High-frequency viscoelasticity of crosslinked actin filament networks measured by diffusing wave spectroscopy

We study the short-time relaxation dynamics of crosslinked and uncrosslinked networks of semi-flexible polymers using diffusing wave spectroscopy (DWS). The networks consist of concentrated solutions of actin filaments, crosslinked with increasing amounts of α-actinin. Actin filaments (F-actin) are long semi-flexible polymers with a contour length 1–100μm and a persistence length of 5–15μm; α-actinin is a small 200kDa homodimer with two actin-binding sites. Using the large bandwidth of DWS, we measure the mean-square-displacement of 0.96μm diameter microspheres imbedded in the polymer network, from which we extract the frequency-dependent viscoelastic moduli via a generalized Langevin equation. DWS measurements yield, in a single measurement, viscoelastic moduli at frequencies up to 10⁵Hz, almost three decades higher in frequency than probed by conventional mechanical rheology. Our measurements show that the magnitude of the small-frequency plateau modulus of F-actin is greatly enhanced in the presence of α-actinin, and that the frequency dependence of the viscoelastic moduli is much stronger at intermediate frequencies. However, the frequency-dependence of loss and storage moduli become similar for both crosslinked and uncrosslinked networks at large frequencies, G′(ω)∝G′′(ω)∝ω^0.75±0.08. This high-frequency behavior is due to the small-amplitude, large-frequency lateral fluctuations of actin filaments between entanglements. Received: 20 January 1998 Accepted: 12 February 1998     

Rheo-dielectric behavior of low molecular weight liquid crystal 2. Behavior of 8CB in nematic and smectic states

Rheo-dielectric behavior was examined for 4−4^′−n-octyl-cyanobiphenyl (8CB) having large dipoles parallel to its principal axis (in the direction of the C≡N bond). In the quiescent state at all temperatures (T) examined, orientational fluctuation of the 8CB molecules was observed as dielectric dispersions at characteristic frequencies ω_c>10⁶ s⁻¹. In the isotropic state at high T, no detectable changes of the complex dielectric constant ɛ^*(ω) were found under slow flow at shear rates ˙γ≫ω_c. In the nematic state at intermediate T, the terminal relaxation intensity of ɛ^*(ω) was decreased under such slow flow. In the smectic state at lower T, the flow effect became much less significant. These results were related to the flow-induced changes of the liquid crystalline textures in the nematic and smectic states, and the differences of the rheo-dielectric behavior in these states are discussed in relation to a difference of the symmetry of molecular arrangements in the nematic and smectic textures. Received: 1 October 1998 Accepted: 13 January 1999     

Unsteady free convection on a vertical cylinder with variable heat and mass flux

The unsteady natural convection boundary layer flow over a semi-infinite vertical cylinder is considered with combined buoyancy force effects, for the situation in which the surface temperature T ^′ _w(x) and C ^′ _w(x) are subjected to the power-law surface heat and mass flux as K(T ^′/r) = −ax ⁿ and D(C ^′/r) = −bx ^m . The governing equations are solved by an implicit finite difference scheme of Crank-Nicolson method. Numerical results are obtained for different values of Prandtl number, Schmidt number ‘n’ and ‘m’. The velocity, temperature and concentration profiles, local and average skin-friction, Nusselt and Sherwood numbers are shown graphically. The local Nusselt and Sherwood number of the present study are compared with the available result and a good agreement is found to exist. Received on 7 July 1998     

New Explicit Expression of Barnett-Lothe Tensors for Anisotropic Linear Elastic Materials

The three Barnett-Lothe tensors H, L, S appear often in the Stroth formalism of two-dimensional deformations of anisotropic elastic materials [1–3]. They also appear in certain three-dimensional problems [4, 5]. The algebraic representation of H, L, S requires computation of the eigenvalues p^v(v=1,2,3) and the normalized eigenvectors (a, b). The integral representation of H, L, S circumvents the need for computing p ^v(v=1,2,3) and (a, b), but it is not simple to integrate the integrals except for special materials. Ting and Lee [6] have recently obtained an explicit expression of H for general anisotropic materials. We present here the remaining tensors L, S using the algebraic representation. They key to our success is the obtaining of the normalization factor for (a, b) in a simple form. The derivation of L and S then makes use of (a, b) but the final result does not require computation of (a, b), which makes the result attractive to numerical computation. Even though the tensor H given in [6] is in terms of the elastic stiffnesses C^μ v while the tensors L, S presented here are in terms of the reduced elastic compliances s′ _μv , the structure of L, S is similar to that of H. Following the derivation of H, we also present alternate expressions of L, S that remain valid for the degenerate cases p ₁ p ₂ and p₁=p₂ = p ₃. One may want to compute H, L, S using either C _μv or s′ _μv v, but not both. We show how an expression in C^μ v can be converted to an expression in s′ _μv v, and vice versa. As an application of the conversion, we present explicit expressions of the extic equation for p in C^μ v and s′ _μv v. This revised version was published online in August 2006 with corrections to the Cover Date.     

Relaxation patterns of long,linear, flexible,monodisperse polymers: BSW spectrum revisited

Theoretical predictions for the dynamic moduli of long, linear, flexible, monodisperse polymers are summarized and compared with experimental observations. Surprisingly, the predicted 1/2 power scaling of the long-time modes of the relaxation spectrum is not found in the experiments. Instead, scaling with a power of about 1/4 extends all the way up to the longest relaxation times near τ/τ _max = 1. This is expressed in the empirical relaxation time spectrum of Baumgaertel-Schausberger-Winter, denoted as “BSW spectrum,” and justifies a closer look at the properties of the BSW spectrum. Working with the BSW spectrum, however, is made difficult by the fact that hypergeometric functions occur naturally in BSW-based rheological material functions. BSW provides no explicit solutions for the dynamic moduli, G ^′(ω), G ^″(ω), or the relaxation modulus G(t). To overcome this problem, close approximations of simple analytical form are shown for these moduli. With these approximations, analysis of linear viscoelastic data allows the direct determination of BSW parameters.     

Uniaxial elongational behavior of poly(vinyl chloride) physical gel

A poly(vinyl chloride) (PVC,  M_w = 102×10³)(\mbox{PVC,}\;{\rm M}_{\rm w} =102\times 10^3) di-octyl phthalate (DOP) gel with PVC content of 20 wt.% was prepared by a solvent evaporation method. The dynamic viscoelsticity and elongational viscosity of the PVC/DOP gel were measured at various temperatures. The gel exhibited a typical sol–gel transition behavior with elevating temperature. The critical gel temperature (T_gel) characterized with a power–law relationship between the storage and loss moduli, G^′ and G^″, and frequency ω, G￠=G^￠￠/tan  ( np/2 ) μ wⁿ{G}^\prime={G}^{\prime\prime}{\rm /tan}\;\left( {{n}\pi {\rm /2}} \right)\propto \omega ^{n}, was observed to be 152°C. The elongational viscosity of the gel was measured below the T_gel. The gel exhibited strong strain hardening. Elongational viscosity against strain plot was independent of strain rate. This finding is different from the elongational viscosity behavior of linear polymer solutions and melts. The stress–strain relations were expressed by the neo-Hookean model at high temperature (135°C) near the T_gel. However, the stress–strain curves were deviated from the neo-Hookean model at smaller strain with decreasing temperature. These results indicated that this physical gel behaves as the neo-Hookean model at low cross-linking point, and is deviated from the neo-Hookean model with increasing of the PVC crystallites worked as the cross-linking junctions.     

Bifurcation From Stability to Instability for a Free Boundary Problem Arising in a Tumor Model

We consider a time-dependent free boundary problem with radially symmetric initial data: σ_t − Δσ + σ = 0 if and σ(r,0)=σ₀(r) in {r < R(0)} where R(0) is given. This is a model for tumor growth, with nutrient concentration (or tumor cells density) σ(r,t) and proliferation rate then there exists a unique stationary solution (σ_S(r), R_S), where R_S depends only on the number . We prove that there exists a number μ_*, such that if μ < μ_* . . . then the stationary solution is stable with respect to non-radially symmetric perturbations, whereas if μ > μ_* then the stationary solution is unstable.     

Effect of the Source Term on Steady Free Convection Boundary Layer Flows over an Vertical Plate in a Porous Medium. Part I

The Darcy free convection boundary layer flow over a vertical flat plate is considered in the presence of volumetric heat generation/absorption. In the present first part of the paper it is assumed that the heat generation/absorption takes place in a self-consistent way, the source term q ^′′′≡ S of the energy equation being an analytical function of the local temperature difference T − T _∞. In a forthcoming second part, the case of the externally controlled source terms S = S(x,y ) will be considered. It is shown that due to the presence of S, the physical equivalence of the up- and downflows gets in general broken, in the sense that the free convection flow over the upward projecting hot plate (“upflow”) and over its downward projecting cold counterpart (“downflow”) in general become physically distinct. The consequences of this circumstance are examined for different forms of S. Several analytical solutions are given. Some of them describe algebraically decaying boundary layers which can also be recovered as limiting cases of exponentially decayingones. This asymptotic phenomenon is discussed in some detail.     

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

Effects of corrugation angle on developing laminar forced convection and entropy generation in a wavy channel

This paper investigates the effects of corrugation angle (β) on the developing laminar forced convection and entropy generation in a wavy channel with numerical methods. The studied cases cover β = 10-, 15-, 20-, 25-, 30- and 35°, whilst Reynolds number (Re) is varied as 100, 200 and 400. The analyzed flow characteristics include recirculating flows, secondary vortices, temperature distributions, and friction factor as well as Nusselt number. In particular, the effects of corrugation angle on the distributions and magnitudes of local entropy generation resulted from frictional irreversibility (S _P ^′′′) and heat transfer irreversibility (S _T ^′′′) are separately discussed in detail in the present paper. Based on the minimal entropy generation principle, the optimal corrugation angle and favorable Re are reported.     

Rheology of SiO2/(acrylic polymer/epoxy) suspensions. I. Linear viscoelasticity

Linear viscoelastic properties of SiO₂/(AP/EP) suspension with various SiO₂ volume fractions (ϕ) in a blend of acrylic polymer (AP) and epoxy (EP) were investigated at various temperatures (T). The AP/EP contained 70 vol.% of EP. The SiO₂ particles were treated with epoxy silane coupling agent. The effects of the SiO₂ particles are more pronounced in the terminal zone: a transition from viscoelastic liquid (ϕ ≤ 30 vol.%) to viscoelastic solid (ϕ ≥ 40 vol.%) was observed which can be interpreted as a critical gelation occurring at a critical particle content and critical gel temperature. The SiO₂/(AP/EP) systems exhibited a critical gel behavior at ϕ ≅ 35 vol.% and T ≅ 100°C characterized with a power–law relationship between the storage and loss moduli (G ^′ and G ^″) and frequency (ω); G ^′ = G ^″/tan(nπ/2) ∝ ω ⁿ . The critical gel exponent (n) was estimated to be about 0.45. The gelation occurred with increasing T.     

Elasticity and dynamics of particle gels in non-Newtonian melts

We investigate the relation between the structure and the viscoelastic behavior of a model polymer nanocomposite system based on a mixture of titanium dioxide (TiO₂) nanoparticles and polypropylene. Above a critical volume fraction, Φ _c, the elasticity of the hybrids dramatically increases, and the frequency dependence of the elastic and viscous moduli reflects the superposition of the independent responses of the suspending polymer melt and of an elastic particle network. In addition, the elasticity of the hybrids shows critical behavior around Φ _c. We interpret these observations by hypothesizing the formation of a transient network, which forms due to crowding of particle clusters. Consistent with this interpretation, we find a long-time, Φ-dependent, structural relaxation, which emphasizes the transient character of the structure formed by the particle clusters. For times below this characteristic relaxation time, the elasticity of the network is Φ-independent and reminiscent of glassy behavior, with the elastic modulus, G^′, scaling with frequency, ω, as G^′∼ω ^0.3. We expect that our analysis will be useful for understanding the behavior of other complex fluids where the elasticity of the components could be superimposed.     

Analytical Solution for Radial Deformations of Functionally Graded Isotropic and Incompressible Second-Order Elastic Hollow Spheres

We analytically analyze radial expansion/contraction of a hollow sphere composed of a second-order elastic, isotropic, incompressible and inhomogeneous material to delineate differences and similarities between solutions of the first- and the second-order problems. The two elastic moduli are assumed to be either affine or power-law functions of the radial coordinate R in the undeformed reference configuration. For the affine variation of the shear modulus μ, the hoop stress for the linear elastic (or the first-order) problem at the point R=(R _ou R _in (R _ou +R _in )/2)^1/3 is independent of the slope of the μ vs. R line. Here R _in and R _ou equal, respectively, the inner and the outer radius of the sphere in the reference configuration. For μ(R)∝R ⁿ , for the linear problem, the hoop stress is constant in the sphere for n=1. However, no such results are found for the second-order (i.e., materially nonlinear) problem. Whereas for the first-order problem the shear modulus influences only the radial displacement and not the stresses, for the second-order problem the two elastic constants affect both the radial displacement and the stresses. In a very thick homogeneous hollow sphere subjected only to pressure on the outer surface, the hoop stress at a point on the inner surface depends upon values of the two elastic moduli. Thus conclusions drawn from the analysis of the first-order problem do not hold for the second-order problem. Closed form solutions for the displacement and stresses for the first-order and the second-order problems provided herein can be used to verify solutions of the problem obtained by using numerical methods.     

St. Venant torsion of composite cylinders

Using a method first applied to some fluid dynamical two-phase flow problems [1], the authors derive expressions for the warping and stress functions describing the St. Venant torsion of some compound cylinders. The construction of the cylinder must have appropriate symmetries, and composites containing both two and four different elastic phases are considered. Effective shear moduli and torsional rigidities are defined, and some tables of numerical values given for various configurations. An analogous solution for the antiplane strain of a two-phase composite is also given.     

Exact analytic solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium

 The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution T _w(x) = T _∞+A·x ^λ and lateral mass flux v _w(x) = a·x ^(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection parameter in detail. The results are compared with the numerical findings of previous authors. Received on 8 March 1999     

Rheological properties of poly(vinyl chloride)/plasticizer systems—relation between sol–gel transition and elongational viscosity

Poly(vinyl chloride) (PVC)/di-isononyl phthalate systems with PVC content of 45.5 (PVC8) and 70.4 wt% (PVC6) were prepared by a hot roller at 150 °C and press molded at 180 °C. The dynamic viscoelasticity and elongational viscosity of PVC8 and PVC6 were measured in the temperature range from 150 to 220 °C. We have found that the storage and loss shear moduli, G′ and G″, of PVC8 and PVC6 exhibited the power-law dependence on the angular frequency ω at 190 and 210 °C, respectively. Correspondingly, the tan δ values did not depend on ω. These temperatures indicate the critical gel temperature T _gel of each system. The critical relaxation exponent n obtained from these data was 0.75 irrespective of PVC content, which was in agreement with the n values reported previously for the low PVC concentration samples. These results suggest that the PVC gels of different plasticizer content have a similar fractal structure. Below T _gel, the gradual melting of the PVC crystallites takes place with elevating temperature, and above T _gel, a densely connected network throughout the whole system disappears. Correspondingly, the elongational viscosity behavior of PVC8 and PVC6 exhibited strong strain hardening below T _gel, although it did not show any strain hardening above T _gel. These changes in rheological behavior are attributed to the gradual melting of the PVC crystallites worked as the cross-linking domains in this physical gel, thereby inapplicability of the of time–temperature superposition for PVC/plasticizer systems.     

Erratum to: Floquet’s Theorem and Stability of Periodic Solitary Waves

This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in L²_per[0, p]{L^2_{{\rm per}}[0, \pi]}. We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic problems.     

Stability and boundedness results for certain nonlinear vector differential equations of the fourth order

We consider the equation X ⁽⁴⁾ + Φ(X″)X‴ + F(X,X′)X″ + G(X′) + H(X) = P(t,X,X′,X″,X‴) in two cases: P ≡ 0 and P ≠ 0. In the case P ≡ 0, the asymptotic stability of the zero solution X = 0 of the equation is investigated; in the case P ≠ 0, the boundedness of all solutions of the equation is proved. Published in Neliniini Kolyvannya, Vol. 9, No. 4, pp. 548–563, October–December, 2006.     

On the Influence of the Kinetic Energy¶on the Stability of Equilibria¶of Natural Lagrangian Systems

Natural Lagrangian systems (T,Π) on R ² described by the equation are considered, where is a positive definite quadratic form in and Π(q) has a critical point at 0. It is constructively proved that there exist a C ^∞ potential energy Π and two C ^∞ kinetic energies T and such that the equilibrium q(t)≡ 0 is stable for the system (T,Π) and unstable for the system . Equivalently, it is established that for C ^∞ natural systems the kinetic energy can influence the stability. In the analytic category this is not true. Accepted: October 20, 1999     

Further Study on Pure Shear

It is known that the Cauchy stress tensor T is a pure shear when trT = 0. An elementary derivation is given for a coordinate system such that, when referred to this coordinate system, the diagonal elements of T vanish while the off-diagonal elements τ ₁, τ ₂, τ ₃, are the pure shears. The structure of τ _i (i = 1, 2, 3) depends on one non-dimensional parameter q = 54(detT)² / [tr(T ²)]³, 0 ≤ q ≤ 1. When q = 0, one of the three τ _i vanishes. A coordinate system can be chosen such that the remaining two have the same magnitude or one of the remaining two also vanishes. When q = 1, all three τ _i have the same magnitude. However, there is a one-parameter family of coordinate systems that gives the same three τ _i . For q ≠ 0 or 1, none of the three τ _i vanishes and the three τ _i in general have different magnitudes. Nevertheless, a coordinate system can be chosen such that two of the three τ _i have the same magnitude. ^★Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.