Relaxation Exponent Research Articles

Mechanical behavior and stimuli-response of nanocomposite hydrogels

Nanocomposite hydrogels (NC gels) have been intensively studied up to now among all the high performance hydrogels due to the facile preparation, unique composition, and excellent properties of mechanical, optical, and swelling/deswelling. We have achieved some significant results in the mechanical behavior of the NC gels and the synthesis of stimuli-response NC gels during several years. Ultrahigh extensibility was found from the as-prepared and swelling-equilibrated poly(N-isopropylacrylamide) (PNIPAm)-hectorite Laponite NC gels with strain hardening at large strain region. Mooney-Rivlin function was found available to describe the stress-strain behavior under compression, but failed to predict the high strain hardening. Creton model was adopted to interpret the stress-strain curves for the NC gels under large deformation well, especially the strain hardening in these gels. For the NC gels, the effective cross-linking density was estimated from the equilibrium shear modulus under small strain, the hysteresis was observed during a deformation-recovery cycle, and the relaxation exponent was determined based on the relaxation modulus. According to these results, we suggest that the ultrahigh extensibility of the NC gels is originated from low cross-linking density and moderate relaxation. In respect of stimuli-response NC gel preparation, a stability window was found from aqueous Laponite suspensions for NIPAm copolymerization with ionic monomers to avoid flocculation and precipitation of Laponite platelets. In this way, the pH response and pH/temperature dual response homogeneous NC gels with ultrahigh extensibility were successfully synthesized. Finally, the unsolved scientific problems concerning the NC gels and the potential applications in the near future were discussed.

Isothermal physical aging of PEEK and PPS investigated by fractional Maxwell model

Generalized fractional Maxwell model is applied to simulate the creep and the relaxation behavior of PEEK and PPS measured at different aging stages and different temperatures. Genetic algorithm and the conjugated gradient method are combined to optimize the model parameters. The results show that the fractional Maxwell model can describe both the creep and relaxation data very well when the model parameters are fitted properly. The momentary creep compliance and the relaxation modulus change with the time at an increasing rate; the fractional orders of the two fractional elements building up the model correspond to the creep or relaxation exponents at the initial glassy stage and the terminal fluid state. The relaxation time τ of the model, which corresponds to the characteristic time of α transition of the sample, shifts towards longer time with increasing the aging time and lowering the aging temperature. In the time domain scaled by the relaxation time τ, all the creep or relaxation curves measured at different aging times and temperatures superpose automatically, which means the time–aging time superposition and the time–temperature superposition present themselves. The shift rates and the temperature shift factors can be then obtained from the model parameters. Good simulation for the creep and relaxation behavior and optimum superposition of the experimental data achieved by the fractional model may offer reliable predictions for the long-term stress-strain response.

Effect of polymer molecular weight on the structural properties of non aqueous ethyl cellulose gels intended for topical drug delivery

Abstract The structural properties of five ethyl cellulose polymers (EC) in mixture with the non aqueous solvent of propylene glycol dicaprylate (PGD) have been investigated with a view to facilitating topical drug delivery. Total concentration of the polymer varied from 12 to 20% (w/w) and its weight average molecular weight was in the range of 64–223 kDa. Experimental techniques utilised were small-deformation dynamic oscillation in shear, attenuated total reflectance Fourier transform infrared spectroscopy (ATR FTIR) and wide-angle X-ray diffraction. The nature of the time and temperature dependence on viscoelastic functions of all samples could be treated with the Chambon and Winter criterion allowing determination of the critical gelation temperature in relation to polymer concentration and molecular weight. A further refinement of this school of thought allowed correlation between the viscoelastic relaxation exponent and fractal dimensions in thermally treated EC/PGD matrices. Tangible evidence on the packing arrangements of the ethyl cellulose network and the molecular nature of its interaction with the non aqueous solvent as a function of polymer concentration and molecular weight was provided by X-ray diffraction and infrared studies.

The dynamics of a complex system: The exchange rate crisis in Southeast Asia

We investigate the dynamics of the exchange rate market just after and prior to the 1997 crisis. The return of the exchange rate is well characterized by a power law, with the relaxation exponent to vary significantly across countries.

Rheology of carbon black suspensions. IV. Effect of suspending media on the sol–gel transition behavior

Linear viscoelastic properties of carbon black (CB) suspensions with various CB concentrations (cCB) in two suspending media, mixtures of a rosin-modified phenol resin-type varnish (Varnish-1)/an alkyd resin-type varnish (Varnish-2) and Varnish-1/petroleum solvent (AF) were investigated. The former medium has higher affinity and the latter has poorer affinity toward CB particles than Varnishi-1, although these two media have almost the same viscoelastic properties. Both CB/(Varnish-1/Varnish-2 = 60/40) and CB/(Varnish-1/AF = 80/20) suspensions exhibited a sol–gel transition on an increase in cCB, as similar to the behavior of CB/Varnish-1 suspensions. But, for the CB/(Varnish-1/Varnish-2 = 60/40) suspensions the critical gel concentration (cgel) was lower and the critical relaxation exponent (n) was higher than that of CB/Varnish-1 suspensions. In contrast, for the CB/(Varnish-1/AF = 80/20) suspensions cgel and n, respectively, were almost the same value as those of CB/Varnish-1 suspensions. These results strongly suggest that cgel and n are attributable to the medium affinity, but not to the medium viscosity, and the CB fractal structure changes from plane-like to straight-line structure with increasing medium affinity.

Effect of alkalis on konjac glucomannan gels for use as potential gelling agents in restructured seafood products

Four dispersions of 3% glucomannan in water, deacetylated with 5% 0.6 N and 1 N KOH (lots L1 and L2) and 0.6 N and 1 N NaOH (lots L3 and L4) as gelling agents, were evaluated for use in raw restructured seafood products. Several properties (pH, moisture content, water binding capacity, cooking loss and lightness) together with puncture data (breaking force and breaking deformation) were determined after 1 and 10 days of chilled storage at 5 °C. All these data were analyzed together with different viscoelastic parameters obtained at small amplitude oscillatory strain (SAOS) after 1 day of chilled storage, showing that L1 and L4 samples were the most suitable gels for incorporation in raw restructured fish products. In both cases the highest stress ( σ max) and strain ( γ max) amplitude values were found in the linear viscoelastic (LVE) range; however, L1 showed both high strain amplitude and breaking deformation values. Moreover, creep and recovery (transient) data showed that L1 was the most time-stable gel with the highest elasticity and the lowest relaxation exponent ( n). L4 gel showed strong rigidity, i.e. the highest values of breaking force and storage moduli ( G′) and the highest n value, making it less gel-like. Both L1 and L4 gels became significantly less gel-like over 10 days of chilled storage due to the loss of gel strength ( S) and a noticeable increase of n. These chilled storage effects were more intense in L4 than in L1.

Relaxation of surface steps after thermal quenches: A numerical study within the terrace-step-kink model

We study the out-of-equilibrium relaxation of surface steps after thermal quenches using numerical simulations of the terrace-step-kink model for a vicinal surface. We analyze both single and interacting steps in a situation where the temperature is suddenly changed at a given quench time. We focus on a physically relevant range of temperatures and show that the relaxation of the roughness is compatible with a power-law behavior with an effective relaxation exponent close to γ = 1/2 in all cases. This value is consistent with a one-dimensional Edwards-Wilkinson equation. In particular, this means that, although the case of interacting steps is effectively a two-dimensional system, its relaxation is dominated by short length-scale fluctuations, where steps are not interacting.

Temperature and pressure effects on stability and gelation properties of silica suspensions

The kinetics of gelation and flow properties of fumed silica suspensions were investigated in pure water and a water/glycerol mixture, as a function of temperature and pressure. In pure water, the gelation time was determined by following the evolution of the storage and loss moduli versus time at several frequencies and temperatures between 18 and 35 °C. Increasing the temperature increased the aggregation kinetics and reduced the gelation time, while the elastic character and relaxation exponent showed more liquid-like viscoelastic properties. The fractal dimensions of the flocs in the gel network decreased when the temperature increased, showing a looser and more open gel network at the higher temperature. In the 50/50 mixture of water/glycerol, the flow behaviour showed a transition from more viscoelastic solid to a shear induced gel and highly Newtonian behaviour when the temperature increased from 18 to 50 °C. The irreversible shear thickening behaviour at the intermediate temperature (25 °C) was explained by the domination of hydrodynamic forces, originating from thermal fluctuations of the medium, in balance with the colloidal interaction forces. When increasing the pressure from 1 to 150 bars at constant temperature, the suspensions showed highly shear thickening effects in the whole pressure range. The onset shear rate of the thickening was accelerated from ambient pressure up to 150 bars. Most anomalies of the temperature and pressure effects on the suspensions stability were attributed to the conformational changes of the water and glycerol molecules and the degree and range of the hydrogen bondings, resulting in a change in the extent of silica bridging during destabilisation of suspensions.

Real-time in situ rheology of alginate hydrogel photocrosslinking

The reaction dynamics of biodegradable, photocrosslinkable sodium alginate hydrogels are studied by in situ, dynamic rheology. Alginate, chemically-modified with methacrylate groups, crosslinks by ultraviolet (UV) light exposure in the presence of a photoinitiator. The gel formation is monitored during UV irradiation from a light emitting diode (LED) bottom plate fixture on the rheometer. Material properties of the hydrogels, including gel points and relaxation exponent, are evaluated using the Winter-Chambon criteria. We also report a new, complementary empirical method for determining the gel point from the reduction in sample strain at the onset of gelation, via monitoring the strain curve. In addition, the crosslinking dynamics and hydrogel moduli are altered by changing the UV irradiation intensities (3–15 mW cm−2) and degree of methacrylation (5–25%). Dynamic rheological measurements of hydrogels as described in this paper are a potentially powerful tool to elucidate the dynamics of gelation and predict mechanical properties. This technique may aid in the design of polymer formulations with light-reactive chemical species, which have tunable properties that can be matched to a range of applications, including regenerative medicine.

Probing equilibrium by nonequilibrium dynamics: Aging in Co/Cr superlattices

Magnetization relaxation is investigated in a structurally ordered magnetic Co/Cr superlattice. Tailored nanoscale periodicity creates mesoscopic spatial magnetic correlations with slow relaxation dynamics when quenching the system into a nonequilibrium state. Magnetization transients are measured after exposing the heterostructure to a magnetic set field for various waiting times. Scaling analysis reveals an asymptotic power-law behavior in accordance with a full aging scenario. The temperature dependence of the relaxation exponent shows pronounced anomalies at the equilibrium phase transitions of the antiferromagnetic superstructure and the ferromagnetic to paramagnetic transition of the Co layers. The latter leaves only weak fingerprints in the equilibrium magnetic behavior but gives rise to a prominent change in nonequilibrium properties. Our findings suggest scaling analysis of nonequilibrium data as a probe for weak equilibrium phase transitions.

Sol–gel transition and gelatinization kinetics of wheat starch

Rheological and optical microscopy experiments were conducted to monitor the sol–gel transition during gelatinization process of starch granules. During the gelatinization phenomenon, the grains grow and dissolve into the plasticizer leading to a drastic increase of the elastic modulus ( G′) and the loss modulus ( G″), and to their crossover as a consequence of the formation of a physical gel. The kinetics of gelation were studied by oscillatory shear experiments and the influence of parameters such as temperature or volume fraction on the gelation process was investigated. The variation of the gel time with temperature for any composition of glycerol/starch suspension was predicted from the Dickinson model. The relaxation exponent ( n = 0.73) was calculated by varying the frequency at constant volume fraction and temperature. The deduced value indicates that the gelation process is described by percolation theory. Apparent activation energies were estimated from the rheological data as well as from the optical microscopy data and were found to be nearly the same. At last the ability of two polyols to act as plasticizer was assessed and compared to that of glycerol.

Viscoelastic and fractal characteristics of a supramolecular hydrogel hybridized with clay nanoparticles

The supramolecular hydrogels derived from low-molecular-mass gelators represent a unique class of soft matters and have important potential applications in biomedical fields, separation technology and cosmetic science. However, they suffer usually from weak mechanical and viscoelastic properties. In this work, we carry out the in situ hybridization of clay nanoparticles (Laponite RD) into the supramolecular hydrogel formed from a low-molecular-mass hydrogelator, 2,6-di[N-(carboxyethyl carbonyl)amino]pyridine (DAP), and investigate the viscoelastic and structural characteristics of resultant hybrid hydrogel. It was found that a small concentration of Laponite RD could lead to a significant increase in the storage modulus, loss modulus or complex viscosity. Compared with neat DAP hydrogel, the hybrid hydrogel has a greater hydrogel strength and a lower relaxation exponent. In particular, the enhancement of the clay nanoparticles to the viscoelastic properties of the DAP hydrogel is more effective in the case of higher DAP concentration. By relating its macroscopic elastic properties to a scaling fractal model, such a hybrid hydrogel was confirmed to be in the strong-link regime and to have a more complex network structure with a higher fractal dimension when compared with neat DAP hydrogel.

Obtaining true transverse relaxation time distributions in high-field NMR measurements of saturated porous media: Removing the influence of internal gradients

It is well known that nuclear magnetic resonance (NMR) transverse relaxation measurements of porous media at high magnetic field strengths provide only an effective relaxation time T(2,eff), as opposed to the true T(2), due to molecular diffusion through magnetic field gradients induced by the magnetic susceptibility contrast between the adsorbent and the adsorbate. Here, we deconvolve the diffusion and surface relaxation contributions to measurements of T(2,eff) and thus obtain the true T(2) relaxation time distribution. This technique is applicable within the short time diffusion regime where the diffusion exponent varies as t(E) (3), while the surface relaxation exponent varies as t(E), where t(E) is the echo time in a standard Carr-Purcell Meiboom-Gill measurement. We demonstrate this technique on measurements of water in contact with glass spheres across a range of magnetic field strengths from B(0)=50 mT to 7.4 T. A direct measurement of T(2,eff) suggests that the transverse relaxation rate increases with field strength, in contrast to theoretical predictions. We show that when the effects of the susceptibility induced gradients, which are known to increase with magnetic field strength, are deconvolved from the T(2,eff) measurement, the true T(2) relaxation rate does indeed decrease with increasing field strength. We also apply the T(2) correction in multidimensional NMR experiments using the example of a T(1)-T(2) relaxation correlation. Here, the correction is essential in order to obtain the true T(1)/T(2) ratio as a function of magnetic field strength, which provides a measure of mobility for surface-adsorbed species; without this correction, we see surface residence times overestimated by up to two orders of magnitude. This novel approach enables the accurate determination of T(2) distributions, and hence T(1)/T(2) ratios, on high-field spectrometers that would have previously been deemed inappropriate for the study of liquids in porous media because of the intrinsic susceptibility effects.

Effects of carbon nanotubes on rheological behavior in cellulose solution dissolved at low temperature

Multi-walled carbon nanotubes (MWNTs) were dispersed, for the first time, in cellulose solution in 9.5wt% NaOH/4.5wt% thiourea aqueous system pre-cooled to −5°C. Dynamic light scattering and transmission electron microscopy results revealed a relatively strong interaction existed between MWNTs and the cellulose macromolecules, leading to a good dispersion of MWNTs in the cellulose solution. Their rheological behaviors, especially the sol–gel transition were investigated by using the advanced rheological expanded system on the basis of Winter and Chambon theory. The gel point and gel concentration of the cellulose/MWNTs solution system were determined, indicating a regularly rheological behavior. The data of loss tangent and relaxation exponent (n) indicated an enhancement in the viscoelasticity of the MWNTs/cellulose system. The results from scaling law before and beyond the sol–gel transition in the MWNTs/cellulose system confirmed that the cluster formation and alteration of the gelation structure occurred at the gel point. Interestingly, the n values calculated by both the Winter and Chambon theory and scaling law were coincident only at relatively low temperature. The predicted gel strength values of the MWNTs/cellulose system were significantly larger than the pure cellulose solution, suggesting a relatively high strength, supported by the mechanical strength of the cellulose/MWNTs material.

Critical behavior of the delay-induced chaos transition in a nonlinear model for the immune response

In this paper we analyze a model for the dynamics of the immune system interacting with a target population. The model consists in a set of two-dimensional delayed differential equations. The model is effectively infinite dimensional due to the presence of the delay and chaotic regimes can be supported. We show that a delayed response induces sustained oscillations and larger delay times implies in a series of bifurcations leading to chaos. The characteristic exponent of the critical power law relaxation towards the stationary state is obtained as well as the critical exponent governing the vanishing of the order parameter in the vicinity of the chaotic transition.

Gelation of Covalently Cross-Linked PEG−Heparin Hydrogels

We study PEG-heparin hydrogels to identify compositions that lead to gel formation and measure the corresponding gelation kinetics. The material consists of a maleimide-functionalized high molecular weight heparin (HMWH) backbone covalently cross-linked with bis-thiol poly(ethylene glycol) (PEG). Using multiple particle tracking microrheology, we investigate a broad composition space, defined by the number of maleimide functional sites per HMWH (f = 3.9-11.8), the molecular weight of the PEG cross-linker (M(n) = 2000, 5000, and 10 000), and the concentrations of the heparin and PEG polymers. Gelation kinetics are characterized by time-cure superposition, yielding the gel time, t(c), and the critical relaxation exponent, n. Gelation times range from 5 < t(c) ≤ 45 min, with the fastest kinetics occurring for the highest HMWH maleimide functionalities. t(c) depends nonmonotonically on the PEG cross-linker molecular weight, suggesting that gelation is affected by the length of the cross-linker relative to intermolecular interactions between heparin molecules. The critical relaxation exponent decreases from n = 0.52 for PEG 2000 to n = 0.39 for PEG 10 000. Finally, 219 equilibrated samples taken over the entire composition space are identified as liquid or solid, defining the "gelation envelope". The boundaries of this empirical gelation envelope are in good agreement with Flory-Stockmayer theory. In all, microrheological measurements enable characterization over a large parameter space and provide crucial insight into the gelation of complex, multifunctional hydrogelators used in therapeutic applications.

Ultrahigh Deformability and Transparence of Hectorite Clay Nanocomposite Hydrogels with Nimble pH Response

Nimble pH response nanocomposite hydrogels (NC gel) with ultrahigh tensibility and high transparency were synthesized via in situ copolymerization of acrylamide and sodium acrylate (SA) in the aqueous suspension of hectorite clay Laponite RDS with a minute amount of N,N′-methylenebisacrylamide (BIS). The stability of the Laponite suspension containing ionic monomers and the tensile properties of the NC gels were investigated. The addition of ionic monomer SA was found to reduce the ζ potential and stability of the suspension. The tensile strength and elongation at break of these ionic NC gels obviously decreased when SA was greater than 10 mol % in the monomers. Interestingly, the addition of a minute amount of BIS (≤0.05 mol %) enhanced the homogeneity of the ionic NC gels and thus improved their transparency (transmittance >90%), tensile strength (>100 kPa), and elongation (>2000%). The relaxation modulus of the ionic NC gels was fit with G(t) = Ge[1 + (t/λ0)−n], where Ge was the equilibrium modulus and λ0 was a material-dependent time constant. The relaxation exponent, n, for the ionic NC gels was 0.10 to 0.18, which is similar to that of the lightly cross-linked nature rubber. This moderate relaxation observed was considered to be the origin of the ultrahigh tensibility of the ionic NC gels. Fortunately, the nimble pH response still remained in the present NC gels containing carboxyl groups. The oscillatory swelling−shrinking circles switched by pH at 7.4 and 3.0 were observed from the present NC gels.

Passive Microrheology of Solvent-Induced Fibrillar Protein Networks

Particle tracking microrheology (PTM) has been used to study the sol-gel transition in solvent-induced fibrillar beta-lactoglobulin gels at room temperature and pH 7. The passive nature of microrheology allowed measurements to be made around and below the critical gelation concentration. The method of superposition introduced by Larsen and Furst (Larsen, T. H.; Furst, E. M. Phys. Rev. Lett. 2008, 100, 146001) was applied to the one-particle mean square displacement (MSD), yielding a critical relaxation exponent of n = 0.58 at concentrations close to the measured critical concentration of 4% (w/v). At a higher concentration of 12% (w/v), n was observed to decrease. The pregel and gel master curves were used to find the viscoelastic moduli over 8 decades of frequency. Combined with the measured shift factors, this allowed cure curves at 1 Hz to be constructed for direct comparison with results from bulk rheology. Time-independent modulus superposition was found for all concentrations. Good agreement for concentration scaling was found between the traditional methods for characterizing gels and the recently described microrheological determination of the gel time and critical behavior.

Particle tracking microrheology of gel-forming amyloid fibril networks

Microrheology is a technique that is increasingly used to investigate the local viscoelastic properties of complex fluids non-invasively, by tracking the motion of micron-sized probe spheres. In this work, passive Particle Tracking Microrheology (PTM) is used to study network formation in the milk protein beta-lactoglobulin at 80 degrees C and pH 2. In these conditions the protein aggregates to form thread-like structures known as amyloid fibrils, which can further aggregate into elastic networks. Using PTM, gels were observed to form at significantly lower concentrations than determined by bulk rheometry, where the oscillatory shear forces may disrupt either fibril or network formation. During incubation, the Mean Square Displacement (MSD) of the probe particles exhibited time-cure superposition, allowing the critical relaxation exponent to be calculated as approximately 0.63, consistent with other biopolymer gels. Combined with the gel-like appearance of the complex modulus at long incubation times, this confirms that a true gel is forming, with physical or chemical crosslinks forming between the fibrils, refining the conclusions of other workers in the field.

Intraday Pattern in Bid-Ask Spreads and Its Power-Law Relaxation for Chinese A-Share Stocks

We use high-frequency data of 1364 Chinese A-share stocks traded on the Shanghai Stock Exchange and Shenzhen Stock Exchange to investigate the intraday patterns in the bid-ask spreads. The daily periodicity in the spread time series is confirmed by Lomb analysis and the intraday bid-ask spreads are found to exhibit $L$-shaped pattern with idiosyncratic fine structure. The intraday spread of individual stocks relaxes as a power law within the first hour of the continuous double auction from 9:30AM to 10:30AM with exponents $\beta_{\rm{SHSE}}=0.19\pm0.069$ for the Shanghai market and $\beta_{\rm{SZSE}}=0.18\pm0.067$ for the Shenzhen market. The power-law relaxation exponent $\beta$ of individual stocks is roughly normally distributed. There is evidence showing that the accumulation of information widening the spread is an endogenous process.