Short-range Elasticity Research Articles

Force Correlations in Disordered Magnets.

We present a proof of principle for the validity of the functional renormalization group, by measuring the force correlations in Barkhausen-noise experiments. Our samples are soft ferromagnets in two distinct universality classes, differing in the range of spin interactions, and the effects of eddy currents. We show that the force correlations have a universal form predicted by the functional renormalization group, distinct for short-range and long-range elasticity, and mostly independent of eddy currents. In all cases correlations grow linearly at small distances, as in mean-field models, but in contrast to the latter are bounded at large distances. As a consequence, avalanches are anti-correlated. We derive bounds for these anticorrelations, which are saturated in the experiments, showing that the multiple domain walls in our samples effectively behave as a single wall.

Correlations between avalanches in the depinning dynamics of elastic interfaces.

We study the correlations between avalanches in the depinning dynamics of elastic interfaces driven on a random substrate. In the mean-field theory (the Brownian force model), it is known that the avalanches are uncorrelated. Here we obtain a simple field theory which describes the first deviations from this uncorrelated behavior in a ε=d_{c}-d expansion below the upper critical dimension d_{c} of the model. We apply it to calculate the correlations between (i) avalanche sizes (ii) avalanche dynamics in two successive avalanches, or more generally, in two avalanches separated by a uniform displacement W of the interface. For (i) we obtain the correlations of the total sizes, of the local sizes, and of the total sizes with given seeds (starting points). For (ii) we obtain the correlations of the velocities, of the durations, and of the avalanche shapes. In general we find that the avalanches are anticorrelated, the occurrence of a larger avalanche making more likely the occurrence of a smaller one, and vice versa. Examining the universality of our results leads us to conjecture several exact scaling relations for the critical exponents that characterize the different distributions of correlations. The avalanche size predictions are confronted to numerical simulations for a d=1 interface with short range elasticity. They are also compared to our recent related work on static avalanches (shocks). Finally we show that the naive extrapolation of our result into the thermally activated creep regime at finite temperature predicts strong positive correlations between the forward motion events, as recently observed in numerical simulations.

Distribution of velocities in an avalanche, and related quantities: Theory and numerical verification

We study several probability distributions relevant to the avalanche dynamics of elastic interfaces driven on a random substrate: The distribution of size, duration, lateral extension or area, as well as velocities. Results from the functional renormalization group and scaling relations involving two independent exponents, roughness ζ, and dynamics z, are confronted to high-precision numerical simulations of an elastic line with short-range elasticity, i.e., of internal dimension d = 1. The latter are based on a novel stochastic algorithm which generates its disorder on the fly. Its precision grows linearly in the time discretization step, and it is parallelizable. Our results show good agreement between theory and numerics, both for the critical exponents as for the scaling functions. In particular, the prediction for the velocity exponent is confirmed with good accuracy.

Quantitative Scaling of Magnetic Avalanches.

We provide the first quantitative comparison between Barkhausen noise experiments and recent predictions from the theory of avalanches for pinned interfaces, both in and beyond mean field. We study different classes of soft magnetic materials, including polycrystals and amorphous samples-which are characterized by long-range and short-range elasticity, respectively-both for thick and thin samples, i.e., with and without eddy currents. The temporal avalanche shape at fixed size as well as observables related to the joint distribution of sizes and durations are analyzed in detail. Both long-range and short-range samples with no eddy currents are fitted extremely well by the theoretical predictions. In particular, the short-range samples provide the first reliable test of the theory beyond mean field. The thick samples show systematic deviations from the scaling theory, providing unambiguous signatures for the presence of eddy currents.

Domain walls in helical magnets: Elasticity and pinning

Recently completely new types of domain walls (DWs) have been discovered in helical magnets, consisting generically of a regular array of pairs of magnetic vortex lines (Li F. et al., Phys. Rev. Lett., 108 (2012) 107203). Only for special orientations DWs are free of vortices. In this paper we calculate their elastic and pinning properties, using the pitch angle θ as a small parameter. In particular we show that vortex-free DWs exhibit long-range elasticity which makes them very stiff and suppresses their pinning by impurities. Their roughening transition temperature is of the order of the Néel temperature. DWs including vortices (either by orientation or due to step formation above their roughening transition) show short-range elasticity and strong pinning by impurities. These results apply both to centrosymmetric as well as to non-centrosymmetric systems. The application to chiral liquid crystals is briefly discussed.

Static fluctuations of a thick one-dimensional interface in the 1+1 directed polymer formulation: Numerical study

We study numerically the geometrical and free-energy fluctuations of a static one-dimensional (1D) interface with a short-range elasticity, submitted to a quenched random-bond Gaussian disorder of finite correlation length ξ>0 and at finite temperature T. Using the exact mapping from the static 1D interface to the 1+1 directed polymer (DP) growing in a continuous space, we focus our analysis on the disorder free energy of the DP end point, a quantity which is strictly zero in the absence of disorder and whose sample-to-sample fluctuations at a fixed growing time t inherit the statistical translation invariance of the microscopic disorder explored by the DP. Constructing a new numerical scheme for the integration of the Kardar-Parisi-Zhang evolution equation obeyed by the free energy, we address numerically the time and temperature dependence of the disorder free-energy fluctuations at fixed finite ξ. We examine, on one hand, the amplitude D[over ̃](t) and effective correlation length ξ[over ̃](t) of the free-energy fluctuations and, on the other hand, the imprint of the specific microscopic disorder correlator on the large-time shape of the free-energy two-point correlator. We observe numerically the crossover to a low-temperature regime below a finite characteristic temperature T(c)(ξ), as previously predicted by Gaussian variational method computations and scaling arguments and extensively investigated analytically in [Phys. Rev. E 87, 042406 (2013)]. Finally, we address numerically the time and temperature dependence of the roughness B(t), which quantifies the DP end point transverse fluctuations, and we show how the amplitude D[over ̃](∞)(T,ξ) controls the different regimes experienced by B(t)-in agreement with the analytical predictions of a DP toy model approach.

Static fluctuations of a thick one-dimensional interface in the 1+1 directed polymer formulation

Experimental realizations of a one-dimensional (1D) interface always exhibit a finite microscopic width ξ>0; its influence is erased by thermal fluctuations at sufficiently high temperatures, but turns out to be a crucial ingredient for the description of the interface fluctuations below a characteristic temperature T(c)(ξ). Exploiting the exact mapping between the static 1D interface and a 1+1 directed polymer (DP) growing in a continuous space, we study analytically both the free-energy and geometrical fluctuations of a DP, at finite temperature T, with a short-range elasticity and submitted to a quenched random-bond Gaussian disorder of finite correlation length ξ. We derive the exact time-evolution equations of the disorder free energy F[over ¯](t,y), which encodes the microscopic disorder integrated by the DP up to a growing time t and an endpoint position y, its derivative η(t,y), and their respective two-point correlators C[over ¯](t,y) and R[over ¯](t,y). We compute the exact solution of its linearized evolution R[over ¯](lin)(t,y) and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder (ξ=0) to justify the construction of a "toy model" leading to a simple description of the DP properties. This model is characterized by Gaussian Brownian-type free-energy fluctuations, correlated at small |y|</~ξ, and of amplitude D̃(∞)(T,ξ). We present an extended scaling analysis of the roughness, supported by saddle-point arguments on its path-integral representation, which predicts D̃(∞)~1/T at high temperatures and D̃(∞)~1/T(c)(ξ) at low temperatures. We identify the connection between the temperature-induced crossover of D̃(∞)(T,ξ) and the full replica symmetry breaking in previous Gaussian variational method (GVM) computations. In order to refine our toy model with respect to finite-time geometrical fluctuations, we propose an effective time-dependent amplitude D̃(t). Finally, we discuss the consequences of the low-temperature regime for two experimental realizations of Kardar-Parisi-Zhang interfaces, namely, the static and quasistatic behavior of magnetic domain walls and the high-velocity steady-state dynamics of interfaces in liquid crystals.

Temperature-induced crossovers in the static roughness of a one-dimensional interface

At finite temperature and in presence of disorder, a one-dimensional elastic interface displays different scaling regimes at small and large lengthscales. Using a replica approach and a Gaussian variational method (GVM), we explore the consequences of a finite interface width $\ensuremath{\xi}$ on the small-lengthscale fluctuations. We compute analytically the static roughness $B(r)$ of the interface as a function of the distance $r$ between two points on the interface. We focus on the case of short-range elasticity and random-bond disorder. We show that for a finite width $\ensuremath{\xi}$ two temperature regimes exist. At low temperature, the expected thermal and random-manifold regimes, respectively, for small and large scales, connect via an intermediate ``modified'' Larkin regime, that we determine. This regime ends at a temperature-independent characteristic ``Larkin'' length. Above a certain characteristic temperature that we identify, this intermediate regime disappears. The thermal and random-manifold regimes connect at a single crossover lengthscale, that we compute. This is also the expected behavior for zero width. Using a directed polymer description, we also study via a second GVM procedure and generic scaling arguments, a modified toy model that provides further insights on this crossover. We discuss the relevance of the two GVM procedures for the roughness at large lengthscale in those regimes. In particular, we analyze the scaling of the temperature-dependent prefactor in the roughness $B(r)\ensuremath{\sim}{T}^{2\th{}}{r}^{2\ensuremath{\zeta}}$ and its corresponding thorn exponent $\th{}$. We briefly discuss the consequences of those results for the quasistatic creep law of a driven interface, in connection with previous experimental and numerical studies.

Anchoring distortions coupled with plane Couette &amp; Poiseuille flows of nematic polymers in viscous solvents: Morphology in molecular orientation, stress &amp; flow

The aim of this work is to model and simulate processing-induced heterogeneity in rigid, rod-like nematic polymers in viscous solvents. We em- ploy a mesoscopic orientation tensor model due to Doi, Marrucci and Greco which extends the small molecule, liquid crystal theory of Leslie-Ericksen-Frank to nematic polymers. The morphology has various physical realizations, all coupled through the model equations: the orientational distribution of the ensemble of rods, anisotropic viscoelastic stresses, and ∞ow feedback. Pre- vious studies in plane Couette & Poiseuille ∞ow (with the exception of (7)) have focused on the coupling between hydrodynamics and the orientational distribution of rigid rod polymers with identical anchoring conditions at solid boundaries; without ∞ow, the equilibrium structure is homogeneous. Here we explore steady structures that emerge with mismatch anchoring conditions at the walls, which couple an equilibrium elastic distortion across the channel, short and long range elasticity potentials, and hydrodynamics. In plane Cou- ette (where moving plates drive the ∞ow) and Poiseuille ∞ow (where a pressure gradient drives the ∞ow), in the regime of weak ∞ow and strong distortional elasticity, asymptotic analysis yields closed-form steady solutions and scaling laws with identical wall conditions. We focus simulations in this regime to expose the efiects due to wall anchoring con∞icts, and illustrate the induced morphology of the orientational distribution, stored viscoelastic stresses, and non-Newtonian ∞ow. A remarkably simple diagnostic emerges in this physical parameter regime, in which salient morphology features are controlled by the amplitude and sign of the difierence in plate anchoring angles of the director fleld at the two plates.

Steady state and transient rheological behavior of mesophase pitch, Part II: Theory

This paper presents a comprehensive nonlinear numerical analysis of the flow modeling of mesophase pitches performed using a previously formulated mesoscopic viscoelastic rheological theory (Singh and Rey (2000)) that takes into account short-range order elasticity, long range elasticity, and flow-induced texture transformations. A complete extra stress tensor equation is developed from first principles for liquid crystal materials under nonhomogeneous arbitrary flow. This mesoscopic viscoelastic model has been adapted to describe the rheology of flow-aligning thermotropic discotic nematic liquid crystals as models of mesophase pitches. Predictions for simple shear flow (under nonhomogeneous conditions) for the shear viscosity, first normal stress differences, and transient shear stress are presented. The accuracy of the numerical results is established by a thorough validation procedure based on [Cato et al. (2004)], which is the companion paper, and permit to validate this mesoscopic viscoelastic theory as model of liquid crystalline mesophase pitch. Very good qualitatively agreement between experiments and simulations is found for all rheological characterizations.

Dynamic interaction between suspended particles and defects in a nematic liquid crystal.

Insertion of spherical particles into a uniform nematic liquid crystal gives rise to the formation of topological defects. In the present work, we investigate how a spherical particle accompanied by its topological defects interacts with neighboring disclination lines. We perform two- and three-dimensional dynamic simulations to analyze the effect of a particle on the annihilation process of two disclination lines. The dynamics of the liquid crystal is described by a time-dependent evolution equation on the symmetric traceless order parameter that includes some of the salient features of liquid crystalline materials: excluded volume effects, or equivalently, short-range order elasticity and long-range order elasticity. At the surface of the particle, the liquid crystal is assumed to exhibit strong homeotropic anchoring. The particle is located between two disclination lines of topological charges +1/2 and -1/2. Two-dimensional simulations indicate that the topological defects bound to the particle mediate an interaction between the two disclination lines which increases the attraction between them. This result is confirmed by three-dimensional simulations that provide a complete description of the director field and of the order parameter around the particle. These simulations indicate that a spherical particle between two disclination lines can be surrounded by a Saturn ring, and suggest that the dynamic behavior of disclination lines could be used to report the structure of a defect around the particle.

Theoretical and Computational Rheology for Discotic Nematic Liquid Crystals

This paper presents an analysis of the role of orientation on the rheology of discotic nematic liquid crystals. The shear rheological properties exhibited by flow-aligning discotic mesophases are calculated by using a complete generalized nonlinear second-order tensor Landau-de Gennes model that takes into account short-range order elasticity, long-range elasticity, and viscous effects. A unified expression for the extra stress tensor is given. The second-order tensor Landau-de Gennes model was reduced to the uniaxial Leslie-Ericksen to obtain limiting rheological material functions valid at low and high shear rates. Analytical results are able to predict the material functions computed by the full Landau-de Gennes model for nonhomogeneous flow-aligning discotic nematic liquid crystals. Experimentally reported changes in the sign of the first normal stress differences with shear rate are captured by the model. A new Carreau-Yasuda liquid crystal model has been used to characterize the shear rheology for characteristic boundary conditions, and a viscosity power law exponent of 0.5 and a normal stress coefficient power law exponent of 0.44 have been obtained.

Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential.

The Doi kinetic theory for flows of homogeneous, rodlike liquid crystalline polymers (LCPs) is extended to model flows of nonhomogeneous, rodlike LCPs through a nonlocal (long-range) intermolecular potential. The theory features (i) a nonlocal, anisotropic, effective intermolecular potential in an integral form that is consistent with the chemical potential, (ii) short-range elasticity as well as long-range isotropic and anisotropic elasticity, (iii) a closed-form stress expression accounting for the nonlocal molecular interaction, and (iv) an extra elastic body force exclusively associated with the integral form of the intermolecular potential. With the effective intermolecular potential, the theory is proven to be well posed in that it warrants a positive entropy production and thereby the second law of thermodynamics. Approximate theories are obtained by gradient expansions of the number density function in the free energy density.

Defect Nucleation and Annihilation in Sheared Polymeric Liquid Crystals

ABSTRACTShear–induced nucleation and annihilation of topological defects due to hydrodynamic instability in nematic liquid crystals is a phenomenon of both scientific interest and practical importance. We use a complete generalized non-linear second order tensor Landau-de Gennes model that takes into account short range order elasticity, long-range elasticity and viscous effects, to simulate the nucleation and annihilation of twist inversion walls in flow-aligning nematic polymers subjected to shear flow. Shearing a homogeneous nematic sample perpendicular to the director results in an linear instability that maybe symmetric at low shear rates, and antisymmetric at higher shear rates. At even higher shear rates the onset of nonlinearities results in the nucleation of a parallel array of twist inversion walls, such that asymmetry prevails. By increasing the shear rate the following director symmetry transition cascade is observed: symmetric → antisymmetric → asymmetric → symmetric. The nucleation of the parallel array of twist inversion walls in the asymmetric mode is due to the degeneracy in reorientation towards the shear plane. The annihilation of twist walls is mediated by twist waves along the velocity gradient direction. Twist walls annihilate by three mechanisms: wall-wall annihilations, wall-wall coalescence, and wall-bounding surface coalescence. The annihilation rate increases with increasing shear rate and at sufficiently high rates the layered structure is replaced by a homogeneously aligned system. The role of short range and long range elasticity on defect nucleation and annihilation is characterized in terms of the Deborah and Ericksen numbers. Close form solutions to approximated equations are used to explain the numerical results of the full Landau-de Gennes equations of nematodynamics.

Effect of long range order on sheared liquid crystalline materials: flow regimes, transitions, and rheological phase diagrams

A generalized theory that includes short-range elasticity, long-range elasticity, and flow effects is used to simulate and characterize the shear flow of liquid crystalline materials as a function of the Deborah (De) and Ericksen (Er) numbers in the presence of fixed planar director boundary conditions; the results are also interpreted as a function of the ratio R between short-range and long-range elasticity. The results are effectively summarized into rheological phase diagrams spanned by De and Er, and also by R and Er, where the stability region of four distinct flow regimes are indicated. The four regimes for planar (two-dimensional orientation) shear flow are (1) the elastic-driven steady state, (2) the composite tumbling-wagging periodic state, (3) the wagging periodic state, and (4) the viscous-driven steady state. The coexistence of the four regimes at a quacritical point is shown to be due to the emergence of a defect structure. The origin, the significant steady and dynamical features, and the transitions between these regimes are thoroughly characterized and analyzed. Quantitative and qualitative comparisons between the present complete model predictions and those obtained from the classical theories of nematodynamics (Leslie-Ericksen and Doi theories) are presented and the main physical mechanisms that drive the observed deviations between the predictions of these models are identified. The presented results fill the previously existing gap between the classical Leslie-Ericksen theory and the Doi theory, and present a unified description of nematodynamics.

Effect of long-range elasticity and boundary conditions on microstructural response of sheared discotic mesophases

A comprehensive analysis of shear flow-induced microstructure phenomena exhibited by discotic mesophases is presented using a complete generalized non-linear theory that takes into account short-range order elasticity, long-range elasticity, and viscous flow effects. The following four distinct shear-induced stable planar non-homogeneous microstructure modes are found: (1) long-range elasticity-induced steady state, (2) bulk tumbling-boundary wagging state, (3) bulk wagging state, and (4) viscous flow induced steady state. The stability of the microstructure modes is presented in terms of a rheological phase plane spanned by the Ericksen number Er (ratio of viscous flow to long-range elasticity), and the ratio of short-range to long-range elasticity ( R). The steady and dynamical features of the various microstructure regimes are thoroughly characterized and analyzed. Two strong surface anchoring conditions, along the velocity gradient direction, and along the flow direction, are employed to investigate their effect on the stability and range of various microstructure regimes on the Er– R phase plane. The average bulk orientation for all the modes is found to be close to the velocity gradient direction. The fixed anchoring along the velocity gradient direction transmits the anchoring conditions into the bulk more strongly than that by the fixed anchoring along the flow direction. The effects of long-range elasticity on the flow-induced microstructure features are characterized. These simulations provide useful information to process carbonaceous mesophases by identifying the principles that govern shear flow-induced orientation in discotic mesophases.

Effect of long range order on sheared liquid crystalline materials Part 1: compatibility between tumbling behavior and fixed anchoring

A viscoelastic constitutive equation for liquid crystalline flows is presented in terms of tensor order parameter dynamics. The presented equation takes into account short range order, long range order (Frank elasticity), and flow effects, and captures the spatio-temporal evolution of the tensor order parameter. The dimensionless parameters arising in the theory are the Deborah (De) number and Ericksen (Er) number, and their ratio is a new dimensionless number R which represent the ratio of short range elasticity to long range elasticity. At Er → ∞ the model converges to the Doi equation, and for De → 0 to the Leslie-Ericksen director equation; the consistency with the Doi theory is shown by the classical shear flow solutions: tumbling, oscillating, and steady state. The model is used to analyze rectilinear simple shear flow subjected to strong anchoring conditions. For the range of governing parameters studied in this paper it is found that for small Er, a spatially nonhomogenous steady state of the tensor order parameters exists. For De corresponding to the tumbling solutions of the Doi equation, the present theory predicts the presence of a composite three layer structure: (a) a center bulk region characterized by director tumbling; and (b) two director oscillating regions next to the bounding surfaces. The amplitude of the director oscillations decrease from a maximum at the boundaries between tumbling-oscillating modes to zero at the bounding surfaces. The boundary between the tumbling-oscillating mode is characterized by the time periodic emergence of an abnormal nematic state, which is characterized by a tensor order parameter with two equal eigenvalues on the shear plane. The emergence of the abnormal nematic state provides for a compatibility mechanism between oscillations and rotations. Thus, we have found through the use of computational modeling a new mechanism that explains the compatibility of Doi's tumbling solutions and fixed surface anchoring conditions. The new flow mechanism is independent of the type of strong anchoring conditions, and arises with planar, homeotropic, and any arbitrary director anchoring.

Absence of mechanical evidence for attached weakly binding cross‐bridges in frog relaxed muscle fibres.

1. Passive force responses to ramp stretches at various velocities were measured in intact and skinned single muscle fibres isolated from the lumbricalis muscle of the frog. Force was measured using a fast capacitance transducer and sarcomere length was measured using a laser light diffraction technique at a point very close to the fixed end so as to avoid effects of fibre inertia. Experiments were performed at 15 degrees C with sarcomere length between 2.13 and 3.27 microns under high (170 mM) and low (20 mM) ionic strength. 2. The analysis shows that the force response is the sum of at least three components: (i) elastic (force proportional to the amount of stretch), (ii) viscous (force proportional to rate of stretch), and (iii) viscoelastic (resembling the response of a pure viscous element in series with an elastic element). 3. The amplitude of all these components increased progressively with sarcomere length in the whole range measured. 4. A further component, attributable to the short-range elasticity (SREC), was present in the force response of the intact fibres. 5. The amplitude of the force response decreased substantially upon skinning at high ionic strength but increased again at low ionic strength. The SREC was completely abolished by skinning. 6. None of the components of the force response was found to have the properties expected from the previously postulated 'weakly binding bridges'.

Elastic and inelastic behaviour of resting frog muscle fibres

Dynamic and static elasticity of the resting skeletal muscle of the frog have been studied as a function of the sarcomere length. Isolated intact fibres as well as glycerol extracted fibres show a resting tension starting at 2.05 microns sarcomere length and increasing approximately exponentially to 2 . 10(4) N/m2 at 3.0 microns sarcomere length. Differences between the two types of preparation were seen in the dynamic experiments. The dynamic Young's modulus of intact fibres (recorded at 1 Hz and small amplitudes) increased from 2 . 10(5) N/m2 at 2.1 microns sarcomere length to 2.5 . 10(6) N/m2 at 2.9 microns while the static modulus varied from 5 . 10(3) N/M2 to 3 . 10(5) N/m2, the dynamic modulus at small amplitudes was equivalent to the modulus of the short range elasticity (SRE). The range of the SRE did not depend on the sarcomere length and amounted to about 5 nm per sarcomere. The dynamic modulus strongly depended on the amplitude: at large amplitudes the muscle became less stiff by a factor of 10 to 20. This tendency levelled off at about 10 Hz by a strain-induced relaxation process. The dynamic modulus of the glycerol extracted fibres were nearly of the same magnitude as the static modulus, there was neither evidence for an SRE nor for a significant amplitude-dependence of the dynamic modulus. For interpreting the results we propose to further develop the meander model of muscle (Pechhold et al. 1977 b).

THE RELATION BETWEEN Z‐DISK LATTICE SPACING AND SARCOMERE LENGTH IN SARTORIUS MUSCLE FIBRES FROM HYLA CERULEA

Sartorius muscles from the green tree frog Hyla cerulea were set at variety of muscle lengths and fixed for electron microscopy using acrolein followed by osmium tetroxide. The sarcomere length, s, was determined in thick sections using laser-diffraction. The Z-disk lattice spacing, z, was measured in electron micrographs of thin sections from the same muscles. The Z-disk lattice was found to expand as sarcomere length decreased such that the quantity sz2 was constant at 1-05 X 10(6) nm3 for all sarcomere lengths in the range 1-9-2-9 mum. Thus, the sarcomere length dependency of the Z-disk lattice is similar to that of the myosin filament lattice. The density of thin filaments per unit cross section of fibril leaving the Z-disk is less than their density in the A band. Thus, fibrils have a smaller cross section in the I band, leaving more inter-fibrillar space there. This may explain why more mitochondria and lipid droplets are located in the I bands than in the A bands. It is suggested that the Z-disk may contributed to the short range elasticity of muscle fibres.