Abstract
Transport coefficients, such as viscosity or diffusion coefficient, show significant dependence on density or temperature near the glass transition. Although several theories have been proposed for explaining this dynamical slowdown, the origin remains to date elusive. We apply here an excessentropy scaling strategy using molecular dynamics computer simulations and find a quasiuniversal, almost compositionindependent, relation for binary mixtures, extending eight orders of magnitude in viscosity or diffusion coefficient. Metallic alloys are also well captured by this relation. The excessentropy scaling predicts a quasiuniversal breakdown of the StokesEinstein relation between viscosity and diffusion coefficient in the supercooled regime. Additionally, we find evidence that quasiuniversality extends beyond binary mixtures, and that the origin is difficult to explain using existing arguments for singlecomponent quasiuniversality.
Introduction
Supercooled liquids approaching the glass transition show significant nonArrhenius temperature or density dependence of their transport coefficients, such as viscosity or diffusion coefficient. Several theories have been proposed to explain this phenomenon, for instance: random firstorder transition theory, entropycontrolled theories, dynamical facilitation, bondorientational order, freevolume theories, elastic models, and more^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. Despite these intriguing theories, a broadlyaccepted and universal picture of what controls the change in transport coefficients near the glass transition has not yet manifested itself, even for the simplest supercooled liquids.
Rosenfeld discovered^{16,17} in 1977 that transport coefficients in the liquid phase are correlated to the excess entropy S_{ex}, where S_{ex} is defined by subtracting the ideal gas contribution from the entropy at the same density ρ and temperature T, i.e., S_{ex}(ρ, T) ≡ S(ρ, T) − S_{id}(ρ, T). The excess entropy is a negative quantity since the liquid is more ordered than the ideal gas. Rosenfeld found by applying appropriate dimensionless units that the viscosity and diffusion coefficient collapse to a univariate function of the excess entropy for singlecomponent atomic liquids. Since then, excessentropy scaling has been the focus of simulation and experimental studies in various contexts^{18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39}, including atomic mixtures, molecular liquids, ionic liquids, networkforming liquids, nanoconfined liquids, nonlinear sheared liquids, and more. For a recent review of excessentropy scaling, see ref. ^{40}.
Explanations for excessentropy scaling have been attempted^{18,19} from frameworks such as modecoupling theory^{12}. The fact that excess entropy correlates to transport coefficients may also be explained in the context of Rsimple liquids and their isomorphs^{41,42,43,44,45,46,47}. Isomorphs are curves in the thermodynamic phase diagram along which structure and dynamics are invariant in appropriate dimensionless units. Some thermodynamic quantities are also invariant along isomorphs, e.g., the excess entropy^{44,46}. Since the excess entropy and the dynamics in dimensionless units, and hence also dimensionless transport coefficients, are invariant along the same curves, one can write \(\widetilde{X}=f({S}_{{\rm{ex}}})\), where \(\widetilde{X}\) is a generic dimensionless transport coefficient and f is some a priori systemspecific function not predicted by isomorph theory^{44,46}.
Rsimple liquids are defined in computer simulations by reference to the correlation coefficient R = 〈ΔWΔU〉/\(\sqrt{\langle {(\Delta W)}^{2}\rangle \langle {(\Delta U)}^{2}\rangle }\) calculated from the virial W and potential energy U constantvolume canonicalensemble fluctuations at a given state point. A pragmatic definition of this class of liquids is R ≥ 0.90 which depends on the state point^{41}. Rsimple liquids include most or all van der Waals and metallic liquids, but exclude networkforming, covalentbonding, and strongly ionic or dipolar liquids. Rsimple liquids have been shown to exist both in experiments and simulations, and the concept is also relevant for the crystal region, under nanoconfinement, in nonlinear shear flow, and more^{26,48,49,50,51,52,53,54}. A review of Rsimple liquids and their isomorphs is given in ref. ^{55}.
Rosenfeld reported in his seminal paper a quasiuniversal relation^{16,17} for singlecomponent atomic liquids given by the expression
in which k_{B} is Boltzmann’s constant, N is the number of particles, and A and B are systemindependent constants, with, e.g., A ≈ 0.6 and B ≈ 0.8 for diffusion^{16} and A ≈ 0.2 and B ≈ −0.8 for viscosity^{17}. Equation (1) enables prediction of unknown transport coefficients for a given system if its excess entropy is known. Later studies revealed that the exponential behavior of the excess entropy does not apply for supercooled liquids whereas excessentropy scaling in the form \(\widetilde{X}=f({S}_{{\rm{e}}x})\) may still apply^{25,31,34,40}. Furthermore, the quasiuniversal relation of singlecomponent atomic liquids was found to break down for, e.g., molecules which in general do not show quasiuniversal behavior^{25,56,57}. The quasiuniversal behavior of singlecomponent atomic liquids may be explained by the exponential (EXP) pair potential^{58,59,60}, which can be used as a basis for expanding other pair potentials under certain conditions.
Notwithstanding the importance of excessentropy scaling for singlecomponent atomic liquids, mixtures of atoms are more often used in simulations and experiments to avoid crystallization and to obtain desirable properties in, e.g., metallic alloys^{61}. Alas, for atomic mixtures one does not expect quasiuniversality; mixtures may involve atoms of various sizes, different compositions, alongside different interactions amongst the constitutent particles. Krekelberg et al.^{22} found poor scaling with excess entropy for binary hardsphere (HS) mixtures with respect to composition and size and formulated a generalized excessentropy scaling to remedy this problem. Banerjee et al.^{34} studied supercooled binary mixtures and found no universal collapse between a tetrahedralforming ionic melt and other simple mixtures. This has also been found for other ionic melts in the supercooled region^{62}. On the other hand, LötgeringLin et al.^{36} found collapse with composition of binary Lennard–Jones (LJ) mixtures in the hightemperature regime over a limited range in viscosity (factor of two). A related result has also been found for the computersimulated metallic alloy AlNi in the hightemperature limit^{32}. On account of the high temperatures simulated these mixtures are expected to behave approximately as singlecomponent atomic liquids, and the results are therefore consistent with the previously mentioned studies and results.
The Stokes–Einstein (SE) relation connects the diffusion coefficient D of a large particle immersed in a solvent with viscosity η, predicting that D ∝ η^{−1}T. The SE relation breaks down in the supercooled regime and explanations have been presented from various theoretical perspectives^{33,38,63,64,65,66,67,68,69,70}. Flenner et al.^{68} obtained a good collapse of the diffusion coefficient plotted against the structural relaxation time (which may be used as a proxy for the viscosity) for supercooled binary mixtures by scaling the diffusion coefficient and the relaxation time. In other words, the authors showed that the breakdown of SE in the supercooled regime occurs at the same scaled relaxation time and in a quasiuniversal manner for these binary mixtures. Flenner et al.^{68} also showed that dynamical heterogeneity exhibits universal features for supercooled liquids. However, it is not clear why a quasiuniversal curve should be observed in the supercooled region as the binary mixtures are very different, and this was also noted by the authors. The focus of the present study is not on the origin behind the SE breakdown, but on the possible quasiuniversality observation of Flenner et al.^{68} related to the SE breakdown.
The above observations motivate us to carry out an indepth study of viscosity and diffusion coefficient going deep into the supercooled regime of a wide range of binary atomic mixtures to investigate whether quasiuniversality applies to mixtures, contrary to the expectation and findings of previous studies. We use molecular dynamics GPUbased computer simulations in the NVT ensemble (the RUMD package^{71}) to study six different binary mixtures: The Kob–Andersen binary Lennard–Jones (KA) mixture, the Wahnström (WS) mixture, the generalized LJ (GLJ) mixture, the KA exponential pair potential (KAEXP) mixture, alloys of copper and zirconium (CuZr), and a size asymmetric (AS) mixture. The systems under study include additive and nonadditive mixtures, different steepness of the pair interactions, effective medium interactions, various size asymmetries, and different compositions. Model and simulation details are found in the Methods section. The Supplementary Tables S1 and S2 include all simulation results in a tabular form. The virial potentialenergy correlation coefficient R is >0.90 at all investigated state points, except for the CuZr_{36:64} and AS mixtures where it is somewhat below 0.90 (see Supplementary Tables S1 and S2); some of these systems have previously been investigated in detail for isomorphs see, e.g., refs. ^{26,41,44,52,53,54}.
The computer models have various degrees of glassforming ability and thus different ranges of supercooling. Throughout the study, we use two different sets of dimensionless units: one using the microscopic parameters of the potentials based on the length and energy scales of the larger (A) particle, which is standard in computer simulations, and another set of dimensionless units using macroscopic quantities with length given in units of ρ^{−1/3}, energy in units of k_{B}T, and time in units of \({\rho}^{1/3}\sqrt{m/{k}_{{\rm{B}}}T}\) (m is the particle mass), as applied in excessentropy scaling and the isomorph theory^{16,44,59}. The macroscopic dimensionless units are termed reduced units and use a tilde above the variable name; microscopic dimensionless units are implicitly assumed when no tilde is given (an exception is the metallic alloys; see “Methods” for their units).
The main findings of the current study are: (1) A nearly compositionindependent excessentropy scaling relation for all studied binary mixtures extending over eight orders of magnitude in viscosity or diffusion coefficient, going three to four orders of magnitude below the modecoupling temperature T_{MCT} (i.e., where the dynamics starts to become landscape dominated). (2) A quasiuniversal excessentropy relation amongst binary atomic mixtures with different interactions (e.g., pair interactions and effective medium interactions), mixing rules, and size asymmetry. We find, additionally, that the departure from universality in the supercooled regime can be rationalized using the socalled densityscaling exponent. As a consequence of these findings, we show that the product of viscosity and diffusion coefficient has virtually the same excessentropy dependence for all mixtures. Our results thus rationalize the observations of Flenner et al.^{68} that SE breaks down at the same scaled relaxation time. The presented simulation results are corroborated by experimental data on metallic alloys from the literature which additionally support the validity of the scalings beyond binary mixtures.
Results
Excessentropy scaling
The study commences by demonstrating deeply supercooled dynamics exemplifed by the selfpart of the intermediate scattering function (ISF; see “Methods” for definition) for the KA mixture at 2:1 composition in Fig. 1a. The value of the wave vector q is that of the first peak of the static structure factor. The 2:1 composition is a much better glass former than the standard 4:1 composition^{72,73,74}, thus giving access to a wider dynamical range. The supercooled dynamics goes three to four decades below T_{MCT} where the standard 4:1 composition would crystallize. The modecoupling temperature for the 2:1 KA mixture is T_{MCT} = 0.55. We find a plateau in the ISF extending over almost five decades with a stretching exponent β = 0.55 at the lowest temperature, i.e., the ISF is well fitted by the stretched exponential function \(\exp [{(t/{\tau }_{\alpha })}^{\beta }]\), where τ_{α} is the αrelaxation time. Figure 1b displays the viscosity η as a function of 1/T for all the studied binary mixtures and shows in all cases strong deviations from a straight line in the supercooled regime, i.e., a significant nonArrhenius behavior.
Figure 1c, d demonstrates excessentropy scaling for the diffusion coefficient in the 4:1 KA and 3:1 WS mixtures for three different densities and several temperatures; Supplementary Fig. S1 provides the corresponding figures for the viscosity. The 4:1 KA mixture is more commonly studied in the literature than the 2:1 KA mixture, and the former model is therefore used to illustrate excessentropy scaling. We focus here on the large Aparticle diffusion coefficient; results for the Bparticle diffusion coefficient are given in Supplementary Fig. S2. Consistent with previous studies^{21,34,69}, an excellent collapse with excess entropy is found, extending here to much lower diffusion coefficients than previously studied. Hereafter we focus on showing results for a fixed density only for each system.
Composition excessentropy scaling
Excessentropy scaling for a fixed composition was demonstrated in the previous section. However, composition is an extra variable besides density and temperature in the phase diagram of mixtures, and the question is therefore whether excessentropy scaling can absorb this extra variable and still collapse data to a univariate function of the excess entropy S_{ex}. As mentioned, in light of the results of Krekelberg et al.^{22} and from the fact that mixtures have rich phase diagrams, one does not a priori expect any collapse for different compositions.
Figure 2 shows the reduced viscosity \(\widetilde{\eta }\) as a function of the excess entropy for the KA mixture, the WS mixture, the GLJ mixture, and the CuZr mixture, each plotted for several compositions. An almost compositionindependent curve is found for all mixtures for a dynamic range extending over eight orders of magnitude in viscosity. This result cannot in an obvious way be explained by the quasiuniversality of singlecomponent atomic liquids or by appealing to high temperatures where binary mixtures are expected to behave approximately as singlecomponent liquids.
Quasiuniversal excessentropy scaling
We proceed to investigate excessentropy scaling relationships by comparing different systems. Figure 3a shows the reduced viscosity as a function of the excess entropy for all mixtures and compositions. Figure 3b shows the reduced Aparticle diffusion coefficient. For reference we have also included data for the singlecomponent LJ (SCLJ) liquid. Additional data for SCLJ are given in ref. ^{75}, demonstrating that the same trend continues into the gaseous region.
For all investigated mixtures and compositions, a quasiuniversal relationship is observed for both viscosity and diffusion coefficient using the excess entropy as the relevant variable. Some deviations are found for the most supercooled states, depending on the mixture, and thus the use of the term quasiuniversal is appropriate as opposed to the nearly universal relationship observed for different compositions in Fig. 2. We conclude that quasiuniversality applies also for binary mixtures, contrary to expectation and previous studies.
To put the magnitude of the observed deviations into perspective, Figure 3b provides as a reference excessentropy scaling for an almost spherelike dumbbell molecule (DB; see grey data points with data taken from ref. ^{26}). This model also has R above 0.90 for all investigated state points. Significant deviations are observed at higher temperatures and no quasiuniversality can possibly be established in the deeply supercooled region, indicating that the deviations between the different binary mixtures are relatively small. The departure from universality in the supercooled region is studied more closely below where it is found to correlate with the value of the densityscaling exponent.
How do the above quasiuniversality observations relate to those of Flenner et al.? Flenner et al.^{68} observed a quasiuniversal breakdown of SE for five different binary atomic mixtures by scaling the relaxation time (a proxy for the viscosity) and the diffusion coefficient and plotting the diffusion coefficient against the relaxation time^{68}. The SE relation in its traditional form is given by
in which σ_{H} is the hydrodynamic diameter and c is a constant. Assuming that the hydrodynamic diameter is not a constant but proportional to ρ^{−1/3}, the SE relation in reduced units becomes^{76}
A hydrodynamic diameter proportional to ρ^{−1/3} was proposed by Zwanzig^{77} and recently shown to be a consequence of the isomorph theory in the sense that Eq. (2) with a constant hydrodynamic diameter is inconsistent with isomorph theory while Eq. (3) is not^{76}. We therefore focus on this expression for SE. For atomic mixtures, a possible generalization of the SE relation is to use the individual diffusion coefficients in Eq. (3), e.g., for the Aparticle the diffusion coefficient D_{A} is used^{33} while the constant c is expected to depend on the particle type.
The SE relation is now investigated for all the binary mixtures. Figure 3 documents a quasiuniversal relation for both \({\widetilde{D}}_{{\rm{A}}}\) and \(\widetilde{\eta }\) as a function of the excess entropy S_{ex}. This result implies that the product is also a quasiuniversal function of S_{ex}. Figure 4a shows \({\widetilde{D}}_{{\rm{A}}}\widetilde{\eta }\) as a function of the excess entropy for all investigated systems. We find a quasiuniversal breakdown of SE (i.e., departure from a constant value) around S_{ex}/k_{B}N ≈ −5.0. A breakdown of SE has been correlated to the crossing of the socalled twoparticle excess entropy and the excess entropy with temperature^{69}. It would be interesting to check whether this observation holds for the systems studied here.
Figure 4b shows \({\widetilde{D}}_{{\rm{A}}}\) versus \(\widetilde{\eta }\) in a plot where the SE relation is a straight line with slope −1 (the full black line). Around \({\widetilde{D}}_{{\rm{A}}}\approx\) 2 × 10^{−2} the SE relation begins to break down for all systems and compositions. These data suggest that the relevant variable is the excess entropy which in a quasiuniversal way correlates to both viscosity and diffusion coefficient and hence also their product, defining the SE relation. Although the departure from universality for viscosity and diffusion coefficient go in opposite directions in Fig. 3, a specific value of S_{ex} corresponds to a specific value of \({\widetilde{D}}_{{\rm{A}}}\) or \(\widetilde{\eta }\) due to the separate quasiuniversality of these two quantities. The breakdown is therefore bound to occur at more or less the same value of \({\widetilde{D}}_{{\rm{A}}}\) or \(\widetilde{\eta }\) for all the studied systems. Supplementary Fig. S2 provides the same figure for the Bparticle diffusion coefficient in which case the same conclusion is reached. We conclude that quasiuniversality for binary mixtures can rationalize the observations of Flenner et al.^{68} that SE breaks down at the same scaled relaxation time.
The excess entropy approach detailed here does not clarify the origin of the SE breakdown, other than it should occur in a quasiuniversal manner. Other theoretical approaches, such as dynamical facilitation or the random firstorder transition theory, have the SE breakdown as a consequence of dynamical heterogeneity^{15,64,65}. These theories provide predictions for the fractional SE exponent observed in Fig. 4b (see the black dashed line) which the excess entropy approach does not provide. We find that the fractional SE exponent for our most supercooled 2:1 KA data is ξ ≈ 0.73, which interestingly is also the number found in simulations of the onedimensional East model in dynamical facilitation^{64}. Similar fractional SE exponents have been noticed before, but in this study we go almost four decades below T_{MCT} and find an excellent agreement with the quasiuniversal excessentropy scaling.
Excessentropy scaling in experiments
A recent experimental study by Blodgett et al.^{78} proposed an interesting universality for metallic liquids by scaling viscosity with the hightemperature limit η_{0} and temperature with the onset of cooperative motion T_{A}. A good collapse of many different alloys was obtained in the hightemperature limit and close to the glass transition, motivated by avoided critical point theory (KKZNT)^{5}. The authors therefore found to a good approximation η/η_{0} = F(T/T_{A}). For the alloys studied the authors noted on average that η_{0} ∝ ρ and T_{A}/T_{l} ≈ 1.075, where T_{l} is the liquidus (freezing) temperature. Recall that for binary mixtures the liquidus temperature specifies the temperature at constant pressure above which the system is completely liquid (the opposite being the socalled solidus temperature). For Rsimple liquids, the temperature is given by T = h(ρ)f(S_{ex}) in which h(ρ) is a function of density^{79}. The freezing line is an approximate isomorph^{44,80}, and since an isomorph is characterized by h(ρ)/T = const., one has h(ρ) ∝ T_{f}(ρ) with the reference isomorph being the freezing line^{81,82}. The quasiuniversality found here explains the quasiuniversality found for metallic alloys since T/T_{A} ≈ T/T_{l} ≈ T/T_{f}(ρ) = f(S_{ex}).
In Fig. 4b we plot quasielastic neutron scattering measurements of the Ni diffusion coefficient against the reduced viscosity for the binary metallic alloy Zr_{64}Ni_{36} using data of Brillo et al.^{83} (see also, e.g., ref. ^{84}) and similar data for Zr_{36}Ni_{64} from ref. ^{85}. The reduced diffusion coefficients and viscosities for both Zr_{64}Ni_{36} and Zr_{36}Ni_{64} collapse nicely onto the quasiuniversal curve, reflecting the underlying quasiuniversal excessentropy scaling relationship. The same figure also shows data for the Vit4 (Zr_{46.8}Ti_{8.2}Cu_{7.5}Ni_{10}Be_{27.5}) fivecomponent metallic glass former from Yang et al.^{86}. The Vit4 glass former also collapses nicely onto the quasiuniversal curve. This shows that quasiuniversality extends beyond the binary mixtures of main focus here. We return to this observation in the “Discussion”.
For testing quasiuniversal excessentropy scaling in experiments as in Fig. 3, the twobody entropy^{32,87} could be used as a proxy, but a hightemperature study indicates that it is not always a good approximation^{33}. For our data the twobody entropy is a somewhat worse correlator than the excess entropy and also weakens the correlation to the densityscaling exponent (see later section). We therefore emphasize that the scaling is correlated to the full excess entropy which is more difficult to calculate, unfortunately. Figure 4b provides an alternative procedure for testing quasiuniversality in experiments which avoids having to evaluate S_{ex} explicitly.
Additional tests for quasiuniversal behavior
Rosenfeld quasiuniversality for singlecomponent atomic liquids can be explained by appealing to the EXP pair potential, in terms of which other pair potentials under certain conditions may be expanded^{59}. For singlecomponent systems quasiuniversality therefore implies not only quasiuniversal Rosenfeld scaling, but also Young and Andersen’s structuredynamics scaling principle^{88,89}, quasiuniversal freezing rules^{90}, invariance of the reduced viscosity along the melting line^{91}, and more. The singlecomponent arguments do not, however, readily generalize to mixtures. In view of this, we proceed to test to which extent quasiuniversality holds for binary mixtures by checking whether the structure is similar amongst state points with similar dynamics, i.e., whether Young and Andersen’s scaling principle applies.
Figures 5a, b compares two different compositions (4:1 and 2:1) for the KA mixture at state points for which the excess entropy and reduced diffusion coefficients are almost identical. For these state points there is less than 9% difference in reduced diffusion coefficient and <0.5% difference in excess entropy. Nevertheless, we find that the AAparticle radial distribution functions (RDFs) show rather large deviations between these two systems, certainly much larger than what is normally found for singlecomponent atomic systems^{88,89}. Even larger deviations are found for the AB and BBparticle RDFs in Supplementary Fig. S3.
This observation implies that twobody correlations do not uniquely determine the supercooled dynamics and thus that manybody correlations are important for the dynamics of the system^{10,92,93}. The rather large difference in RDFs between the two compositions might also be anticipated from the relevance of the locally favored structures (bicapped square antiprisms) for the dynamics in these mixtures^{94}. Furthermore, this anticipation is supported by a connection between decoupling of component dynamics, dynamical heterogeneity, and development of different local mediumrangelike ordering in the supercooled regime for certain alloys where the local ordering is directly detectable in the RDFs^{95}.
Figure 5c, d compares AAparticle RDFs and MSDs amongst the KA and WS mixtures at the same 3:1 composition. The state points have <1% difference in reduced diffusion coefficient and <0.3% difference in excess entropy. We find also here rather large variations of the AAparticle RDFs and even larger ones for the AB and BBparticle RDFs (Fig. 5e, f).
Supplementary Fig. S4 compares the distribution of Voronoi volumes in the liquid for the same systems and state points as above, showing also here clear differences. The quasiuniversality found in supercooled binary mixtures thus appears to be more subtle than the quasiuniversality observed in singlecomponent atomic liquids at high temperatures. Future work should focus on clarifying the nature behind this observation in the supercooled regime which could be related to local orderings in the supercooled liquid^{95}.
Departure from universality
Figure 3 displayed some deviations from universality in the scaling in the supercooled regime. This section considers these deviations in more detail. Figure 6 shows the reduced viscosity and diffusion coefficient as a function of the excess entropy, where each data point is colored after its value for the densityscaling exponent^{44}.
The departure from universality in the supercooled regime correlates with the value of the densityscaling exponent, with a smaller value of γ moving the curve up for viscosity and down for diffusion, the opposite being the case for larger γvalues. More similar γvalues—irrespective of mixing rules, interaction types, etc.—therefore conform to a more universal scaling in the supercooled regime. Based on this observation, an empirical correction using only knowledge of γ can be formulated.
In order to investigate the departure from universality more closely, we use an empirical referencecurve fit to the viscosity data for the SCLJ and 4:1 GLJ systems representing the leftmost part of the data set. The following functional form is used
with the bestfit coefficients c_{0} = −0.601 ± 0.0541, c_{1} = 0.267 ± 0.0657, c_{2} = 0.256 ± 0.028, c_{3} = 0.0568 ± 0.005, and c_{4} = 0.00448 ± 0.000319, where the number after the ± indicates the estimated standard deviation. A plot of the viscosities for the SCLJ and 4:1 GLJ systems along with the reference curve is shown in Fig. 7a. Figure 7b displays for all binary mixtures the ratio of the viscosity to that obtained from the referencecurve fit \({\widetilde{\eta }}_{{\rm{r}}ef}\) of Eq. (5). There is clearly a systematic trend in γ, though a few exceptions can also be found.
The excessentropy dependence of the viscosity is superArrhenius. A pragmatic approach for linearizing the data is therefore to consider \({\mathrm{log}\,}_{10}({\mathrm{log}\,}_{10}(\widetilde{\eta }/{\widetilde{\eta }}_{{\rm{ref}}}))\); these values are shown in Fig. 8a. The slope in these coordinates, for a given value of γ, is approximately constant with a value of −0.8. The intercept value b(γ) was found to be acceptably modeled by linear interpolation between the intercept values for \({\gamma }_{\min }\) = 1.9 and \({\gamma }_{\max }\) = 6.1
yielding the overall correction of
Figure 8b shows the corrected data for viscosity using this threeparameter expression; to apply the correction only knowledge of γ is needed. A better collapse is obtained compared to Fig. 6a.
Discussion
For both the diffusion coefficient and the viscosity, the current study has detailed an almost compositionindependent relation to the excess entropy for a given system, as well as a quasiuniversal relation amongst different systems. As the viscosity and diffusion coefficient both show quasiuniversality, their product is also quasiuniversal. The SE relation for viscosity and diffusion coefficient must then break down at the same reduced relaxation time or, equivalently, the same value of the excess entropy. Our observations therefore rationalize the universal SE breakdown results of Flenner et al.^{68}. The departure from universality correlates with the densityscaling exponent γ with more similar γvalues exhibiting a more similar scaling. This may provide a hint towards explaining the observed deviations in the future.
The isomorph theory states that certain quantities in reduced units are invariant along constant excessentropy curves in the thermodynamic phase diagram. This fact leads immediately to excessentropy scaling as described in the Introduction. Does this necessarily imply a causal link between the excess entropy and transport coefficients? The answer is no because one can in principle take the opposite view and posit that transport coefficients control the excess entropy.
Quasiuniversality is often explained by referring to the HS model^{96}. The HS model was recently questioned as a good reference system as this model cannot account for all quasiuniversality observations^{58,59,60,96}. Likewise, we do not believe that the HS model can explain our observations, even by introducing two different spheres, as we considered both very soft and very harsh repulsive pair potentials, highly nonadditive and exothermic mixtures, and mixtures with effective medium interactions. It is also not obvious how the EXP pairpotential arguments for singlecomponent atomic liquid’s quasiuniversality can be extended to explain our observations. The fact that the RDFs and Voronoi volumes were observed not to be the same for state points with very similar dynamics and excess entropy, points to a possibly more complex kind of quasiuniversality than that of singlecomponent systems.
An open question is why quasiuniversality is only observed in atomic mixtures but not also in, e.g., singlecomponent molecular systems, even for small molecules^{25}. A conjecture is that by removing certain degrees of freedom (e.g., vibrational degrees of freedom) one might be able to unravel quasiuniversality in molecular systems^{52,97}. Another relevant question is how a large mass or size ratio would influence the scalings. We studied up to a factor of two in mass ratio and up to a factor of three in size ratio between the constituent particles. A recent study^{98} has shown that both cases can have a nontrivial effect on the dynamics of supercooled liquids. Binary mixtures with very large size ratios are not expected to be Rsimple^{53}. A possible explanation for the lack of scaling in some of the results of Krekelberg et al.^{22} is then that these systems are not Rsimple.
A limitation of the current study is the focus on binary atomic mixtures. Figure 4b included data for the Vit4 (Zr_{46.8}Ti_{8.2}Cu_{7.5}Ni_{10}Be_{27.5}) fivecomponent glass former^{86} and showed a good collapse for the diffusion coefficient of Ni/Ti/Cu. We therefore anticipate that mixtures with several components are also covered by the current quasiuniversality relation discovered for binary mixtures. However, it has been observed in some metallic glass formers that SE can apply for one specific component but not for others (see, e.g., ref. ^{99}). This behavior could be related to the development of different local orderings in the liquid as seen for a Crbased alloy^{95}. Fundamental questions are therefore: For which component does quasiuniversality hold—and why?
Related to this topic, a recent study found similar structure and dynamics for weakly polydisperse systems sharing the same repulsion when compared at the same T/T_{g} value^{100}, where T_{g} is the glass transition temperature. These results are consistent with our conjecture that quasiuniversality extends beyond binary mixtures. Due to the extremely timeconsuming simulations of this paper, this intriguing topic is left for future research.
A longstanding issue in the study of supercooled liquids is what controls the dynamics. We find in this study that the excess entropy correlates well with the viscosity and diffusion coefficient for a wide range of binary mixtures, including metallic alloys. Furthermore, evidence has been presented that these results may extend beyond binary mixtures. The novel multicomponent metallic alloys being designed today cannot be comprehensively studied in experiments because of the immense number of possible mixture compositions^{101,102}. The approach proposed in this paper offers a means of providing predictive guidance for the transport properties of novel alloys since the quasiuniversal excessentropy scaling is expected to hold for these liquids. As a result, it is a realistic hope that excessentropy scaling may facilitate the design of future metallic glasses.
Methods
Simulation details
Molecular dynamics computer simulations were carried out using Nvidia Geforce GTX 1080 graphics cards and the Roskilde University Molecular Dynamics (RUMD) package, version 3.4, in single precision^{71}. Very long equilibration runs (the longest ones lasting more than 12 months) were used to ensure equilibrium before initiating production runs. The equilibrium and production runs were in the NVT ensemble with Nosé–Hoover thermostatting^{103}. Possible crystallization was checked using various order parameters, potential energy, etc. It was confirmed after equilibration that the results are reproducible by running the productionrun simulations at least twice.
Binary mixtures
We studied six different binary mixtures: the Kob–Andersen binary Lennard–Jones (KA) mixture, the Wahnström (WS) mixture, the GLJ mixture, the KA exponential pair potential (KAEXP) mixture, alloys of copper and zirconium (CuZr), and a size asymmetric (AS) mixture. One or several compositions were studied for each mixture. We focused mainly on one density and varied the temperature, but for the 4:1 KA and 3:1 WS mixtures density was additionally varied. For reference the SCLJ liquid was also simulated (ρ = 0.850 and N = 1024). All pair potentials used a shiftedpotential cutoff, except for KAEXP which used a shiftedforce cutoff^{47,60,104,105}.
The KA mixture^{106,107} uses the LJ pair potential v_{αβ}(r) = \(4{\epsilon }_{\alpha \beta }[{({\sigma }_{\alpha \beta }/r)}^{12}{({\sigma }_{\alpha \beta }/r)}^{6}]\) with parameters: σ_{AA} = 1, σ_{BB} = 0.88, σ_{AB} = 0.80, and ϵ_{AA} = 1, ϵ_{BB} = 0.50, ϵ_{AB} = 1.50. The mass is unity for both particles and the cutoff is r_{cut} = 2.50σ_{αβ}. The density of interest was for 4:1 KA: ρ = 1.204, 3:1 KA: ρ = 1.400, 2:1 KA: ρ = 1.400, 1:1 KA: ρ = 1.450. The density was changed to avoid negative pressure upon cooling, and we note the composition for 4:1 KA is very slightly nonstandard with 4.019:1. The particle numbers were N = 1024, 10000, 10002, 10000 for 4:1, 3:1, 2:1, 1:1 KA, respectively. The time step was in the range Δt = 0.001–0.0035, depending on temperature. The longest production runs were 69 billion time steps.
The WS mixture^{108} uses the same pair potential and cutoff as the KA mixture but applies the Lorentz–Berthelot mixing rules with σ_{AA} = 1.0, σ_{BB} = 0.833, and ϵ_{AA} = ϵ_{BB} = ϵ_{AB} = 1. The masses are m_{A} = 2.0 and m_{B} = 1.0. The density of interest was for 4:1 WS: ρ = 1.000, 3:1 WS: ρ = 1.100, 2:1 WS: ρ = 1.100, 1:1 WS: ρ = 1.296. The particle numbers were N = 1024, 1000, 1002, 1024 for 4:1, 3:1, 2:1, 1:1 WS, respectively. The time step was in the range Δt = 0.001–0.0025. The longest production runs were 268 million time steps.
The GLJ mixture varies the exponents of the LJ pair potential but keeps the location of the minimum fixed; in our case m = 12 and n = 10, where m and n are the repulsive and attractive exponents of the GLJ pair potential, respectively. The GLJ pair potential is given by v_{αβ}(r) = \({\epsilon }_{\alpha \beta }/(mn)[n{({\sigma }_{\alpha \beta }/r)}^{m}m{({\sigma }_{\alpha \beta }/r)}^{n}]\). The parameters are: σ_{AA} = 2^{1/6}, σ_{BB} = 0.88 × 2^{1/6}, σ_{AB} = 0.80 × 2^{1/6}, and ϵ_{AA} = 1, ϵ_{BB} = 0.50, ϵ_{AB} = 1.50 with r_{cut} = (2.50/2^{1/6})σ_{αβ}. The density of interest was for 4:1 GLJ: ρ = 1.200, 2:1 GLJ: ρ = 1.350. The particle numbers were N = 1000, 1002 for 4:1, 2:1 GLJ, respectively. The time step was in the range Δt = 0.001–0.0025. The longest production runs were 1.1 billion time steps.
The KAEXP mixture uses the same parameters as the KA mixture but replaces the LJ pair potential with repulsive exponential pair potentials given by v_{αβ}(r) = \({\epsilon }_{\alpha \beta }\exp [r/{\sigma }_{\alpha \beta }]\). The cutoff is r_{cut} = 4.50ρ^{−1/3}, i.e., the cutoff depends on density. The density was ρ = 0.001 for 4:1 composition. The particle number was N = 1024 and the time step was \(\Delta \tilde{t}\) = 0.0025 in (macroscopically) reduced units. The longest production runs were 4.3 billion time steps. The singlecomponent EXP pairpotential liquid was studied in refs. ^{60,105}.
The CuZr mixture was simulated using the Effective Medium Theory (EMT) for metallic alloys^{109,110}. EMT is a semiempirical manybody potential derived from DFT that offers a significant advantage over, e.g., most standard Embedded Atom Method (EAM) potentials since EMT does not require a tabulated format for the potential. The unit system used for CuZr is with the length scale of angstrom Å, mass dimension of atomic mass unit u, and energy scale of electron volt eV. We studied the compositions CuZr_{64:36} and CuZr_{36:64} at the density ρ = 0.08 Å^{−3}, and the particle number was N = 1000 for both compositions. The time step was Δt ≈ 7.13 fs. The longest production runs were 400 million time steps.
The AS mixture is governed by the LJ pair potential with σ_{AA} = 1.00, σ_{AB} = 0.65, σ_{BB} = 0.30. ϵ_{AA} = 1.00, ϵ_{AB} = 1.40, ϵ_{BB} = 0.80. m_{A} = 2.0 and m_{B} = 1.0. r_{cut} = 2.50σ_{αβ} and ρ = 1.100 at 3:1 composition. For this kind of size disparity (more than a factor of three) it is difficult to avoid crystallization by phase separation even with a negative heat of mixing. The particle number was N = 1000 and the time step was Δt = 0.0025. The longest production runs were 67 million time steps.
Analysis
The diffusion coefficients of each particle type were obtained from fitting their respective longtime meansquare displacements to the Einstein relation. The shear viscosities were obtained from integrating the shearstress time autocorrelation function via the Green–Kubo relation
where S_{αβ} is the αβcomponent of the stress tensor (α ≠ β = x, y, z), V is the volume, and 〈. . . 〉 denotes an ensemble average. All three offdiagonal stress tensor components were averaged for better statistics. The value of the viscosity was extracted from the first maximum of the integral which corresponds to the plateau value obtained in the running integral of Eq. (8). The selfpart of the ISF was evaluted from \({{F}_{{\rm{s}}}}({\bf{q}},t)=\langle \exp [{\rm{i}}{\bf{q}}\cdot \Delta {{{\bf{r}}}_{i}}]\rangle\), where r_{i} is the position of particle i, and q is the wave vector. The length of the wave vector is given by the position of the first peak of the static structure factor.
The excess entropy S_{ex} was calculated from the thermodynamic relation
using thermodynamic integration, where F_{ex} is the excess Helmholtz free energy, and U_{ex} ≡ U is the potential energy. Application of thermodynamic integration to supercooled liquids is standard^{6,34,38}. First, a path at a high temperature T_{ref} above the critical point was chosen, integrating the equation
from low density (the ideal gas) to the density of interest ρ in order to obtain F_{ex}(ρ, T_{ref}), in which W is the virial defined by W = PV − Nk_{B}T. Both U and W were obtained from the actual simulations. Afterwards, a path at the constant density ρ was simulated, integrating from T_{ref} to T to obtain F_{ex}(ρ, T) using the identity
We confirmed that the results for S_{ex}, within half a percent, are independent of the thermodynamic path as well as of the applied discretization of density and temperature. Larger error bars on S_{ex} were found for the CuZr mixtures than for the other systems due to a nonmonotonic behavior at very low densities.
Data availability
The data in csv file format that support the findings of Figs. 1–4 and 6–8 are available as Supplementary information. Equilibrated starting configurations and RUMD scripts are available at https://doi.org/10.18434/M32244.
Code availability
The RUMD package is available online at http://rumd.org/.
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Acknowledgements
T.S.I. and J.C.D. are supported by the VILLUM Foundation’s Matter grant (No. 16515). We are indebted to Nick Bailey, Lorenzo Costigliola, Harold W. Hatch, David Heyes, Ken Kelton, Mohammed H. Mousazadeh, Thomas B. Schrøder, Thomas Voigtmann, and Fan Yang for many useful discussions and suggestions.
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T.S.I. designed research; I.H.B., and T.S.I. performed research; I.H.B., J.C.D., and T.S.I. analyzed data; and T.S.I. wrote the paper with input from J.C.D.
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Bell, I.H., Dyre, J.C. & Ingebrigtsen, T.S. Excessentropy scaling in supercooled binary mixtures. Nat Commun 11, 4300 (2020). https://doi.org/10.1038/s41467020179481
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