Adam-Gibbs model in the density scaling regime and its implications for the configurational entropy scaling

To solve a long-standing problem of condensed matter physics with determining a proper description of the thermodynamic evolution of the time scale of molecular dynamics near the glass transition, we have extended the well-known Adam-Gibbs model to describe the temperature-volume dependence of structural relaxation times, τα(T, V). We also employ the thermodynamic scaling idea reflected in the density scaling power law, τα = f(T−1V−γ), recently acknowledged as a valid unifying concept in the glass transition physics, to differentiate between physically relevant and irrelevant attempts at formulating the temperature-volume representations of the Adam-Gibbs model. As a consequence, we determine a straightforward relation between the structural relaxation time τα and the configurational entropy SC, giving evidence that also SC(T, V) = g(T−1V−γ) with the exponent γ that enables to scale τα(T, V). This important findings have meaningful implications for the connection between thermodynamics and molecular dynamics near the glass transition, because it implies that τα can be scaled with SC.

The phenomenon of glass transition is an important and intriguing area of research in condensed matter physics, which is continuously attracting a lot of the researchers attention. A key problem in this field is to develop the physical model that will be able to describe the evolution of the structural relaxation time, τ α , or alternatively viscosity, η, on approaching the glass transition. So far, the most efforts have been devoted to the analysis and the correct description of the temperature dependence of τ α at ambient pressure. In this context, the question is often raised whether the structural relaxation dynamics diverges at some finite temperature. However, the major challenge is to deliver the appropriate equation of state, i.e., the analytical expression for τ α in the full (temperature-pressure-volume) thermodynamic space. This challenge is driven by the fact that the experimental studies of the molecular dynamics of glass-forming systems at elevated pressure are now possible in many laboratories around the world.
Among variety of models proposed for description of the temperature dependence of τ α , the entropy-based model formulated by G.Adam and J.H.Gibbs (AG) 1 has become one of the most discussed as reflected in the number of citations (cited more than 4000 times). This seminal work provides a connection between thermodynamic and dynamic quantities. According to this model, the structural relaxation time of supercooled liquid is controlled by the configurational entropy S C (T) which determines the size of cooperatively rearranging regions (CRR) and represented by the following formula Here, S C (T) is defined as the configurational entropy, and estimated as the difference between the entropy of the melt and the vibrational contribution to the entropy (rather from the glass than from the crystal) 2 , S C (T) = S melt − S vib . The constant A is related to the intermolecular potential and is proportional to the free energy barrier (per molecule in CRR) for rearrangement Δ μ while τ AG is the value of structural relaxation time in the limit of high temperatures. CRRs are defined as the smallest volume elements that can relax to the new configurations independently of their environment. In the last decade, a lot of interest has been directed toward the analysis of molecular dynamics of supercooled liquids in terms of thermodynamic scaling [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] . This alternative approach is very appealing due to the possibility of universal description of relaxation phenomena for all supercooled liquids based on the generalized Lennard-Jones potential 18 . According to the thermodynamic scaling, some dynamic quantities can be scaled into a single master curve if they are plotted versus T −1 V −γ , where T is the system temperature; V is the system specific volume and γ is the scaling exponent 13,14 The variable x denotes one of the dynamic quantities, such as the structural relaxation time τ α , viscosity η, or other dynamic property. The key quantity is here the scaling exponent γ. At the first stage of development of the thermodynamic scaling approach, it was postulated 19 that the value of scaling exponent should be equal to 4 in accordance with the initial finding for OTP 19,20 . However, it has been subsequently demonstrated by a number of research groups that the value of γ can significantly differ from 4 for other glass formers. For example, for van der Waals liquids: PDE: γ = 4.5, BMPC: γ = 7.0, BMMPC: γ = 8.5 [12][13][14][15]17,21,22 ; for polymers: 1.9 ≤ γ ≤ 5.6 [23][24][25][26][27] ; for ionic liquids: 2.25 ≤ γ ≤ 3.7 12,16,28 ; for substances with hydrogen bonds: sorbitol: γ = 0.13 12,13 , salol: γ = 5.2 12,13,17,21 .
Major advances in understanding of the molecular basis of thermodynamic scaling and its relation to macroscopic thermodynamic properties of viscous systems have been possible to achieve by performing molecular dynamics (MD) simulations. Assuming that a short range effective intermolecular potential can be approximated by a combination of dominating repulsive inverse power law and small attractive background, the validity of the thermodynamic scaling was demonstrated on the basis of MD simulations 7,16,[29][30][31][32][33][34] where ε, σ are respectively the potential well depth and the finite distance of the zero potential, which are the typical parameters of the Lennard-Jones potential, and A t is a small attractive background. Moreover, it was pointed out that the parameter γ IPL can be identified with the scaling exponent γ in the thermodynamic scaling law (Eq. (2)). This straightforward connection between both exponents made the thermodynamic scaling very attractive approach.
In this context, it is natural to ask how the thermodynamic scaling is incorporated into the AG model. Answering this question requires converting the temperature-dependent AG model (AG(T)) to its T-V representation AG(T, V). Furthermore, the generalization of the AG(T) model to T-V variables might be essential for testing its validity in general.
In this paper, we propose an extension of the original AG model (Eq. (1)) taking into account the combined effect of temperature and volume changes on τ α . Consequently, we aim to verify whether or not the concept of thermodynamic scaling is consistent with the modified Adam-Gibbs model. These considerations lead us to very important implications for the temperature-density scaling rule for the configurational entropy S C and the well-grounded relation between τ α and S C .

Theory
In order to apply the AG model, presented by Eq. (1), it is essential to know the form of S C . The temperature-dependent form of S C , originally proposed by Adam and Gibbs, is given by The difference between the isobaric heat capacity of the liquid and crystalline (or the glass) phase varies inversely with temperature, Δ C P = K P /T, which was found for several glass-formers by R. Richert and C. A. Angell 35 by comparing the behavior of the dielectric relaxation time with the experimentally obtained configurational entropy. K P is a constant parameter, T k is Kauzmann's temperature and S ∞ = K P /T k is the limiting value of S C at very high temperature.
The AG(T) expression (Eq. (1)) can also be derived by considering both temperature and pressure dependence of S C . The configurational entropy decreases on cooling or with an increase in pressure. Thus, the consideration of the dependence of S C also on pressure, not only on temperature, is Scientific RepoRts | 5:13998 | DOi: 10.1038/srep13998 very essential. This problem was investigated by Casalini et al. 36 by adding the term S isoth to Eq. (4), S C (T, P) = S isobar + S isoth , which explicitly involves the isothermal pressure variation of thermal expansion, If the dependence of the configurational entropy on volume and temperature, i.e., S C (T, V) is known, a much more direct way to test the connection between AG(T, V) and the thermodynamic scaling is to study the molecular dynamics by T − V representation of Eq. (1).
To determine the T − V version of Eq. (1), we consider the system entropy as a function of temperature and volume, the total differential of which is given as follows Using the well-known Maxwell's thermodynamic relationship, (∂S/∂V) T = (∂P/∂T) V , and (∂S/∂T) V = C V /T, Eq. (6) can be rewritten as which leads to a temperature-volume function for configurational entropy, where the first integral with respect to temperature is calculated from the difference between the isochoric heat capacity of the melt and the vibrational contribution to the isochoric heat capacity (rather from the glass than from the crystal) 2 It also can be approximately described over a limited range by K V /T, with a constant K V , similarly to its isobaric counterpart 36 . The second integral with respect to volume constituting S C (T, V) is calculated from the difference between the temperature derivatives of pressure of the melt and the contribution related to the vibrational modes (rather from the glass than from the crystal) 2 , Here, we assume that the vibrational part of the difference between the temperature derivatives of pressure is constant and can be regarded as a fitting parameter. The assumed lower limits of the integrals are respectively Kauzmann's temperature, T k , and the volume at Kauzmann's temperature for the examined material, V k . The pressure dependence of temperature at a constant volume can be estimated by using an equation of state (EOS) 37 is the isothermal bulk modulus at a reference pressure P 0 , parameterized by an exponential temperature function as is the volume at the reference pressure parameterized by a quadratic temperature function, is the glass transition temperature at P 0 , and γ EOS is a material constant independent of thermodynamic conditions. It is worth noting that the exponent γ EOS is also related 8,[37][38][39][40] to the exponent γ IPL in the effective short-range intermolecular potential given by Eq. (3). However, in case of real materials, we have observed 8,9,28,[37][38][39] where γ is the density scaling exponent for dynamic quantities in terms of Eq. (2). This discrepancy can be argued 8,41 by considerably different values of the density scaling exponent for the dynamic quantities and the total system entropy, which have been established for real glass formers.
In Eq. (9), A 0 , A 1 , A 2 , b 0 , b 2 , γ EOS are fitting parameters. Defining new quantities δ = − b 2 B T (P 0 )/γ EOS and ω = A 1 + 2A 2 (T − T 0 ), the melt part of difference of the temperature derivative of pressure is then given by Consequently, the integral of Eq. (10) takes the following form Finally, taking into account the classical AG equation (Eq. (1)) and the expression S C (T, V), we obtain the AG(T, V) representation for τ α (T, V) If the structural relaxation time and the configurational entropy obey the thermodynamic scaling law in the form of the power law density scaling, τ α = F(TV γ ) and S C = G(TV γ ), then a consequence of the thermodynamic scaling hypothesis for the elementary activation energy in the material-specific coefficient A of Adam-Gibbs approach (Eq. (1)) is that it is expected to be not a constant but to comply with a power law dependence of volume in the form A = A(V) → A′ V −γ . The scenario for the volume (or density) dependence of A in the AG equation was postulated by C. Alba-Simionesco et al. 42 , but it was not tested. Using the Kob-Andersen binary Lennard-Jones mixture, an explicit simulation tests of TV γ -scaling of S C and τ α in terms of the AG model as well as the scaled volume dependent change in A(V) was successfully performed by S. Sengupta et al. 43 . Following this approximation, we propose the second formula for τ α (T, V), which is a modified Eq. (13) by involving the volume contribution to the parameter A where the scaling exponent γ is computed from standard methods. The AG(T, V) model, in this form, is a good candidate to be a TV γ -scaling model. It should be noted that Eq. (14) with the density dependent parameter A also satisfies some earlier suggestions (e.g. that made by Johari 44 ) that the parameter A should depend on pressure.

Results and Discussion
In order to verify the equations (13) and (14), we have carried out the high pressure dielectric spectroscopy studies of simple van der Waals liquid, Tributyl-2-acetylcitrate (TBAC) with the aim to determine the temperature and pressure dependence of structural relaxation times. Dielectric spectra were measured both at isobaric (0.1 and 200 MPa) and isothermal conditions (199.0 K, 202.5 K, 205.9 K, 209.0 K, 212.9 K, 216.5 K, 225.9 K and 240.7 K) over a wide frequency range from 10 −2 to 10 6 Hz. In Fig. 1a and Fig. 1b, we show a number of representative dielectric loss spectra obtained at various temperatures at ambient pressure and as a function of pressure at constant temperature, T = 216.5 K, respectively. Lowering temperature has a similar effect as increasing pressure, i.e., in both cases, the relaxation peaks moves to lower frequencies. By analyzing all the measured spectra, we have determined the temperature and pressure dependence of the structural relaxation times, which were calculated from the inverse of the frequency corresponding to peak maxima, τ π = ( ) α − f 2 max 1 . Eventually, using structural relaxation times in various thermodynamic conditions (T and P), it was possible to construct the 3D relaxation map depicted in Fig. 2.
As a next step toward the experimental verification of the equations (13) and (14), it is necessary to convert the T-P data to their T-V representation. Therefore, apart from the high pressure dielectric studies, we additionally performed PVT measurements. Figure 3a displays the experimentally obtained temperature dependences of specific volume V(T) isobars at labeled pressures, in the range of 10 MPa-200 MPa. The experimental PVT data for TBAC were satisfactorily parameterized by means of the EOS equation of state (Eq. (9) -solid lines) with the following values of its fitting parameters as well as errors and relative errors of their determination: .09 ± 0.02 [Δ r γ EOS = 0.2%], assuming the reference state at a fixed glass transition temperature T 0 = 186.06 K at ambient pressure. The value of adjusted R 2 is equal to 0.99998. The above set of data enables us to convert τ α (T, P) to τ α (T, V), and finally to construct 3D or 2D plots of the structural relaxation times versus T and V, required to perform the test for the validity of the AG(T, V) model. The best 3D numerical fit of τ α (T, V) for TBAC data to Eq.  Here, for TBAC, T k = 156.29 K and V k (T k , P 0 ) = 0.8482 cm 3 /g. In this context, it is worth noting that this equation basically has the same number of free fitting parameters as the previous one, because the additional parameter γ in Eq. (14) was determined from the criterion for the density scaling 45,46 . The value of the scaling exponent γ, required to construct the thermodynamic scaling plot, was determined from the linear regression of τ α T log 10 against τ α V log 10 at a few constant structural relaxation times (Fig. 3b). As can be seen in Fig. 3b, the best linear fit was achieved for the value γ = 3.17 ± 0.01. Using this value of the exponent γ, we constructed the scaling curve by plotting the structural relaxation   Eq. (9)). The inset (b) presents double logarithmic plot of the temperature τ α T versus the volume τ α V at τ α = const for several relaxation times τ α , which has been used to found the value of the scaling exponent γ = 3.17 ± 0.01 as a slope of the linear dependences.
times versus the product of the temperature T and the specific volume V raised to the exponent γ. It is obvious from Fig. 3 that all the scaled experimental isobars and isotherms collapse onto a single master curve. This result is in accord with a general observation of the validity of thermodynamic scaling for van der Waals liquids.
The volume dependence of isothermal and isobaric structural relaxation times determined from dielectric measurements and the best fitting curves obtained using Eq. (14) are displayed in As can be seen, a satisfactory agreement between fits and the experimental points has been achieved. The quality of the fits to Eqs. (13) and (14), imply that the TV-generalized AG model, represented by both the two equations, provides a satisfactory description of experimental data. Analyzing errors of determination for each fitting parameter, we can also confirm the statement that a satisfactory description of experimental data is achieved, because most relative errors of determination of the values of the fitting parameters are less than 3%, and only the error of determination for the value (∂ /∂ ) P T V vib exceeds the value of this parameter in case Eq. (13), while Eq. (14) is devoid of such a problem. These facts can have influence on further discussion about correctness and applicability of both equations. Nevertheless, a comparison of the values of the adjusted R 2 obtained from fitting experimental data to Eq. (13) (Adj. R 2 = 0.99894) and Eq. (14) (Adj.R 2 = 0.99927) seems to indicate that both the equations lead to the same outcome. Does it indeed mean that both the equations are internally consistent with the thermodynamic scaling concept?
To answer this question we refer to our recent findings reported in ref. 47. As we pointed out there, one can formulate some general rules for isobaric m P T and isochoric m V T fragilities, i.e., (i) compression brings about the decrease in the isobaric fragility m P T and (ii) the isochoric fragility m V T is an invariant parameter with pressure. They are both valid if the density scaling is satisfied. Consequently, on the basis of above rules, we will be able to check the correctness of the derived equations, because if they work the appropriate trend in the fragilities behavior should be reproduced. The isobaric and isochoric fragilities can be defined in the following way where x stands for either P or V, depending on the thermodynamic conditions. Analyzing the temperature dependences of the structural relaxation time at constant pressures, we found for TBAC that m P , calculated from Eq. (13), systematically increases with increasing pressure. It is shown in Fig. 6 (solid  squares). Similarly, we calculated the isochoric fragility m V from Eq. (13) and tested it as a function of   to m P = 82.17 at P = 200 MPa, which agrees with the general trend found in case of van der Waals liquids. In addition, we have established that m V is pressure-independent within error bars and has a constant value equal to 59.34 at investigated pressure range (open circles in Fig. 6). From the comparison of the isobaric and isochoric fragilities, obtained from Eq. (13) and Eq. (14), we can see that these equations lead to both quantitatively and qualitatively different results. The above analysis unambiguously shows that the appropriate form of AG-model transformed to the T-V thermodynamic space is that given by Eq. (14), which complies with the following compact representation It should be stressed that the AG(T, V) model expressed by Eq. (14) has turned out to be consistent with the thermodynamic scaling idea, although Eq. (14) has been derived without any scaling assumptions for the configurational entropy S C (T, V), because the latter has been employed in Eq. (16) by using Eq. (12). This finding raises a subsequent important question concerning the scaling of the configurational entropy. According to this, S C calculated from Eq. (12) should be possible to collapse onto a single curve by plotting it as a function of TV γ . Thus, our next step is to check whether or not the configurational entropy S C , similarly to the structural relaxation time τ α , satisfies the TV γ -scaling rule. Prior to doing that, we verify the results given by Eq. (12) with the values of its parameters taken from the fitting experimental dependence τ α (T, V) to Eq. (14). In Fig. 7, we present the temperature dependence of isobaric heat capacity at ambient pressure obtained from the differential scanning calorimetry (DSC) measurements with stochastic temperature modulation (TOPEM). Based on this data, we determined the configurational entropy (see the inset in Fig. 7) by taking the definite integral ∫ (Δ / ′) ′ , where the value of T k is the same as that assumed to fit the dielectric structural relaxation times to Eqs (13) and (14), and Δ C P is taken as the difference between two linear functions describing respectively temperature behavior of C P liquid and C P glass . In the inset in Fig. 7, we also show a very satisfactory agreement between the dependence S C (T) determined from the heat capacity measurements (solid squares) at ambient pressure and that obtained from Eq. (12) (open circles) at ambient pressure, based on the same values of its fitting parameters as those which very well describe the dependence τ α (T, V) in terms of Eq. (14). After this additional confirmation of the validity of Eq. (12), we analyze the temperature and volume dependences of the configurational entropy (see Fig. 8a), and plot the dependences C at a few constant S C (Fig. 8b). As can be seen, these dependences have a linear character. From the simple linear regression, we have found that the value of the slope of all the isoentropic lines (γ = . ± .  The inset shows comparison of the dependence S C (T) determined from the heat capacity measurements at ambient pressure with that calculated from Eq. (12) with the values of its parameters found from fitting τ α (T, V) to Eq. (14).
that we are able to scale the configurational entropy (see Fig. 8) with the value of the scaling exponent, which very well corresponds to that established for the structural relaxation time (γ γ = S C ).

Conclusions
The latter finding has very important implications for making a final identification of the role of entropy in the thermodynamic evolution of the time scale of molecular dynamics near the glass transition. An important consequence of the found equivalence of the values of the scaling exponents γ and γ S C for the structural relaxation time τ α and the configurational entropy S C should be a subsequent scaling of τ α vs S C . As can be seen in Fig. 9, this scaling indeed occurs, because the structural relaxation times of TBAC can be plotted onto a single master curve as a function of the configurational entropy. This meaningful result clearly shows that the structural relaxation time can be a single variable function of the configurational entropy, τ α = h(S C ), although the more complex formula (Eq. (16)) is required to meet the power law density scaling criterion in terms of the AG model originally based on Eq. (1). An essential impact of the configurational entropy on the thermodynamic evolution of the time scale of molecular dynamics near the glass transition has been anticipated for many years. For instance, Wolynes and coworkers suggested 48,49 a function τ α = h(S C ) based on the random first-order transition theory and showed 50 the power law density scaling of S C using simulation data in a simple model based on the Lennard-Jones potential. To achieve the power law density scaling = ( ) for real glass formers, Casalini and Roland proposed 51-53 an alternative way to calculate S C , which requires determining reference values of S C along a chosen isochrone τ α = const. In this paper, for the first time based on experimental data analyses and without making any limiting assumptions for the configurational entropy calculations, we show that both the structural relaxation time and the configurational entropy follow the same pattern of the power law density scaling behavior, which relies on the same value of the scaling exponent γ, i.e., τ α = F(TV γ ) and S C = G(TV γ ). In this way, we solve a long-standing problem with determining the proper effect of thermodynamics on molecular dynamics near the glass transition. The found single variable function, τ α = h(S C ), which is a consequence of the density scaling laws, τ α = F(TV γ ) and S C = G(TV γ ), has a decreasing character (see Fig. 9) which implies that a decrease in the configurational entropy straightforwardly causes the dramatic slowdown in the molecular dynamics (reflected in the rapid increase in its time scale) near the glass transition. Thus, the configurational entropy seems to be sufficient to govern the structural relaxation of supercooled liquids without any contributions from additional factors. For comparison, we have very recently established 41 that such an exclusive impact is not made on the structural relaxation by the total system entropy S and the excess entropy S ex (defined as the difference between the total system entropy and the entropy of an ideal gas at the same density and temperature), although both S and S ex obey the density scaling law. For S and S ex , the values of the scaling 3 17

S C
. The insets present (a) temperature and volume dependences of S C , and (b) double logarithmic plot of the temperature T S C versus the volume V S C at S C = const for several configurational entropies S C , which has been used to found the value of the scaling exponent γ = . ± .
3 17 0 03 S C as a slope of the linear dependences. exponents have been found by us to be considerably different from that valid for τ α of a given glass former, and consequently the different values of the scaling exponents rationalize the decoupling observed by us between τ α and S (or S ex ) and imply that the relation between τ α and S (or S ex ) requires supplementing with an additional density factor 41 . In this context, our findings reported herein become especially useful for further investigations, because they suggest a way to formulate an optimal model of the thermodynamic evolution of the time scale of molecular dynamics of supercooled liquids, which is expected to be able to take a form of a single variable function of the configurational entropy S C or the scaling variable TV γ in the power law density scaling regime.