Dynamical anisotropy and heterogeneity in a bidimensional gel

Molecular mobility determines many of the notable properties of soft matter systems, particularly in interfacial systems which are natural constituents of products and are exploited in a wide variety of technologies, from paints to food science, and from oil recovery to personal care products. Molecular mobility may be slowed down by a transition to dynamical arrest, often happening without any associated structural transition. To date, there is no unifying theory describing these dynamical transitions, in marked contrast with "classic" phase transitions, e.g. crystallisation or evaporation, where the underlying mechanisms are well understood. However, there is a broad consensus that the transition to dynamical arrest is characterised by heterogeneous, non-Gaussian dynamics [1]. This has become evident in numerical simulation work [2], but direct experimental evidence on real systems is scarce.

Figure 1. Sketch of the setup used for XPCS experiments on Langmuir monolayers.

Dimensionality is known to play a key role in arrested systems: in a well-known example, hard sphere caging occurs in three dimensions at volume fractions of about 60 – 64%, while in two dimensions random close packing is only attained at a much higher area fraction of about 82%. The system we studied here is a two-dimensional gel formed by gold nanoparticles at the air-water interface. The XPCS [3] experiments were performed by a joint group of researchers based at the Physics Department of the University of Parma, at IMEM-CNR (Parma), at the European XFEL in Hamburg, and at the ESRF. The high brilliance and stability of the coherent X-ray beam at ESRF beamline ID10A was necessary to obtain data with excellent signal-to-noise ratio from a sample consisting of only one single Langmuir monolayer (see Figure 1). The intensity autocorrelation function, g⁽²⁾, could be retrieved with high accuracy in order to assess the anisotropy of the dynamics, which turned out to be completely confined at the air/water interface plane (see the feature image). Moreover, as the interfacial mechanical properties [4] of the same interfacial gel system have been independently characterised, a direct comparison between the rheological properties and the dynamical properties was made. A model advanced a few years ago by Bouchaud and Pitard [5] proved particularly successful in describing the data.  In this model, a relation between the elastic modulus G’ and the characteristic relaxation time of g⁽²⁾ in elastic soft solids is postulated. Our findings clearly corroborate this hypothesis, as shown in Figure 2.The good signal-to-noise ratio of the data allowed measurement of the four-times correlation functions g⁽⁴⁾(q,t₁,t₂) for the first time by XPCS, as displayed in Figure 3, where it is compared with the usual second-order time correlation function g⁽²⁾(q,t) in the left panel. g⁽⁴⁾ allows a determination of the lifetime and nature (avalanches, intermittency,…) of the dynamical heterogeneity that characterises the dynamics of this 2D gel. The right panel of Figure 3 shows the q- dependence of the lifetime of dynamical heterogeneity (open symbols) compared with the relaxation time of the dynamics obtained by g⁽²⁾.  g⁽⁴⁾ features a broad peak centred at a characteristic time which is one order of magnitude smaller than the relaxation time of g⁽²⁾. Both times follow the same qualitative law τ ≈ q^-1 which is an experimental finding that challenges the various theoretical frameworks attempting to describe dynamical arrest transitions.

Figure 2. Comparison of experimental results with theory: the relaxation time (τ, blue squares) and the elastic modulus (G’, empty circles) as a function of gel concentration follow the theoretical prediction (solid line) in this 2D gel system.

Figure 3. a) First experimental determination of the fourth order correlation function g(4),  which turns out to be a decade faster than the usual second order g(2). b) The characteristic  time scales for both types of correlation functions share the same dependence on q||, also at different surface concentrations of the gel.

Principal publication and authors
Heterogeneous and anisotropic dynamics of a 2D gel, D. Orsi (a), L. Cristofolini (a), G. Baldi (a,b), A. Madsen (c,d), Phys. Rev. Lett. 108, 105701 (2012).
(a) Physics Department, Parma University (Italy)
(b) CNR-IMEM Institute, Parma (Italy)
(c) ESRF
(d) European X-Ray Free Electron Laser, Hamburg (Germany)

References
[1] A. Madsen, R.L. Leheny, H. Guo, M. Sprung, and O. Czakkel, New Journal of Physics 12, 055001 (2010).
[2] L. Berthier, Physics 4, 42 (2011).
[3] G. Grübel, A. Madsen, and A. Robert, in Soft-Matter Characterization, edited by R. Borsali and R. Pecora (Springer, 2008), pp. 935-995.
[4] Interfacial Rheology, edited by R. Miller, L. Liggieri (V S P International Science Publishers, 2009).
[5] J.-P. Bouchaud and E. Pitard, The European Physical Journal E 6, 231-236 (2001). 

Top image: The observed dynamics is anisotropic: the relaxation times depend only on the parallel component, the momentum transfer is perpendicular to the water surface. In this q-map the solid lines are isochrones.