Quantitative imaging of heterogeneous dynamics in drying and aging paints

Drying and aging paint dispersions display a wealth of complex phenomena that make their study fascinating yet challenging. To meet the growing demand for sustainable, high-quality paints, it is essential to unravel the microscopic mechanisms underlying these phenomena. Visualising the governing dynamics is, however, intrinsically difficult because the dynamics are typically heterogeneous and span a wide range of time scales. Moreover, the high turbidity of paints precludes conventional imaging techniques from reaching deep inside the paint. To address these challenges, we apply a scattering technique, Laser Speckle Imaging, as a versatile and quantitative tool to elucidate the internal dynamics, with microscopic resolution and spanning seven decades of time. We present a toolbox of data analysis and image processing methods that allows a tailored investigation of virtually any turbid dispersion, regardless of the geometry and substrate. Using these tools we watch a variety of paints dry and age with unprecedented detail.


Calibration of prefactor γ
To determine the open time of paint films, we extract the characteristic relaxation time τ 0 by fitting the field autocorrelation functions to a single-exponential decay: g 1 (τ) = exp −γ(τ/τ 0 ) α , where γ is a numerical constant that we have calibrated. γ = 1 + ∆ is the distance in units of transport mean free path after which the ballistic light impinging onto the sample is converted into diffusive light inside the sample. The value of ∆ depends on factors including the polarisation of the incident and detected light, particle size, and refractive index ratio between sample and surroundings. For isotropic scattering by particles dispersed in a medium whose refractive index equals that of the surroundings, the Milne theory predicts that ∆ = 0.710. However, most situations involve deviatory values of ∆ and consequently of γ. 1 Calibration of γ for highly concentrated, complex systems such as paints is unfeasible and therefore requires a different approach. To estimate the value of γ in our experiments, we measure the diffusion coefficients of polystyrene particles suspended in water-glycerol mixtures of different ratios, and fit the resulting D values to those measured by dynamic light scattering (DLS). The samples used for LSI contain 1 wt% of polystyrene and are enclosed in a sealed, glass chamber; the samples used for DLS contain 1 × 10 −3 wt% of polystyrene and are measured in standard polycarbonate capillaries of 1.9 mm diameter, on an ALV instrument equipped with an ALV-7002 external correlator and a Cobolt Samba 300 mW DPSS laser operating at a wavelength of 532 nm (detection angle = 90 • , T = 23 ± 1 • C).
We use two different sizes of the particles, 0.5 and 1 µm, density-matched in 45:55 H 2 O-D 2 O, and four different glycerol concentrations for the large particles: 0, 25, 30 and 55 wt%. For each sample we measure the multi-speckle averaged autocorrelation function using our LSI set-up and extract the mean square displacement ∆r 2 (τ) according to: g 1 (τ) = exp −γ k 0 ∆r 2 (τ) , where k 0 = 2πn/λ is the wave vector. The refractive indices n of the different water-glycerol mixtures are obtained from the literature. The resulting mean square displacements versus the correlation time τ are shown in Fig. S1. Their linear scaling confirms the absence of evaporation and sedimentation during the measurements, implying that Brownian motion is the only type of dynamics occurring, hence the diffusion coefficients can be determined by: ∆r 2 (τ) = 6Dτ. We fit these diffusion coefficients to the 'true' values measured using DLS, considering that D LSI ∝ 1/γ 2 , and find the optimal fit for γ ≈ 1.5.
We note that this value is strictly valid only for samples of polystyrene particles in water-glycerol mixtures and will not apply perfectly to drying paint films. In fact, the strong heterogeneity of paint drying precludes using a single value of γ for the entire drying process. Nevertheless, the actual deviation of γ from 1.5 will be limited and will change only the absolute values of ∆r 2 (τ) slightly.

Differentiation between different types of dynamics
For samples displaying a single type of motion, such as the polystyrene suspensions in sealed sample chambers shown in Fig. S1, where Brownian motion is the only form of dynamics, interpreting the autocorrelation curves and associated mean square displacements is relatively straightforward. Most practical applications, however, involve different types of processes in parallel. An example is the drying droplet in Fig. 4 of the main text. Evaporation of water from the droplet causes not only changes in the diffusion of particles but also a directional flow of particles to the contact line and droplet surface. We here demonstrate the differentiation between these processes in a drying polystyrene suspension droplet using a high-speed camera at 600 fps. The spatially resolved drying dynamics of this droplet are depicted in Fig. S2a. The coffee-ring effect causes the deposition of a dense ring of colloids at the contact line of the droplet, which grows inward as time evolves. A distinct asymmetry appears in the coffee ring after approximately 10 minutes (t 2 -t 4 ). The location where the last bulk water evaporates is delineated by a green circle. To elucidate the governing dynamics over time, we measure the multi-speckle averaged mean square displacements for the circular region (Fig. S2b). The dependence of ∆r 2 on τ shows a clear transition from linear (t 1 ), indicative of diffusive dynamics, to quadratic (t 3 -t 4 ), indicative of ballistic motion. Intermediate times display a mixture of diffusive and ballistic transport (t 2 ). This shift signifies a decrease in D with increasing concentration and simultaneous increase in advective velocity v due to the coffee-ring effect. We fit the mean square displacements to ∆r 2 (τ) = 6Dτ + (vτ) 2 , assuming that at t 1 the v term is negligible with respect to the D term, and vice versa at t 3 -t 4 . This gives a decrease in D from 6 · 10 −4 µm 2 /s at t 1 to 4 · 10 −5 µm 2 /s at t 2 , while simultaneously v increases from 0.12 µm/s at t 2 to 0.21 µm/s at t 3 to 0.32 µm/s at t 4 . After evaporation of the bulk water, all fast motion has vanished (t 5 ) and ∆r 2 is negligible in the measured τ range. Open time analysis: evolution of spatial dynamics and film mass, g 2 and d 2 curves, and sum of squared errors of the fits  Fig. 2b-c of the main text), still a considerable amount of water is present in the films. (c) Evolution of the multi-speckle averaged intensity autocorrelation function g 2 (τ) and (d) intensity structure function d 2 (τ) during drying of a 100 µm thick film. The two graphs display the same drying times as shown in Fig. 2a of the main text: t = 0 min (•), 3.3 min ( ), 6 min ( ), 7.3 min (×), 9.3 min ( ), 11.5 min ( ), 15 min ( ) and 20 min (•). All g 2 curves converge towards the intercept g 2 (0) = 1.40 ± 0.005, which equals β + 1. (e) Sum of squared errors SSE corresponding to the single-exponential fits to the g 1 curves, defined as: SSE = m i=1 g 1 (τ i ) − f (τ i ) 2 with m = 4500 the number of data points per g 1 curve, τ i the i th correlation time, g 1 (τ i ) the i th g 1 value to be predicted, and f (τ i ) the predicted value of g 1 (τ i ).

Information about Supplementary Movies
Movie S1) Speckle movie of a drying white paint droplet on white paper Raw data corresponding to Fig. 1b of the main text. First 4 minutes of drying, 20× real time.
Movie S2) Primary and secondary cracking in the centre of a paint film on glass Full time series corresponding to Fig. 3c-d of the main text. 5× real time.
Movie S3) Delamination of the centre of a paint film from glass Full time series corresponding to Fig. 3e of the main text. 5× real time.
Movie S4) Coffee-ring effect in a drying dispersion droplet on glass Full time series corresponding to Fig. 4a of the main text. t = 5-7.6 min after droplet deposition, real time.
Movie S5) Drying of a paint film on plywood, applied by brush at high pressure Full time series corresponding to Fig. 5a, left column, of the main text. First 6 minutes of drying, 20× real time.
Movie S6) Drying of a paint film on plywood, applied by brush at low pressure Full time series corresponding to Fig. 5a, right column, of the main text. First 7 minutes of drying, 20× real time.
Movie S7) Propagation of an imbibition front through paper Full time series corresponding to Fig. 6a of the main text. 5× real time.