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2009, 113, 7419–7422 Published on Web 05/04/2009
Consequences of an Equation of State in the Thermodynamic Scaling Regime A. Grzybowski,* M. Paluch, and K. Grzybowska Institute of Physics, Silesian UniVersity, Uniwersytecka 4, 40-007 Katowice, Poland ReceiVed: February 4, 2009; ReVised Manuscript ReceiVed: April 11, 2009
An equation of state following from the generalized Lennard-Jones potential has been successfully applied to describe isothermal volumetric data for several glass-forming liquids on the assumption of the thermodynamic scaling validity. The scaling exponents γ evaluated from the fitting procedure for each tested material have turned out at least 2 times larger than those earlier found from relaxation data. The discrepancy indicates that the appealing idea of thermodynamic scaling, which has been recently intensively explored, requires more efforts to find a satisfying theoretical grounds for it. Thermodynamic scaling of molecular dynamics near the glass transition has been intensively studied by several groups1-7 for the last years. The attractive idea comes from an interesting finding established for many glass-forming liquids that all relaxation data for a given material can be plotted on one master curve described by a function f (T -1V -γ) unique to the material; e.g., different isobaric and isothermal structural relaxation times τ for a given material can be expressed in accordance with the scaling equation
log10(τ) ) f(T -1V -γ)
(1)
where γ is the scaling exponent which is a material constant. The scaling behavior of relaxation dynamics in the supercooled region, where cooperative and local dynamic processes play a main role, is usually related to a generalized Lennard-Jones potential8
[( σr ) - ( σr ) ]
ULJ(r) ) 4ε
m
n
(2)
r and especially to its repulsive part ULJ ) 4ε(σ/r)m,9,10 making an assumption that m ) 3γ. Although even some attempts4,11,12 have been already made to determine an analytical form of the scaling function f, new facts are still reported, which demand to look closer at the relation between eq 1 and the potential eq 2 in context of a state equation considering the thermodynamic scaling. Recently, Coslovich and Roland13 have shown by performing a molecular dynamics simulation of a model supercooled Lennard-Jones liquid that the scaling exponent γ found from the thermodynamic scaling procedure for the diffusion coefficient yields is slightly larger than the exponent of the repulsive term of the Lennard-Jones potential. Even more interesting results have been obtained by Dyre’s group.14-16 First of all, the authors have demonstrated in computer simulations of the Lennard-Jones liquid and other simple van der Waals liquids strong correlations between equilibrium fluctuations of the configurational parts of pressure and energy, which are not obeyed for strongly hydrogen-bonded liquids in accordance with the thermodynamic scaling tests based on experimental data for dipropylene glycol17 and water.9 Moreover, the correlations yield
10.1021/jp9010235 CCC: $40.75
an elegant way to determine the scaling exponent γ as a function of the exponent of the repulsive term of the Lennard-Jones potential. As a result of the procedure for ortho-terphenyl (OTP) typical van der Waals liquid, the value γ ≈ 8.0 has been evaluated, which is about 2 times larger than that suggested from inelastic neutron scattering18 and later established from the thermodynamic scaling of different relaxation data. From the reports, an important question arises of whether an equation of state directly following from the homogeneous repulsive part of the Lennard-Jones potential also gives a similar discrepancy in the value of the exponent γ in comparison with the scaling exponent previously found from relaxation or viscosity data. In this work, we exploit some state equation proposed by Bardik and Shakun,19 which is appropriate to investigate the problem. Their20 as well as Dyre’s group15,16 ideas originate from the commonly known expression for instantaneous pressure p ) NkT/V + W/V, where N is the number of species of momenta p1, ..., pN and positions r1, ..., rN in volume V, k is the Boltzmann constant, T ) T(p1, ..., pN) is treated as the “kinetic temperature” proportional to the kinetic energy per particle, and W ) W(r1, ..., rN) is the virial which yields a configurational contribution to pressure. In general, the virial can be calculated for a translationally invariant function of the potential energy U ) U(r1, ..., rN) by using the formula W ) -(1/3)∑i ri · ∇iU, as it is usually done in computer simulations.21 However, even in the case of simulations, we may take into account only the virial values which are averages in an appropriate thermodynamic ensemble practically evaluated as some averages in time. Within the framework of the thermodynamic perturbation theory, a configurational contribution to pressure Pr ) 〈W〉/V0 can be expressed analogically by means of the average perturbed virial with the canonical distribution function of a reference state whose volume equals V0. Starting from that and assuming that a repulsive part of the intermolecular potential is a homogeneous function with the homogeneity exponent m, like it is in eq 2, one can find an equation of state derived from such a molecular potential, which herein is presented for specific volume υ. The Euler theorem 2009 American Chemical Society
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Letters
Figure 1. Plots of isothermal PVT data for phenylphthalein-dimethylether taken from ref 24. Solid lines denote fits of the data to the equation of state given by eq 6. (a) Isotherms are of the uniform pressure range. (b) Isotherms that temperatures correspond to the temperature range of isotherms earlier taken into account in ref 3 to determine the scaling exponent typically from eq 1 by using dielectric data.
Figure 2. Plot of isothermal PVT data for ortho-terphenyl, which are generated by using the Tait equation with its parameters proposed in ref 2 and retaining the original measurement range reported in ref 25. Solid lines denote fits of the data to the equation of state given by eq 6.
on homogeneous functions applied to the potential U reveals a useful property of the intermolecular potential
U(sr1, ..., srN) ) s-mU(r1, ..., rN)
(3)
If a similarity r f sr is additionally considered, whose scale, s )(υ/υ0)1/3 ) (F0/F)1/3, can be expressed by specific volume υ or density F, where υ0 and F0 are, respectively, the specific volume and density in a reference state, one can propose a state equation for the elevated pressure region,20 P ) (RT/Mυ) + (υ0/υ)m/3Pr, where R is the gas constant, Mis the molar mass of substance, and Pr denotes in this case a configurational contribution to pressure given by a species of total mass equal to a unit mass of substance, which interact with each other by repulsive forces. However, the state equation is rather of low use for fitting PVT experimental data due to an undefined density-temperature dependence of the parameter Pr in the case of most real materials. Therefore, some generalized form of the state equation has been introduced,19 which herein is presented for specific volume υ
P)
[ ( ( ) )]
1 υ0 RT 1+φ Mυ kT υ
m/3
(4)
The unknown form of the φ-function can be derived by expanding in a series about a reference point with specific
Figure 3. Plot of isothermal PVT data for propylene carbonate, which are generated by using the Tait equation with its parameters proposed in ref 7. Temperatures of isotherms and the range of pressure are chosen in accordance with those included in volumetric data of PC reported in ref 26. Solid lines denote fits of the data to the equation of state given by eq 6.
volume υ0, which is assumed to be characterized by a low isothermal compressibility. Then, taking into account only terms of the series up to the first order, we obtain some local expression for pressure
P)
[ ( ( ) )| ( ) )|
(
1 υ0 m/3 1 RT 1 1+φ φ′ υ)υ0 + Mυ kT υ kT kT υ0 m/3 υ)υ0 × υ υ0 m/3 -1 υ
(( )
)]
(5)
Remembering that m ) 3γ and making additional assumptions of the configurational pressure in the reference state, P0 ) φ[(1/ kT)(υ0/υ)m/3]|υ)υ0(RT/Mυ), and that the first-order coefficient for the low compressibility region, where its density dependence can be neglected, is only temperature dependent, B(T) ) φ′[(1/ kT)(υ0/υ)m/3]|υ)υ0(R/Mυk), we can rewrite eq 5 as follows
P)
[( ) ]
υ0 RT + P0 + B(T) Mυ υ
γ
-1
(6)
which is the equation of state convenient to test the hypothesis
Letters
Figure 4. Plot of isothermal PVT data for meta-fluoroaniline obtained from the corresponding pressure dependences of density presented in ref 27. In order to keep the isothermal dependences sufficiently smooth, the plot is limited to the range of pressure where the Tait equation reported in ref 26 correctly describes volumetric data for m-FA. Solid lines denote fits of the data to the equation of state given by eq 6.
about one scaling parameter γ by using isothermal PVT data. We would like to mention that we know only two earlier attempts10,22 to discuss an equation of state in the context of the thermodynamic scaling but both of them did not involve any explicit form of term coming from a contribution of repulsive interactions. As theoretical investigations10,15,16,23 lead to a conclusion that an inverse power law approximation of intermolecular potential is appropriate to describe dynamic properties of dense, highly viscous liquids, it is well-founded to apply the equation of state (eq 6) to study the thermodynamic behavior of such materials. In our analysis, we consider volumetric data earlier published for phenylphthalein-dimethylether (PDE),24 ortho-terphenyl (OTP),25,2 propylene carbonate (PC),26,7 meta-fluoroaniline (m-FA),27,26 and 1,2-polybutadiene (1,2-PBD).28 The thermodynamic scaling of relaxation times on one master plot according to eq 2 has been validated for all of the materials which represent van der Waals liquids (PDE,2,3 OTP,1,2 PC7), weakly hydrogenbonded liquids (m-FA26), and polymers (1,2-PBD3). Values of the scaling exponent γ obtained from fitting relaxation data to
J. Phys. Chem. B, Vol. 113, No. 21, 2009 7421 eq 2 for most of those and several other materials have been collected in a review article by Roland et al.29 We have applied the equation of state (eq 6) derived in the thermodynamic scaling regime to fit isothermal PVT data for the mentioned materials. In accordance with the small compression assumption made for the reference state of the equation, the parameters P0 and υ0 have been established from the volumetric data, taking the smallest value of specific volume for each isotherm as the reference specific volume υ0 which corresponds to the highest pressure Pmax for the given isotherm, and then calculating the reference pressure P0 from eq 6 where P ) Pmax and υ ) υ0. In that way, the fitting procedure by means of eq 6, which has been applied to simultaneously approximate all sets of isotherms for a material, becomes more reliable because the only parameters required for each material are the parameters such as B(T) for each isotherm and the shared scaling exponent γ. Thus, a possibility of finding another satisfactory set of parameters is significantly reduced because the prior choice of (υ0, P0) for each isotherm strictly based on experimental data strongly determines the isotherm fits. As a result, which can be seen in Figures 1-5, the isothermal PVT data have been successfully described by the equation of state given by eq 6, and only in the case of m-FA the quality of fits is slightly lower, which is likely caused by not fully smooth experimental dependences P(υ). The values γ found for all of the considered materials by fitting the isothermal PVT data to the equation of state (eq 6) derived on the assumption of the thermodynamic scaling validity and those earlier evaluated1,3,7,26 from relaxation data according to the typical scaling procedure based on eq 1 are collected in Table 1 and denoted as γes and γr, respectively. The comparison of the values γes and γr shows an interesting but also worrying dependence on each other. The exponent of the state equation γes has turned out to be at least 2 times larger than the scaling exponent γr, and in the case of polymer 1,2-PBD, the ratio γes/ γr even reaches almost 5. To remove any doubts that our result could be strongly dependent on the temperature range of isotherms included in our analysis in the supercooled region, we have considered for PDE and 1,2-PBD besides isotherms of a uniform set of pressures (see Figures 1a and 5a) also isotherms (see Figures 1b and 5b) from the same temperature ranges as those used in determining the values γr from dielectric relaxation data,3 whose corresponding pressure range varies
Figure 5. Plot of isothermal PVT data for 1,2-polybutadiene taken from ref 28. Solid lines denote fits of the data to the equation of state given by eq 6. (a) Isotherms are of the uniform pressure range. (b) Isotherms that temperatures correspond to the temperature range of isotherms earlier taken into account in ref 3 to determine the scaling exponent typically from eq 1 by using dielectric data.
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TABLE 1: Comparison of the Values of Scaling Exponent γ Earlier Established from Relaxation Data by Using eq 1 with Those Evaluated Herein from Volumetric Data by Fitting to the Equation of State Given by eq 6a γr (γ found from eq 1) PDE OTP PC m-FA 1,2-PBD
b
4.5 4.0c 3.7d 2.7e 1.9b
γes (γ found from eq 6)
γes/γr
9.2 ( 0.2 10.0 ( 0.1 9.4 ( 0.1 7.2 ( 0.1 9.4 ( 0.1
2.04 2.50 2.54 2.67 4.95
a
The values γes for PDE and 1,2-PBD shown herein are determined only by using isotherms of the uniform pressure set presented in Figures 1a and 5a. b From ref 3. c From ref 1. d From ref 7. e From ref 26.
according to the isothermal data due to their measuring in the large vicinity of the glass transition. The values γes obtained for the T-P range for which dielectric data have been earlier scaled are even slightly larger than those found for isotherms of higher temperatures and uniform pressure range; i.e., there are 9.8 and 9.2 for PDE as well as 9.5 and 9.4 for 1,2-PBD. Such a result can be predicted, because intermolecular distances become smaller with decreasing temperature; hence, the steepness of the repulsive part of the potential is expected to become slightly larger. Additionally, one can observe that eq 6 correctly describes the sensitivity of materials to pressure, which is usually higher for van der Waals liquids than it is for polymers, because the larger difference in the values γes exhibits for PDE. Finally, it is worth noting that such a big discrepancy between the values γes and γr can originate neither from fitting errors, which are really small, nor the previously mentioned local character of the state equation eq 6, especially that our results correspond to the scaling exponent γ obtained for OTP from molecular dynamics simulation15,16 where no form of state equation was assumed. Of course, it should be expected that values of the scaling exponent γ evaluated from different data and by means of different methods but based on the generalized Lennard-Jones potential (eq 2), whose repulsive part exponent equals 3γ, should be the same or close. Therefore, the opposite findings throw the thermodynamic scaling idea into confusion. This work is aimed at paying attention to the existence of the problem. We believe that its solution will considerably enhance our understanding of genuine bases of the thermodynamic scaling, which seem to be still unclear. Acknowledgment. The work is a part of the theoretical aspect of the project entitled From Study of Molecular Dynamics in Amorphous Medicines at Ambient and EleVated Pressure to NoVel Applications in Pharmacy, which is operated within the
Foundation for Polish Science Team Programme cofinanced by the EU European Regional Development Fund. The authors are deeply thankful for this financial support. The authors appreciate a valuable discussion with Dr. Vitalij Yu. Bardic from the National Taras Shevchenko University of Kyiv in Ukraine on the equations of state proposed by him and his co-workers. References and Notes (1) Dreyfus, C.; Aouadi, A.; Gapinski, J.; Matos-Lopes, M.; Steffen, W.; Patkowski, A.; Pick, R. M. Phys. ReV. E 2003, 68, 011204. (2) Dreyfus, C.; Le Grand, A.; Gapinski, J.; Steffen, W.; Patkowski, A. Eur. Phys. J. B 2004, 42, 309. (3) Casalini, R.; Roland, C. M. Phys. ReV. E 2004, 69, 062501. (4) Casalini, R.; Roland, C. M. Phys. ReV. B 2005, 71, 014210. (5) Alba-Simionesco, C.; Cailliaux, A.; Alegria, A.; Tarjus, G. Europhys. Lett. 2004, 68, 58. (6) Reiser, A.; Kasper, G.; Hunklinger, S. Phys. ReV. B 2005, 72, 094204. (7) Pawlus, S.; Casalini, R.; Roland, C. M.; Paluch, M.; Rzoska, S. J.; Ziolo, J. Phys. ReV. E 2004, 70, 061501. (8) Hoover, W. G.; Ross, M. Contemp. Phys. 1971, 12, 339. (9) Roland, C. M.; Bair, S.; Casalini, R. J. Chem. Phys. 2006, 125, 124508. (10) Alba-Simionesco, C.; Tarjus, G. J. Non-Cryst. Solids 2006, 352, 4888. (11) Casalini, R.; Mohanty, U.; Roland, C. M. J. Chem. Phys. 2006, 125, 014505. (12) Casalini, R.; Roland, C. M. J. Non-Cryst. Solids 2007, 353, 3936. (13) Coslovich, D.; Roland, C. M. J. Phys. Chem. B 2008, 112, 1329. (14) Pedersen, U. R.; Bailey, N. P.; Schroe der, T. B.; Dyre, J. C. Phys. ReV. Lett. 2008, 100, 015701. (15) Bailey, N. P.; Pedersen, U. R.; Gnan, N.; Schroe der, T. B.; Dyre, J. C. J. Chem. Phys. 2008, 129, 184507. (16) Bailey, N. P.; Pedersen, U. R.; Gnan, N.; Schroe der, T. B.; Dyre, J. C. J. Chem. Phys. 2008, 129, 184508. (17) Grzybowski, A.; Grzybowska, K.; Zioło, J.; Paluch, M. Phys. ReV. E 2006, 74, 041503. Grzybowski, A.; Grzybowska, K.; Zioło, J.; Paluch, M. Phys. ReV. E 2007, 76, 013502. (18) To¨lle, A. Rep. Prog. Phys. 2001, 64, 1473. (19) Bardik, V. Yu.; Shakun, K. S. Ukr. J. Phys. 2005, 50, 404 (online access: http://www.ujp.bitp.kiev.ua/files/papers/500415p.pdf). (20) Bardic, V. Yu.; Malomuzh, N. P.; Sysoev, V. M. J. Mol. Liq. 2005, 120, 27. (21) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, U.K., 1987. (22) Roland, C. M.; Feldman, J. L.; Casalini, R. J. Non-Cryst. Solids 2006, 352, 4895. (23) Coslovich, D.; Roland, C. M. J. Chem. Phys. 2009, 130, 014508. (24) Paluch, M.; Casalini, R.; Best, A.; Patkowski, A. J. Chem. Phys. 2002, 117, 7624. (25) Naoki, M.; Koeda, S. J. Phys. Chem. 1989, 93, 948. (26) Reiser, A.; Kasper, G.; Hunklinger, S. Phys. ReV. B 2005, 72, 094204. Reiser, A.; Kasper, G.; Hunklinger, S. Phys. ReV. B 2006, 74, 019902(E). (27) Kasper, G.; Reiser, A. J. Non-Cryst. Solids 2006, 352, 4900. (28) Roland, C. M.; Casalini, R.; Santangelo, P.; Sekula, M.; Ziolo, J.; Paluch, M. Macromolecules 2003, 36, 4954. (29) Roland, C. M.; Hensel-Bielowka, S.; Paluch, M.; Casalini, R. Rep. Prog. Phys. 2005, 68, 1405.
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