A new view of the Lindemann criterion
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov A new view of the Lindemann criterion
U. Buchenau ∗ and R. Zorn J¨ulich Center for Neutron Science, Forschungszentrum J¨ulichPostfach 1913, D–52425 J¨ulich, Federal Republic of Germany
M. A. Ramos
Laboratorio de Bajas Temperaturas, Departamento de Fisica de la Materia Condensada,Condensed Matter Physics Center (IFIMAC) and Instituto Nicolas Cabrera,Universidad Autonoma de Madrid, Cantoblanco, E-28049 Madrid, Spain (Dated: November 19, 2013)The Lindemann criterion is reformulated in terms of the average shear modulus G c of the meltingcrystal, indicating a critical melting shear strain which is necessary to form the many differentinherent states of the liquid. In glass formers with covalent bonds, one has to distinguish betweensoft and hard degrees of freedom to reach agreement. The temperature dependence of the picosecondmean square displacements of liquid and crystal shows that there are two separate contributions tothe divergence of the viscosity with decreasing temperature: the anharmonic increase of the shearmodulus and a diverging correlation length . PACS numbers: 63.50.+x, 64.70.Pf
According to the hundred years old Lindemann cri-terion [1], melting occurs when the thermal motion ofthe atoms of the crystal reaches a critical mean squaredisplacement of about one tenth of the interatomic dis-tance. It is not very accurately fulfilled [2, 3], but it isan intriguing unexplained relation between dynamics andthermodynamics which has always fascinated the physi-cists working in the field.Here we reformulate the Lindemann criterion in termsof the average shear modulus G c of the melting crystal.If one takes the crystal to be a Debye solid and uses thehigh temperature approximation, the Lindemann crite-rion reads h u c i ( T m ) = 3 k B T m M ω D ≡ (0 . a ) , (1)where the value 0.072 has been fitted to the data collec-tion of Grimvall and Sj¨odin [3], h u c i is the mean squaredisplacement in one direction, the atomic distance a isdefined by the atomic volume v = a , M is the averageatomic mass and ω D is the Debye frequency.The Debye frequency is given by the longitudinal soundvelocity v l and the transverse sound velocity v t ω D = 18 π v (1 /v l + 2 /v t ) . (2)Taking an average ratio v l /v t of 1.8, one gets the meansquare displacement h u c i = 0 . k B TG c v a (3) ∗ Electronic address: [email protected] and can express the Lindemann criterion in terms of
M v t = G c v G c v = 31 k B T m = 51 k B T g , (4)with the approximate relation T g = 0 . T m between melt-ing temperature T m and glass temperature T g .The formulation suggests a new view of the Lindemanncriterion: it does not indicate an instability of the crys-tal, but it is rather a necessary condition for the entropyof the liquid. In order to obtain the two stable inher-ent structures per atom which together with the excessvibrational entropy supply the melting entropy of about k B per atom, one needs to shear the atomic volumes inthe melt by about ten percent. Obviously, construct-ing a stable solid out of flexible strained units providesmuch more possibilities than a construction out of rigidunits, though one has to pay the price of a considerablestrain energy. Above T m , the entropy gain overcompen-sates this shear energy and makes the liquid the stablethermodynamic state. substance ρ M T g G Gvk B T g f s Gvf s k B T g kg/m a.u. K GPavit-4 a O Gv/k B T g in seven glass formers.vit-4 is the metallic glass vitralloy-4, CKN stands for the ionicglass former K Ca (NO ) and PVC is polyvinylchloride. a ref. [4]; all other data from reference [5]. The glass shear modulus G is usually smaller than G c .Table I lists experimental values of Gv/k B T g . In themetallic glass vitralloy-4, the value is even higher thanthe Lindemann prediction of eq. (4), showing that a mix-ture of atoms with different sizes melts more easily thana pure substance. In a large data collection on metal-lic glasses [4], the values for Gv/k B T g range from 48 to91. The same tendency is seen in numerical calculationsfor the Lennard-Jones potential, where one only reachesthe glass state with atoms of different sizes [6, 7], whilea pure Lennard-Jones crystal has the same Lindemannmean square displacement at its melting point as puremetals [8].Selenium has a much lower Gv/k B T g -ratio than themetallic glasses. But selenium has covalent bonds. Eachselenium atom is covalently bonded to two neighbors.This implies that one has two hard degrees of freedomper atom (the Se-Se distance and the Se-Se-Se angle) andone soft van-der-Waals degree of freedom; the fraction f s of soft degrees of freedom per atom is 1/3. In selenium,the frequency of the Se-Se bond stretching vibration isabout eight times higher than the van-der-Waals bandat low frequencies [9]. The Se-Se-Se covalent bending vi-bration is lower, but still about a factor of three higherthan the low frequency band. This implies that 90 %of the mean square displacement is due to the van-der-Waals bonds. One degree of freedom supplies virtuallythe whole mean square displacement and eq. (3) is de-rived for three equivalent degrees of freedom per atom.Consequently, Gv/k B T g has to be corrected by a factor of3. Indeed, Gv/f s k B T g = 33 . O , one excludes the B-O stretching as wellas the O-B-O bending. This leaves three van-der-Waalsdegrees of freedom per B O -unit, arriving at f s = 1 / unit as a rigid molecule with only six degrees of freedom,reducing the number of degrees of freedom by 19/33. Thepolyvinylchloride monomer C H Cl has only the two C-C rotations as soft degrees of freedom, which implies f s = 1 /
9, again in reasonable agreement with the Lin-demann criterion. The average value for the seven glassformers in Table I is
Gv/f s = 41 k B T g , about 80 % ofthe crystalline value of eq. (4).It is tempting to look for a connection between thisvalue and the effective energy barrier for the flow at T g ,which happens to have a value nearby, about 36 k B T g .But in the Lindemann relation Gv is a force constant,not a barrier. Converting it into a barrier requires amicroscopic consideration:Consider the four neighboring atoms shown in Fig. 1,undergoing a shear transformation from the stable con-figuration on the left to another stable configuration onthe right. The second derivative of their potential in theshear angle is 4 Gv . The difference between the shear an-gle of the two stable configuration is 60 degrees, in radian units close to 1. For a cosine potential, the correspondingbarrier height E is 2 Gv/π .Of course, such a structural jump is not possible withina stable solid. In fact, the energy maximum configurationof the square in the middle is even stable in an fcc crys-tal, because the elastic matrix around the square com-pensates the negative spring inside. The same is true forthe gliding triangle motion of six atoms [12], which con-verts an octahedron into a bitetrahedron. In this case,the barrier is only 3 Gv/ π (6 k B T g for Gv = 41 k B T g ),because the shear angle from one stable configuration tothe other is smaller.While a single four-atom or six-atom jump is not possi-ble, it seems likely that the real structural transitions at T g are combinations of several such jumps within a cen-tral core, leading to a new core which again fits reason-ably well into the surrounding elastic matrix. To obtainthe total barrier E b of about 36 k B T g needed to inflatethe microscopic time constant τ = 10 − seconds to therelaxation time of hundred or thousand seconds at theglass transition, one has to postulate a combination ofabout six elementary six-atom jumps, consistent with aninner core of twenty to forty atoms.In this picture, the energy barrier for the flow isnot only proportional [13] to the temperature-dependentmodulus G , but also to the number n s of four-atom orsix-atom jumps within the central core (the cooperativelyrearranging region [14]), which might increase with de-creasing temperature. An increase and even a divergenceof a dynamic correlation length with decreasing tempera-ture has been first postulated by Adam and Gibbs [14]. Itis a tempting idea: the structural entropy extrapolates tozero at the Kauzmann temperature, so one has less andless possibilities to jump into another structural state.Then the structural reorganization requires larger andlarger cooperatively rearranging volumes. The Adam-Gibbs concept is supported by numerical calculations,which have been able to see the increase of the correla-tion length with decreasing temperature in various ways[15, 16].Following Jeppe Dyre [13], we define the fragility in E b FIG. 1: The potential energy of a group of four atoms in closepacking as a function of the local shear. terms of the negative logarithmic derivative I of the flowbarrier with respect to temperature ( I = ( m − /
16 interms of the usual measure m of the fragility [17]). Theflow barrier (taking six-atom units) E b = 3 n s π Gv (5)is proportional both to n s and to T / h u i (via Gv and eq.(3)). Therefore it has the negative logarithmic derivative − ∂ ln E b ∂ ln T = − ∂ ln n s ∂ ln T + (cid:18) ∂ ln h u i ∂ ln T − (cid:19) (6)At first sight, this separation in two contributionsseems to contradict the experimental finding of a pro-portionality of the flow barrier to the shear modulusalone in many substances [13]. This discrepancy, how-ever, has been resolved by a recent thorough investiga-tion of mean square displacements at different energy res-olution [18]. One needs a resolution corresponding to atime scale of at least several nanoseconds to see the meansquare displacement of the macroscopic shear modulus.The temperature dependence of the quantity h u i /T onthe picosecond level is always too weak to explain thefull fragility. This has also been seen in a recent datacollection [19], which postulated a proportionality of thelogarithm of the viscosity to α + β/ h u i + γ/ h u i . Theterm γ/ h u i introduces a similar effect as the temper-ature dependence of n s and dominates the behavior at T g .The best experiment for a check of these ideas is atwenty-year old determination [20] of the mean squaredisplacements in glassy, liquid and crystalline selenium(Fig. 2). The logarithmic derivative of the liquid h u i at T g = 304 K is 3.1, explaining 2.1 units of the totalfragility I = 4 . n s of selenium crystalliquidglass T m T g < u > ¯ temperature (K) FIG. 2: The mean square displacements in crystalline, glassyand liquid selenium [20] Note that the liquid data extrapolateto the crystalline ones at 252 K. η/η ( η high temperature viscosity, which in seleniumis 3.1 · − P a s ) to the inverse difference between liquidand crystalline mean square displacements [20]. It holdsover eighteen decades of viscosity variation, from the ag-ing regime below T g up to a temperature high above themelting temperature, providing a much better fit of theviscosity than any Vogel-Fulcher law.The same proportionality between the logarithm of the
50 100 150 200 250 3000.00.10.2 250 300 3500246810121416 T g (a) mean square displacements orthoterphenyl liquid crystal < u > ¯ temperature (K) T m T g (b) viscosity data 0.165/( - ) Vogel-Fulcher l og / temperature (K) FIG. 3: (a) The mean square displacements in crystalline,glassy and liquid ortoterphenyl [25]. The lines are the corre-sponding fits. (b) Proportionality of the logarithm of the vis-cosity [24] η/η with η = 0 . P a s to the inverse difference be-tween liquid and crystalline mean square displacements (thecontinuous line). The dashed line is the Vogel-Fulcher rela-tion obtained by linearizing the mean square displacementsaround T g . viscosity ratio [24] and the inverse difference between liq-uid and crystal mean square displacement is found inorthoterphenyl (OTP). The mean square displacements[25] in Fig. 3 (a) show a much stronger curvature at T g than those of selenium. Nevertheless, if one fits h u i interms of a third order function (the continuous line inFig. 3 (a)) and h u c i in terms of a second order functionin temperature (the dashed line in Fig. 3(a)), one findsagain the proportionality shown by the continuous linein Fig. 3 (b). The fit with η = 0 . P a s fails above T m ,but still covers twelve decades of viscosity variation.One obtains a clearer understanding of these two ex-perimental findings assuming that n s is proportional tothe inverse difference between crystal and liquid shearmodulus G c − G . In the Adam-Gibbs reasoning [14], n s isinversely proportional to the structural entropy differenceof liquid and crystal. A proportionality of the shear mod-ulus difference to the entropy difference sounds plausible.The assumption provides the extrapolated divergence of n s at G c = G which one obviously needs to understandthe strong tendency to a divergence of the viscosity closeto T g . Since E b is proportional to the product of n s and Gv log η/η ∝ E b T ∝ GvT ( G c − G ) ∝ h u i − h u c i , (7)if one neglects the weak temperature dependence of thecrystal shear modulus G c . Note that a linearization ofboth h u i and h u c i around T g converts this relation intothe Vogel-Fulcher law log η/η ∝ / ( T − T ), thus iden- tifying the Vogel-Fulcher temperature T with the pointwhere the extrapolated mean square displacement of theliquid reaches the crystalline one.To summarize, the Lindemann criterion has its physi-cal basis in the entropy of the liquid: a temperature ableto reach a mean square displacement of one tenth of theinteratomic distance in the crystal is also able to distortthe structural units of the liquid by a shear angle of aboutone tenth. The shear flexibility allows to form a muchlarger number of stable inherent structures in the liquidthan those accessible to rigid units, enhancing the liq-uid entropy to the point where it compensates the shearenergy and makes the liquid the stable phase.One can use the Lindemann criterion to estimate theflow barrier in the liquid. The estimate requires a geo-metrical consideration of the elementary relaxing units,of which several have to combine to form a cooperativelyrearranging region. On the basis of this concept, onefinds a physical explanation for the proportionality of thelogarithm of the viscosity ratio η/η ( η high temperatureviscosity) to the inverse of the difference of the picosec-ond mean square displacements of crystal and liquid. Ameasurement of these two quantities allows to determinethe volume of the cooperatively rearranging region. 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