Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers

We focus on the linear viscoelastic response of heterogeneous nematic polymers to small amplitude oscillatory shear, paying special attention to the macroscopic influence of strong plate anchoring conditions. The model consists of the Stokes hydrodynamic equations with viscous and nematic stresses, coupled to orientational dynamics and structure driven by the flow gradient, an excluded-volume potential, and a two-constant distortional elasticity potential. We show that the dynamical response simplifies when plate anchoring is either tangential or homeotropic, recovering explicitly solvable Leslie–Ericksen–Frank behavior together with weakly varying order parameters across the plate gap. With these plate conditions, we establish “model consistency” so that all experimental driving conditions (plate-controlled velocity [strain] or shear stress, imposed oscillatory pressure) yield identical dynamic moduli for the same material parameters and anchoring conditions, eliminating the culpability of device influence in scaling behavior. Two physical predictions emerge that imply significant macroscopic elastic and viscous effects controlled by plate anchoring relative to flow geometry: (1) The storage modulus is enhanced by two to three orders of magnitude for homeotropic relative to parallel anchoring, across all frequencies. (2) The loss modulus exhibits enhancement of a factor of two to three for homeotropic over tangential anchoring, restricted to low frequencies. We further deduce a scaling law for the dynamic moduli versus anisotropy of the distortional elasticity potential.

Mechanical and structural investigation of isotropic and anisotropic thermoplastic magnetorheological elastomer composites based on poly(styrene-b-ethylene-co-butylene-b-styrene) (SEBS)

Novel smart thermoplastic magnetorheological elastomer composites containing micron-sized magnetic carbonyl iron (CI) particles were prepared with a poly(styrene-ethylene-butylene-styrene) (SEBS) triblock copolymer utilized as the thermoplastic matrix rubber, and the structures and properties of the CI-SEBS composites were examined. The CI particles were uniformly dispersed in the composites prepared in the absence of the magnetic field at high temperatures T (>T $_{\rm g}^{\rm S})$ , and this isotropic composite exhibited a larger storage modulus G^?? compared to the SEBS matrix at room temperature (<?<T $_{\rm g}^{\rm S})$ where the EB phase therein was rubbery while the PS phase was in the glassy state. In contrast, the SEBS composite prepared under the magnetic field (with the intensity ???< 2.5?T) at high T (>T $_{\rm g}^{\rm S})$ contained a chain structure of CI particles. This chain structure became longer and better aligned on an increase of ?? up to a saturation of the particle magnetization and on an increase of the time interval of applying the field (that allowed the particles to move and equilibrate their aligned structure). The modulus G^?? of this ??pre-structured?? composite measured for both cases of ?? = 0 and ???> 0 in the direction perpendicular to the chain structure at room temperature was enhanced compared to G^?? of the isotropic composites. This difference of the filler effect (for ???=?0) and the magnetorheological effect (for ???> 0) between the pre-structured and isotropic composites was enhanced when the chain structure of the CI particles in the pre-structured composites became longer and better aligned. A mechanism(s) of this enhancement was discussed in relation to the morphologies (particle distribution) in the composites with the aid of a filler model and a molecular expression of the stress due to magnetically interacting particles.     

Extensional and shear rheometry of oriented triblock copolymers

 The effects of extensional flow orientation on the rheological properties of two poly(styrene)-poly(ethylene-co-butylene)-poly (styrene) (PS-PEB-PS) triblock copolymers containing either spherical or cylindrical PS microdomains were studied by oscillatory shear and oscillatory extensional experiments. Extensional measurements revealed that below the PS block glass transition temperature pre-oriented triblocks display highly anisotropic mechanical properties. For both polymers, the storage modulus E ′ is higher along the flow direction. Above the PS glass transition temperature the materials are no longer anisotropic and the same storage moduli are obtained along the flow direction and perpendicular to it. Above the PS glass transition temperature the rheological behaviour parallel and perpendicular to the flow direction was also probed in pre-oriented and non-oriented samples by oscillatory shear rheometry. At high frequencies, the mechanical response of the triblocks was found to be independent of the orientation for both copolymers while at low frequencies a strong effect of the flow orientation could be observed. For both polymers the value of the storage modulus was found to be lower along the flow direction that perpendicular to it. This was explained by the ability of PS blocks to relax more easily along the flow direction. Received: 10 September 1999/Accepted: 1 October 1999     

Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers II: oblique anchoring angles

We extend previous work on the linear viscoelastic moduli of heterogeneous nematic polymers in a small-amplitude oscillatory shear flow, focusing on the role of the orientational anchoring conditions at the plates. When tangential or normal anchoring conditions are applied, the Doi–Marrucci–Greco orientation tensor-flow model effectively reduces to the Leslie–Ericksen director-flow model, predicting that director distortions control the dynamic moduli with negligible contributions from tensor-order parameters. In this paper, we examine oblique anchoring angles. We use a combination of analysis and numerical simulation on the generalized tensor-flow system for arbitrary anchoring conditions to show that any oblique anchoring condition induces a nontrivial order parameter contribution to the dynamic moduli, which vanishes only in the limit of tangential or normal anchoring. Our approach reveals that the storage and loss moduli admit an approximate decomposition in terms of two reduced problems that are exactly solvable: the heterogeneous director–flow response plus the monodomain tensor response to an imposed shear. The importance of this result is that we gain scaling properties of the moduli with respect to material parameters and experimental conditions without having to compute and assimilate across the full parameter space. These results provide insight into the relative importance of the distortional vs bulk nematic elastic stress in determining the viscoelastic moduli, predicting that anchoring conditions tune the relative contributions.     

Rheology of carbon black suspensions. V. Heat-induced gelation and its reversibility

Linear viscoelastic properties of carbon black (CB) suspensions in a mixture of a rosin-modified phenol resin-type varnish (Varnish-1)/an alkyd resin-type varnish (Varnish-2), which exhibited a sol?Cgel transition on an increase in CB concentration, were investigated from 30°C to 80°C. The viscoelastic properties were reversible from 30°C to 60°C. In contrast, at temperatures above 60°C, the storage (G??) and loss (G??) moduli were irreversible and increased significantly with increasing temperature. This increase in the moduli is due to a change of the dispersion state to agglomerated state by heating. The agglomerated state was held, when the suspensions were lowered at 30°C. However, the G?? and G?? recovered to the original values upon shearing. This heat-induced gelation should be a universal feature for suspensions of weakly attractive particles. The temperature and shearing histories of the suspensions were discussed in relation to adsorption of polymeric component in the varnish on the CB particles.     

Conoscopic observations of shear-induced rotations in nematic liquid crystals

Orientational changes in monodomains of flow-aligning liquid crystals, 4-n-pentyl-4′-cyanobiphenyl and N-(4-methoxybenzylidene)-4-butylaniline, were studied during shear and recovery in a linear shearing device fitted to an optical microscope. Planar alignment (director in the shear plane) allows the study of twist effects and was generated by strong planar anchoring at the walls with orientations in a range of 0–90° with the shear direction. While being held back by the anchoring walls, shear caused the bulk director to rotate towards a steady-state alignment angle in the shear direction (Leslie angle θ_L). The transient director rotation was observed with conoscopy. It was found that increasing the initial alignment towards the vorticity direction increased the measured θ_L. Upon stopping the flow, the bulk director returned to its initial state. With initial alignment orientation changing from parallel to perpendicular to the flow direction, the rate of the twist-driven recovery process increases. This rate increase is not seen in the splay-driven recovery which is constant and consistently faster than twist-driven recovery at all orientations studied. Received: 10 December 1998/Accepted: 7 June 1999     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

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     

Rheological behaviour of vitreous humour

The vitreous humour (VH) is a complex biofluid that occupies a large portion of the eyeball between the lens and the retina, and exhibits non-Newtonian rheological properties that are key for its function in the eye. It is often possible to distinguish two different phases in VH, known as liquid and gel phases (Sebag J Eye 1: 254–262, 1987). In this work, we present a detailed rheological characterisation of the two phases of the VH under shear and extensional flow conditions. Healthy New Zealand rabbit eyes were used to measure the surface tension and the shear and extensional rheological properties of VH in different phase conformations and at different times after dissection. The results show that VH liquid phase exhibits a surface tension of 47.8 mN/m, a shear thinning behaviour reaching a viscosity plateau around 10^?3 Pa s for shear rates above ~1000 s^?1, and an average relaxation time of 9.7 ms in extensional flow. Interestingly, both VH phases present higher storage modulus than loss modulus, and the measurements performed with VH gel phase 4?±?1 h after dissection exhibit the highest moduli values. The compliance measurements for the gel phase show a viscoelastic gel behaviour and that compliance values decrease substantially with time after dissection. Our results show that the two VH phases exhibit viscoelastic behaviour, but with distinct rheological characteristics, consistent with a gel phase mostly composed of collagen entangled by hyaluronan and a second phase mainly composed of hyaluronan in aqueous solution.     

Dynamics of three coupled limit cycle oscillators with?vastly?different frequencies

A system of three coupled limit cycle oscillators with vastly different frequencies is studied. The three oscillators, when uncoupled, have the frequencies ?? ₁=O(1), ?? ₂=O(1/??) and ?? ₃=O(1/?? ²), respectively, where ???1. The method of direct partition of motion (DPM) is extended to study the leading order dynamics of the considered autonomous system. It is shown that the limit cycles of oscillators 1 and 2, to leading order, take the form of a Jacobi elliptic function whose amplitude and frequency are modulated as the strength of coupling is varied. The dynamics of the fastest oscillator, to leading order, is unaffected by the coupling to the slower oscillator. It is also found that when the coupling strength between two of the oscillators is larger than a critical bifurcation value, the limit cycle of the slower oscillator disappears. The obtained analytical results are formal and are checked by comparison to solutions from numerical integration of the system.     

Effect of ionic interaction on linear and nonlinear viscoelastic properties of ethylene based ionomer melts

The effect of ionic interaction on linear and nonlinear viscoelastic properties was investigated using poly(ethylene-co-methacrylic acid) (E/MAA) and its ionomers which were partially neutralized by zinc or sodium. Dynamic shear viscosity and step-shear stress relaxation studies were performed. Stress relaxation moduli G(t, y) of the E/MAA and its sodium or zinc ionomers were factorized into linear relaxation moduli G°(t) and damping functions h(y). The relaxation modulus at the smallest strain in each ionomer agreed with the linear relaxation modulus calculated from storage modulus G and loss modulus G. In the linear region, the ionic interaction shifted the relaxation time longer with keeping the same relaxation time distribution as E/MAA. In the nonlinear region, the ionic interaction had no influence on h(y) when the ion content was low. At higher ion content, however, the ion bonding enhanced the strain softening of h(y).     

Effective elastic properties of porous materials: Homogenization schemes vs experimental data

In this paper, we focus on the prediction of elastic moduli of isotropic porous materials made of a solid matrix having a Poisson's ratio vm of 0.2. We derive simple analytical formulae for these effective moduli based on well-known Mean-Field Eshelby-based Homogenization schemes. For each scheme, we find that the normalized bulk, shear and Young's moduli are given by the same form depending only on the porosity p. The various predictions are then confronted with experimental results for the Young's modulus of expanded polystyrene (EPS) concrete. The latter can be seen as an idealized porous material since it is made of a bulk cement matrix, with Poisson's ratio 0.2, containing spherical mono dispersed EPS beads. The Differential method predictions are found to give a very good agreement with experimental results. Thus, we conclude that when vm=0.2, the normalized effective bulk, shear and Young's modulus of isotropic porous materials can be well predicted by the simple form (1 − p)² for a large range of porosity p ranging between 0 and 0.56.     

Determination of the molecular weight distribution of linear polymers by inversion of a blending law on complex viscosities

The method presented in this paper allows to calculate the molecular weight distribution (MWD) of linear homopolymer melts from the complex shear modulus data measured in a wide frequency domain. An empirical blending law on complex viscosities is first developed; as a consequence, the variations of the storage and loss modulus as a function of MWD are presented. This simulation demonstrates also the role of the shape of the MWD itself, and shows that one should not postulate a priori the shape of the MWD. An efficient numerical approach based on a Tikhonov regularization method with constraint is used to solve this ill-posed problem; the MWD is hence derived without any assumption on its shape. This method is first applied on simulated data to prove its numerical efficiency. Then the inversion method is applied on complex moduli data of various commercial polymers (polypropylene, polyethylene and polystyrene) and on an artificial mixture of polystyrene that have been presented in the literature. For amorphous polymers, the coupling of the terminal relaxation domains with the transition region at higher frequency leads to errors in the low molecular weight tail: one way to solve this problem is to cut off the experimental data at the high frequencies. This general method needs only a few physical parameters, namely the scaling law for the Newtonian viscosity η₀=f(M _w ) and the plateau modulus G _N ⁰, and leads to reasonable results with respect to the simplicity of the viscoelastic model used. Received: 27 October 1997 Accepted: 24 February 1998     

Generalized Rouse model for a dilute solution of clustered polymers

A theory analogue to tha of Rouse is given, to describe the rheological behavior of dilute solutions consisting of clusters of crosslinked polymers. The frequency-dependent behavior of the dynamic moduli of these fluids differs substantially from that of the well-known Rouse-like fluid (GG^1/2). In our case the storage modulus becomes proportional to ^3/2, while the loss modulus is proportional to . The loss modulus dominates the dynamic behavior for frequencies smaller than the largest normal frequency of the clusters.     

The rheology of strongly-flocculated suspensions

The rheology of strongly-flocculated dispersions of colloidal particles has been investigated at particle concentrations where a continuous network is formed rather than a collection of discrete flocs. Such networks are shown to possess a true yield stress in both shear and in uniaxial compression (as realised in a centrifuge). Properties measured as a function of particle concentration and particle size include the yield stresses in shear (σ_y) and compression (P_y); the limiting and strain-dependent, instantaneous shear moduli G_O and G(γ); the elastic recovery at finite strains, and the rate of centrifugally-driven compaction. The yield stresses and moduli appear to show a power-law dependence on particle concentration with G_O and P_y, having the same power-law index and σ_y a somewhat lower one. The data are in part consistent with predictions based on the idea that the networks have a heterogeneous structure comprising a collection of interconnected fractal aggregates. The behaviour as a function of particle size and concentration is however not completely scaleable as might be expected on this basis. Thus, whereas the shear yield stress could be scaled to remove its dependence on particle radius a and volume fraction φ (over the measured range 0.25 μm ⩽ a ⩽ 3.4 μm; 0.05 ⩽ φ ⩽ 0.25) as could the strain dependent modulus (0.25 ⩽ a ⩽ 1.3 μm; 0.08 ⩽ 0.25), the particle-size and concentration dependence of P_y and G_O could only be scaled for particles with radii between 0.16 and 0.5 μm, smaller and larger particles having different and much higher power-law index in respect of their concentration dependencies. In the case of the smaller particles the failure of the scaling is thought to be due to an anomaly since these particles distort significantly under the influence of the strong van der Waals forces and this causes the aggregates to be more compact then they otherwise would be. The reasons for the failure at larger sizes is not clear.     

Molecular orientation in sheared molten thermotropic random copolyester

The macromolecular alignment and texture orientation in sheared thermotropic copolyester were investigated using in situ wide-angle X-ray scattering (WAXS) and polarizing optical microscopy (POM). The molecular behavior was correlated with viscoelastic properties. The polymer is a random copolyester based on 60 mol% 1,4-hydroxybenzoic acid (B) and 40 mol% ethylene terephthalate (ET) units. X-ray scattering showed that the molecular chains were aligned along the flow direction. The degree of molecular orientation, , is an increasing function of the applied shear rate. However, rheo-optics showed that shear flow could not orient the polydomain texture, i.e., neither defect stretching nor elimination of defects was observed. Instead, shear compressed the microdomains and gave rise to long-range orientation correlations. Rheology showed that the nematic melt is viscoelastic, the loss modulus G″ dominates the elastic modulus G′, and the dynamic viscosity η* is shear thinning. Moreover, the steady shear viscosity, η, also behaved shear thinning, while the first normal stress difference N ₁ remained positive. The empirical Cox–Merz rule did not hold, , within the shear rate range studied. The microscopic and rheological properties suggest that B–ET is a flow-aligning nematic polymer.     

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     

On Anisotropic Elastic Materials for which Young''s Modulus E ( n ) is Independent of n or the Shear Modulus G ( n , m ) is Independent of n and m

It is shown that, among anisotropic elastic materials, only certain orthotropic and hexagonal materials can have Young modulus E(n) independent of the direction n or the shear modulus G(n,m) independent of n and m. Thus the direction surface for E(n) can be a sphere for certain orthotropic and hexagonal materials. The structure of the elastic compliance for these materials is presented, and condition for identifying if the material is orthotropic or hexagonal is given. We also study the case in which n of E(n) and n, m of G(n,m) are restricted to a plane. When E(n) is a constant on a plane so are G(n,m) and Poisson's ratio ν(n,m). The converse, however, does not necessarily hold. A plane on which E(n) is a constant can exist for all anisotropic elastic materials. In particular, existence of such a plane is assured for trigonal, hexagonal and cubic materials. In fact there are four such planes for a cubic material. For these materials, not only E(n) is a constant, two other Young's moduli, the three shear moduli and the six Poisson's ratio on the plane are also constant.