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Strongly correlating liquids and their isomorphs Department of Chemistry, University of California, Berkeley, California 94720, USA Nicoletta Gnan, Nicholas P. Bailey, Thomas B. Schrøder, and Jeppe C. Dyre DNRF Center "Glass and Time", IMFUFA, Dept. of Sciences, Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark (Dated: April 9, 2010) This paper summarizes the properties of strongly correlating liquids, i.e., liquids with strong correlations between virial and potential energy equilibrium fluctuations at constant volume. Weproceed to focus on the experimental predictions for strongly correlating glass-forming liquids. Thesepredictions include i) density scaling, ii) isochronal superposition, iii) that there is a single functionfrom which all frequency-dependent viscoelastic response functions may be calculated, iv) thatstrongly correlating liquids are approximately single-parameter liquids with close to unity Prigogine-Defay ratio, and v) that the fictive temperature initially decreases for an isobaric temperature upjump. The "isomorph filter", which allows one to test for universality of theories for the non-Arrhenius temperature dependence of the relaxation time, is also briefly discussed.
defined by Eq. and the temperature is defined fromthe kinetic energy in the usual fashion After the initial reports in early 2008 of the existence If ∆U is the instantaneous potential energy minus its of a class of strongly correlating liquids these liq- average and ∆W the same for the virial at any given uids were described in four comprehensive publications state point, the W U correlation coefficient R is defined by that appeared later in 2008 and in 2009 in the Journal of (where sharp brackets denote equilibrium NVT ensemble Chemical Physics This paper briefly summarizes the properties and characteristics of strongly correlatingliquids as detailed in Refs. and present a number ofnew computer simulations. We list a number of experi- mental predictions for strongly correlating liquids, focus- ph(∆W )2ih(∆U )2i ing on glass-forming liquids since this volume constitutesthe proceedings of the Rome conference held in Septem- By the Cauchy-Schwarz inequality the correlation coeffi- ber 2009 (6IDMRCS). The main message is that the class cient obeys −1 ≤ R ≤ 1. We define strongly correlating of strongly correlating liquids, which includes the van liquids by the condition R > 0.9 The correlation coef- der Waals and metallic liquids, are simpler than liquids ficient is state-point dependent, but for all of the several in general. This explains, for instance, the long known liquids we studied by simulation R is either observation that hydrogen-bonded liquids have several above 0.9 in a large part of the state diagram, or not at Figure shows two examples of constant-volume ther- mal equilibrium fluctuations of virial and potential en- STRONG VIRIAL / POTENTIAL ENERGY ergy for two model systems, the standard Lennard-Jones CORRELATIONS IN LIQUIDS (LJ) liquid and the Wahnstr¨ om binary Lennard-Jones mixture In both cases there are strong virial / po- Consider a system of N particles in volume V at tem- tential energy correlations. In (b) one sees striking dips perature T . The virial W is defined by writing the pres- in the potential energy; these dips reflect the existence sure p is a sum of the ideal gas term N kBT /V and a term of transient clusters in the liquid characterized by the arXiv:1004.1182v1 [cond-mat.soft] 7 Apr 2010 reflecting the interactions as follows same short range order as the crystal During thedips virial and potential energy also correlate strongly.
Actually, the correlation even survives crystallization pV = N kBT + W .
Thus the property of strong virial / potential energy cor- relations is quite robust; even complex systems like bio- 1, ., rN ) is the potential energy function, the virial, which has dimension of energy, is given by logical membranes may exhibit strong correlations One way to illuminate the correlations is to plot in- stantaneous values of virial and potential energy versus W (r1, ., rN ) = −1/3 1, ., rN ) .
one another in so-called scatter plots. Figure shows an example of this with data taken from a simulation of Equation describes thermodynamic averages, but it the Kob-Andersen binary Lennard-Jones (KABLJ) liquid also applies for the instantaneous values if the virial is This has become the standard liquid for studying FIG. 2: (a) Virial / potential energy correlations for the Kob-Andersen binary Lennard-Jones liquid (1000 particles studiedby Monte Carlo simulation, ρ = 1.264, T = 1.24 in standardLJ units). (b) Inherent state energies and virials of the simu- FIG. 1: (a) Instantaneous normalized equilibrium fluctua- lation in (a); the correlation is still high. The slope γ defined tions of virial and potential energy in the standard single- = γ∆U (t) is slightly different, but comparable to component Lennard-Jones liquid at constant volume (NVT that of the true dynamics.
simulation). W (t) and U (t) correlate strongly. (b) The samefor the supercooled Wahnstr¨ om binary Lennard-Jones mix- ture here W (t) and U (t) correlate strongly even during the inherent dynamics analogue of Fig. This is the formation of a so-called Frank-Kaspers cluster done in Fig. which gives the same simulation dataafter minimizing the configurations' potential energy us-ing the conjugate gradient method. The correlations are viscous liquid dynamics, because it is difficult to crystal- still present and the "slope" γ doesn't change very much lize (this requires simulating for more than 100 microsec- – even though the virial decreased by more than 60% go- onds (Argon The "slope" γ of the scatter plot ing from (a) to (b). This confirms the robustness of virial gives the proportionality constant of the fluctuations ac- / potential energy correlations.
A convenient way to get an overview of a liquid's W U thermal equilibrium fluctuations at constant volume is = γ∆U (t) .
to collect scatter plots for several state points in a com-mon diagram. Figure (top) shows such a plot for the The number γ, which varies slightly with state point, is standard LJ liquid. Each state point is represented by roughly 6 for the standard LJ liquid, roughly 5 for the one color. As in Fig. the strong correlation is reflected KABLJ liquid, and roughly 8 for the OTP model studied in the fact that the ovals are highly elongated. For each value of the density the ovals form almost straight lines Since viscous liquid dynamics consists of long-time vi- with slope close to 6 (in Ref. it was shown that during brations around potential energy minima – the so-called and after constant-volume crystallization the system's inherent states – followed by rapid transitions be- scatter plots fall on the extension of the line). The bot- tween the inherent states it is interesting to study tom three figures show the correlation coefficient R (Eq.
we reported simulations of 13 different model liquids. Allliquids with van-der-Waals type interactions were foundto be strongly correlating (R > 0.9), whereas modelsof the two hydrogen-bonding liquids water and methanolwere not. Although much remains to be done by means oftheory and simulation, it has now been established with-out reasonable doubt that liquids can be classified intotwo classes: (i) The class of strongly correlating liquids,which includes the van der Waals and metallic liquids;this liquid class has a number of regularities and simpleproperties.
(b) All remaining liquids – the hydrogen- bonded, the covalently bonded, and (strongly) ionic liq-uids – which are much more complicated.
CAUSE OF STRONG VIRIAL / POTENTIAL ENERGY CORRELATIONS Before discussing the consequences of strong virial / potential energy correlations we briefly reflect on thecause of the correlations. The starting point is the well-known fact that for any liquid in which the parti-cles (of one or more types) interact with purely repulsiveinverse power-law forces, v(r) ∝ r−n, there is 100% cor-relation between W and U : W (t) = γU (t) where From the values of γ close to 6 observed for the LJ liquidone would expect that, if the LJ liquid somehow corre-sponds to an IPL liquid, the exponent n is close to 18.
Although at first sight this may seem strange given ther−6 and r−12 terms that enter into the definition of theLJ potential, a potential proportional to r−18 does indeedgive a good fit to the repulsive part of the LJ potential(Fig. (a)). The reason that a much larger exponentthan 12 is required is that the attractive r−6 term makes FIG. 3: (a) Scatter plot of the W U thermal equilibrium the LJ repulsion much steeper than that of the r−12 term fluctuations at constant volume for the standard single- alone. Figure (b) shows that both potential energy and component LJ liquid, and (b) plots of various quantities as virial fluctuations of the LJ liquid are well represented by functions of temperature for the different densities studied.
those of an r−18 IPL potential.
The full black line marks state points of zero average pres- In our first publications on strongly correlating liquids it was suggested that the strong correlations de-rive from particle close encounters taking the intermolec-ular distance to values below the LJ potential minimum, the "slope" γ, and the average pressure as func- at which the IPL potential is a good approximation. It tions of temperature for the different densities. Clearly, quickly became clear, however, that this is not the full both R and γ are somewhat state-point dependent. At explanation; thus this can explain neither the existence a given density R increases with temperature whereas of strong correlations in the crystal (above 99% at low γ decreases; at a given temperature R increase with in- temperatures nor the existence of correlations at creasing density. The thick black lines mark state points low pressures at which nearest-neighbor interparticle dis- of zero average pressure. Note that the density effect of tances fluctuate around the LJ potential's minimum dis- increasing R "wins" over the temperature effect of de- tance. Also, the original explanation is a single-pair ex- creasing R upon cooling at constant low pressure. Thus planation, which would imply that the strong correlations one expects higher correlations upon supercooling a liq- should be present as well in constant pressure ensembles.
uid, which is an important observation when it comes to This contradicts our finding that switching from constant focusing on glass-forming liquids.
volume to constant pressure reduces R from values above How common are strong W U correlations? In Ref. 0.9 to values around 0.1 little at constant volume Thus as regards fluctua-tions, the pure IPL gives representative results. This ex-plains why the IPL approximation works so well and whythe strong correlations disappear when going to constantpressure ensembles. This also explains why several IPLliquid properties are not shared by LJ-type liquids (e.g.,the IPL equation of state is generally quite wrong anddoes not allow for low-pressure stable liquid states, andthe IPL free energy and bulk modulus are quite wrong).
While the eIPL approximation explanation of strong W U correlations for physically realistic cases, there arealso strong correlations in the purely repulsive Weeks-Chandler-Andersen version of the KABLJ liquidThe slope γ here varies quite a lot (from 5.0to 7.5) over the range of densities and temperatures inwhich γ is fairly constant for the KABLJ liquid. Our sim-ulations show that the strong correlations for the WCAcase is a single-particle-pair effect, not the cooperativeeffect that only applies at constant volume conditions,observed for LJ-type liquids. More work is needed toilluminate the correlation properties of this interesting(but physically unrealistic) potential.
ISOMORPHS: CURVES OF INVARIANCE IN THE PHASE DIAGRAM This section defines isomorphs and summarizes their FIG. 4: (a) Approximation of the LJ potential by an effective invariants. As shown in Ref. a liquid has isomorphs if inverse power law (IPL) potential ∝ r−18. The blue dotted and only if the liquid is strongly correlating. An isomorph curve marks the IPL potential, which approximates the LJ is a curve in the phase diagram along which a number of potential well at small interparticle spacing. The red open properties are invariant.
circles mark the radial distribution function at a typical low- For any microscopic configuration (r1, . , rN ) of a pressure state point. The difference between the LJ potential thermodynamic state point with density ρ, the "reduced" and the IPL potential is approximately linear in r; this fact coordinates are defined by ˜ ri ≡ ρ1/3ri. State points (1) forms the basis for the "extended inverse power law" (eIPL) and (2) with temperatures T1 and T2 and densities ρ1 approximation (Eq. (b) Two figures demonstrating that the LJ potential and its virial in their thermal equilib- 2 are said to be isomorphic if, whenever two mi- rium fluctuations correlate strongly to the same quantities for croscopic configurations (r the r−18 IPL potential.
have identical reduced coordinates, to a good approxima-tion they have proportional configurational NVT Boltz-mann probability factors: References and detail the more complete expla- nation of the cause of strong correlations. The differ- ence between the IPL potential and the LJ potential is plotted in Fig. (a) as the red dashed curve. Thegreen dashed curve is a straight line, which approximates The constant C12 here depends only on the state points the red dashed curve well around the LJ minimum (i.e., (1) and (2), not on the microscopic configurations. Iso- over the entire first peak of the structure factor). Thus morphic curves in the state diagram are defined as curves over the most important intermolecular distances an "ex- for which any two state points are isomorphic. The prop- tended" inverse power law potential, eIPL, defined by erty of having isomorphs is generally approximate – onlyIPL liquids have exact isomorphs. For this reason Eq. should be understood as obeyed to a good approximation for the physically relevant configurations, i.e., those that eIPL(r) = Ar−n + B + C r do not have negligible canonical probabilities gives a good approximation to the LJ potential, vLJ(r) ∼ Figure illustrates Eq. by checking the logarithm veIPL(r). It has been shown by simulation that the lin- of this equation, where (a) gives simulation data for ear "quark confining" term of the eIPL potential gives a the KABLJ liquid. We consider a number of configu- contribution to the total potential energy that fluctuates rations of the state point with density and temperature FIG. 5: Direct check of the isomorph condition for the KABLJliquid (8000 particles), which is strongly correlating (a), andfor the SPC water model (5120 molecules), which is not (b).
For both liquids the consistency of the isomorph condition ischecked by jumping from one to a different density and back.
This works well for the KABLJ liquid but not for SPC water;details are given in the text.
FIG. 6: (a) AA particle radial distribution function of theKABLJ liquid for two isomorphic state points (left) and for the same temperature at the two densities (right). The iso- 1, T1) = (1.258, 0.628) in standard LJ units. For these configurations the total potential energy was evaluated.
morphic state points have the same radial distribution func- In order to investigate whether the state point has an tions. (b) The same for the AA incoherent intermediate scat-tering function at the k-vector corresponding to the first peak isomorphic state point at density ρ2 = 1.228 we scaled of the radial distribution function. The two isomorphic state the simulated configurations of the first state point to points have the same dynamics (in reduced units, as used density ρ2. For the scaled configurations the potential energies are plotted against the original energies of statepoint 1 (top figure). According to the isomorph defini-tion Eq. the best fit slope gives the ratio between thetemperatures of the isomorphic state points; in this way determined by the contribution to the partition function we estimate that T coming from the linear term of the eIPL, and thus C The right panel of Fig. investigates the consis- reflects the deviation from true IPL behavior).
tency of this procedure by reversing it in order to check Figure makes this "direct isomorph test" for the whether the original temperature T1 is arrived at. In- non-strongly correlating liquid SPC water, starting from deed, when this is done one does find the original tem- temperature T1 = 200 K. From the slope of the left panel perature to be 0.628. Two things should be noted. The we find T2 = 179.6 K. When the reverse jump is per- first is the very strong correlation between scaled con- formed, however, one does not come back to the initial figurations, as required for having good isomorphs. The state point, but to a predicted temperature of 166.36 K.
second notable fact is that the best fit lines do not pass This shows that water does not have isomorphs, consis- through (0, 0). This shows that the constant C12 of Eq.
tent with the fact that it is not a strongly correlating is not unity, as it would be for an IPL liquid (C12 is For the practical identification of an isomorph in the phase diagram the above method may be used. Alterna-tively, it has been shown that to a good approximationisomorphs are characterized by ργ = Const.
Here γ is the above discussed "slope" characterized by∆W (t) ∼ = γ∆U (t). As shown in Ref. this quantity may be calculated to a good approximation from equilibriumfluctuations via the expression (giving the least-squaredlinear-regression best-fit slope of W U scatter plots, com-pare Appendix B of Ref. Several physical quantities are invariant along a strongly correlating liquid's isomorphs to a good approx-imation. These include: 1) Thermodynamic propertieslike the excess entropy (i.e., in excess of the ideal gas en-tropy at same density and temperature) and the excess FIG. 7: cV per particle for the Lewis-Wahnstr¨ isochoric specific heat, 2) static averages like radial dis- ortho-terphenyl consisting of three LJ spheres arranged with tribution function(s) in reduced coordinates, 3) dynamic fixed bond lengths. The left panel shows the raw simulation quantities like the reduced diffusion constant, viscosity, data, The right panel shows the same data replotted as func- and heat conductivity, time-autocorrelation functions in tion of ρ7.9/T in which the exponent 7.9 was determined fromthe proportionality between equilibrium virial and potential properly reduced units, average reduced relaxation times, energy fluctuations This model follows the Rosenfeld- Tarazona prediction of c Figure shows results of simulations of the KABLJ liq- V varying with temperature as T −2/5 uid at two isomorphic state points (left subfigures) andisothermal state points (right subfigures) of the AA par-ticle radial distribution functions and the AA incoherentintermediate scattering functions, respectively. These fig- that the effective temperature concept for a strongly ures confirm the prediction that isomorphic state points correlating liquid makes good sense physically. On the have identical static distribution functions and identical x-axis the inherent state energies of given state points are shown as the system fell out of equilibrium. The ar- Since the isochoric specific heat is an isomorph invari- rested phase is characterized by an effective temperature ant, this quantity should be a function of ργ /T for a Teff , which can be calculated in the standard way from strongly correlating liquid.
Figure confirms this for the violation of the fluctuation-dissipation theorem the Lewis-Wahnstr¨ om OTP model consisting of three LJ On the y-axis is shown the inherent energies found from an equilibrium simulation with temperature equal to Teff The theory further predicts that jumps between two for the corresponding arrested phase.
isomorphic state points should take the system instanta- give data for the strongly correlating KABLJ liquid, the neously to equilibrium, because the Boltzmann statisti- green points give data for the non-strongly correlating cal factors of two isomorphic state points by definition monatomic Lennard-Jones Gaussian liquid. Clearly, the are proportional More generally, isomorphic state latter system fell out of equilibrium by freezing into a points are equivalent during any aging scheme. We re- part of phase space that is not characterized by Teff .
cently showed that the isomorph concept can be used tothrow light on the concept of an effective temperatureIn particular, the theory implies that for strongly THE EQUATION OF STATE OF A correlating liquids the effective temperature after a jump STRONGLY CORRELATING LIQUID to a new (low) temperature and a new density, dependsonly on the new density (Fig. We showed alsothat this does not apply for the non-strongly correlat- This section shows that the Helmholtz free energy for ing monatomic Lennard-Jones Gaussian liquid, confirm- any strongly correlating liquid is of the form ing the general conjecture that strongly correlating liq-uids have simpler physics than liquids in general. Fig-ure (b) shows a result confirming the finding of Ref.
Fex/N = T ψ T f (ρ)
+ g(ρ) .
Since CV = T (∂S/∂T )V = T φ0 (x)(∂x/∂T ) fine m(x) ≡ φ2(x)/φ0 (x), one has T (∂x/∂T ) Along an isochore this implies that dx/m(x) = d ln T .
When integrated along the isochore this gives h(x) ≡R x dx0/m(x0) = ln T + α(V ), implying exp(h(x)) = T f (V ) where f (V ) = exp(α(V )). In other words, linesof constant x have constant T f (V ). Since x is merelyused for labeling the isomorphs, this means that wecan redefine x as follows x ≡ T f (V ). Integrating nowS = φ1(x) = −(∂F/∂T )V along an isochore gives φ1(x0)dT 0+g(V ) = − φ1(x0)dx0+g(V ) .
The integral is some function of x, and we have thusderived Eq. As mentioned the slope γ may vary with state point.
The equation of state gives information about which vari-ables γ may depend on: FIG. 8: (a) Virial versus potential energy for the KABLJ liquid during a temperature quench and a crunch crunch increases the density and keeps the temperature con- stant; this is equivalent to first an instantaneous jump along = −T 2ψ0 T f (V )
f 0(V ) − g0(V ) .
an isomorph (green curve) to the right density followed by a temperature quench (black curve). The inset shows that the crunch and the quench have the same fluctuation-dissipation Combining these equations leads to violation factors, i.e., result in the same effective tempera-ture. (b) Inherent state energies for several state points ofthe KABLJ liquid as the system falls out of equilibrium and U − g(V ) − V g0(V ) .
freezes versus the equilibrium inherent state energy at the effective temperature (details are given in Ref. For simplicity we shall not indicate "excess" quantitiesexplicitly, V is used as variable instead of the ρ, and the N is ignored since it is fixed; thus we shall prove that Suppose a given strongly correlating liquid's isomorphsare labeled by the variable x, i.e., that its isomorphs are curves of constant x for which x = x(T, V ). Isomorphsare curves of constant (excess) entropy, as well as curves of constant (excess) C This means that for some functions φ1(x) and φ2(x) one may write Thus if γ varies with state point, it may only depend onvolume As shown in Ref. this result is consistent S = φ1(x) , CV = φ2(x) .
with the original experimentally based formulation of the so-called "density scaling" due to Alba-Simionesco and This has not yet been tested experimentally, but it co-workers Figure shows that γ is not rigor- is consistent with the finding that density scaling ously constant along isochores as predicted by Eq. works well for van der Waals liquids but not for although it varies only of order 10% when temperature hydrogen-bonded liquids Meanwhile, den- is tripled. This serves to emphasize that isomorphs are sity scaling has been shown to apply in computer only approximate constructs and so are their predicted simulations of strongly correlating liquids with γ invariants (only IPL liuqids have exact isomorphs).
given by Eq. to a good approximation The equation of state Eq. is of the Mie-Gr¨ form for the excess variables (excess pressure: W/V , and 2. Isochronal superposition excess energy: U ) Isochronal scaling is the further, fairly recent find-ing that for varying pressures and temperaturesthe dielectric loss as a function of frequency de- pends only on the loss-peak frequency U + ψ(V ) .
This is trivial if the liquid obeys time-temperature-pressure superposition (TTPS) in which case noth- uneisen equation of state for solids γ(V ) = ing changes. But many liquid do not obey TTPS, d ln ω/d ln V where ω is a phonon mode eigenfrequency, and for such liquids isochronal superposition is a U is the vibrational potential energy, and ψ(V ) relates new and striking regularity that works generally to the volume derivative of the energy per atom (i.e., for van der Waals liquids, but rarely for hydrogen- of the energy of the force-free configuration about which bonding liquids Since both the relaxation time the vibrational motion occurs). Further discussion of the and the entire relaxation spectrum are isomorph relation of strong W U correlations to the Mie-Gr¨ invariants, isochronal superposition must apply for equation of state is given in Ref. any strongly correlating liquid: If temperature andpressure for two state points are such that their re- SOME EXPERIMENTAL PREDICTIONS laxation times are the same, the two points must FOR STRONGLY CORRELATING belong to the same isomorph and thus have same GLASS-FORMING LIQUIDS relaxation time spectra for any observables, in par-ticular the dielectric loss as function of frequency Strongly correlating liquids have a number of since long should be the same.
described properties. For instance, since the melting line 3. Frequency-dependent viscoelastic response func- in the phase diagram is an isomorph strongly corre- lating liquids have several invariants along their melting There are eight fundamental complex, frequency- lines, including the radial distribution function and di- dependent linear thermoviscoelastic response func- mensionless transport coefficients. Such regularities have tions like, e.g., the frequency-dependent isochoric been observed in simulation and experiment; we refer to or isobaric specific heat, the frequency-dependent the reader to Ref. for more details. This section focuses isobaric expansion coefficient, and the frequency- on predictions for highly viscous liquids, for which the dependent adiabatic or isothermal compressibility strong-correlation property implies several experimental Standard linear irreversible thermodynamic arguments, where the Onsager relations play the 1. Density scaling role of the Maxwell relations of usual thermody-namics, show that there are only three independent In the last decade, in particular since 2005, many frequency-dependent response functions. If more- papers appeared dealing with the so-called density over stochastic dynamics is assumed as is realis- scaling, which is the finding that for several glass- tic for highly viscous liquids there are only forming liquids the relaxation time τ at varying two independent response functions For pressure and temperature is some function of the strongly correlating liquids the further simplifica- quantity ργ /T , in which the exponent γ is an em- tion appears that there is just a single indepen- pirical fitting parameter: dent response function Since there areexplicit expressions linking the different responsefunctions (depending on the ensemble considered τ = F (ργ /T ) .
this can be tested experimentally. Unfortu-nately it is difficult to measure thermoviscoelastic Neither the function F nor γ are universal. The functions properly; to the best of our knowledge isomorph theory shows that all strongly cor- there are yet no reliable data for a complete set of relating glass formers obey density scaling with the three or more such response functions on any liquid.
exponent γ given by the equilibrium fluctuationsat one state point (Eq. (provided γ is fairly 4. The Prigogine-Defay ratio: Strongly correlating liq- constant over the relevant part of phase space).
uids as approximate single-parameter liquids After many years of little interest the Prigogine- morph filter.
Defay (PD) ratio has recently again comeinto focus in the scientific discussion about glass- 6. Fictive temperature variations following a temper- From a theoretical point of view the PD ratio is poorly defined since Any jump from equilibrium at some density and it involves extrapolations from the liquid and glass temperature to another density and temperature phases to a common temperature It is pos- proceeds as if the system first jumped along an iso- sible to overcome this problem by modifying the PD morph to equilibrium at the final density and then, ratio by referring exclusively to linear response ex- immediately starting thereafter, jumped to the fi- periments; here the traditional difference between nal temperature (Fig. The first isomorphic liquid and glass responses is replaced by a difference jump takes the system instantaneously to equilib- between low- and high-frequency values of the rel- rium. This property, which applies for all strongly evant frequency-dependent thermoviscoelastic re- correlating liquids, means that glass-forming van sponse function In this formulation, the prop- der Waals and metallic liquids are predicted to erty of strong virial / potential energy correlations have simpler aging behavior than, e.g., covalently manifests itself as a PD ratio close to unity. Actu- bonded liquids like ordinary oxide glasses ally, an extensive compilation of data showed that In traditional glass science the concept of "fictive van der Waals bonded liquids and polymers have temperature" is used as a structural characteristic PD ratios close to unity – even though as men- that by definition gives the temperature at which tioned the traditional PD ratio is poorly defined, the structure would be in equilibrium . For there are good reasons to assume that it approxi- any aging experiment, in glass science one assumes mates the rigorously defined "linear" PD ratio.
that the fictive temperature adjusts itself mono- The theoretical developments of Refs. show tonically from the initial temperature to the final that in any reasonable sense of the old concept Consider, however, a sudden tem- "single-order-parameter liquid", strongly correlat- perature increase applied at ambient pressure. In ing liquids are precisely the single-order-parameter this case there is first a rapid thermal expansion The isomorph concept makes this even before any relaxation takes place.
more clear: State points along an isomorph have so taneous isomorph" takes the system initially to a many properties in common that they are identi- state with canonical (Boltzmann) probability fac- cal from many viewpoints. In the two-dimensional tors corresponding to a lower temperature. In other phase diagram this leaves just one parameter to words, immediately after the temperature up jump classify which isomorph the state point is on; thus the system has a structure which is characteristic a liquid with (good) isomorphs is an (approximate) of a temperature that is lower than the initial tem- single-parameter liquid. Note that this is consis- perature. With any reasonable definition of the fic- tent with the old viewpoint that single-parameter tive temperature, this quantity thus initially must liquids should have unity PD ratio decrease during an isobaric positive temperaturejump – at least for all strongly correlating liquids.
5. Cause of the relaxation time's non-Arrhenius tem- perature dependence: The isomorph filter Since the relaxation time τ is an isomorph invariant for any strongly correlating liquid, any universallyvalid theory predicting τ to depend on some phys- The class of strongly correlating liquids includes ical quantity must give τ as a function of another the van der Waals and metallic liquids, but excludes isomorph invariant (we do not distinguish between the hydrogen-bonded, the covalently bonded, and the the relaxation time and the reduced relaxation time (strongly) ionic liquids. Due to their "hidden scale in- since their temperature dependencies are virtually variance" – the fact that they inhering a number of identical). This gives rise to an "isomorph filter" IPL properties – strongly correlating liquids are sim- showing that several well-known models can- pler than liquids in general. Strongly correlating liquids not be generally valid. For instance, the entropy have isomorphs, curves along which a number of physical model cannot apply in the form usually used by properties are invariant when given in properly reduced experimentalists: τ ∝ exp(C/Sconf T ) where C is a units. In particular, for glass-forming liquids the prop- constant and Sconf is the configurational entropy; erty of strong virial / potential energy correlations in the it can only be correct if C varies with density as equilibrium fluctuations implies a number of experimen- C ∝ ργ . Likewise, the free volume model does not tal predictions. Some of these, like density scaling and survive the isomorph filter.
If the characteristic isochronal superposition, are well-established experimen- volume Vc of the shoving model (predicting that tal facts for van der Waals liquid and known not apply τ ∝ exp(VcG∞/kBT ) varies with density for hydrogen-bonded liquids. This is consistent with our as Vc ∝ 1/ρ, this model is consistent with the iso- predictions. Some of the predicted properties have not yet been tested, for instance that the density scaling ex- ponent can be determined by measuring the linear ther-moviscoelastic response functions at a single state point,or that jumps between isomorphic state points take thesystem instantaneously to equilibrium, no matter how URP is supported by the Danish Council for Indepen- long is the relaxation time of the liquid at the relevant dent Research in Natural Sciences. The centre for vis- state points. – We hope this paper may inspire to new cous liquid dynamics "Glass and Time" is sponsored by experiments testing the new predictions.
the Danish National Research Foundation (DNRF).
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