Abstract
If liquids, polymers, biomaterials, metals and molten salts can avoid crystallization during cooling or compression, they freeze into a microscopically disordered solidlike state, a glass^{1,2}. On approaching the glass transition, particles become trapped in transient cages—in which they rattle on picosecond timescales—formed by their nearest neighbours; the particles spend increasing amounts of time in their cages as the average escape time, or structural relaxation time τ_{α}, increases from a few picoseconds to thousands of seconds through the transition. Owing to the huge difference between relaxation and vibrational timescales, theoretical^{3,4,5,6,7,8,9} studies addressing the underlying rattling process have challenged our understanding of the structural relaxation. Numerical^{10,11,12,13} and experimental studies on liquids^{14} and glasses^{8,15,16,17,18,19} support the theories, but not without controversies^{20} (for a review see ref. 21). Here we show computer simulations that, when compared with experiments, reveal the universal correlation of the structural relaxation time (as well as the viscosity η) and the rattling amplitude from glassy to lowviscosity states. According to the emerging picture the glass softens when the rattling amplitude exceeds a critical value, in agreement with the Lindemann criterion for the melting of crystalline solids^{22} and the freevolume model^{23}.
Main
In the solid state atoms oscillate with mean square amplitude 〈u^{2}〉 around their equilibrium positions (henceforth to be referred to as the Debye–Waller (DW) factor). With increasing temperature, solids meet different fates depending on the structural degree of order. In the crystalline state the ordered structure melts at T_{m}, whereas in the amorphous state the disordered structure softens at the glass transition temperature T_{g}, above which flow occurs with viscosity η. The empirical law T_{g}≃2/3T_{m} (refs 1, 2, 7) suggests that the two phenomena have a common basis. In fact, this viewpoint motivated extensions to glasses^{24} of the Lindemann melting criterion for crystalline solids^{22} and pictures the glass transition as a freezing in an aperiodic crystal structure (ACS)^{5}.
According to the ACS model, the viscous flow is due to activated jumps over energy barriers ΔE∝k_{B}T a^{2}/〈u^{2}〉, where a is the displacement to overcome the barrier, k_{B} is the Boltzmann constant and T the temperature. The usual rate theory leads to the Hall–Wolynes (HW) equation^{5,21} τ_{α},η∝exp(a^{2}/2〈u^{2}〉). 〈u^{2}〉 is the DW factor of the liquid, that is, it is the amplitude of the rattling motion within the cage of the surrounding atoms. This vibrational regime is assumed to occur on short timescales largely separated by those of the brownian diffusion. The ACS model is expected to fail when τ_{α} becomes comparable to the typical rattling times corresponding to picosecond timescales, a condition that is met at high temperatures (for example, in selenium it occurs at T_{m}+104 K (ref. 14)).
Several tests of the HW equation have been carried out^{21}. However, either the crystal or the glass contributions after extrapolation in the liquid regime are usually subtracted from 〈u^{2}〉. In selenium, the curve log η versus 1/〈u^{2}〉 is concave, whereas if the glass or the crystal contribution is removed a convex curve or a straight line—the latter agreeing with the HW equation—is seen, respectively^{14}. The fact that many glassformers have no underlying crystalline phases, as well as the fact that in many studies removing the glass contribution, unlike selenium, leads to the HW equation, raises some ambiguities about the above subtractions.
It seems natural to generalize the HW equation by adopting a suitable distribution p(a^{2}) of the square displacement to overcome the energy barriers, independent of state parameters, for example the temperature or the density. This is motivated by the observation that, irrespective of the relaxation time, the distance a particle moves during the structural relaxation is about the same, comparable to the molecular diameter^{1}. Differently, the amplitude of the vibrational dynamics 〈u^{2}〉 is expected to be affected by state parameters^{7,9}. We choose a gaussian form for p(a^{2}) with average and σ_{a}^{2}^{2} variance. Averaging the HW equation over this distribution leads to
Equation (1) yields the leading dependence on 〈u^{2}〉 even if the gaussian is truncated to account for a minimum displacement. In addition to the central limit theorem, other motivations support the gaussian form of p(a^{2}). For example, if the kinetic unit is undergoing a harmonic motion due to an effective spring with constant k, 〈u^{2}〉∝k_{B}T/k, equation (1) reduces to a form reported for both supercooled liquids^{25} and polymers^{26}. Furthermore, along the same line of reasoning, we may reinterpret the gaussian form of p(a^{2}) as a gaussian distribution of the energy barriers ΔE∝k a^{2} (ref. 27).
We show that the dependence of the structural relaxation time on the DW factor collapses to a universal master curve provided by equation (1). We devised a twostep strategy. First, a master curve is constructed using extensive moleculardynamics (MD) numerical simulations for a bead–spring model^{28} of a polymer melt. Then a suitable scaling of both the numerical and experimental data is introduced to convert the MD master curve into the universal one including both strong and fragile glasses^{1} and polymers; the latter, apart from a few studies^{17,19}, have been little considered.
We now discuss the MD simulations. They involved changes in the temperature T, the density ρ, the chain length M and the interaction potential U_{p,q}(r). To characterize the shorttime dynamics and the structural relaxation we use the monomer mean square displacement (MSD) 〈r^{2}(t)〉 and the incoherent intermediate scattering function (ISF) F_{s}(q_{max},t), q_{max} being the qvector of the maximum of the static structure factor (see the Methods section).
Figure 1 shows typical MSD and ISF curves. At very short times (ballistic regime), MSD increases according to with μ the monomer mass and t the time and ISF starts to decay. The repeated collisions with the other monomers slow the displacement of the tagged one, evinced by a knee of MSD at , where Ω_{0} is an effective collision frequency: Ω_{0} is the mean smalloscillation frequency of the monomer in the potential well produced by the surrounding ones kept at their equilibrium positions^{29}. At later times a quasiplateau region, also found in the ISF, occurs when the temperature is lowered and/or the density increased. This signals the increased caging of the particle. The latter is released after an average time τ_{α}, defined by the relation F_{s}(q_{max},τ_{α})=e^{−1} (other definitions differ by an overall constant due to the superposition of the ISF curves at long times by a suitable logarithmic shift; see the inset of Fig. 1b). For t≳τ_{α} MSD increases more steeply. The monomers of short chains (M≲3) undergo diffusive motion, 〈r^{2}(t)〉∝t^{δ}, with δ=1. For longer chains, owing to the increased connectivity, the onset of the diffusion is preceded by a subdiffusive region (δ<1, Rouse regime)^{23}.
The dynamics of the model polymer depends in a complex way on the state parameters. Nonetheless, we systematically found that, if two states (labelled by multiplets {T,ρ,M,p,q}) have equal relaxation times τ_{α}, the corresponding MSD and ISF curves coincide from times rather longer than τ_{α} down to the crossover to the ballistic regime and even at shorter times if the states have equal temperatures. Examples are shown in Fig. 1. Notice that the coincidence of MSD and ISF curves of states with equal τ_{α} at intermediate times (t≲τ_{α}) must not be confused with the customary superposition of ISF curves at long times (t≳τ_{α}) following a suitable logarithmic time shift (see the inset of Fig. 1b).
The above findings clearly show that a correlation between the structural relaxation and the fast dynamics sets in. We assess it by correlating τ_{α} and the DW factor 〈u^{2}〉.
Prior to the discussion of the DW factor, we have to clarify whether the fast dynamics of the monomers takes place in cages. In this respect, we point out that the product Ω_{0}τ_{α}∼20 for states with the fastest relaxation and much larger for states with slower relaxation; that is, the structure lifetime is at least one order of magnitude longer than the collision time. Furthermore, in the present study we always observe (not shown) that the time velocity correlation function (VCF), after a first large drop due to pair collisions, reverses the sign because the monomer rebounds from the cage wall^{29}.
We now show that the DW factor is a characteristic length scale of the rattling motion into the cage. The measure of the DW factor must take place in a time window where both the inertial and the relaxation effects are not present. To clearly identify this time window we consider the quantity Δ(t)≡∂log〈r^{2}(t)〉/∂logt; representative plots of Δ(t) are given in the inset of Fig. 1a. Δ(t) shows a clear minimum at (corresponding to an inflection point in the log–log plot of 〈r^{2}(t)〉), which separates two regimes. At short times, , the VCF always exceeds the noise floor (not shown) and the inertial effects become apparent. At long times, , relaxation sets in. The short and the longtime limits of Δ(t) correspond to the ballistic (Δ(0)=2) and the diffusive regimes (), respectively. To observe a minimum of Δ(t) requires that the VCF shows a negative tail at long times. A monotonically decreasing VCF, that is, no cage effect, leads to a monotonically decreasing Δ(t). Therefore, the MSD at is a mean localization length and the DW factor is (the same definition has been adopted, with no justification, in ref. 11). Notice that , corresponding to about 1–10 ps (ref. 28), is consistent with the timescales of the experimental measurement of the DW factor; see, for example, ref. 14.
Figure 2 shows the dependence of the structural relaxation time τ_{α} on the DW factor. The data collapse on a welldefined master curve well fitted by equation (1). States with different densities, chain lengths and interaction potentials are included in Fig. 2, corresponding to different degrees of anharmonicity of the rattling motion in the cage (that is, nonlinear temperature dependence of the DW factor) and then to different fragilities^{4,7,10,12,15,18,19,21}. The scaling shows that for our model system both the average value and the spread σ_{a} ^{2} of the square displacement needed to overcome the energy barriers are not affected by the anharmonicity. The bestfit value of the average is , consistent with both the observation that 〈r^{2}(t=τ_{α})〉^{1/2}≲0.5 (see Fig. 1) and the wellknown result that the atomic MSD during the structural relaxation is less than one atomic radius (∼0.5 in MD units)^{1}.
The concavity of the master curve in Fig. 2 is a signature of the heterogeneity of the structural relaxation. The best fit of our MD data with equation (1) gives σ_{a}^{2}∼0.25, indicating a distribution of the displacement required to overcome the energy barriers. The magnitude of the ratio of the quadratic and the linear terms of equation (1) with respect to 〈u^{2}〉^{−1}, , discriminates two different regimes. If the DW factor 〈u^{2}〉 is small enough that R is larger than unity, the distribution of the displacement required to overcome the energy barriers shows up. In this case, because different a values produce different monomer mobility, a heterogeneous mobility distribution is expected. On the other hand, if R is smaller than unity, the dynamics is homogeneous. To support this scenario we recall that, on approaching the glass transition, a spatial distribution of mobilities develops with increasing nongaussian features of the molecular displacement^{2,7,30}. The features are characterized by the maximum α_{2max} of the timedependent nongaussian parameter α_{2}(t) (see the Methods section)^{30}. States with coinciding ISF and MSD shown in Fig. 1 have coinciding α_{2}(t) curves too (not shown). For these states we sketch the relation between α_{2max} and R in the inset of Fig. 2. It is seen that, when R exceeds the unit value, α_{2max} increases exponentially. Notably, the inset of Fig. 2 reduces to an activated law for strong glassformers, where 〈u^{2}〉 is nearly proportional to T; this law has been observed for silica^{30}.
Equation (1) with the bestfit parameters from Fig. 2 offers the opportunity to find the DW factor 〈u_{g}^{2}〉 at the glass transition of the model polymer system. At the glass transition τ_{α}=τ_{α g}≡10^{2} s in laboratory units^{1}, which corresponds to τ_{α g}=10^{13}–10^{14} in dimensionless MD units (the time unit corresponds to 1–10 ps (ref. 28)). Equation (1) yields 〈u_{g}^{2}〉^{1/2}=0.129(1). This amplitude corresponds to the ratio v_{0}∼(2〈u_{g}^{2}〉^{1/2})^{3}=0.017 between the volume that is accessible to the monomer centre of mass and the monomer volume. It has been proposed that the glass transition takes place under isofreevolume conditions with the universal value (ref. 23). Furthermore, an extension of the ACS model (leading to the HW equation) predicts that, just as for a crystalline solid^{22}, there is a Lindemann criterion for the stability of glasses: the ratio f=〈u_{g}^{2}〉^{1/2}/d, d being the average nextneighbour distance of the atoms in the lattice, is a quasiuniversal number^{24} (). Our data yield f∼0.12–0.13 (d is taken from the monomer radial distribution function), which is close to f=0.129 for the melting of a hardsphere facecentred cubic solid^{22}.
We are now in a position to show that equation (1) with the bestfit parameters from Fig. 2 may be recast as a universal curve by considering the reduced variable . For MD data we set 〈u_{g}^{2}〉^{1/2}=0.129. Figure 3 shows scaling for several glassformers and polymers in a wide range of the fragility m, which measures the rapidity with which viscosity and structural relaxation change as the glassy state is approached^{1}. The scaling in Fig. 3 must not be ascribed to 〈u_{g}^{2}〉, which does not correlate with m (compare m from Fig. 3 and 〈u_{g}^{2}〉 from the Supplementary Information, Table S1). Instead, it shows that both the reduced mean square displacement to overcome the energy barriers and the spread σ_{a}^{2}/〈u_{g}^{2}〉 are fragility independent, and then also the curvature of the master curve, which indicates the heterogeneity of the structural relaxation.
The experimental data in Fig. 3 were collected by changing the temperature. In this respect, the universal scaling of Fig. 3 proves that the wellknown increasing deviation of 〈u^{2}(T)〉 from the linear temperature dependence of the harmonic behaviour by increasing the fragility index m (refs 4, 7, 9, 18, 19) just mirrors the corresponding increasing bending of τ_{α}(T) versus T_{g}/T in the Angell plot^{1} from the glass transition region to the liquid state. However, the glass transition may be reached under isothermal conditions by increasing the density or the connectivity (here expressed by the chain length) as well. Our MD results highlight the correlation of structural relaxation and vibrational dynamics also for these alternative routes. This prediction awaits experimental confirmation.
Methods
In our numerical simulations each chain is pictured as a freely jointed linear sequence of M soft spheres, the monomers, with M=2, 3, 5, 10. For details see Supplementary Information and ref. 28. The nonbonded monomers belonging to the same or different chains interact via the potential with (refs 12, 29). All quantities are in reduced units: length in units of σ, temperature in units of ε/k_{B} and time in units of , where μ is the monomer mass. The potential is cut and shifted to zero by U_{cut} at r=2.5. The bond length is b=0.97. Changing the p and q parameters does not affect the position or the depth ε of the potential minimum but only the steepness of the repulsive and the attractive wings. The monomer MSD 〈r^{2}(t)〉 is defined as , where the sum runs over the total number of N monomers and the brackets denote a suitable ensemble average. The incoherent ISF F_{s}(q_{max},t) is defined as , q_{max} being the qvector of the maximum of the static structure factor. The timedependent nongaussian parameter is defined as α_{2}(t)=(3〈r^{4}(t)〉/5〈r^{2}(t)〉^{2})−1 and vanishes if the displacement r is gaussian.
To prepare Fig. 3, data about the structural relaxation (in seconds) and the viscosity (in Pa s) were scaled to the MD master curve by logarithmic vertical shifts +11.5(5) and +1.5(5), respectively, apart from B_{2}O_{3} (+8.4(5) and −2.2(5)). Data of polymers refer to τ_{α}. Data related to B_{2}O_{3},OTP and ferrocene/dibutylphthalate include two independent sets, one for τ_{α}, the other for η, which for simplicity’s sake are presented with the same symbol. See Supplementary Information for the data sources and further details.
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Acknowledgements
The authors acknowledge discussions with C. A. Angell, S. Capaccioli, G. Carini, P. G. Debenedetti, J. Dyre, A. Fontana, K. L. Ngai, G. Ruocco, F. Sciortino and S. N. Shore. A. Wischnewski is thanked for providing the DW data of silica. Computational resources by ‘Laboratorio per il Calcolo Scientifico’ (Physics Department, University of Pisa) and financial support from MIUR within the PRIN project ‘Aging, fluctuation and response in outofequilibrium glassy systems’ are acknowledged.
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Larini, L., Ottochian, A., De Michele, C. et al. Universal scaling between structural relaxation and vibrational dynamics in glassforming liquids and polymers. Nature Phys 4, 42–45 (2008). https://doi.org/10.1038/nphys788
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