aa r X i v : . [ c ond - m a t . d i s - nn ] M a r Kauzmann’s paradox
U. Buchenau ∗ Forschungszentrum J¨ulich GmbH, J¨ulich Centre for Neutron Science(JCNS-1) and Institute for Complex Systems (ICS-1), 52425 J¨ulich, Germany (Dated: March 2, 2021)The rapid structural and vibrational entropy decrease with decreasing temperature in undercooledliquids is explained in terms of the disappearance of local structural instabilities, which freeze in atthe glass temperature as boson peak modes and low temperature tunneling states. At the Kauzmanntemperature, their density extrapolates to zero.
In 1948, Walter Kauzmann [1] realized that the excessentropy S exc of an undercooled liquid over the crystal ex-trapolates to zero at the nonzero Kauzmann temperature T K . Kauzmann’s paradox lies in the difficulty of findinga theory for the disappearance of the structural entropyof the liquid at a finite temperature [2].Of course, the undercooled liquid undergoes a kineticfreezing into the glass phase at the glass transition tem-perature T g > T K , thus avoiding Kauzmann’s paradox.But in fragile glass formers, T K lies closely below T g ,thus encouraging speculations about a true thermody-namic second order phase transformation [2] at T K . Thedisappearance of the excess entropy is well described by[3–5] S exc = S ∞ (cid:18) − T K T (cid:19) , (1)leading to an excess heat capacity proportional to 1 /T .Later, it became clear that not only the excess entropyextrapolates to zero at the Kauzmann temperature, butalso two other important quantities, namely the inverseviscosity [3, 5, 6] and the excess mean square displace-ment [7–10]. The increase of the viscosity with decreasingtemperature can be described by a thermally activatedprocess with an energy barrier increasing with 1 /T I − ,where I , the fragility index [11], takes values between 1for strong glass formers and up to 10 for fragile ones. Formany glass formers, this barrier increase is closely equalto the increase of the shear modulus [11, 12], an increasewhich is more than a decade stronger than the one cal-culated from the thermal expansion of the liquid and theGr¨uneisen parameter of the transverse sound waves inthe glass phase.The present paper ascribes the rapid structural entropyloss to the decrease in the density of the soft modes re-sponsible for the low temperature glass anomalies.The low temperature glass anomalies are well describedby the soft potential model [13–15], an extension of thetunneling model [16] to include soft vibrations and lowbarrier relaxations. New numerical results confirm threesoft potential model predictions, namely the existence ofa strong fourth order term in the mode potential [17], ∗ Electronic address: [email protected] a density of soft vibrational modes increasing with thefourth power of their frequency [17–21], and a densityof low barrier relaxations increasing proportional to thepower 1/4 of the barrier height [22]. A strong decreaseof the number density of the soft modes with decreas-ing glass temperature was established [21] using the newswap mechanism [23] to cool numerical glass formers tovery low freezing temperatures.Another crucial numerical finding [24–26] is the unsta-ble core of the soft modes (”soft spot”), stabilized by thesurroundings. The unstable core of the soft mode is a lo-cal saddle point configuration between two minima, lowerthan the saddle point in energy by the creation energy E s of the soft mode. E s was found to be about 2.5 k B T g in a binary glass former [26]. Since the modes couplestrongly to an external shear [26, 27], the core is a localshear instability, and the shear modulus (measured attimes shorter than the shear relaxation time) decreaseswith an increasing number of soft modes. This explainsthe strong decrease of the shear modulus with increasingtemperature [11, 12].With an increasing number of soft modes, the entropyincreases in two ways. The first is an increase of thestructural entropy (you have the choice where to put thelocal instability). The second is the increase in vibra-tional entropy [28, 29], because a softer mode has a largernumber of occupied vibrational levels between which itcan choose.The increase of the vibrational entropy has been ana-lyzed in detail from inelastic neutron scattering data inselenium [28]. It was found that it is a fraction 0.3 ofthe total entropy increase, having the same temperaturedependence as the total. Selenium is a short chain poly-mer [30] with one anharmonic Van-der-Waals, one bondbending, and one bond stretching degree of freedom peratom.The heat capacity data of crystalline, glassy and liquidselenium [31] are shown in Fig. 1. The crystalline valueincreases above the Dulong-Petit expectation for a per-fectly harmonic solid by about 0.3 k B per atom at themelting temperature, a factor of two to three higher thanthe value c p − c V = T V m α c κ c , (2)expected from the molar volume V m , the thermal volumeexpansion α c , and the compressibility κ c of the crystal[32]. The dashed line in Fig. 1 is the calculation fromthe vibrational density of states of the crystal plus ananharmonic linear increase which reaches 0.3 k B per atomat the melting point.At the glass temperature, the liquid heat capacity peratom is ∆ c p = 1 . k B higher than the one of the glass,suggesting a structural strain energy of about k B T / E s of the soft modes. For higher energies, the vibra-tional degree of freedom can get rid of its high energyby the creation of a soft mode in its neighborhood. Thepostulate is supported by numerical evidence in a binaryglass described in the first section of the SupplementalMaterial.To determine the cutoff energy E s , one first calcu-lates ∆ c V by subtracting the thermal expansion con-tribution of eq. (2) for both liquid and glass. With α g = 1 .
41 10 − /K and κ g = 1 .
09 10 − m /J for theglass and α l = 3 .
63 10 − /K and κ l = 1 .
46 10 − m /Jfor the liquid [37], one finds ∆ c V = 1 . k B , requiring acutoff energy E s /k B T g = 4 .
6. With this limitation, eachof the two covalent degrees of freedom contributes 0.44 k B to ∆ c p , and the van-der-Waals degree of freedom re-sponsible for the thermal expansion contributes the rest,0.95 k B , about twice as much as each harmonic degree offreedom.The continuous line in Fig. 1 was calculated with E s /k B T g = 4 . T g , assuming the decrease of E s withincreasing temperature proportional to 1 /T . expectedfrom the average Gr¨uneisen parameter of the glass phase[32] and the thermal expansion of the liquid.A value close to E s /k B T g = 4 .
200 300 400 500 600203040 T m Dulong-PetitT g T K crystal glass+melt E s =4.6 k B T g eq. (1) selenium C p ( J / m o l e K ) temperature (K) FIG. 1: Heat capacities of crystalline, glassy and liquid se-lenium [31] compared to the calculation with a cutoff energyof 4.6 k B T g (continuous line) and with eq. (1) (dotted line).The dashed line is the fit of the crystalline data. The concept of a structural strain energy in each vibra-tional degree of freedom is supported by the Prigogine-Defay ratio of the glass transition [34–36]Π = ∆ c p ∆ κ (∆ α ) T g V m = ∆ H ∆ V (∆ H ∆ V ) , (3)which relates the increases of the heat capacity at con-stant pressure ∆ c p , of the compressibility ∆ κ and ofthe thermal volume expansion ∆ α at the glass temper-ature T g to the additional enthalpy and volume fluctu-ations ∆ H and ∆ V in the liquid, respectively. V m isthe molar volume. If the enthalpy and volume fluctua-tions are completely correlated, the Prigogine-Defay ratiois one. But if there are contributions from different vi-brational modes i with different Gr¨uneisen parameters γ i = ∆ V i /κ ∆ H i , then Π = γ γ , (4)with the γ -values weighted with their contribution to∆ c p .By far the strongest Gr¨uneisen parameter in seleniumis the positive van-der-Waals one, but the bond bendingmodes have a small negative one which is a factor of 8weaker [32]. Inserting these values and their respectiveweight into eq. (4), one calculates a Prigogine-Defay ratioof 2.19, in agreement with the measured values 1.7, 2.0[37] and 2.4 [36] in the literature within their error bars.Since the two harmonic covalent degrees of freedomcontribute only to ∆ c p and practically nothing to ∆ α nor to ∆ κ , one expects (and finds, not only in selenium[37], but also in many other glass formers [3]) the validityof the fourth Ehrenfest equation [35] dT g dp = ∆ κ ∆ α , (5)while the first Ehrenfest equation gives a dT g /dp whichis a factor 1 / Π too small.Of course, the calculated curve in Fig. 1 also reachesthe equality of crystal and liquid entropy at a temper-ature which is only slightly lower than the Kauzmanntemperature T K = 245 ± OH ) and 3-bromopentane(C BrH ) ∆ c p corresponds to 8.91 and 9.1 k B permolecule [5], indicating 18 participating degrees of free-dom, in the simplest possible interpretation the degreesof freedom of the six non-hydrogen atoms per molecule.The same interpretation works in n-propanol (C OH )with a ∆ c p corresponding to 6.35 k B per molecule [5]and in glycerol (C O H ) with a ∆ c p corresponding to9.6 k B per molecule [38]. In this last case, ∆ c p is as largeas the c p of the glass, showing that a degree of freedomcan take up distortion energy without the quantum lim-itation which holds for the uptake of vibrational energy.There are other even more drastic examples like sulfuricacid with water [39].A much smaller participation of additional degrees offreedom is found in polymers [39], where ∆ c p correspondson the average to 1.4 k B per ”bead”. Here every chainatom counts as a bead, so the polystyrene monomer C H consists of two beads, where half of the beads carry aphenylene ring. But these phenylene rings do not seemto contribute; polystyrene is in the middle of the polymerspectrum [39], while selenium with its ∆ c p of 1.83 k B per bead is at the upper end. Obviously, in polymersthe essential things happen only between beads, eitherin the two covalent bonds connecting neighboring beadsor in the one van-der-Waals bond of a bead to many otherbeads in neighboring chains.Both findings, molecular glasses and polymers, providethe common criterion that a bond must be able to take upstructural distortion energy in order to participate. Anatom or atomic group which is covalently fixed to onlyone other atom and feels the disorder around exclusivelyby the weak van-der-Waals forces, does not contribute. Inthese examples, the real glass transition occurs when ∆ c p reaches k B T / and GeO , where ∆ c p is small [41], corresponding onlyto 0.96 k B per SiO -unit and 0.66 k B per GeO -unit,respectively. In these two glass formers, the soft modesare connected tetrahedra librations [42, 43] with a neg- -1 0 1012 core quartic double-well E ( A ) / E s A/A min
FIG. 2: Mode potentials in units of the creation energy E s of the soft mode as a function of the normal coordinate A :continuous line core potential, dashed line quartic soft modepotential for perfect compensation, dotted line tunneling statepotential. ative Gr¨uneisen parameter, responsible for the densitymaximum in SiO around 150 K [44], together with thepositive Gr¨uneisen parameter of the higher frequency vi-brations. The very large Prigogine-Defay ratios of morethan hundred for SiO and 6.85 for GeO [41] demon-strate again that the slow excess enthalpy fluctuations ofthe liquid occur not only in the soft modes, but also inthe high frequency modes.The low ∆ c p -values in SiO and GeO require muchlower cutoff energies E s /k B T g than in selenium, 1 . k B T g in SiO and k B T g in GeO . This interpretation is sup-ported by a high density of tunneling states at low tem-peratures [46] and a much higher boson peak [43] in SiO than in selenium [28]. If one adds NaO to SiO , one re-trieves the k B T / E ( A ) as a function of the nor-mal coordinate A . The mode is soft because the negativecontribution to the restoring force from the unstable corepotential (the continuous line in Fig. 2, described here bya cosine function), is compensated by the positive elas-tic restoring forces from the surroundings. One gets thepurely quartic potential (the dashed line in Fig. 2) forperfect compensation and if the zero point of the har-monic outside forces happens to lie exactly on the saddlepoint. If the outer forces happen to be weaker or E s happens to be higher, one gets a low-barrier double-wellpotential (the dotted line in Fig. 2), such as one needsto explain low temperature tunneling states [16].It is clear that in this situation one expects a constantdistribution of linear and quadratic terms adding to thepurely quartic potential, the central assumption of thesoft potential model. These modes freeze in at the glasstemperature and provide the boson peak and the tunnel-ing states, which dominate the heat capacity, the thermalconductivity and the sound absorption at low tempera-tures.To summarize, the undercooled liquid looses its struc-tural and vibrational entropy as it approaches the glasstemperature by reducing the number of local structuralinstabilities, which freeze in as boson peak vibrations orlow temperature tunneling states at the glass transition.Their density extrapolates to zero at the Kauzmann tem-perature. Kauzmann’s paradox is explained if the liquidowes its many structural choices exclusively to local in-stabilities.The creation energy of the soft modes limits the dis-tortion energy of the vibrational degrees of freedom. Vit-reous silica has a low soft mode creation energy and, con-sequently, a small heat capacity jump at the glass tran-sition, and a low Kauzmann temperature. 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