Characterization of the non–Arrhenius behavior of supercooled liquids by modelingnon–additive stochastic systems

A. C. P. Rosa Jr.,1 C. Cruz,1 W. S. Santana,1 and M. A. Moret2, 3

1Grupo de Informacao Quantica, Centro de Ciencias Exatas e das Tecnologias,Universidade Federal do Oeste da Bahia. Rua Bertioga,

892, Morada Nobre I, 47810-059 Barreiras, Bahia, Brazil.2Programa de Modelagem Computacional - SENAI - CIMATEC, 41650-010 Salvador, Bahia, Brazil

3Universidade do Estado da Bahia - UNEB, 41150-000 Salvador, Bahia, Brazil(Dated: May 10, 2019)

The characterization of the formation mechanisms of amorphous solids is a large avenue for re-search, since understanding its non-Arrhenius behavior is challenging to overcome. In this context,we present one path toward modeling the diffusive processes in supercooled liquids near glass tran-sition through a class of non-homogeneous continuity equations, providing a consistent theoreticalbasis for the physical interpretation of its non-Arrhenius behavior. More precisely, we obtain thegeneralized drag and diffusion coefficients that allow us to model a wide range of non-Arrhenius pro-cesses. This provides a reliable measurement of the degree of fragility of the system and an estimationof the fragile–to–strong transition in glass–forming liquids, as well as a generalized Stokes-Einsteinequation, leading to a better understanding of the classical and quantum effects on the dynamics ofnon–additive stochastic systems.

I. INTRODUCTION

The dynamic response of a wide class of materials canbe achieved using the so-called Arrhenius law [1–5]. Ba-sically, it consists of an exponential decay with the in-verse of the temperature characterized by the so-calledtemperature independent activation energy [1–3]. Thesearch for a physical interpretation for the activation en-ergy established the fundamentals of the transition statetheory [6–8] since it associates an Arrhenius-like behaviorwith diffusive processes in several systems [4, 9–12].

However, from the development of new technologiesand advances in materials preparation techniques, a widevariety of new compounds could be synthesized, leadingto the improvement of experimental techniques for thestudy of chemical reactions and diffusive processes. Inthis scenario, several systems have revealed deviationsfrom Arrhenius behavior, evidenced through the tem-perature dependence of the activation energy [1]. Inrecent years, the characterization of non-Arrhenius be-haviors has received considerable attention, since it wasobserved in water type models SPC/E (extended simplepoint charge) [13, 14], food systems [15], diffusivity in su-percooled liquids near glass transition [2, 10, 11, 16, 17],chemical reactions [8, 18, 19] and several biological pro-cesses [20, 21]. Therefore, modeling these non-Arrheniussystems is a large avenue for research and an actual chal-lenge to overcome.

The non-Arrhenius behaviors manifest themselves asconcave curves (sub-Arrhenius behavior), associated withnon-local quantum effects [8, 19, 26], or convex curves(super-Arrhenius behavior), associated with the predom-inance of classical transport phenomena [3, 20, 26, 27].Despite much effort by the scientific community, thereare only a few phenomenological relationships proposedto model non-Arrhenius processes, such as the Vogel-

Tamman-Fulcher equation [22–24] and the Aquilanti-Mundim d-Arrhenius model [3, 19, 20, 25–28]. Otherphenomenological expressions have recently been pro-posed [5, 10, 17]. However, there is a need to establish awide class of equations that characterize non-Arrheniusprocesses in a consistent theoretical basis for the physicalinterpretation of the characteristic non-Arrhenius behav-ior of several diffusive processes.

Nevertheless, Aquilanti-Mundim equation can be de-rived from the stationary process of the non-linearFokker-Planck equation, and the diffusivity dependencewith the temperature is consistent with experimental re-sults [5]. Non-linear Fokker-Planck equations, especiallythose whose stationary solutions maximizes non-additiveentropies [29], such as the Tsallis entropy [30], has beensuccessfully employed for modeling non-Markovian pro-cesses [31, 32], anomalous diffusion [33, 34], astrophysi-cal systems [35], sunspots [36] and pitting corrosion [37],suggesting that this class of equations can also be an al-ternative way to describe the non-Arrhenius behavior ofnon-additive stochastic systems.

In this context, we show in this letter a class of non-homogeneous continuity equations whose the general-ized coefficient allow the modeling of a wide range ofnon-Arrhenius processes. We modeled the characteris-tic super-Arrhenius behavior of diffusivity and viscos-ity in supercooled liquids, determining a characteristicthreshold temperature associated with the discontinu-ities in its dynamic properties, such as the viscosity andthe activation energy. In addition, we define a general-ized exponent that characterizes the non-Arrhenius pro-cess and serves as an indicator of the level of fragilityin glass-forming systems, whereas the threshold temper-ature indicates a fragile–to–strong transition, the gen-eral behavior of metallic glass–forming liquids [38]. Ourmodel also derives a generalized version for the Stokes-Einstein equation, where we obtain a characteristic tem-

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perature independent behavior (at low temperatures) forsub-Arrhenius processes, and a sudden death behavioraround the threshold temperature for super-Arrheniusprocesses. Our results pave the way for the character-ization of the breakdown of the standard Stokes-Einsteinrelation [39–42], mainly in supercooled liquids [43], pro-viding one path toward understanding the dynamic evo-lution of non-Arrhenius processes, leading to the estab-lishment of a theoretical interface between a macroscopicand microscopic perspective of the matter through a non-equilibrium statistical mechanics.

II. GENERALIZED REACTION–DIFFUSIONMODEL

Let us consider a concentration ρ(r, t) of a substancemeasured in volume V at time t, the total amount ofsubstance for the same volume is given by the non-homogeneous continuity equation. In this context, wepropose the following conditions:

(i) f(r, t) = ~∇ · ~η(r, t) is a volumetric density perunit time associated with dissipative processes and~η(r, t) is a field of non-zero divergence;

(ii) ~η(r, t) = −κ−1m ρm~∇φ, where κm is a positive con-

stant parameterized by the exponent m and φ is ageneralized potential;

(iii) for the steady state ~η(r, t)→ ~ηS(r), which is a fieldof zero divergence;

(iv) ~J = −D(r, t; ρ)~∇ρ is a diffusion flux, for a gener-alized version of Fick’s first law [12, 44] in whichD(r, t; ρ) is a generalized diffusion coefficient;

(v) D(r, t; ρ) = (Γ/2)ρn−1 [29], where Γ is a positivedefinite parameter, related to a class of nonlinearequations associated with anomalous diffusive pro-cesses [45].

In this circunstances, the non-homogeneous continuityequation becomes a particular class of nonlinear Fokker-Planck equations [29] whose non-linearity of the gener-alized drag coefficient involves the information of thedissipative or exchange processes, such as phase tran-sitions or chemical reactions. The equations are definedin such a way that leads to the generation of solutionsthat compose a class of rapidly decreasing functions [46]that maximizes non-additive entropies, such as the Tsal-lis entropy [30], since this guarantees the possibility offundamental solutions for the diffusion equation. Fromthese conditions we obtain an alternative way to describethe non-Arrhenius behavior of the diffusion processes ofnon-additive stochastic systems such as supercooled liq-uids, from a consistent theoretical basis.

III. DIFFUSIVITY AND VISCOSITY OFGLASS–FORMING LIQUIDS

The characterization of diffusivity and viscosity in su-percooled liquids are effective to understand the glasstransition and the formation mechanisms of amorphoussolids. In order to establish a wide class of equationsthat characterize non–Arrhenius behavior of supercooledliquids from a theoretical perspective, we define the dif-fusion coefficient in (v) for the particular case n = 2.In this context, the generalized potential φ can be rein-terpreted as a potential energy U (r) associated with aconservative force field, in dynamic equilibrium. Thus,we obtain the non-homogeneous continuity equation:

∂ρ(r, t)

∂t= κ−1

m~∇ ·[(~∇U (r)

)ρm]

+Γ

2∇2[ρ2]. (1)

Because the stationary solution of Eq. (1) is a gen-eralized exponential, the dependence of the generalizeddiffusion coefficient with the temperature can be writtenas,

D(T ) = D0

[1− (2−m)

E

kBT

] 12−m

, (2)

where D0 = ΓC0 (C0 is a normalization constant of the

stationary concentration), E = −∫~∇U (r) · r is a gener-

alized energy and C2−m0 κmΓ = kBT [47, 48]. From Eq.

(2), the Arrhenius standard behavior is recovered whenthe coefficient m → 2, then the activation energy E, inthis limit, corresponds to a temperature independent en-ergy.

Figure 1 shows the diffusivity of a supercooled liq-uid as a function of the reciprocal temperature. Underthe condition m < 2 the proposed model encompassesa class of super-Arrhenius diffusive processes, associatedwith the predominance of classical transport phenomena[3, 20, 26, 27], predominantly according to experimentalreports [5, 10, 16, 17]. In addition, the model also coversa wide class of sub-Arrhenius diffusive processes, charac-terized by the condition m > 2, associated with non-localquantum effects [8, 19, 26], and less sensitive to the ex-ponent variations than the super-Arrhenius processes.

It is also possible to verify the existence of a thresholdtemperature for super-Arrhenius processes, from whichthe diffusivity goes to zero, given by

Tt =(2−m)E

kB(3)

From Eq. (2) we can obtain the temperature depen-dence of the activation energy as

EA (T ) =E

1− (2−m) EkBT

, (4)

the main feature of non–Arrhenius processes. Further-more, from Eq. (4), for the m→ 2 the activation energy

3

D

(T)/D

0

10−4

10−2

100

1000/T0 2 4 6

F2F3F4F5F7

Sub-Arrhenius

Super-Arrhenius

m = 2.2m = 2.1

m = 2.0

m = 1.9

m = 1.8

Figure 1. Monolog plot of the diffusivity as a function ofthe reciprocal temperature. The curves m > 2 characterizea class of sub-Arrhenius processes, while the curves m < 2characterizes a class of super-Arrhenius processes. The m = 2curve corresponds to the usual Arrhenius plot. The curveswere simulated for the E/kB = 1000K condition

achieves a temperature independent behavior EA(T ) →E corresponding to the Arrhenius law, as previously men-tioned.

Figure 2 shows the activation energies, correspondingto the diffusivity curves presented in Figure 1, calculatedfrom Eq. (4). The activation energy is an increasingfunction of the reciprocal temperature for sub-Arrheniusprocesses and decreasing for super-Arrhenius processes.In addition, for the super-Arrhenius processes, when thethreshold temperature, Eq. (3), is achieved the activationenergy diverges to infinity, indicating that this temper-ature is related to the viscosity divergence in the glasstransition.

From Eq. (1), we can define the viscosity from thegeneralized mobility of the fluid [47] as

η (T ) = ακmρ1−m , (5)

where α is a positive definite constant. From Eq. (5) theArrhenius model from the viscosity is recovered for thelimit case m→ 2.

Figure 3 shows the viscosity as a function of the recip-rocal temperature. For super-Arrhenius processes (m <2) the threshold temperature characterizes the regimefrom which the viscosity diverges to infinity. Thus, thethreshold temperature, Eq. (3), serves as an indicationof how close the system is to the glass transition regionbecause it involves discontinuities in the dynamic proper-ties, such as the activation energy, Eq. (4), and viscosity,Eq. (5). The glass-liquid transition occurs in a rangeof temperatures for which the viscosity assumes a largevalue, but still does not diverge. In most glass-formingliquids, the glass transition temperature is established atthe viscosity reference value of 1012 Pa.s, thus Tt ≤ Tg.

EA(T

) / k

B

0

103

2·103

3·103

1000/T0 2 4 6

F1F2F3F4F5

m = 2.2m = 2.1

m = 2.0

m = 1.8

m = 1.9

Super-Arrhenius

Sub-Arrhenius

Figure 2. The activation energy as a function of the recipro-cal temperature. The curves where the activation energy isan decreasing function of the reciprocal temperature charac-terize a class of sub-Arrhenius processes, while the increasingcurves characterize a class of super-Arrhenius processes. Inaddition, the m = 2 curve corresponds to the Arrhenius ac-tivation energy, characterized by a temperature independentbehavior. The curves were simulated using the scale factorE/kB = 1000K.

η(T)

/ α κ

(m)

100

102

104

1000/T0 1 2 3 4 5 6

F1F2F3F4F5

Sub-Arrhenius

Super-Arrheniusm = 1.8

m = 1.9

m = 2.0m = 2.1m = 2.2

Figure 3. The viscosity as a function of the reciprocal temper-ature. The curves m > 2 characterize a class of sub-Arrheniusmodels for viscosity, while the curves m < 2 characterize aclass of super-Arrhenius models. The m = 2 curve corre-sponds to the Arrhenius model for the viscosity.

This model can also be used to calculate the level offragility Mη in glass-forming systems [38, 49, 50] by ourexponent m as

Mη =

(m− 1

2−m

)(1

1− Tt

Tg

), (6)

For the usual Arrhenius diffusive processes, the condi-

4

tion m = 2 characterizes a strong glass system, whereasfor a wide class of super-Arrhenius diffusive processes thecondition m < 2 characterizes a fragile glass [49, 50]. Inaddition, another important feature that arises from ourmodel is the distinguishability between strong and fragilesystems for super-Arrhenius processes (m < 2), since howfar further the glass transition temperature Tg is from thethreshold temperature, Eq. (3), more fragile the systemwill be. In this way, the ratio Tt/Tg (Eq. 6) indicatesa fragile–to–strong transition [38] usually found in somewater and silica systems, which is possibly a general be-havior of metallic glass–forming liquids [38], where an ini-tially fragile supercooled liquid can be transformed intoa strong liquid upon supercooling toward Tg. Therefore,the dynamics around the glass transition region, charac-terized by Eq. (3) provide a measurement of how fragile asystem is, establishing the theoretical basis understand-ing the intrinsic features of the formation mechanisms ofamorphous solids.

Moreover, a remarkable result can be extracted fromour model. The product between the generalized diffu-sion coefficient, Eq. (2) and the viscosity, Eq. (5), ob-tained from our generalized model for reaction–diffusionprocesses, provides a generalized Stokes-Einstein relationfor any non–Arrhenius diffusion process, given by

Dη = αkBT

[1− (2−m)

E

kBT

](7)

Figure 4 shows the temperature dependence of the gen-eralized Stokes-Einstein relation, Eq. (7), for differentvalues of the coefficient m. For the super-Arrhenius dif-fusive processes (m < 2) the relation gives an estimateof the glass transition temperature, since the generalizeddiffusion coefficient, Eq. (2), goes to zero faster than theviscosity, Eq.(6), diverges to infinity. Thus, the regionin which the generalized Stokes-Einstein goes to zero isequivalent to the threshold temperature of glass transi-tion, Eq. (3). In addition, as demonstrated in Figure4, the usual form of the Stokes-Einstein relation is re-covered from Eq. (7) under two conditions: (i) for anyArrhenius-like process (m→ 2); and (ii) for the conditionE << kBT , i.e., thermal fluctuations predominate in theprocess, to the detriment of the concentration gradient.

On the other hand, for the sub-Arrhenius diffusiveprocesses, it is worth noting that, from the conditionE >> kBT , the generalized Stokes-Einstein equation,Eq. (7), presents a temperature independent behavior,enabling the differentiation of the classical and quantumregimes, paving the way for the characterization of sub-Arrhenius processes through Eq. (7). This provides onepath toward understanding the quantum effects in thedynamics of the non–additive stochastic systems.

IV. CONCLUSIONS

In summary, our main result was to provide an alter-native way to describe the non–Arrhenius behavior of

Dη/

α k B

0.1

1

10

100

Temperature (K)0.1 1 10 100

Sub-Arrhenius

Super-Arrhenius

m = 2.2

m = 2.1

m = 2.0

m = 1.9

m = 1.8

Figure 4. The temperature dependence of the generalizedStokes-Einstein relation, Eq. (7), for different values of the co-efficient m. For super-Arrhenius processes (m < 2) the regionin which the generalized Stokes-Einstein rapidly goes to zerois equivalent to the threshold temperature of glass transition,Eq. (3). For sub-Arrhenius processes, (m > 2), from the con-dition E >> kBT the generalized Stokes-Einstein equationpresents a temperature independent behavior. The straightline corresponds to the usual form of the Stokes-Einstein re-lation, recovered for any Arrhenius-like process (m = 2) andfor the condition E << kBT , that separates the super andsub-Arrhenius regimes.

diffusive processes in glass-forming liquids. Our modelwas characterized by a generalized exponent m that de-fines the class of non-Arrhenius processes and serves asan indicator of the degree of the fragility in these sys-tems. In addition, we determine the threshold tempera-ture, Eq. (3), from which the dynamic properties, suchas the activation energy and viscosity diverges, and givesus a reliable estimate of the degree of fragility, since theratio Tt/Tg (Eq. 6) indicates a fragile–to–strong transi-tion, establishing the theoretical basis for understandingthe intrinsic features of amorphous solids.

Also interesting is the realization of a generalizedStokes-Einstein equation, Eq. (7), which allows usto characterize the breakdown of the standard Stokes-Einstein relation in supercooled liquids. For sub-Arrhenius processes, the generalized relation presents acharacteristic temperature independent behavior at lowtemperatures while, for the class of super-Arrhenius dif-fusive processes, rapidly goes to zero around the thresh-old temperature. Moreover, the usual form of the Stokes-Einstein relation is recovered for any Arrhenius-like pro-cess and when the thermal fluctuations predominate inthe process to the detriment of the concentration gradi-ent (E << kBT ). Our results provide one path towardthe differentiation of the super and sub-Arrhenius pro-cesses, leading to a better understanding of the classicaland quantum effects on the dynamics of non–additivestochastic systems, paving the way for the characteriza-

5

tion of the formation mechanisms of amorphous solidsthrough the study of non-Arrhenius diffusive processesin these systems.

ACKNOWLEDGMENTS

The authors thank James C. Phillips for his helpful

comments. This study was financed in part by the CNPqand the Coordenacao de Aperfeicoamento de Pessoal deNıvel Superior - Brasil (CAPES) - Finance Code 001.

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