In search of physical meaning: defining transient parameters for nonlinear viscoelasticity

A complete set of model-independent viscoelastic functions for understanding responses to transient nonlinear rheological tests is presented, using large-amplitude oscillatory shear strain as a model nonlinear protocol. The derivation makes no assumptions about symmetries, and is therefore applicable to the responses to any input, allowing researchers to unambiguously define time-dependent moduli, viscosities, compliances, fluidities, and normal stress coefficients. A legend for interpreting the dynamic trajectories in modulus space is provided, along with explicit definitions of the rates at which the moduli change. These provide a quantitative mechanism to identify when, and by how much, a material response stiffens, softens, thickens, or thins while being deformed. In addition to providing analytical expressions for the moduli, the derivation requires the definition of a conceptually new term. This means there exist three, not two, time-dependent nonlinear viscoelastic functions by which any response can be fully described. The third function accounts for nonlinear properties such as yield stresses and the shifting of the strain equilibrium. This complete analysis scheme is unique in making a distinction between the strains in the lab and material frames. The quantitative sequence of physical process analysis, which is fully developed in this work, allows for comprehensive physical interpretations of responses to transient deformations of any kind to be made, including the steady alternance responses to large-amplitude oscillatory shear (LAOS), time-dependent oscillatory shear startup responses, and thixotropic and anti-thixotropic responses.     

Elastomers in large-amplitude oscillatory uniaxial extension

In this work, we subject elastomers to a fixed pre-stretch in uniaxial extension, ε _p , upon which a large-amplitude, ε ₀, oscillatory uniaxial extensional (LAOE) deformation is superposed. We find that if both ε _p and ε ₀ are large enough, the stress responds with a rich set of higher harmonics, both even and odd. We further find the Lissajous-Bowditch plots of our measured stress responses versus uniaxial strain to be without twofold symmetry and, specifically, to be shaped like convex bananas. Our new continuum model for this behavior combines a new nonlinear spring, in parallel with a Newtonian dashpot, and we call this the Voigt model with strain-hardening. We consider this three-parameter (Young’s modulus, viscosity, and strain-hardening coefficient) model to be the simplest relevant one for the observed convex bananas. We fit the parameters to both our LAOE measurements and then to our uniaxial elongation measurements at constant extension rate. We develop analytical expressions for the Fourier components of the stress response, parts both in-phase and out-of-phase with the extensional strain, for the zeroth, first, second, and third harmonics. We find that the part of the second harmonic that is out-of-phase with the strain must be negative for proper banana convexity.     

Controlling the yield behavior of fat-oil mixtures using cooling rate

The rheological behavior such as yielding of fat crystal networks are dictated by many variables. Among these variables, the shape of the constituent fat cluster is important yet relatively unexplored. In this work, we describe the rheological investigations of a fat-oil system which can be formulated to either contain bundles of needles or spherical clusters by controlling the cooling rate and fat concentration. Fat-oil mixtures containing high-fat concentrations exhibited weak frequency dependence of storage modulus (G ^′) and loss modulus (G ^″). The yielding behavior of the mixtures were investigated by large amplitude oscillatory shear (LAOS) rheology using strain and stress controlled modes. Lissajous-Bowditch plots and orthogonal set of Chebyshev polynomials were used to analyze the non-linearities associated with the yielded fat-oil mixtures. For a given fat concentration, the yield stress of fat networks obtained at low cooling rates (bundles of needles) were similar to that of networks obtained at high cooling rates (spherical clusters). However, after yielding, Lissajous-Bowditch plots suggested that the mixtures comprising of bundles of needles exhibited viscous-like behavior while the spherical clusters exhibited a plastic-like behavior. This was further supported by microscopy images of yielded fat-oil mixtures. Overall, for a given fat concentration, the two different shapes of fat clusters can give rise to networks of similar yield stress values but different behaviors after yielding.     

Filling the gap between transient and steady shear rheology of aqueous graphene oxide dispersions

Even though the rheological behavior of aqueous graphene oxide (G-O) dispersions has been shown to be strongly time-dependent, only few transient measurements have been reported in the literature. In this work, we attempt to fill the gap between transient and steady shear rheological characterizations of aqueous G-O dispersions in the concentration range of 0.004 < ? <?3.5 wt%, by conducting comprehensive rheological measurements, including oscillatory shear flow, transient shear flow, and steady shear flow. Steady shear measurements have been performed after the evaluation of transient properties of the G-O dispersions, to assure steady-state conditions. We identify the critical concentration ? _c =?0.08 wt% (where G-O sheets start to interact) from oscillatory shear experiments. We find that the rheology of G-O dispersions strongly depends on the G-O concentration ?. Transient measurements of shear viscosity and first normal stress difference suggest that G-O dispersions behave like nematic polymeric liquid crystals at ?/? _c =?25, in agreement with other work reported in the literature. G-O dispersions also display a transition from negative to positive values of the first normal stress difference with increasing shear rates. Experimental findings of aqueous graphene oxide dispersions are compared and discussed with models and experiments reported for nematic polymeric liquid crystals, laponite, and organoclay dispersions.     

Wall slip and multi-tier yielding in capillary suspensions

Impact of wall slip on the yield stress measurement is examined for capillary suspensions consisting of cocoa powder as the dispersed phase, vegetable oil as the continuous primary fluid, and water as the secondary fluid using smooth and serrated parallel plates. Using dynamic oscillatory measurements, we investigated the yielding behavior of this ternary solid-fluid-fluid system with varying particle volume fraction, ?, from 0.45 to 0.65 and varying water volume fraction, ?_w, from 0.02 to 0.08. Yield stress is defined as the maximum in the elastic stress (G^′γ), which is obtained by plotting the product of elastic modulus (G^′) and strain amplitude (γ) as a function of applied strain amplitude. With serrated plates, which offer minimal slippage, capillary suspensions with ? ≥?0.45 and a fixed ?_w =?0.06 showed a two-step yielding behavior as indicated by two peaks in the plots of elastic stress as a function of strain amplitude. On the other hand with smooth plates, the capillary suspensions showed strong evidence of wall slip as evident by the presence of three distinct peaks and lowered first yield stresses for all ? and ?_w. These results can be interpreted based on the fact that a particle-depleted layer, which is known to be responsible for slip, is present in the vicinity of the smooth surfaces. The slip layer presents itself as an additional “pseudo-microstructure” (characteristic length scale) besides the two microstructures, aqueous bridges and solid particle agglomerates, that may occur in the system. With serrated plates, both the yield stresses (σ₁σ₂) and storage moduli plateau at lower strain (before the first yield point) and at higher strain (before the second yield point) (G\(^{\prime }_{p1}\), G\(^{\prime }_{p2}\)) were found to increase with ? (at a fixed ?_w =?0.06) following power-law dependences. Similarly with increasing ?_w (0.02 – 0.08) at a fixed ? =?0.62, the system behaved as a solid-like material in a jammed state with particles strongly held together as manifested by rapidly increasing σ₁ and σ₂. The usage of smooth surfaces primarily affected σ₁ which was reflected by an approximately 70–90% decrement in the measured σ₁ for all values of ?. By contrast, σ₂ and G\(^{\prime }_{p2}\) were found to be unaffected as shown by close agreement of values obtained using serrated geometry due to vanishing slip layers at higher strain amplitudes.     

Rheological behavior of composites made from linear medium-density polyethylene and hemp fibers treated by surface-initiated catalytic polymerization

This work presents different rheological methods to determine the effect of fiber surface treatment on their interaction with a polymer matrix. In particular, surface-initiated catalytic polymerization was investigated on hemp fibers to improve their adhesion with linear medium-density polyethylene (LMDPE). The selected rheological tests (creep-recovery (solid state), small and large amplitude oscillation shear, and transient rheology (melt state)) were used to compare the treated and untreated fiber composites with the neat matrix. The results showed a significant improvement of the treated hemp composite (LPHC) creep modulus with respect to its untreated counterpart (LNHC) leading to a reduction of the creep strain, especially as temperature increases. The transient viscosity was modeled using a modified Kohlrausch-Williams-Watt (KWW) equation showing an increase in the transient viscosity (\( {\eta}_0^{+} \)) and relaxation time (τ) with fiber addition and surface treatment. These results were confirmed by large amplitude oscillatory shear (LAOS) through the reduction of the relative third harmonic (I_3/1), intrinsic nonlinearity parameter (Q₀), and nonlinear viscoelastic ratio (NRL). The results clearly show that catalytic polymerization is a good surface modification technique to increase the compatibility between natural fibers and polymer matrices as to improve all their final properties.

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Graphical Abstract ?

     

Shear flow of a Newtonian fluid over a quiescent generalized Newtonian fluid

The flow of an upper shear-driven Newtonian fluid above an otherwise still non-Newtonian fluid is considered. The lower fluid is modelled as a generalized Newtonian fluid and set into motion by interfacial shear. By means of similarity transformations, the governing partial differential equations for the two-fluid problem transform exactly into two sets of ordinary differential equations coupled only at the interface. The successful transformation of the two-fluid problem is applied to the particular case when the lower fluid obeys power-law rheology. The resulting three-parameter problem is solved numerically for some different parameter combinations by means of a direct integration approach with the density ratio fixed to unity. We observed that the interfacial velocities decreased with increasing values of the power-law index n in the range from 0.6 to 1.4 whereas the shear-induced motion of the lower fluid penetrates far deeper into a shear-thinning (n < 1) than into a shear-thickening (n > 1) fluid. This phenomenon is ascribed to a corresponding increase of the non-linear viscosity function with lower n-values.     

Instabilities of a thin viscoelastic liquid film flowing down an inclined plane in the presence of a uniform electromagnetic field

The effect of a uniform electromagnetic field on the stability of a thin layer of an electrically conducting viscoelastic liquid flowing down on a nonconducting inclined plane is studied under the induction-free approximation. Long-wave expansion method is used to obtain the surface evolution equation. The stabilizing role of the magnetic parameter M and the destabilizing role of the viscoelastic parameter Γ as well as the electric parameter E on this flow field are established. A novel result which emerges from our analysis is that the stabilizing effect of M holds no longer true for both viscous and viscoelastic fluids in the presence of electromagnetic field. It is found that when E exceeds a certain critical value depending on Γ, magnetic field exhibits the destabilizing effect on this flow field. Indeed, this critical value decreases with the increase of the viscoelastic parameter Γ since it has a destabilizing effect inherently. Another noteworthy result which arises from the weakly nonlinear stability analysis is that both the subcritical unstable and supercritical stable zones are possible together with the unconditional stable and explosive zones for different values of Γ depending on the wave number k.     

Rheology and structural changes of plasticized zeins in the molten state

Zeins, storage proteins from maize, are suitable for making biobased thermoplastic materials. The rheological behavior of a commercial zein plasticized with 20 w% glycerol was studied in the molten state by steady-state flow experiments in extrusion conditions and oscillatory rheometry. For low residence times, a shear-thinning viscoelastic behavior was observed, with G″ exceeding G′. After 300 s at 130 °C, the complex viscosity |η ^?|?=?7?×?10³ ω ^?0.46 was found to be similar to that of thermoplastic polymer melts used in fused deposition modeling. However, the ratio between the exponents of the power laws describing G′(ω) and G″(ω) did not meet the typical value of 2 for entangled polymer melts. Moreover, for longer residence times, the viscosity increased and a gelation phenomenon was observed with a crossing over of G′(ω) and G″(ω). Gel times ranged from 6000 s at 120 °C to 1700 s at 150 °C. The evolution of the macromolecular structure assessed by sodium dodecyl sulfate-polyacrylamide gel electrophoresis and high-performance size exclusion chromatography suggested that this gelation phenomenon involves various types of covalent and non-covalent cross-links. Disulfide bonds played a significant role in gelation kinetics despite a very low cysteine residue content in the protein primary structure (about 1 mol%). These results suggested that plasticized zeins initially behave like a low-viscosity non-entangled polymer melt, before cross-linking progressively led to a continuous network.     

Dual yielding in capillary suspensions

Rheological measurements were performed to examine the yielding behavior of capillary suspensions prepared by mixing cocoa powder as dispersed phase, vegetable oil as the continuous primary fluid, and water as the secondary fluid. Here, we investigated the yielding behavior of solid-fluid-fluid systems with varying particle volume fraction, ?, spanning the regime from a low volume fraction (? = 0.25) to a highly filled regime (? = 0.65) using dynamic oscillatory measurements. While for ? ≤ 0.4 with a fixed water volume fraction (? _w ) of 0.06 as the secondary fluid, capillary suspensions exhibited a single yield point due to rupturing of aqueous capillary bridges between the particles, while capillary suspensions with ? ≥ 0.45 showed a two-step yielding behavior. On plotting elastic stress (G^′ γ) as a function of applied strain (γ), two distinct peaks, indicating two yield stresses, were observed. Both the yield stresses and storage modulus at low strains were found to increase with ? following a power law dependence. With increasing ? _w (0 – 0.08) at a fixed ? = 0.65, the system shifted to a frustrated, jammed state with particles strongly held together shown by rapidly increasing first and second yield stresses. In particular, the first yield stress was found to increase with ? _w following a power law dependence, while the second yield stress was found to increase exponentially with ? _w . Transient steady shear tests were also performed. The single stress overshoot for ? ≤ 0.4 with ? _w = 0.06 reflected one-step yielding behavior. In contrast, for high ? (≥ 0.45) values with ? _w = 0.06, two stress overshoots were observed in agreement with the two-step yielding behavior shown in the dynamic oscillatory measurements. Experiments on the effect of resting time on microstructure recovery demonstrated that aggregates could reform after resting under quiescent conditions.     

Application of scaling transformation to characterizing complex rheological behaviors and fractal derivative modeling

It has been long observed that cumbersome parameters are required for the traditional viscoelastic models to describe complex rheological behaviors. Inspired by the relationship between normal and anomalous diffusions, this paper tentatively employs t ^α to replace t, called as the scaling transformation, in the traditional creep compliance and relaxation modulus. With this methodology, the relaxation modulus is found to agree with the well-known Kohlrausch-Williams-Watts (KWW) stretched exponential function. The fitting results confirm that the proposed models accurately characterize rheological behaviors only with one more parameter α. Moreover, it is noted that the present formulations are directly related to the fractal derivative viscoelastic models and the index α is actually the order of the fractal derivative.     

Global Attractor for a Nonlinear Oscillator Coupled to the Klein–Gordon Field

The long-time asymptotics is analyzed for all finite energy solutions to a model\(\mathbf{U}(1)\)-invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a single point: each finite energy solution converges as t→ ± ∞ to the set of all “nonlinear eigenfunctions” of the form ψ(x)e^{?iω t}. The global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.We justify this mechanism by the following novel strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time spectrum in the spectral gap [ ? m,m] and satisfies the original equation. This equation implies the key spectral inclusion for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of each omega-limit trajectory to a single harmonic \(\omega\in[-m,m]\).The research is inspired by Bohr’s postulate on quantum transitions and Schrödinger’s identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled\(\mathbf{U}(1)\)-invariant Maxwell–Schrödinger and Maxwell–Dirac equations.     

Experimental Study on Three-Dimensional Deformation Field of Portevin–Le Chatelier Effect Using Digital Image Correlation

In view of its high precision and high efficiency, three-dimensional digital image correlation (3D-DIC) is widely used to accurately measure full-field deformation. A spatiotemporal experimental study using 3D-DIC to explore the Portevin–Le Chatelier (PLC) deformational behavior, provides a new insight into the whole 3D deformation field, including the out-of-plane displacement, and in particular the relationship between the serrations and the strain field in the deformation bands corresponding to individual serrations. Specimens 1, 2 and 3 mm thick of 5456 Al-based alloy were tested in uniaxial tension at room temperature at strain rates from 1.8 × 10^?4 to 9.1 × 10^?3s^?1. The spatial and temporal characteristics of the strain localization were quantitatively analyzed. The out-of-plane displacement increment field (w) of the localized bands was observed by 3D-DIC, and found to be related to the specimen thickness and the in-plane strain increment. The largest displacement increments were respectively 15, 10 and 5 μm for 3, 2 and 1 mm specimens at maximum strain increment of about 12000 με. The elastic shrinkage outside the deformation bands was found to be an essential characteristic of the PLC effect. The width of the PLC band (w_band) increased with increasing thickness; the angle of the PLC band (??_band) was not affected by either specimen thickness or serration amplitude. Temporally, the serrations in the plots both of in-plane strain and out-of-plane displacement vs. time coincided throughout the entire loading procedure. When PLC banding occurred, the serration amplitude within the bands was found to be proportional to the maximum strain increment in the direction of the applied tensile force (??_max).     

Infinitely Many Stability Switches in a Problem with Sublinear Oscillatory Boundary Conditions

We consider the elliptic equation \(-\Delta u +u =0\) with nonlinear boundary condition \(\frac{\partial u}{\partial n}= \lambda u + g(\lambda ,x,u), \) where \(\frac{g(\lambda ,x,s)}{s} \rightarrow 0, \hbox { as }|s|\rightarrow \infty \) and g is oscillatory. We provide sufficient conditions on g for the existence of unbounded sequences of stable solutions, unstable solutions, and turning points, even in the absence of resonant solutions.     

Large amplitude oscillatory shear behavior of graphene derivative/polydimethylsiloxane nanocomposites

Rheological properties of three different nanocomposites, consisting of graphene oxide (GO), reduced graphene oxide (rGO), and polyhedral oligomeric silsesquioxane grafted reduced graphene oxide (rGO-POSS) as nanofillers and polydimethylsiloxane (PDMS), were investigated by large amplitude oscillatory shear (LAOS). The viscoelastic nonlinearity of the three nanofluids groups was studied by Lissajous curves, local nonlinear viscoelastic moduli of an oscillatory shear cycle, and Fourier transform rheology as a function of filler concentration and increasing and decreasing strain magnitude. The nonlinear behavior of the nanofluids was compared to understand the variation of internal microstructures. Firstly, GO/PDMS composites behave with higher moduli and smaller linear viscoelastic range comparing to that of other two composites. Secondly, the elastic stress Lissajous curves of these composites changed from elliptic to rectangular with round the corner with increasing the filler level and strain amplitude. Thirdly, all these three nanofluids exhibited intra-cycle strain stiffening with increasing strains and shear thickening at intermediate strain and then shearing thinning with increasing strain further. Fourthly, higher harmonic intensity of rGO/PDMS increased with increasing strain and came to a plateau, while that of other two nanofluids reached a maximum and then decreased. It suggested that different surface functionalization of nanoparticles will present different rheological behavior due to formed different network and LAOS could be used as a potential helpful method to characterize rheological properties of nanocomposites, especially at higher shear strain.     

Regularity of Solutions to the Navier-Stokes Equations for Compressible Barotropic Flows on a Polygon

The Navier-Stokes system for a steady-state barotropic nonlinear compressible viscous flow, with an inflow boundary condition, is studied on a polygon D. A unique existence for the solution of the system is established. It is shown that the lowest order corner singularity of the nonlinear system is the same as that of the Laplacian in suitable L ^q spaces. Let ω be the interior angle of a vertex P of D. If \(\) and \(\), then the velocity u is split into singular and regular parts near the vertex P. If α < 2 and \(\) or if α > 2 and 2 < q < ∞&;, it is shown that u∈ (H ^{2, q} (D))².     

Hydrodynamic properties of squirmer swimming in power-law fluid near a wall

Hydrodynamic properties of squirmer swimming in power-law fluid near a wall considering the interaction between squirmer and wall are numerically studied with an immersed boundary-lattice Boltzmann method. The power-law index, Reynolds number, initial orientation angle of squirmer, and initial distance of squirmer from the wall are all taken into account to investigate the swimming characteristics for pusher (β?<?0), neutral squirmer (β?=?0), and puller (β?>?0) (three kinds of swimmer types) near the no-slip boundary. Four new kinds of swimming modes are found. Results show that, for the pushers and pullers, the wall displays an increasing attraction with increasing power-law index n, which differs from the neutral squirmer who always departs from the wall after the first collision with the wall. Both the initial orientation angle and initial distance from the wall only affect the moving situations rather than the moving modes of the squirmers. However, the squirmers depart from the wall as the Reynolds number increases and chaotic orbits appear for some squirmers at Re?=?5. Several typical flow fields are analyzed and the power consumption and torque for different kinds of flows are also studied. It is found that, as the absolute value of β increases, the power consumption generally increases in shear-thinning (n?=?0.4), Newtonian (n?=?1), and shear-thickening (n?=?1.6) fluids. Moreover, the pushers (β?<?0) and the pullers (β?>?0) expend almost the same power if the absolute value of β remains the same. In addition, the power consumption of the squirmers is highly dependent on the power-law index n.     

Experimental study on rheological and thermophysical properties of seawater with surfactant additive—part I: rheological properties

The rheological properties of seawater with the addition of surfactant additive (cetyltrimethyl ammonium chloride (CTAC)/sodium salicylate (NaSal)) are measured at different temperatures, including shear viscosity and first normal stress difference (N₁). The effects of the temperature, the salts, and CTAC/NaSal concentration on the rheological properties of test solutions are investigated, and the corresponding influence mechanisms are analyzed. It shows that the addition of salt can decrease the shear viscosities of the solutions, and also decrease N₁ and even eliminate the sharp augment of N₁ above a certain shear rate. The growing elasticity can be characterized by the increase of the initial shear rate for shear-thickening inception. High temperature can also remove the sharp increase of N₁ with salt. Nevertheless, the increase of CTAC/NaSal concentration can withstand the elimination of the sharp augment of N₁.     

Power-law creep and residual stresses in a carbopol gel

We report on the interplay between creep and residual stresses in a carbopol microgel. When a constant shear stress σ is applied below the yield stress σ _y, the strain is shown to increase as a power law of time, γ(t) = γ ₀ + (t/τ) ^α , with an exponent α = 0.39 ± 0.04 that is strongly reminiscent of Andrade creep in hard solids. For applied shear stresses lower than some typical value σ _c ? 0.2σ _y, the microgel experiences a more complex, anomalous creep behavior, characterized by an initial decrease of the strain, that we attribute to the existence of residual stresses of the order of σ _c that persist after a rest time under a zero shear rate following preshear. The influence of gel concentration on creep and residual stresses are investigated as well as possible aging effects. We discuss our results in light of previous works on colloidal glasses and other soft glassy systems.