Infinite-frequency Shear Modulus Research Articles

Upper bound of fragility from spatial fluctuations of shear modulus and boson peak in glasses.

It is shown that the normalized rms fluctuation of the shear modulus on the medium-range order scale in glasses correlates with fragility: the higher fragility, the smaller the fluctuation amplitude. The latter is calculated within the heterogeneous elasticity theory using the data on the boson peak in glasses. On a smaller scale corresponding to cooperative structural relaxation, the normalized rms fluctuation of the infinite-frequency shear modulus was estimated using the data on the decoupling of viscosity and diffusion in supercooled liquids. These fluctuations are much smaller in amplitude, and, in contrast, they increase with increasing fragility. Extrapolation predicts intersection of both rms fluctuations and disappearing of the boson peak at the upper limit to fragility ≈180.

Modeling of liquid internal energy and heat capacity over a wide pressure–temperature range from first principles

Recently, there have been significant theoretical advances in our understanding of liquids and dense supercritical fluids based on their ability to support high frequency transverse (shear) waves. Here, we have constructed a new computer model using these recent theoretical findings (the phonon theory of liquid thermodynamics) to model liquid internal energy across a wide pressure–temperature range. We have applied it to a number of real liquids in both the subcritical regime and the supercritical regime, in which the liquid state is demarcated by the Frenkel line. Our fitting to experimental data in a wide pressure–temperature range has allowed us to test the new theoretical model with hitherto unprecedented rigor. We have quantified the degree to which the prediction of internal energy and heat capacity is constrained by the different input parameters: the liquid relaxation time (initially obtained from the viscosity), the Debye wavenumber, and the infinite-frequency shear modulus. The model is successfully applied to output the internal energy and heat capacity data for several different fluids (Ar, Ne, N2, and Kr) over a range of densities and temperatures. We find that the predicted heat capacities are extremely sensitive to the values used for the liquid relaxation time. If these are calculated directly from the viscosity data, then, in some cases, changes within the margins of the experimental error in the viscosity data can cause the heat capacity to exhibit a completely different trend as a function of temperature. Our code is computationally inexpensive, and it is available for other researchers to use.

Shear modulus of two-dimensional Yukawa or dusty-plasma solids obtained from the viscoelasticity in the liquid state.

Langevin dynamical simulations of two-dimensional (2D) Yukawa liquids are performed to investigate the shear modulus of 2D solid dusty plasmas. Using the known transverse sound speeds, we obtain a theoretical expression of the shear modulus of 2D Yukawa crystals as a function of the screening parameter κ, which can be used as the candidate of their shear modulus. The shear relaxation modulus G(t) of 2D Yukawa liquids is calculated from the shear stress autocorrelation function, consisting of the kinetic, potential, and cross portions. Due to their viscoelasticity, 2D Yukawa liquids exhibit the typical elastic property when the time duration is much less than the Maxwell relaxation time. As a result, the infinite frequency shear modulus G_{∞}, i.e., the shear relaxation modulus G(t) when t=0, of a 2D Yukawa liquid should be related to the shear modulus of the corresponding quenched 2D Yukawa solid (with the same κ value), with all particles suddenly frozen at their locations of the liquid state. It is found that the potential portion of the infinite frequency shear modulus for 2D Yukawa liquids at any temperature well agrees with the shear modulus of 2D Yukawa crystals with the same κ obtained from the transverse sound speeds. Thus, we find that the shear modulus of 2D Yukawa solids can be obtained from the motion of individual particles of the corresponding Yukawa liquids using their viscoelastic property.

Short-time elasticity of polymer melts: Tobolsky conjecture and heterogeneous local stiffness

An extended Molecular-Dynamics study of the short-time "glassy" elasticity exhibited by a polymer melt of linear fully-flexible chains above the glass transition is presented. The focus is on the infinite-frequency shear modulus $G_\infty$ manifested in the picosecond time scale and the relaxed plateau $G_p$ reached at later times and terminated by the structural relaxation. The local stiffness of the interactions with the first neighbours of each monomer exhibits marked distribution with average value given by $G_\infty$. In particular, the neighbourhood of the end monomers of each chain are softer than the inner monomers, so that $G_\infty$ increases with the chain length. $G_p$ is not affected by the chain length and is largely set by the non-bonding interactions, thus confirming for polymer melts the conjecture formulated by Tobolsky for glassy polymers.

Properties of liquid iron along the melting line up to Earth-core pressures

We report a molecular dynamics study of the transport coefficients and the infinite frequency shear modulus of liquid iron at high temperatures and high pressures. We observe a simultaneous rise of both the shear viscosity and the diffusion coefficient along the melting line and estimate whether liquid iron can vitrify under Earth-core conditions. We show that in the conditions of the model studied in our work iron demonstrates a moderate increase of viscosity along the melting line. It is also demonstrated that at the limit of high temperatures and high pressures the liquid iron behaves similarly to the soft sphere system with exponent n ≈ 4.6.

Density dependence of the stress relaxation function of a simple fluid

We present accurate molecular dynamics calculations of the shear stress relaxation modulus of a simple atomic fluid over a wide range of densities. The high accuracy of the data enables us to study changes in the functional form of the shear relaxation modulus, and the properties that are derived from it, as the density is increased from the ideal gas limit to the upper limit of the fluid range. We show that the shear relaxation modulus of a dilute atomic fluid can be accurately described with a simple functional form consisting of a Gaussian plus a single exponential, whereas the dense fluid exhibits a more complicated relaxation function. The infinite-frequency shear modulus, the zero-shear viscosity, and the zero-shear first normal stress coefficient are calculated from the stress autocorrelation function. The ratio of the first normal stress coefficient to the viscosity is used to calculate the viscoelastic relaxation time. While the viscosity and the infinite-frequency shear modulus both increase monotonically with increasing density, the first normal stress coefficient and the viscoelastic relaxation time both decrease to a minimum at intermediate densities before increasing again. Our results for the viscosity and the first normal stress coefficient in the low-density limit both agree well with the predictions of kinetic theory.

Many-body and quantum effects in some thermodynamic properties and infinite shear modulus of HFD-like fluid using the radial distribution function

We have calculated pressure, internal energy, and infinite-frequency shear modulus of two-body Hartree–Fock dispersion (HFD)-like potential using the statistical mechanical formula for neon. Molecular dynamics simulations have been also applied to generate the required radial distribution function (RDF) for HFD-like fluid in the range 1.1≤T⁎≤4.4 and 0.43≤ρ⁎≤0.82. Quantum corrections have been considered using the Feynman–Hibbs (FH) approach. To take many-body forces into account for the HFD-like system, the simple three-body potentials of Wang and Sadus (2006) and Hauschild and Prausnitz (1993) were used with the two-body potential without requiring an expensive three-body calculation. The results show that the quantum and three-body effects improve the prediction of the pressure and internal energy of the HFD-like fluid.

Evaluation of the Dyre shoving model using dynamic data near the glass temperature

The temperature dependence of the dynamics of glass-forming systems remains an important fundamental problem in glass physics. Here we use literature data [S. A. Hutcheson and G. B. McKenna, J. Chem. Phys. 129, 074502 (2008)] reanalyzed with the Baumgärtel-Schausberger-Winter (BSW) [M. Baumgärtel, A. Schausberger, and H. H. Winter, Rheol. Acta 29, 400 (1990); M. Baumgärtel and H. H. Winter, ibid. 28, 511 (1989); M. Baumgärtel and H. H. Winter, J. Non-Newtonian Fluid Mech. 44, 15 (1992)] model of complex fluid dynamics to evaluate the Dyre shoving model [J. C. Dyre, N. B. Olsen, and T. Christensen, Phys. Rev. B 53, 2171 (1996); J. C. Dyre, Rev. Mod. Phys. 78, 953 (2006)] that relates the temperature dependence of viscosity to the infinite-frequency shear modulus and its temperature dependence. In Dyre's model, the free-energy barrier for a "flow event" is attributed to the work done in shoving aside the surrounding molecules; the free-energy barrier is proportional to the glassy modulus, which increases as the temperature decreases. In the present work, the glassy modulus was obtained by the extrapolation to zero time or infinite frequency of the Kohlrausch-Williams-Watts (KWW) [G. Williams and D. C. Watts, Trans. Faraday Soc. 66, 80 (1970); F. Kolrausch, Pogg. Ann. Phys. 12, 393 (1847)] and BSW [M. Baumgärtel, A. Schausberger, and H. H. Winter, Rheol. Acta 29, 400 (1990); M. Baumgärtel and H. H. Winter, ibid. 28, 511 (1989); M. Baumgärtel and H. H. Winter, J. Non-Newtonian Fluid Mech. 44, 15 (1992)] functions to experimental data for m-toluidine and sucrose benzoate. It was found that the glassy modulus obtained from the KWW function for m-toluidine and sucrose benzoate [S. A. Hutcheson and G. B. McKenna, J. Chem. Phys. 129, 074502 (2008)] provides a consistent picture of the temperature-dependent dynamics within the framework of the shoving model. A similar analysis using a BSW description of the dynamics provides consistency for the sucrose benzoate but not for the m-toluidine.

Density and pressure dependence of the equation of state and transport coefficients of soft-sphere fluids

Molecular Dynamics, MD, simulations were used to compute physical properties of model fluids in which the particles interacted via the soft-sphere or inverse power pair potential, φ(r) = ε(σ/r) n . n dictates the steepness or stiffness of the potential, and ε and σ are a characteristic energy and distance, respectively. A wide range of n values were considered, from the hard-sphere (n → ∞) limit down to (the latter for the first time). A linear isotherm relationship for dense fluids observed by Parsafar and Mason for supercritical compressed gases [J. Phys. Chem. 97, 9048 (1993)], was found to apply to the data for n ∼ 12 (values typical of simple fluids). For smaller n, there is a change in sign of the slope, and the data exhibited more curvature. The self-diffusion coefficient, D, and shear viscosity, ηs, were also calculated. At intermediate to high densities, D −1 and ηs depend to a very good approximation linearly on pressure, as was found by van der Gulik [Physica A 256, 39 (1998)] on treatment of experimental shear viscosity data for simple molecules. Values for D and ηs at fluid–solid coexistence are given as a function of n. We refine further simple formulae for D and ηs proposed in our previous publication [Phys. Chem. Chem. Phys. 10, 4036 (2008)]. The glass transition packing fraction and pressure for the fluid are estimated by extrapolation of the self-diffusion coefficient data. In contrast to the n = 12 case, the shear stress correlation function correlation time shows only a weak density dependence near coexistence for the very soft interactions (e.g. ). It is shown that for the very soft interactions close to n = 3, the increase in viscosity is largely determined by the infinite frequency shear modulus rather than the relaxation time, which hardly changes with density at high density.

A new expression for radial distribution function and infinite shear modulus of Lennard-Jones fluids

Molecular dynamics simulations are applied to generate an expression for the radial distribution function (RDF) of Lennard-Jones fluids for 165 reduced temperature-isochors in the range of 1.4 ⩽ T ∗ ⩽ 4.2 and 0.35 ⩽ ρ ∗ ⩽ 0.85. Due to the impossibility of presenting a single general expression with few adjustable parameters for the whole distance range, one function is presented for the first peak of the RDF and a second function for the other peaks. The calculated pressure is very sensitive to the shape of RDF and is used as a criterion for the investigation of convergence of the simulation and the accuracy of the simulated RDF. The parameters in the function are adjusted to precisely reproduce of the first peak and thus to accurately predict the pressure of the Lennard-Jones fluids with a relative mean error about 1.6%. Based on the simulated RDF, the infinite-frequency shear modulus, G ∞ of the Lennard-Jones fluid is calculated and a simple expression is presented for G ∞ in the range of 1.0 ⩽ T ∗ ⩽ 4.2 and 0.35 ⩽ ρ ∗ ⩽ 0.85.

Viscoelasticity and Metastability Limit in Supercooled Liquids

A supercooled liquid is said to have a kinetic spinodal if a temperature Tsp exists below which the liquid relaxation time exceeds the crystal nucleation time. We revisit classical nucleation theory taking into account the viscoelastic response of the liquid to the formation of crystal nuclei and find that the kinetic spinodal is strongly influenced by elastic effects. We introduce a dimensionless parameter lambda, which is essentially the ratio between the infinite frequency shear modulus and the enthalpy of fusion of the crystal. In systems where lambda is larger than a critical value lambda(c) the metastability limit is totally suppressed, independently of the surface tension. On the other hand, if lambda<lambda(c) a kinetic spinodal is present, and the time needed to experimentally observe it scales as exp([omega/(lambda(c)-lambda)2], where omega is roughly the ratio between surface tension and enthalpy of fusion.

Nonideality in Binary Mixtures: Correlations between Excess Volume, Excess Viscosity, and Diffusion Coefficients

Although many experimental studies over the years have shown strong (inverse) correlation between excess viscosity and excess volume of a binary mixture, there does not seem to exist any microscopic theoretical study of such behavior. In this work, we introduce and study two new models of binary mixture with the aim of removing this lacuna. Our models consist of a mixture of two molecular species (1 and 2) with the same diameter and mass but varying solute−solvent Lennard−Jones interactions. In model I, the two different species are strongly attractive, while in model II, the attraction is weaker than that between the pure components. That is, model I encourages structure formation, while the reverse is true for model II. Extensive constant pressure−constant temperature (NPT) molecular dynamics (MD) simulations have been carried out which show that these models can indeed reproduce the strong inverse correlation between the calculated excess volume and excess viscosity. The two models show opposite trends in the composition dependence. Another interesting result is the observation of a re-entrant behavior in the excess volume dependence of total viscosity of the mixturethe same re-entrance is also observed in the dependence of excess viscosity on excess volume. In addition, we derive the first two terms in the virial expansion for the composition dependence for the infinite frequency shear modulus, G∞. These terms can explain the composition dependence at low composition in terms of partial radial distribution functions and interaction potentials. The calculated self-diffusion coefficients have nonmonotonic behavior and show considerable deviations from Stokes−Einstein relation.

Raman free-induction-decay measurements in low viscosity and supercooled toluene: Vibrational dephasing by shear fluctuations

Total dephasing decay profiles of the ν12 (1002 cm−1) ring-stretching mode of toluene have been measured in the time domain in a range extending from the low viscosity, normal liquid (380 K, 0.26 cP) to the high viscosity, supercooled liquid (140 K, 4600 cP). In the region from 1 to 5 cP (250 to 200 K), the profile makes a transition from exponential to partially Gaussian, consistent with a loss of motional narrowing. In contrast to many interpretations of dephasing in terms of collisional dynamics, these data clearly indicate an important role for diffusive dynamics in vibrational dephasing. Above 10 cP, oscillations appear, and their period decreases with increasing viscosity. An analysis of the low to moderate viscosity region with a Kubo model shows that the frequency modulation time is dependent on the viscosity. The dephasing decay profiles are used to test a recent viscoelastic theory of dephasing [Chem. Phys. 233, 257 (1998)], which attributes the perturbation of the vibrational frequency to shear fluctuations in the liquid. A second dephasing process must be included to obtain reasonable fits in the viscosity range below 10 cP. The second process is consistent with a Gaussian frequency modulation whose modulation time is weakly viscosity dependent or has no viscosity dependence. We speculate that this process is due to rapid rotation about the toluene z axis and that the same process also causes the oscillations seen at higher viscosity. There is a good correspondence between the value of the infinite frequency shear modulus found from fitting the Raman free-induction-decay data and the value recently found by Larsen, Ohta, and Fleming by fitting photon-echo-peak-shift measurements in toluene [J. Chem. Phys. 111, 8970 (1999)]. This correspondence supports the idea that both vibrational dephasing and nonpolar electronic solvation are caused by the same physical mechanism.

The shear modulus of bulk amorphous Pd 40Ni 40P 20 and its relation to viscosity and specific heat

We have measured the shear modulus and its temperature dependence of Pd 40Ni 40P 20 using an EMAT technique. Measurements were made of the temperature dependence at constant heating rate of the infinite frequency shear modulus G. Values of the shear softening parameter γ are estimated and compared with those obtainable from frequency dependent shear moduli, specific heat and viscosity measurements. There is overall agreement with the Interstitialcy Theory of Condensed Matter States.

The Shear Modulus of Vit 1 in the Supercooled Liquid and Glassy State

AbstractUsing a non-contact electromagnetic-acoustic transformation (EMAT) technique, we have measured the temperature dependence of the infinite frequency shear modulus of Vit 1 at constant heating rate in the glassy and supercooled liquid states. Values of the shear softening fragility parameter -dln(G/Gg/d(T/Tg) are compared with those obtainable from specific heat and viscosity measurements, using the interstitialcy theory of condensed matter states. There is overall agreement found between these independently measured values.

Brownian dynamics simulations of Lennard-Jones gas/liquid phase separation and its relevance to gel formation

Brownian dynamics computer simulations have been used to follow the phase separation behaviour of model adhesive colloidal systems quenched in temperature from a supercritical state into the two-phase coexistence region, CR, of the phase diagram. The systems studied consisted of monodisperse spherical particles interacting through either 12–6, 24–12 or 36–18 pair potentials. Systems quenched to state points just inside the two-phase boundary initially exhibit negligible domain growth or a period of ‘latency’ before slow decomposition, a feature associated with phase separation by nucleation. Systems quenched further into the two-phase region are characterised by immediate decomposition and more rapid growth of a characteristic length scale, behaviour characteristic of phase separation via a spinodal decomposition mechanism. Both domain size and peak height followed a power law growth with time over a period of the separation and some systems exhibit dynamic scaling of the structure factors. For systems at a volume fraction in excess of ca. 0.1 percolating interconnecting networks formed which manifested gel-like properties such as an increasing infinite frequency shear modulus, the appearance of a low-frequency shear modulus and more pronounced power-law behaviour of the stress time autocorrelation functions. The LJ clusters consist of compact regions separated by thin branches, whereas the shorter ranged potentials formed more tenuous gel-like structures. Phase separation was slower for the shorter range attractive potentials.

Diffusivity and Nuclear Spin Relaxation Measurements at High Pressure in Methanol

ABSTRACTDiffusivity D and nuclear spin relaxation times T1 and T2 have been measured by NMR to 4.0 GPa in methanol, using a diamond anvil cell probe. In pure MeOH, D−1 and T2 show essentially identical activation volumes. However, these are ∼18% larger than the activation volume of viscosity. By relating these observations to an average molecular correlation time a pressure-dependent infinite-frequency shear modulus G∞ can be inferred, using two independent approaches. The relation between diffusivity and viscosity shows increasing departure from Stokes-Einstein behavior with increasing pressure, if a constant hydrodynamic radius is assumed. This departure is attributed to the pressure dependence of G∞, and can be described empirically by a simple modification of the Stokes-Einstein relation.

Isomerisation dynamics of a thiacarbocyanine dye in different electronic states and in different classes of solvents

A study of the isomerisation dynamics of 3,3′-diethylthiacarbocyanine in the excited state in alkanenitriles as well as in the ground state in both alkanenitriles and n-alcohols, using ps transient grating spectroscopy and flash photolysis is reported. The intrinsic activation energy for isomerisation has been determined from isoviscosity plots. In the ground state, the activation energy amounts to 12.9 kcal/mol in both nitriles and alcohols, while it is equal to 3.0 kcal/mol in the excited state. The reduced isomerisation rate constant shows a fractional dependence on the solvent viscosity, with a fractional exponent α of 0.20 and 0.38 for the ground state isomerisation in alcohols and nitriles and of 0.65 for the excited state isomerisation. The data have been analysed within the framework of the Kramers theory using different models for friction. The best agreement was obtained with the Grote-Hynes model, which takes into account the frequency dependence of the friction. The solvent dependence of α for the ground state isomerisation is accounted for by a different average value of the infinite frequency shear modulus in alcohols and nitriles. The barrier height dependence of α is explained by a decrease of the barrier's top frequency by going from the ground to the excited state.

A 1H NMR study of glycerol at high pressure

Relative proton chemical shifts and spin relaxation times T 1 and T 2 have been measured at 400 MHz and room temperature, as functions of pressure using the diamond anvil cell. Hydroxyl proton chemical shifts indicate a substantial increase in hydrogen bonding with increasing pressure, to 10 kbar. From T 1 data, a minimum is inferred at a negative pressure of ∼ − 2.5 kbar, yielding an average molecular correlation time in excellent agreement with a value based independently upon extensive recent measurements of glycerol viscosity. It is concluded that over the observed pressure range NMR correlation times are essentially identical to shear (viscosity) relaxation times in glycerol, and that a single, well-defined average structural correlation time, τ, determines all relaxation phenomena in this liquid. Dielectric susceptibility and Brillouin scattering line width measurements support this conclusion, the former to very high (∼ 20 kbar) pressure. It is also found that values of the spin-spin relaxation rate T 2 −1 yield approximately 2.2 × 10 11 s −2 for the coupling constant between this rate and the correlation time, τ, under slow-hopping conditions, and that differences between observed pressure dependences of viscosity and spin relaxation rates are well accounted for by the pressure dependence of the infinite-frequency shear modulus, G ∞.

Physical properties of liquid water by molecular dynamics simulations

We Calculate the temperature dependence of the thermodynamic, structural and dynamic properties of the simple point-charge model, SPC/E, water. We compute for the first time the infinite-frequency shear modulus and the shear viscosity of the model water. The self-diffusion coefficients are approximately twice the experimental value and the shear viscosities are correspondingly about half the experimental value, indicating that the model is under-estimating the strength of hydrogen-bond formation within the liquid.