Abstract
Understanding glass formation is a challenge because the existence of a true glass state, distinct from liquid and solid, remains elusive: Glasses are liquids that have become too viscous to flow. An old idea, as yet unproven experimentally, is that the dynamics becomes sluggish as the glass transition approaches because increasingly larger regions of the material have to move simultaneously to allow flow. We introduce new multipoint dynamical susceptibilities to estimate quantitatively the size of these regions and provide direct experimental evidence that the glass formation of molecular liquids and colloidal suspensions is accompanied by growing dynamic correlation length scales. Why does the viscosity of glass-forming liquids increase so dramatically when approaching the glass transition? Despite decades of research a clear explanation of this phenomenon, common to materials as diverse as molecular glasses, polymers, and colloids is still lacking [1, 2]. The conundrum is that the static structure of a glass is indistinguishable from that of the corresponding liquid , with no sign of increasing static correlation length scales accompanying the glass transition. Numerical simulations performed well above the glass temperature, T g , reveal instead the existence of a growing dynamic length scale [3, 4, 5, 6, 7] associated to dynamic hetero-geneities [8]. Experiments [8, 9, 10, 11, 12] have indirectly suggested a characteristic length scale of about 5 to 20 molecular diameters at T g , but its time and temperature dependencies, which are crucial for relating this finding to the glass transition, were not determined. We present quantitative experimental evidence that glass formation in molecular liquids and colloids is accompanied by at least one growing dynamic length scale. We introduce experimentally accessible multipoint dynamic susceptibilities that quantify the correlated nature of the dynamics in glass formers. Because these measurements can be made using a wide variety of techniques in vastly different materials, a detailed characterization of the microscopic mechanisms governing the formation of amorphous glassy states becomes possible. Supercooled liquids are believed to exhibit spatially heterogeneous dynamics over length scales that grow when approaching the glass state [1, 13, 14, 15]. This het-erogeneity implies the existence of significant fluctuations of the dynamics because the number of independently relaxing regions is reduced. Numerical simulations have focused on a " four-point " dynamic susceptibility χ 4 (t), which quantifies the amplitude of spontaneous fluctuations around the average dynamics [3, 4, 5, 6, 7]. The latter is usually measured through ensemble-averaged cor-relators, F (t) = δA(t)δA(0) = C(t), where δA(t) = A(t) − A represents the spontaneous fluctuation of an observable A(t), such as the density. Dynamic correlation leads to large fluctuations of C(t), measured by χ 4 (t) = N δC 2 (t), where N is the number of particles in the system. The susceptibility χ 4 (t) typically presents a nonmonotonic time dependence with a peak centered at the liquid's relaxation time [16]. The height of this peak is proportional to the volume within which correlated motion takes place [4, 5, 15, 16]. Unfortunately, numerical findings are limited to short timescales (∼ 10 −7 s) and temperatures far above T g. Experimentally, detecting spontaneous fluctuations of dynamic correlators remains an open challenge, because dynamic measurements have to be resolved in both space and time [17]. Induced fluctuations are more easily accessible experimentally than spontaneous ones and can be related to one another by fluctuation-dissipation theorems. We introduce a dynamic susceptibility defined as the response of the correlator F (t) to a perturbing field x: χ x (t) = ∂F (t) ∂x (1) The relaxation time of supercooled liquids increases abruptly upon cooling, so a relevant perturbing field is temperature, in which case Eq. 1 becomes χ T (t) = ∂F (t)/∂T. Density also plays a role in supercooled liquids , although a less crucial one [18]. Hence, another interesting susceptibility is χ P (t) = ∂F (t)/∂P , where P is the pressure. Colloidal hard spheres undergo a glass transition [19] at high particle volume fraction ϕ. Thus, the appropriate susceptibility for colloids is χ ϕ (t) = ∂F (t)/∂ϕ. Equation 1 also applies in the frequency domain, χ x (ω) = ∂ ˜ F (ω)/∂x, where˜Fwhere˜ where˜F (ω) can be the dielectric susceptibility. We will show below that linear response formalism and fluctuation theory can be used to relate χ x (t) to the spontaneous fluctuations of C(t), and thus to χ 4 (t). Thus, χ x (t) is an experimentally accessible multi-point dynamic susceptibility that