## Abstract

We studied the free fluid-fluid interface in a phase-separated colloid-polymer dispersion with laser scanning confocal microscopy and directly observed thermally induced capillary waves at the interface in real space. Experimental results for static and dynamic correlation functions validate the capillary wave model down to almost the particle level. The ultralow interfacial tension, the capillary length, and the capillary time are found to be in agreement with independent measurements. Furthermore, we show that capillary waves induce the spontaneous breakup of thin liquid films and thus are of key importance in the process of droplet coalescence.

At rest, the free interface between any two fluids, like that between a liquid and its vapor, appears to be smooth. Yet thermal motion inevitably gives rise to statistical fluctuations of the local interface position, leading to a rough interface. This phenomenon was first predicted by von Smoluchowski in 1908 (*1*); 5 years later, Mandelstam quantitatively described the interface roughness in terms of thermally excited capillary waves (*2*). These capillary waves have been studied with light (*3*) and x-ray scattering (*4*–*8*). They play an important role in modern theories of interfaces (*9*–*11*) and have been argued to be essential in the rupture of thin liquid films (*12*, *13*), as occurs in droplet coalescence. We show how to tune length and time scales with the use of colloidal suspensions, such that the fluctuating fluid-fluid interfaces can be seen directly in real space with a resolution comparable to the particle size. Moreover, the droplet coalescence event can be observed down to the scale of capillary fluctuations in quasi slow motion.

In the capillary wave spectrum, each Fourier component *h*** _{k}** of the interface displacement contributes according to the equipartition theorem (1) leading to an interface roughness proportional to . Here, γ is the interfacial tension between the two fluids,

*k*

_{B}is the Boltzmann constant,

*T*is absolute temperature,

*L*is the lateral system size,

*k*is the absolute value of the wave vector

**k**, the angle brackets denote a thermal average, and ξ is the lateral correlation length (capillary length), defined as (2) where

*g*is Earth's acceleration and Δρ is the mass density difference between the two phases. In molecular fluids, γ is on the order of 10 to 100 mN/m and Δρ is about 10

^{2}to 10

^{3}kg/m

^{3}. Therefore, the interface roughness is ∼0.3 nm, whereas the correlation length is ∼3 mm, resulting in extreme roughness–correlation length ratios of 10

^{–7}; such ratios are only accessible through scattering techniques. Here we exploit the scaling up of lengths when going from molecules to mesoscopic colloidal particles (diameter ∼100 nm) to directly observe capillary waves in real space.

Adding polymer to a colloidal suspension may induce a fluid-fluid demixing transition that is widely accepted to be the mesoscopic analog of the liquid-gas phase transition in atomic substances (*14*). The coexisting phases are a colloidal liquid (rich in colloid and poor in polymer) and a colloidal gas (poor in colloid and rich in polymer). The origin of the phase separation lies in the entropy-driven attraction between the colloids, which is mediated by the polymers (*15*, *16*). It is known from experiment (*17*–*19*) and theory (*20*–*22*) that in such systems the interfacial tension scales as γ ∼ *k*_{B}*T*/*d*^{2}, where *d* is the particle size, leading to ultralow values for γ (∼1 μN/m and below). This in turn implies that using colloidal suspensions scales up the interface roughness and simultaneously scales down the correlation length. With the current system (see below), we successfully bring both into the μm regime. In addition, the interplay between ultralow interfacial tension and relatively large viscosity, η, sets the capillary velocity γ/η [see, e.g., (*23*)] in the range of μm/s, as opposed to typical velocities on the order of 10 m/s in molecular fluids. The associated characteristic time for the decay of interfacial fluctuations, which we refer to as the capillary time (3) becomes on the order of seconds in the case of colloids. The capillary velocity also sets the time in droplet coalescence (*24*). Thus, through the appropriate choice of the colloid diameter, we can trace both the statics and dynamics of the capillary waves at a free interface and fully explore the coalescence events with optical microscopy.

We used fluorescently labeled poly(methylmethacrylate) (PMMA) colloidal spheres (*25*) with radius *R*_{c} = 71 nm (obtained from static light scattering) and size polydispersity of less than 10%. The polymer was commercially available polystyrene (Fluka) with molecular weight *M*_{w} = 2 × 10^{6} g mol^{–1} (*M*_{w}/*M*_{n} < 1.2, where *M*_{n} is number-average molecular weight) and radius of gyration *R*_{g} ∼ 44 nm [estimated from data in the literature (*26*)]. Thus, the size ratio, *R*_{g}/*R*_{c} = 0.6, was sufficiently large to obtain stable fluid-fluid demixing (*27*). Both species were dispersed in cis/trans-decalin, and because all densities were known, mass fractions could be directly converted to packing fractions of colloids, , and polymers, , where *n*_{c} and *n*_{p} are the number densities of colloids and polymers, respectively. Samples were prepared by mixing colloid and polymer stock dispersions and diluting with decalin to control the overall packing fractions ϕ_{c} and ϕ_{p}. Large glass cuvettes (volume ∼1 cm^{3}) with very thin cover glass walls (0.17 mm) were used. A confocal scanning laser head (Nikon C1) was mounted on a horizontally placed light microscope (Nikon Eclipse E400). Each measurement was done after 1 day of equilibration. We checked that the system was well equilibrated by following the recovery of intensity after bleaching a space region in the gas and/or liquid phase. The recovery appeared to be governed solely through diffusion of particles without any indications of drift (e.g., through convection). Data sets were acquired at many different state points following several dilution lines. The underlying phase diagram is shown in Fig. 1.

Pictures such as those in Fig. 2 represent an intensity distribution of fluorescent light, *I*(*x, z, t*) at a certain time *t*, with *x* the horizontal (along the interface) and *z* the vertical (opposite to gravity) components of the space vector. The microscope records the fluorescence of excited dye within the colloids, hence the colloid-rich (liquid) phase appears bright and the colloid-poor (gas) phase appears dark. *I*(*x, z, t*) is a direct measure of the local and instantaneous distribution of colloidal particles and provides the starting point for a statistical analysis. Because of the finite resolution (*28*), we can access length scales of ∼2*R*_{c}; we neglect effects induced by the finite time needed to scan each frame, and we take *I* as an instantaneous snapshot (justified by comparing the colloid self-diffusion time with the scanning time). Thus, the real-space pictures in Fig. 2 show the structure of a gas-liquid interface practically at the particle scale. We rely on the concept of a local interface between both phases. In the spirit of a Gibbs dividing surface, we define an interface position *h*(*x, t*) such that in one column of vertical length *L*_{z} the total intensity can be written as (4) where *I*_{gas}(*x*) and *I*_{liq}(*x*) are the average bulk intensities in the gas and liquid phase, respectively, and are taken to be functions of *x* to account for the microscope objective properties. In practice, integrals in the notation are sums over pixels, and we have checked that the results of the subsequent analysis do not depend sensitively on the precise definitions of *I*_{gas} and *I*_{liq}. The resulting “height” function *h*(*x*) (yellow line in Fig. 2) describes the interface position quite accurately. From top to bottom in Fig. 2 we approach the critical point, and both the capillary waves and density fluctuations increase while the density (intensity) difference between the two phases decreases.

For each frame the average interface position is *h̄*(*t*) ≡ 〈*h*(*x*′, *t*)〉, in which the angle brackets denote averages over primed quantities. The time-dependent height-height correlation function is constructed as (5) The corresponding static correlation function, *g*_{h}(*x*) ≡ *g*_{h}(*x, t* = 0), is obtained (*5*) by Fourier-transforming Eq. 1 as (6) where *K*_{0} is the modified Bessel function of the second kind as a function of *x*/ξ. Equation 6 holds for distances *x* larger than a small-distance cutoff on the order of the particle size. Capillary wave theory in the overdamped regime (*29*, *30*) predicts modes with wave vector **k** to decay according to exp[–*t*(γ*k* + *g*Δρ/*k*)/η]. Introducing *k̄* = ξ*k*, the dynamical correlation function at fixed position, *g*_{h}(*t*) ≡ *g*_{h}(*x* = 0, *t*), can be expressed as (7) where τ is as given in Eq. 3 with η = η_{liq} + η_{gas}, the sum of the viscosities of the (colloidal) liquid and gas phase (*29*, *30*).

Equations 6 and 7 describe the experimental data points very well, as can be clearly seen in Fig. 3, A and B, for various state points with only two physical parameters (γ, ξ in the static case and γ, τ in the dynamic case). Results for the interfacial tension, capillary length, and capillary time are displayed in Fig. 3, C to E, respectively. The relation of these quantities through Eq. 3 allows for an independent check of the consistency of our measurements. For example, for state point I, γ= 100 nN/m, ξ= 15 μm, and η_{liq} + η_{gas} = (30 + 12.6) mPa·s, leading via Eq. 3 to a capillary time of 6 s. From the dynamical correlation function we find τ= 7 s. The quality of the fits and the internal agreement validate the capillary wave model down to the particle level.

The role of thermal capillary waves in droplet coalescence has long been a topic of speculation (*12*, *13*). The coalescence event is known to follow three steps: (i) film drainage of the continuous phase between the droplet and the free interface, (ii) breakup of the film, and (iii) extrusion of the droplet material into its bulk phase. Figure 4 displays these steps on the time and length scale of capillary fluctuations. We show two cases: that of a droplet far away from the critical point (top row of Fig. 4) and that of a droplet close to the critical point (bottom row of Fig. 4). In each row, the first and second panels show the drainage step, the third panel the film breakup, and the fourth panel the growth of the neck of the liquid bridge. The well-known Reynolds equation (*31*) describes the rate of thinning of the film (step 1) and the film drainage becomes very slow at small film thicknesses. The actual breakup of the film (step 2) is elusive in molecular fluids; here it is evident that capillary waves induce the spontaneous breakup, which occurs when two opposite bulges at the two interfaces meet. The probability for such an event depends on the interface roughness and on the interface correlation length and time. Typically, the connection is made at film thicknesses of ∼1 μm. A rough estimate shows that a fluctuation of 0.5 μm occurs in a couple of seconds. Hence, the breakup itself is a stochastic process dominated by capillary waves, but before and after the breakup, hydrodynamics apply. We observe that the neck of the liquid bridge (step 3) initially grows linearly with time. This initial regime is driven by the interfacial tension (*24*) and is extremely short in the case of molecular fluids because of the large capillary velocity, but in the present case the typical velocity is many orders of magnitude smaller. We thus see that the extremely low interfacial tension not only reveals the role of capillary waves in coalescence events, but also permits us to probe hydrodynamic regimes that were previously difficult to access in the laboratory.