Fluctuations and Shape of Cooperative Rearranging Regions in Glass-Forming Liquids

We develop a theory of amorphous interfaces in glass-forming liquids. We show that the statistical properties of these surfaces, which separate regions characterized by different amorphous arrangements of particles, coincide with the ones of domain walls in the random field Ising model. A major consequence of our results is that supercooled liquids are characterized by two different static lengths: the point-to-set ${\xi }_{\mathrm{PS}}$, which is a measure of the spatial extent of cooperative rearranging regions, and the wandering length ${\xi }_{\perp }$, which is related to the fluctuations of their shape. We find that ${\xi }_{\perp }$ grows when approaching the glass transition but slower than ${\xi }_{\mathrm{PS}}$. The wandering length increases as $s_c^{-1/2}$, where $s_c$ is the configurational entropy. Our results strengthen the relationship with the random field Ising model found in recent works. They are in agreement with previous numerical studies of amorphous interfaces and provide a theoretical framework for explaining numerical and experimental findings on pinned particle systems and static lengths in glass-forming liquids.

Popular Summary

Glass-forming liquids appear in many fields of physics, chemistry, and biology. As these liquids are cooled below their freezing point, or supercooled, they become brittle and behave more like glass than liquid. Understanding how this transition happens is a big challenge for physicists. One of the fundamental ingredients in the theory of how liquids become glassy is what is known as the cooperative rearranging region (CRR)—an area in which all of the particles behave as a collective. While several theoretical and experimental studies in recent years have determined how the size of these regions varies with temperature, little work has been done on understanding their shape.

We have developed a theory to characterize the shape of CRRs in glass-forming liquids. Our theory shows that two different lengths determine the physics of supercooled liquids: the spatial extent of the CRRs, which we call the point-to-set length, and the size of fluctuations along the CRR edge, which we call the wandering length. We show that when the liquid is cooled as the CRRs grow, the boundary of these regions becomes rougher and more fluctuating.

Our findings allow us to fully characterize CRRs. In the future, this framework could help develop new simulations and experiments on glass-forming liquids, including how the CRR shape is altered by quantum fluctuations.

Figure 1

Illustration of a two-dimensional cut of a cooperative rearranging region: the linear spatial extent is of the order ${\xi }_{\mathrm{PS}}$, whereas the external shape is rough and fluctuating over the length scale ${\xi }_{\perp }$.

Figure 2

Pictorial representation of the protocol used to study amorphous interfaces: pink particles on the left of the plane are pinned and green ones on the right are reequilibrated in the presence of this amorphous boundary condition.

Figure 3

Overlap profiles $p^{h_1}(z)$, $p^{h_2}(z)$, $q^{h_1}(z)$, $q^{h_2}(z)$, $q_{12}(z)$ for $h_1=10$, $h_2=30$ and $s_c=0$ (the unit of length is $\sqrt{c}$); as explained in the text, $q^{h_1}(z)=I(z-h_1)$. Note that $p^{h_1}(z)$ also represents the profile $p^h(z)$ obtained for a single replica. For the Landau action considered in the text, $q_{EA}=1$ when $s_c=0$. Similar results are obtained for a generic small value of $s_c$.