Impact of microphysics on the growth of one-dimensional breath figures

Droplet patterns condensing on solid substrates (breath figures) tend to evolve into a self-similar regime, characterized by a bimodal droplet size distribution. The distributions comprise a bell-shaped peak of monodisperse large droplets and a broad range of smaller droplets. The size distribution of the latter follows a scaling law characterized by a nontrivial polydispersity exponent. We present here a numerical model for three-dimensional droplets on a one-dimensional substrate (fiber) that accounts for droplet nucleation, growth, and merging. The polydispersity exponent retrieved using this model is not universal. Rather it depends on the microscopic details of droplet nucleation and merging. In addition, its values consistently differ from the theoretical prediction by Blackman and Brochard [Phys. Rev. Lett. 84, 4409 (2000)]. Possible causes of this discrepancy are pointed out.

Figure 1

Sketch of the model adopted for the numerical simulation of the growth of droplets $i$ and $i+1$ with radii $R_i$ and $R_{i+1}$, respectively. The droplets reside on a fiber (solid gray line). They grow due to a constant water flux per unit length, $\mathrm{\Phi }$ $\left[\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}\right]$, due to the supersaturated vapor impinging on the fiber. There is a precursor film of radial thickness $e_i$ in the region between adjacent droplets, which has a length $l_i$. When the precursor film collects sufficient water volume, new droplets are nucleated. When two droplets approach each other within their interaction ranges $\varepsilon R_i$ and $\varepsilon R_{i+1}$, they merge.

Figure 2

Time evolution of the space occupied by the droplets for $\varepsilon =2\%,\mathrm{\Phi }=1000\mu {\mathrm{m}}^2/\mathrm{s},N_0=1.5\times {10}^5$, and $R_f=R_{\text{min}}=1\mu \mathrm{m}$. The different colors represent the different ages of the droplets at the considered time instant, and they vary from dark (blue online) for the youngest and still very small droplets to light gray (yellow [red online]) for the oldest and hence largest droplets. The central panel is a magnification of the area indicated by the black frame in the left panel, and the right panel shows a magnification of the frame in the central panel. They are both displayed on a logarithmic scale with the origin corresponding to the instant when the free area started to be populated (i.e., the bottom side of the black frame in the previous panel).

Figure 3

Droplet number density for $\varepsilon =2\%,R_f=R_{\text{min}}=1\mu \mathrm{m},\mathrm{\Phi }=1000\mu {\mathrm{m}}^2/\mathrm{s},L_f=1$ m, $N_0=1.5\times {10}^5$. The solid black line represents $n(s,t)$. The dashed black line is a fitting line for the polydisperse scaling range of the droplets, and it has a slope $-\tau$ on the log-log scale. The gray solid line represents the cutoff for the large droplets $\widehat{f}[s/\mathrm{\Sigma }\left(t\right)]$. The gray dashed line represents the cutoff for the small droplets: $\widehat{g}(s/s_0)$. The two vertical dash-dotted lines delimit the scaling range. For small droplets $\left(s<{\lambda }^{*}\right)$ the distribution is dominated by $\widehat{g}(s/s_0)$. For large droplets $\left(s/\mathrm{\Sigma }\left(t\right)>1\%\right)$, the distribution is dominated by $\widehat{f}[s/\mathrm{\Sigma }\left(t\right)]$. In this range, we can identify a monodisperse bump, corresponding to the oldest droplets, and a gap, corresponding to the times where the spaces released by merging droplets were not large enough to allow the nucleation of new droplets. The present graphs are derived by averaging the droplet size distributions over 10 time instants and over five simulations with different but equivalent initial conditions. The considered instants are in the self-similar regime $(t>1000$ s), and they are chosen in such a way to have 100 points per time decade.

Figure 4

Sketch of the simplified model adopted for deriving the time growth law of the size of the largest droplet $\mathrm{\Sigma }\left(t\right)$. The exponent of such a time growth is calculated based on the assumptions that the droplets are monodisperse and in touch with each other. The considered mechanisms governing the growth are the droplet coalescence two by two and the deposition of water from a uniform external flux per unit length $\mathrm{\Phi }$.

Figure 5

Time evolution of the maximum droplet size (same parameters as in Fig. 3). The black solid line represents the values inferred from our simulations. The black dashed line has a slope of $3/2$ on the double logarithmic axes, proving that $\mathrm{\Sigma }\sim t^{3/2}$, and it represents the function $y={(3\mathrm{\Phi }t/2\pi )}^z$. The gray solid line is a reduced plot, obtained by dividing $\mathrm{\Sigma }$ by the right-hand side of Eq. (8).

Figure 6

Time evolution of the effective porosity (same parameters as in Fig. 3). The lower lines (black) represent the total effective porosity, as defined in Eq. (14), for five simulations with different but equivalent initial conditions; the upper lines (gray, solid) represent the partial effective porosity ${\overline{p}}^{*}=1-A^{*}/A_{\text{tot}}$, where $A^{*}$ is the sum of the areas (lengths) covered by the droplets of size $s>{\lambda }^{*3}$. The gray dashed line represents the line used to fit the porosity ${\overline{p}}^{*}$, and it has a slope $-k=-0.332\pm 0.003$, on the log-log axes. From this we estimate $\tau =1.112\pm 0.002$.

Figure 7

Time evolution of the number of droplets per unit length (same parameters as in Fig. 3). The dark gray thin lines (blue online, uppermost right, denoted as “TOTAL” in the legend) represent the total number of droplets $N\left(t\right)$, for five simulations with different but equivalent initial conditions. The light gray thin lines (green online) represent the number of the small droplets, with $s\le {\lambda }^{*3}$ and its large oscillations. The medium-gray thick lines (red online) represent the number of large and medium size droplets, with $s>{\lambda }^{*3}$; upon fitting, we derive a decaying exponent $z^{\prime }=0.329\pm 0.002$. The black lines represent the number of the large droplets, with $s/\mathrm{\Sigma }\left(t\right)>0.{75}^3$; their decaying exponent is $0.503\pm 0.026\simeq 1/2$, in agreement with the theoretical prediction for a monodisperse droplet population.

Figure 8

Droplet number density $n(s,t)$ (same parameters as in Fig. 3). The black line represents $n(s,t)s^{\theta }$. The gray solid line represents the same quantity rescaled: $n(s,t)s^{\theta }{(s/\mathrm{\Sigma })}^{\tau -\theta }$, with $\tau =1.112$, from the porosity fit: such an estimate seems to be consistent, since the plotted curve is horizontal, in the self-similar range (medium size droplets). The gray dashed line represents the same rescaled quantity with ${\tau }_{\text{theor}}=7/6$ [31]: such a value is not consistent, since the plotted curve is not horizontal, in the range of self-similar sizes. The present graphs are derived by averaging the droplet size distributions over 10 time instants and over five simulations with different but equivalent initial conditions. The considered instants are in the self-similar regime $\left(t>1000s\right)$, and they are chosen in such a way to have 100 points per time decade.

Figure 9

The exponent $\tau$ as a function of (a) the interaction range, $\varepsilon$, and (b) the ratio among the radius of the nucleated droplets $R_{\text{nucl}}$ and the characteristic length of the perturbation originating the droplets chain ${\lambda }^{*}$. The solid lines represent the values of $\tau$ inferred from the fit of the effective porosity. The dashed line is the theoretical value predicted by the classical theory for breath figures [31] ${\tau }_{\text{theor}}=7/6$. The parameters in (a) are $R_f=R_{\text{min}}=1\mu \mathrm{m},\mathrm{\Phi }=1000\mu {\mathrm{m}}^2/\mathrm{s}$, and $N_0=1.5\times {10}^5$, while for (b) we used $\varepsilon =0\%,\mathrm{\Phi }=1000\mu {\mathrm{m}}^2/\mathrm{s}$, and $N_0=1.5\times {10}^5$.

Figure 10

Reduced number density of collisions per unit length $\mathrm{\Delta }{n}_{\text{coll}}^{*}$, during two different time intervals $\mathrm{\Delta }{t}_1=t_1-t_0$ (left) and $\mathrm{\Delta }{t}_2=t_2-t_1$ (right), with $t_0=480.5$ s, $t_1=1035.3$ s, $t_2=1308.9$ s (same parameters as in Fig. 3). To a good approximation, the plotted quantities are independent of time. However, they show a nontrivial dependence on $s_1$ and $s_2$.

Figure 11

Cross sections of the data displayed in Fig. 10 (a) along the main diagonal and (b) perpendicularly to it, during two different time intervals $\mathrm{\Delta }{t}_1$ (dashed lines) and $\mathrm{\Delta }{t}_2$ (solid lines). (a) Along the main diagonal, sharply defined maxima appear at sizes $s_i^{*}$ (with $i=0,\frac{1}{2},1,2,3)$. (b) Perpendicularly to the main diagonal, along the lines $s_1{s}_2={\left(s_0^{*}\right)}^2$ (black lines) and $s_1{s}_2={\left(s_2^{*}\right)}^2$ (light gray lines, green online), there are maxima at the diagonal $s_1/s_2=1$, dips to its left and right, and a plateau for $|{log}_{10}{s}_1/s_2|\gtrsim 3$.