been widely studied because of their periodic featured curvatures under various external forces. Generally, their curvatures are on the micron-scale due to the bulk elasticity of smectic LC materials. In this review, a combination of sublimation and recondensation of smectic LC materials generates a variety of curvatures at the nanometer scale, which cannot be achieved by self-assembly under thermal equilibrium conditions. In particular, we have focused on the change of curvatures in focal conic domains under non-equilibrium conditions, in which negative and zero Gaussian curvatures in the micron-scale transform to positive Gaussian curvatures in the micron- and nanometer scales. Finally, the review closes with applications using such non-equilibrium self-assembly of smectics.

Electric-field-mediated instability modes and Fréedericksz transition of thin nematic films

Instabilities at the deformable free surface of a thin nematic liquid crystal film can develop interesting patterns when exposed to an external electrostatic field. A general linear stability analysis is performed involving the Ericksen–Leslie governing equations for the dynamics of the nematic film coupled with the anisotropic Maxwell stresses for the electric field to uncover the salient features of these instabilities. The study reveals the coexistence of twin instability modes: (i) long-wave interfacial mode – stimulated when the sole destabilizing influence of the electric field overcomes the Frank bulk elasticity and surface tension force, and (ii) finite-wavenumber mode – engendered by the combined destabilizing influence originating from the anisotropic electric field and Ericksen stress, for the films with positive dielectric anisotropy and weaker Frank bulk elasticity. The results reported here are in contrast with the same obtained from the more frequently employed long-wave approach. The air-to-liquid-crystal filling ratio between the electrodes as well as thermodynamic parameters such as the dielectric anisotropy, Frank elasticity, and director orientations across the film and boundaries are found to play crucial roles in the selection of modes, whereas kinetic parameters such as Leslie viscosity coefficients influence only the time scale of instability. Importantly, at higher field intensities a symmetry-breaking Fréedericksz-type transition of director orientations is found to happen, which also causes the transition of the dominant mode of instability from the long-wave to the finite-wavenumber mode for films with relatively lower values of Frank bulk elasticity and positive dielectric anisotropy.

Strain-Dependent Solid Surface Stress and the Stiffness of Soft Contacts

Surface stresses have recently emerged as a key player in the mechanics of highly compliant solids. The classic theories of contact mechanics describe adhesion with a compliant substrate as a competition between surface energies driving deformation to establish contact and bulk elasticity resisting this. However, it has recently been shown that surface stresses provide an additional restoring force that can compete with and even dominate over elasticity in highly compliant materials, especially when length scales are small compared to the ratio of the surface stress to the elastic modulus, $\Upsilon/E$. Here, we investigate experimentally the contribution of surface stresses to the force of adhesion. We find that the elastic and capillary contributions to the adhesive force are of similar magnitude, and that both are required to account for measured adhesive forces between rigid silica spheres and compliant, silicone gels. Notably, the strain-dependence of the solid surface stress contributes significantly to the stiffness of soft solid contacts.

Surface elastic properties in silicon nanoparticles

The elastic behavior of the external surface of a solid body plays a key role in nanomechanical phenomena. While bulk elasticity enjoys the benefits of a robust theoretical understanding, many surface elasticity features remain unexplored: some of them are here addressed by blending together continuum elasticity and atomistic simulations. A suitable readdressing of the surface elasticity theory allows to write the balance equations in arbitrary curvilinear coordinates and to investigate the dependence of the surface elastic parameters on the mean and Gaussian curvatures of the surface. In particular, we predict the radial strain induced by surface effects in spherical and cylindrical silicon nanoparticles and provide evidence that the surface parameters are nearly independent of curvatures and, therefore, of the surface conformation.

General Relations of Indentations on Solids With Surface Tension

The conventional contact mechanics does not account for surface tension; however, it is important for micro- or nanosized contacts. In the present paper, the influences of surface tension on the indentations of an elastic half-space by a rigid sphere, cone, and flat-ended cylinder are investigated, and the corresponding singular integral equations are formulated. Due to the complicated structure of the integral kernel, it is difficult to obtain their analytical solutions. By using the Gauss–Chebyshev quadrature formula, the integral equations are solved numerically first. Then, for each indenter, the analytical solutions of two limit cases considering only the bulk elasticity or surface tension are presented. It is interesting to find that, through a simple combination of the solutions of two limit cases and fitting the direct numerical results, the dependence of load on contact radius or indent depth for general case can be given explicitly. The results incorporate the contribution of surface tension in contact mechanics and are helpful to understand contact phenomena at micro- and nanoscale.

Defect driven shapes in nematic droplets: analogies with cell division.

Building on the striking similarity between the structure of the spindle during mitosis in living cells and nematic textures in confined liquid crystals, we use a continuum model of two-dimensional nematic liquid crystal droplets to examine the physical aspects of cell division. The model investigates the interplay between bulk elasticity of the microtubule assembly, described as a nematic liquid crystal, and surface elasticity of the cell cortex, modeled as a bounding flexible membrane, in controlling cell shape and division. The centrosomes at the spindle poles correspond to the cores of the topological defects required to accommodate nematic order in a closed geometry. We map out the progression of both healthy bipolar and faulty multi-polar division as a function of an effective parameter that incorporates active processes and controls centrosome separation. A robust prediction, independent of energetic considerations, is that the transition from a single cell to daughters cells occurs at critical value of this parameter. Our model additionally suggests that microtubule anchoring at the cell cortex may play an important role for successful bipolar division. This can be tested experimentally by regulating microtubule anchoring.

Cellulose Nanofiber as a Distinct Structure-Directing Agent for Xylem-like Microhoneycomb Monoliths by Unidirectional Freeze-Drying.

Honeycomb structures have been attracting attention from researchers mainly for their high strength-to-weight ratio. As one type of structure, honeycomb monoliths having microscopically dimensioned channels have recently gained many achievements since their emergence. Inspired by the microhoneycomb structure that occurs in natural tree xylems, we have been focusing on the assembly of such a structure by using the major component in tree xylem, cellulose, as the starting material. Through the path that finally led us to the successful reconstruction of tree xylems by the unidirectional freeze-drying (UDF) approach, we verified the function of cellulose nanofibers, toward forming xylem-like monoliths (XMs). The strong tendency of cellulose nanofibers to form XMs through the UDF approach was extensively confirmed with surface grafting or a combination of a variety of second components (or even a third component). The resulting composite XMs were thus imparted with extra properties, which extends the versatility of this kind of material. Particularly, we demonstrated in this paper that XMs containing reduced graphene oxide (denoted as XM/rGO) could be used as strain sensors, taking advantage of their penetrating microchannels and the bulk elasticity property. Our methodology is flexible in its processing and could be utilized to prepare various functional composite XMs.

Sensing surface mechanical deformation using active probes driven by motor proteins.

Studying mechanical deformation at the surface of soft materials has been challenging due to the difficulty in separating surface deformation from the bulk elasticity of the materials. Here, we introduce a new approach for studying the surface mechanical deformation of a soft material by utilizing a large number of self-propelled microprobes driven by motor proteins on the surface of the material. Information about the surface mechanical deformation of the soft material is obtained through changes in mobility of the microprobes wandering across the surface of the soft material. The active microprobes respond to mechanical deformation of the surface and readily change their velocity and direction depending on the extent and mode of surface deformation. This highly parallel and reliable method of sensing mechanical deformation at the surface of soft materials is expected to find applications that explore surface mechanics of soft materials and consequently would greatly benefit the surface science.

Quantifying the impact of molecular defects on polymer network elasticity.

Elasticity, one of the most important properties of a soft material, is difficult to quantify in polymer networks because of the presence of topological molecular defects in these materials. Furthermore, the impact of these defects on bulk elasticity is unknown. We used rheology, disassembly spectrometry, and simulations to measure the shear elastic modulus and count the numbers of topological "loop" defects of various order in a series of polymer hydrogels, and then used these data to evaluate the classical phantom and affine network theories of elasticity. The results led to a real elastic network theory (RENT) that describes how loop defects affect bulk elasticity. Given knowledge of the loop fractions, RENT provides predictions of the shear elastic modulus that are consistent with experimental observations.

Liquid drops attract or repel by the inverted Cheerios effect

Solid particles floating at a liquid interface exhibit a long-ranged attraction mediated by surface tension. In the absence of bulk elasticity, this is the dominant lateral interaction of mechanical origin. Here, we show that an analogous long-range interaction occurs between adjacent droplets on solid substrates, which crucially relies on a combination of capillarity and bulk elasticity. We experimentally observe the interaction between droplets on soft gels and provide a theoretical framework that quantitatively predicts the interaction force between the droplets. Remarkably, we find that, although on thick substrates the interaction is purely attractive and leads to drop-drop coalescence, for relatively thin substrates a short-range repulsion occurs, which prevents the two drops from coming into direct contact. This versatile interaction is the liquid-on-solid analog of the "Cheerios effect." The effect will strongly influence the condensation and coarsening of drops on soft polymer films, and has potential implications for colloidal assembly and mechanobiology.

The Boussinesq–Mindlin problem for a non-homogeneous elastic halfspace

Boussinesq’s problem for the indentation of an isotropic, homogeneous elastic halfspace by a rigid circular punch constitutes a seminal problem in the theory of contact mechanics as does Mindlin’s problem for the action of a concentrated force at the interior of an isotropic, homogeneous elastic halfspace. The combined action of the surface indentation in the presence of the interior loading is referred to as the Boussinesq–Mindlin problem, which has important applications in the area of geomechanics. The Boussinesq–Mindlin problem, which represents a self-stressing loading configuration, serves as a useful model for interpreting the mechanics of indentation of geologic media for purposes of estimating their bulk elasticity properties. In this paper, the analysis of the problem is extended to include an exponential variation in the linear elastic shear modulus of the halfspace region.

Estimates of thermoelastic characteristics of composites reinforced by short anisotropic fibers

We construct a mathematical model describing thermomechanical interaction between composite structure elements (isotropic particles of the matrix and anisotropic short fibers) and the macroscopically isotropic elastic medium with desired thermoelastic characteristics. At the first stage of this model, the self-consistency method is used to obtain relations determining the elasticity moduli of the composite, and at the second stage, the model permits determining its linear thermal expansion coefficient. The dual variational statement of the linear thermoelasticity problem in an inhomogeneous solid permits obtaining two-sided estimates for the bulk elasticity modulus, shear modulus, and linear thermal expansion coefficient of the composite under study. The calculated dependencies presented in the paper permit predicting the thermoelastic characteristics of a composite reinforced by anisotropic short fibers (including those in the form of nanostructure elements).

Interconnection of defect entropies and enthalpies in BaF2 revisited

Here, we investigate the following key prediction of a thermodynamical model that interrelates the defect parameters with the bulk elastic and expansivity data: for various defect processes in a given matrix material, a proportionality exists between defect entropies and enthalpies. The investigation is focused on BaF2 for which ab initio calculations within density functional theory and the generalized-gradient approximation have been recently made as far as the formation and migration of intrinsic defects are concerned, as well as for the elastic constants. Four defect processes have been studied in BaF2: anion Frenkel formation, fluorine vacancy migration, fluorine interstitial motion and electrical relaxation associated with a single tetravalent uranium. For these processes, the entropies and enthalpies vary by almost two orders of magnitude and reveal a proportionality between them. We find that this proportionality is solely governed by the bulk elasticity and expansivity data, which conforms to the aforementioned thermodynamical model.

Wetting and phase separation in soft adhesion

In the classic theory of solid adhesion, surface energy drives deformation to increase contact area whereas bulk elasticity opposes it. Recently, solid surface stress has been shown also to play an important role in opposing deformation of soft materials. This suggests that the contact line in soft adhesion should mimic that of a liquid droplet, with a contact angle determined by surface tensions. Consistent with this hypothesis, we observe a contact angle of a soft silicone substrate on rigid silica spheres that depends on the surface functionalization but not the sphere size. However, to satisfy this wetting condition without a divergent elastic stress, the gel phase separates from its solvent near the contact line. This creates a four-phase contact zone with two additional contact lines hidden below the surface of the substrate. Whereas the geometries of these contact lines are independent of the size of the sphere, the volume of the phase-separated region is not, but rather depends on the indentation volume. These results indicate that theories of adhesion of soft gels need to account for both the compressibility of the gel network and a nonzero surface stress between the gel and its solvent.

Chemomechanical evolution of pore space in carbonate microstructures upon dissolution: Linking pore geometry to bulk elasticity

AbstractOne of the challenges faced today in a variety of geophysical applications is the need to understand the changes of elastic properties due to time‐variant chemomechanical processes. The objective of this work is to model carbonate rock elastic properties as functions of pore geometry changes that occur when the solid matrix is dissolved by carbon dioxide. We compared two carbonate microstructures: porous micrite (“mudstone”) and grain‐supported carbonate (“packstone”). We formulated a mathematical model that distinguishes the effects of microporosity and macroporosity on stiffness changes. We used measures of mechanical and chemical porosity changes recorded during injection tests to compute elastic moduli and compare them to moduli obtained from wave velocity measurements. In mudstones, both experimental and numerical results indicate that bulk moduli change by less than 5%. The evolution of elastic moduli is controlled by macropore enlargement. In packstones, model predictions underestimate changes of elastic moduli with total porosity by 10% to 80%. The total porosity variation is 60% to 75% smaller than the chemical porosity variation, which indicates that pore expansion due to dissolution is counterbalanced by pore shrinkage due to compaction. Packstone elastic properties are controlled by grain sliding. The methodology presented in this paper can be generalized to other chemomechanical processes studied in rocks, such as dislocations, glide, diffusive mass transfer, recrystallization, and precipitation.

Three-dimensional coating flow of nematic liquid crystal on an inclined substrate

We consider a coating flow of nematic liquid crystal (NLC) fluid film on an inclined substrate. Exploiting the small aspect ratio in the geometry of interest, a fourth-order nonlinear partial differential equation is used to model the free surface evolution. Particular attention is paid to the interplay between the bulk elasticity and the anchoring conditions at the substrate and free surface. Previous results have shown that there exist two-dimensional travelling wave solutions that translate down the substrate. In contrast to the analogous Newtonian flow, such solutions may be unstable to streamwise perturbations. Extending well-known results for Newtonian flow, we analyse the stability of the front with respect to transverse perturbations. Using full numerical simulations, we validate the linear stability theory and present examples of downslope flow of nematic liquid crystal in the presence of both transverse and streamwise instabilities.

Nanoparticle self-assembly at the interface of liquid crystal droplets

Nanoparticles adsorbed at the interface of nematic liquid crystals are known to form ordered structures whose morphology depends on the orientation of the underlying nematic field. The origin of such structures is believed to result from an interplay between the liquid crystal orientation at the particles' surface, the orientation at the liquid crystal's air interface, and the bulk elasticity of the underlying liquid crystal. In this work, we consider nanoparticle assembly at the interface of nematic droplets. We present a systematic study of the free energy of nanoparticle-laden droplets in terms of experiments and a Landau-de Gennes formalism. The results of that study indicate that, even for conditions under which particles interact only weakly at flat interfaces, particles aggregate at the poles of bipolar droplets and assemble into robust, quantized arrangements that can be mapped onto hexagonal lattices. The contributions of elasticity and interfacial energy corresponding to different arrangements are used to explain the resulting morphologies, and the predictions of the model are shown to be consistent with experimental observations. The findings presented here suggest that particle-laden liquid crystal droplets could provide a unique and versatile route toward building blocks for hierarchical materials assembly.

Elastocapillarity can control the formation and the morphology of beads-on-string structures in solid fibers.

Beads-on-string patterns have been experimentally observed in solid cylinders for a wide range of material properties and structural lengths, from millimetric soft gels to nanometric hard fibers. In this work, we combine theoretical analysis and numerical tools to investigate the formation and nonlinear dynamics of such beaded structures. We show that this morphological transition is driven by elastocapillarity, i.e., a complex interplay between the effects of surface tension and bulk elasticity. Unlike buckling or wrinkling, the presence of an axial elongation is found here to favor the onset of fiber beading, in agreement with existing experimental results on electrospun fibers, hydrogels, and nerves. Our results also prove that the applied stretch can be used in fabrication techniques to control the morphology of the emerging beads-on-string patterns. Such quantitative predictions open the way for several applications, from tissue engineering to the design of stretchable electronics and the microfabrication of functionalized surfaces.

Quantitative characterization of viscoelasticity of microbubbles in ultrasound contrast agent

Ultrasound contrast agent (UCA) microbubbles have been commonly used in clinic to enhance the acoustic backscattering signals in ultrasound imaging diagnosis. With increasing demand for the continuous improvement of imaging resolution and sensitivity, new type UCAs (e.g., targeted microbubbles and multifunctional microbubbles) have attracted growing interest in both medical and scientific communities. Many efforts have been made to modify microbubble shell properties, which can strongly affect microbubble dynamic behaviors, so as to enable to create some new functionalities of UCAs. However, accurate characterization of the shell mechanical properties of UCAs has been recognized to be rather challenging. In previous work, microbubble’s mechanical properties are normally estimated by fitting measured dynamic response signals with coated-microbubble models. Inevitable uncertainty will be introduced in fitting results because there are more than one unknown shell parameters are adopted in these dynamic models. In the present paper, a comprehensive approach is developed to quantitatively characterize the visco-elasticity of the encapsulated microbubbles. By combining the techniques of atomic force microscopy (AFM), single particle optical sensing (SPOS), acoustic attenuation measurement, and the coated-bubble dynamics simulation, the size distribution, shell thickness, shell elasticity and viscosity of UCA microbubbles are determined one by one in sequence. To examine the validity of this approach, a kind of albumin-shelled microbubbles with diameters ranging from 1 to 5 μm are fabricated in our lab. Based on AFM technology, the microbubble effective shell stiffness and bulk elasticity modulus are measured to be 0.149±0.012 N/m and 8.31±0.667 MPa, respectively. It is noteworthy that the shell elastic property is shown to be independent of the initial size of microbubbles. Furthermore, the size distribution and acoustic attenuation measurements are also performed of these bubbles. Then, combined with microbubble dynamic model simulations, the UCA shell viscosity is calculated to be 0.374±0.003 Pa·s. Compared with previous estimation method, the current technology can be used as an effective tool to assess UCA shell visco-elasticity with improved accuracy and certainty. It is also shown that the feasibility to optimize the design and fabrication of UCAs can satisfy different requirements in ultrasound diagnostic and therapeutic applications.

Surface tension and the mechanics of liquid inclusions in compliant solids.

Eshelby's theory of inclusions has wide-reaching implications across the mechanics of materials and structures including the theories of composites, fracture, and plasticity. However, it does not include the effects of surface stress, which has recently been shown to control many processes in soft materials such as gels, elastomers and biological tissue. To extend Eshelby's theory of inclusions to soft materials, we consider liquid inclusions within an isotropic, compressible, linear-elastic solid. We solve for the displacement and stress fields around individual stretched inclusions, accounting for the bulk elasticity of the solid and the surface tension (i.e. isotropic strain-independent surface stress) of the solid-liquid interface. Surface tension significantly alters the inclusion's shape and stiffness as well as its near- and far-field stress fields. These phenomena depend strongly on the ratio of the inclusion radius, R, to an elastocapillary length, L. Surface tension is significant whenever inclusions are smaller than 100L. While Eshelby theory predicts that liquid inclusions generically reduce the stiffness of an elastic solid, our results show that liquid inclusions can actually stiffen a solid when R<3L/2. Intriguingly, surface tension cloaks the far-field signature of liquid inclusions when R=3L/2. These results are have far-reaching applications from measuring local stresses in biological tissue, to determining the failure strength of soft composites.