Experimental evidence of a state-point dependent scaling exponent of liquid dynamics

A large class of liquids have hidden scale invariance characterized by a scaling exponent. In this letter we present experimental evidence that the scaling exponent of liquid dynamics is state-point dependent for the glass-forming silicone oil tetramethyl-tetraphenyl-trisiloxane (DC704) and 5-polyphenyl ether (5PPE). From dynamic and thermodynamic properties at equilibrium, we use a method to estimate the value of $\gamma$ at any state point of the pressure-temperature plane, both in the supercooled and normal liquid regimes. We find agreement between the average exponents and the value obtained by superposition of relaxation times over a large range of state-points. We confirm the state-point dependence of $\gamma$ by reanalyzing data of 20 metallic liquids and two model liquids.

Decreasing the temperature (T ) or increasing the density (ρ) by applying pressure of liquids lead to slowing down of the molecular dynamics and eventually a glass transition if crystallization is avoided [1,2].It has been demonstrated that for numerous low molecular weight liquids and polymers, the relaxation time or other dynamic quantities can be superimposed onto a master curve within the experimental uncertainty when plotted as a function of ρ γ /T .The scaling exponent γ is thus sometimes referred to as a material constant [3][4][5].In this letter, we show that γ is in fact state-point dependent and can be measured at a single state point.In the following we will use subscripts to distinguish definitions of γ's.
Let τ be the structural relaxation time measured from the loss-peak frequency of the electric permittivity.We will express τ in reduced units of m/k B T ρ 2/3 where m is a atomic/molecular/polymer-segment mass and define the scaling exponent of τ as [6] In this general definition the exponent depends on the state point and the fixed quantity (τ in the above case).
A physical interpretation of γ τ is that it quantifies the relative contribution of volume and thermal energy to the temperature evolution of the molecular mobility [7].Similar to the structural relaxation time, configurational adiabats can also be associated with a scaling exponent: In general, two scaling exponents of different observables (structural, dynamical or thermodynamic) will have different values.However, if the system has so-called "hidden scale-invariance" [6,8] then all scaling exponents of different properties will have the same value.In Ref. [9] it was experimentally shown that γ τ = γ Sex at one state point of the silicone oil DC704.This result is spectacular since hidden scale-invariance can only be present in a class of systems [10,11].This class is believed to include systems where van der Waals (vdW) interactions dominate, but exclude systems where hydrogen-bondes (HB) dominates the Hamiltonian [12].The isomorph theory [6,8,10,12] is a framework for describing systems where the potential energy function U (R) possesses hidden scale invariance (here, R is the collective coordinate of the system).Formally hidden scale invariance can be formulated as the following criterion for two configurations a and b: if [8].For these systems, scaling exponents of many properties have the same state-point dependence.Thus, there is only one scaling exponent that give the slope of the so-called isomorphs [6] along which dynamical, structural and many thermodynamic properties are constant in reduced units.As an example of applications, the framework of the isomorph theory explains Rosenfeld's excess entropy scaling law [13][14][15][16], i.e. that relaxation time is a function of entropy: τ (S ex ), by equating Eqs. 1 and 2, and recently the framework has been used to make predictions for properties of the melting line [17].For the latter, the state point dependence of the scaling exponent is an essential ingredient.
As a special case, Alba-Simionesco, Kivelson and Tarjus (AKT) have investigated the validity of a scaling law for activated dynamics where the scaling exponent only depends on density [18][19][20], but not on temperature.Regarding γ a material constant is an even more constraining assumption, which is only valid if the potential part of the Hamiltonian can be approximated by a sum of inverse power-laws (IPL) r −n pair interactions plus an arbitrary constant (the IPL hypothesis) [3,[21][22][23][24][25]. Then the scaling exponent is independent of state-point (γ IPL = n/3) and the relaxation time falls on a master curve when plottet along ρ γ /T .In this letter we wish to investigate the state point dependence of γ without any assumptions and emphasize error estimates.This information can be used to determine whether: a) γ is constant (the IPL hypothesis), b) γ is only a function of ρ (the AKT hypothesis), or c) γ is a function of two thermodynamical variables (the generalized framework of the isomorph theory).
To this aim, we first give an expression for γ τ in terms of quantities that can be measured at isobaric or isothermal conditions where most experiments are performed.We define a generalized fragility [5,7,20,[26][27][28], inspired by Angell's suggestion [26]: where A and B are thermodynamic variables such as T , ρ or pressure p.We note that the temperature-fragility m T p (which is negative) is related to the apparent activation enthalpy [5] , and when the temperature-fragility m T p is evaluated at the glass transition temperature T g at ambient pressure (0.1 MPa) it is related to the dimensionless fragility index originally proposed by Angell [29]: p=0.1 MPa .The isochoric activation energy E V [5] can also be expressed via the generalized fragility: ρ .Note that we have not defined the generalized fragility as a dimensionless Angell-type index.
Below we consider the pressure-fragility m p T and temperature-fragility m T p as the quantities that are directly experimentally accessible.We rewrite γ τ , Eq. 1, in terms of the ratio between two generalized fragilities by using the thermodynamic identity However, typically the pressure, and not the density is controlled in an experiment.Thus, we wish to have an expression involving m p T and m T p .From the chain rule , it follows that where , and inserting the thermal-expansion Finally, by combining the three numbered equations above we arrive at an expression that relates γ τ to directly measurable properties at equilibrium without needing the superposition of relaxation times: Thus, the state point dependence of the scaling exponent γ τ can be obtained from the thermal-expansion coefficient (α p ), pressure and temperature fragilities (m p T and m T p ) and isothermal bulk modulus (K T ).The idea of computing γ from two fragilities has been applied in other studies [5,7,18,28].In this study we suggest to generalize this approach to the entire liquid state away from the glass transition.Due to the measuring methods favored in our lab [30], we choose to get the value of the isothermal bulk modulus from the adiabatic modulus K S measured by the speed of sound [30]: where C p is the isobaric heat capacity measured with differential scanning calorimetry.
We focus our investigation to two well-studied van der Waals liquids, tetramethyl-tetraphenyl-trisiloxane (DC704) and 5-polyphenyl ether (5PPE), with values of T g at atmospheric pressure of 211 and 245 K respectively [32].As an example, in Fig. 1 we collect the quantities we need to calculate γ τ by using Eq.7 where K T is computed from K s and C p as mentioned above.Figure 2(a) show the relaxation times at atmospheric pressure for the silicone oil DC704, and Fig. 2(b) displays γ τ at four temperatures along the 0.1 MPa isobar.
To evaluate to what extent the measured quantities contribute to the final error of γ τ and, in this way, to predict where one should give particular attention to reduce as much as possible the uncertainty of the data, we use statistical tools for analyzing the results.A large population of values for each variable (N = 10 5 ) are sampled by using a Monte Carlo approach, assuming a normal distribution centered about its mean within an interval determined by the corresponding standard deviation.The error is calculated by sampling a random collection of different scenarios for the variables in Eq. 7.
The P γ , i values in the panels of Fig. 1 represent the pairwise Pearson correlation coefficients between γ τ and the variables involved in the computation of Eq. 7. If P γ , i equals 100%, there is a total positive correlation and 0 would indicate absolute lack of correlation.The property that has the strongest correlation with γ τ is the pressure fragility, m p T , followed by the temperature fragility, m T p .It is therefore recommended to measure the generalized fragilities with high accuracy in order to reduce the uncertainty in γ τ .On the contrary, slight variations in the density ρ and in the thermal-expansion coefficient α has a minor effect on the resulting values of γ τ .The final error estimate on τ are shown as errorbars on Fig. 2(b) suggesting an increase of γ τ as temperature increases or the density is decreases.
To challenge the robustness of our approach, we collect in Table I the values of γ τ for other systems, including two hydrogen-bonded liquids.We also compare our results with other procedures, such as the superposition of relaxation times (γ IPL ), the expression proposed by Casalini and Roland (γ * ), and an experimental prediction of the isomorph theory (γ isom ) mentioned in the introduction.The route to the scaling exponent proposed by Casalini and Roland, in which continuity of the entropy at the glass transition is assumed, has been shown to work effectively in several van der Waals and hydrogenbonded glass-forming liquids, as well as in polymeric materials [5,25].From Table I, one observes that the values of γ's obtained through these three methods are in fair agreement with the average value we would obtain using Eq.7 from the results at different state points.Table I includes information on the class of liquid and the method utilized for obtaining the bulk modulus.
Figure 3 shows that the scaling exponents is also statepoint dependent for model liquids and metals.Figure 3(a) shows γ Sex computed from molecular dynamics simulations [41] of the Lennard-Jones (LJ) liquid [42] and a Lennard-Jones trimer suggested by Lewis and Wahnström [43] as a coarse-grained model for ortho-terphenyl (LW-oTP).In line with the experimental findings, the LW-oTP model show an increase of the exponent with  temperature, though less dramatic.In Fig. 3(b) we reanalyze experimental data for 20 metallic liquids [44], including metals where ab initio density functional theory calculations show hidden-scale invariance [45].For the mono atomic metallic liquids, the scaling exponent of excess entropy is estimated using , where γ G = α p K T /ρc v is the thermodynamic Grüneisen parameter [46] (it is assumed that the material is above the Debye temperature [11,45]).The exponents have significant temperature dependencies with both positive and negative slopes along the p = 0.1 MPa isobar.
These results are in line with the state point dependence of the scaling exponent γ by simulations of Kob-Andersen binary Lennard-Jones liquids [47], and also consistent with the generalized scaling equation of state reported in ref. [48].The latter study proposes a density dependence function for the scaling exponent with two parameters that can be estimated from their generalized density-scaling equation of state [48].In this way, it is possible to determine the evolution of γ with density from PVT measurements as it was recently reported for several organic liquids [49].
An advantage of Eq. 7 is that it allow the determination of γ τ for any kind of liquid regardless of its vitrification ability, and the formalism is not assuming activated dynamics [18].Thus, we suggest a procedure that can potentially be used at any given point in the temperaturepressure plane.We note that the average γ τ found here  (6.1 ± 1.1 for the investigated state points of the silicone oil DC704) is in good agreement with the exponent estimated by other methods: i) the results obtained by superposing relaxation times [9], ii) the experimental prediction of the isomorph theory [9], and iii) the value obtained through the equation derived by Casalini and Roland [5], confirming the validity of Eq. 7 for the universal prediction of γ τ .In experimental studies, changes in the density are usually small, so in the scaling relation, X = f (ρ γIPL /T ), the exponent is generally assumed to be state-point independent, with X a measure of the molecular mobility and f a function that is a priori unknown.Nonetheless, within the framework of the isomorph theory, the exponent may depend on the state point as it was corroborated by computer simulations of strongly correlating liquids (see Fig. 3), that is, those exhibiting hidden scale invariance such as van der Waals molecules and metals [12,47].
In summary, we have shown that it is possible to determine the scaling parameter γ τ from dynamic and thermodynamic properties of liquids at a single state-point, and that γ τ can be state point dependent.The expression presented in Eq. 7 connects γ τ to measurable quantities that can be potentially estimated in a wide range of thermodynamic conditions, from state points near the glass transition to well above the melting point (including elevated pressures).This new route is free of assumptions and can be utilized for both glass-and non-glass-forming materials.

I. ACKNOWLEDGMENTS
This study was initiated after a series of discussion between Niels Boye Olsen and URP.We thank Jeppe C. Dyre for valuable input to the manuscript.This work is supported by the VILLUM Foundations VKR-023455 and Matter (16515) grants.

FIG. 1 .
FIG. 1. Experimental dynamic and thermodynamic properties measured by our group and used for calculating the exponent γτ at two different state points, (228 K, 0.1 MPa) and (243 K, 0.1 MPa), for the silicone oil DC704.(a) Pressure fragility at two isotherms as a function of pressure from dielectric relaxation measurements.(b) Temperature fragility at 0.1 MPa as a function of temperature from dielectric relaxation measurements.(c) Adiabatic bulk modulus from the standing waves in a bulk transducer.Red crosses highlight the values of Ks used for estimated γτ at four different temperatures.(d) Thermal expansion coefficient extracted from PVT measurements.(e) Temperature evolution of density for liquid DC704 from the Tait equation (solid line) and from the extrapolation of the values at room conditions by using the expansion coefficient (dashed line).(f) Heat capacity as a function of temperature for liquid DC704 from DSC measurements.Blue circles included in panel (f) corresponds to literature data [31].The Pγ,i values in each panel indicate the contributions to the statistical error on the estimate of γτ at T = 288 K.

FIG. 2 .
FIG. 2. (a)The structural relaxation time τ of DC704 measured by the dielectric loss.(b) Density scaling exponent (Eq.7) as a function of temperature at atmospheric pressure (0.1 MPa).Black line indicates the average value of γτ and the red line is a guide to the eye.The inset displays γτ as a function of density.Blue line is a guide to the eye.

FIG. 3 .
FIG. 3. (a) Comparison of experimentally determined scaling exponents (open symbols) with values of two model liquids (filled symbols).(b) The scaling exponent of the configurational adiabat γS ex for liquid metals.

TABLE I .
Density-scaling exponent γτ at p = 0.1 MPa for selected substances obtained by using Eq.7 and its comparison with other methods.Scaling exponent computed in this study via Eq.7.b Scaling exponent computed with superposition of relaxation times (assuming the IPL hypothesis for pair interactions).c A state-point independent scaling exponent calculated by Casalini et al. from static ambient-pressure quantities.d Scaling exponent calculated from static properties using an expression derived from isomorph theory (DC704 at 214 K, 0.1 MPa). a