Phonons in Disordered Solids

In recent years, experimental calculations of the vibrational modes of colloidal materials have become a useful tool for investigating phase behavior, bulk properties and local structure in ordered and disordered packings [18–20, 42, 43, 47, 54, 72, 101, 147, 179, 180]. We discuss the details of this technique in Section 3.3.2. Here, we briefly touch on the motivations for applying this analysis to attractive disordered packings.

3.2.4.1 Conventional Low-Frequency Phonon Behavior: The Debye Model

A traditional theory for describing vibrational modes in a solid material is the so-called De- bye model. This model assumes a continuous elastic medium, where vibrational modes at low frequencies are described as plane waves with a dispersion relation𝜔 = 𝑐_∣⇀𝑘_∣; here 𝜔 is the frequency of a given mode, ⇀𝑘 is its wavevector, and 𝑐 is the speed of sound at that particu- lar frequency. In a𝑑-dimensional system, one can determine the vibrational density of states

(vDOS) by calculating the number of vibrational modes in an infinitesimal frequency interval [𝜔, 𝜔 +𝑑𝜔]. For example, given a periodic array of constituent particles, the Debye model assumes an even distribution of modes in𝑘-space. Then, for a given frequency𝑤=𝑐_∣⇀𝑘_∣, there exists a degeneracy of modes proportional to the area of a surface with constant_∣⇀𝑘_∣. For a𝑑- dimensional system, this yields a density of states _∝ 𝜔𝑑−1. This prediction turns out to be an accurate descriptor for many solid materials at low frequencies, and it is typically used to model the temperature dependence of specific heat (𝐶𝑉) in crystalline solids at low temperature (i.e.,

𝐶𝑉 ∝𝑇3).

3.2.4.2 Low-Frequency Phonons in Disordered Solids

A puzzling property of amorphous solids is that the density of low-frequency vibrational modes can significantly exceed the Debye prediction. When a vDOS (or𝐷(𝜔)) curve is normalized by the expected Debye behavior, i.e. 𝐷(𝜔)/𝜔𝑑−1_{, this excess is apparent as a “bump” commonly}

referred to as the “Boson peak.” Though the height and position of this Boson peak has been used as an indicator of the glass transition in disordered materials [18, 169], the exact nature of this excess of low-frequency modes remains a current topic of theoretical and experimental investi- gation. Besides being observed in dense glassy systems, for example, features resembling the Boson peak have been observed in sparser gel-like systems [139], as well as structural crystals with bond-strength disorder [47, 48]. Recent studies have made progress in clarifying properties of this Boson peak, if not a universal mechanism behind it. It appears that many modes near and below this frequency tend to be quasilocalized [18, 170]. Additionally, the Boson peak has been observed to evolve into the Van Hove singularity of crystals in disordered packings that are

evolving into a more ordered configuration [22]. Several recent studies have come to the con- clusion that that Boson peak frequency represents the high-frequency limit of transverse modes propagating in a disordered material [143, 153].

Figure 3.2: (Left) Illustration of the difference in low-frequency density of states 𝐷(𝜔) in an ordered Debye solid (black solid line) and a disordered glassy solid (dashed red line) with 𝑑

dimensions. Note the boson peak exhibited at low frequencies exhibited by the disordered solid. (Right) vDOS calculated from particle trajectories in quasi-2D disordered packings of thermal colloidal microgel spheres with soft repulsive interactions at various area fractions 0.840 < 𝜙 <0.885, adapted from [18]. Note low-frequency peak which changes height and frequency location with area fraction.

Low-frequency modes in disordered packings also highlight otherwise unobservable struc- tural features. Specifically, experimental [19, 42] and numerical [12, 13, 101, 165] studies have observed that quasi-localized low-frequency modes show enhanced participation in areas that are particularly prone to irreversible rearrangements with the application of shear. These ar- eas, termed “soft spots”, are thus considered the amorphous equivalent of dislocation or grain boundary defects in ordered media which (in polycrystalline materials) move in response to me- chanical forces. Though the exact structural properties of these “soft spots” are not entirely clear, previous work has observed that particles in “soft spot” regions exhibit increased local motion and are to some degree associated with local structural order and free volume [71].

While an excess of low-frequency vibrational modes have been observed in gel-like materi- als, the nature of these modes is complicated by the heterogeneous structure of the gel. Recent simulations of sparse, stringy gels of patchy particles observe an excess of low-frequency modes which are tentatively labeled a “boson peak” and are associated with long-wavelength transverse modes of long linear particle chains and related rotational/translational mode coupling [139]. Direct acoustic measurements of ultrasonic vibrational modes in particle gels predict non-Debye behavior in a “diffusive” regime where plane mode wavelengths exist on the size scale of struc- tural heterogeneities and thus scatter diffusively [25, 186]. Additionally, it has been shown that structural heterogeneity in gels has varying correlations to dynamic heterogeneity depending on its overall morphology [30, 31]. One such specific effect is that particles at surfaces in sparse gels will exhibit enhanced motion compared to more closely packed regions. One could imagine that such relationships between structure and dynamics would be reflected in vibrational modes.

Our experimental work is motivated, in large part, by an effort to make connections (or dis- tinctions) between the low-frequency vibrational behavior observed in dense glassy phases, i.e., boson peaks and soft spots, and the structure-induced prevalence of low-frequency modes ob- served in sparse gels. If the low-frequency behavior were to change continuously from attractive glasses to gels, than perhaps an obvious structural feature of low-frequency modes in gels could be extrapolated to denser packings and could further elucidate the structural nature of soft spots or the boson peak. If, on the other hand, one were to observe very different low-frequency behav- ior in sparse and dense colloidal packings, then one might surmise that there exists a distinction between the low-frequency effects caused by structural heterogeneities and those caused by local disorder. In this case, one might become disinclined to apply the broad term of “boson peak” to

any deviations from Debye behavior.