3.1 Marangoni stress singularity due to weak surfactant leakage

Because superspreading generally occurs at the early stage of spreading, if the amount of surfactant exchange with the bulk is small compared to the initial amount of the surfactant on the interface, the surfactant spreader might behave like an insoluble surfactant with $j_{D}^{\prime }=0$ . Karapetsas et al. (Reference Karapetsas, Craster and Matar2011) have shown in their simulations that superspreading can occur to insoluble surfactant through direct transfer of surfactant from the interface onto the substrate without having mass interchange with the bulk. Below we theoretically show that it is possible to realize superspreading with insoluble surfactant in principle. As will also be shown subsequently, although actual superspreaders are not insoluble, this part of the analysis is able to elicit key features of superspreading that will appear again in the more realistic situation using soluble surfactant in § 4.

At $j_{D}^{\prime }=0$ , equation (2.9) leads to

(3.1) $$\begin{eqnarray}[HG_{z}+2V]G=4j_{s},\end{eqnarray}$$

with boundary condition $G(z\rightarrow 1)=1$ . Here $j_{s}\equiv J_{s}/u_{M}\unicode[STIX]{x1D6E4}_{0}=J_{s}\unicode[STIX]{x1D702}/\unicode[STIX]{x1D703}_{0}\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}_{0}\unicode[STIX]{x1D6E4}_{0}({>}0)$ measures the strength of surfactant leakage relative to Marangoni convection. The actual strength of the leakage flux is represented by $J_{s}$ , accounting for the surfactant transfer from the interface to the substrate through the contact line. This flux must be balanced by the convective surface flux along the interface: $u_{s}\unicode[STIX]{x1D6E4}=-J_{s}$ , the original dimensional form of (3.1) from an integration of (2.5) with $J_{D}=0$ . Assume that this surfactant leakage is small so that it can be sustained by a sufficiently high surfactant concentration away from the contact line. Then we can let $-J_{s}$ be a negative constant to represent a ‘sink’ constantly drawing interfacial surfactant molecules toward the contact line. As can also be clearly seen from (3.1), to establish $G_{z}>0$ and hence the Marangoni stress $\unicode[STIX]{x1D6F4}\equiv -G_{z}<0$ to aid in the contact line’s advancement, it can only be achieved by a local surfactant leak with $j_{s}>0$ (see also figure 1 a).

However, the contact line’s advection $V$ term in (3.1) tends to sweep interfacial surfactant molecules toward the contact line. This leads to an accumulation of surfactant near the contact line and hence diminishes $G_{z}$ established by the $j_{s}$ term. So whether the resulting Marangoni stress can be established toward the contact line will be determined by the competition between these two terms. If the surfactant leakage is too slow $(j_{s}\ll V/2)$ , equation (3.1) reduces to $-G_{z}=2V/H$ , making $\unicode[STIX]{x1D6F4}>0$ oppose the contact line motion. Obviously, it is impossible to have superspreading in this case.

A similar competition between surfactant depletion and contact line sweeping has been shown previously by Joanny (Reference Joanny1989) and Karapetsas et al. (Reference Karapetsas, Craster and Matar2011). In particular, Karapetsas et al. (Reference Karapetsas, Craster and Matar2011) showed in their simulations that the drop spreading rate can be promoted at a sufficiently high $Pe$ but not too high (see their figure 5), where $Pe$ is the Péclet number measuring the strength of surface convection relative to surface diffusion in the interfacial surfactant transport. Requiring a large enough $Pe$ to promote spreading by Marangoni stress is simply because a too small $Pe$ will smooth out the surfactant concentration gradient by diffusion. On the other hand, if $Pe$ gets too high, strong contact line sweeping will make surfactant accumulate at the contact line, which will slow down the spreading unless surfactant depletion is fast enough to reverse the trend.

In fact, to achieve superspreading, $j_{s}>V/2$ is necessary for preventing surfactant accumulation at the contact line. If the surfactant leakage is fast ( $j_{s}\gg V/2$ ), equation (3.1) reduces to

(3.2) $$\begin{eqnarray}HG_{z}G=4j_{s}.\end{eqnarray}$$

To determine $G$ from (3.2), we use Voinov’s approximation (Voniov Reference Voniov1976; Snoeijer Reference Snoeijer2006) by assuming the wedge shape $H$ to be a slowly varying function with respect to the linear one:

(3.3) $$\begin{eqnarray}H=z\unicode[STIX]{x1D6E9},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E9}\equiv \unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}_{0}$ measures the departure of the point slope $\unicode[STIX]{x1D703}=h/x$ from $\unicode[STIX]{x1D703}_{0}$ and $|\unicode[STIX]{x1D6E9}_{z}|\ll 1$ . Because $\unicode[STIX]{x1D6E9}$ varies slowly with $z$ , $G$ can be determined by a straightforward integration of (3.2):

(3.4) $$\begin{eqnarray}G=[1+(8j_{s}/\unicode[STIX]{x1D6E9})\times \ln (z)]^{1/2}.\end{eqnarray}$$

This indicates that $G$ increases with the distance $z$ to the contact line until it reaches the far-field value at $z=1$ (i.e. reaching the scale of the macroscopic length $L$ where the interfacial surfactant concentration is nearly constant $\unicode[STIX]{x1D6E4}_{0}$ ). Since surfactant molecules are constantly leaking out through the contact line due to the leakage flux $j_{s}$ , $G$ has to decrease on approaching the contact line (i.e. decreasing $z$ ). It is clear that the larger $j_{s}$ is, the stronger the leakage and hence the greater the reduction of $G$ in the direction toward the contact line. As also indicated by (3.4), since $G$ starts from a constant level away from the contact line, as its decline continues toward the contact line, at some point $G$ must eventually vanish. We call this point the complete depletion point at which $G(z^{\ast })=0$ :

(3.5) $$\begin{eqnarray}z^{\ast }=\exp (-\unicode[STIX]{x1D6E9}^{\ast }/8j_{s}),\end{eqnarray}$$

with $\unicode[STIX]{x1D6E9}^{\ast }\equiv \unicode[STIX]{x1D6E9}(z^{\ast })$ . Equation (3.5) indicates that $z^{\ast }\rightarrow 0$ as $j_{s}\rightarrow 0$ , meaning that for a small value of $j_{s}$ there must exist a small surfactant-free zone of size $z^{\ast }$ .

Equation (3.4) yields the Marangoni stress as

(3.6) $$\begin{eqnarray}\unicode[STIX]{x1D6F4}=-G_{z}=-(4j_{s}/\unicode[STIX]{x1D6E9})\times (1/z)\times [1+(8j_{s}/\unicode[STIX]{x1D6E9})\times \ln (z)]^{-1/2}.\end{eqnarray}$$

As a result, $\unicode[STIX]{x1D6F4}$ will be acting toward the contact line and thereby promote the contact line advancement. Note that both (3.4) and (3.6) only hold up to the critical point $z^{\ast }$ given by (3.5). Below this point, surfactant molecules are completely emptied out by the leak and hence the Marangoni stress vanishes. As also indicated by (3.5), the surfactant-free zone of size $z^{\ast }$ will shrink very rapidly on lowering $j_{s}$ . In other words, $G$ with small $j_{s}$ will exhibit a sharp decline near the contact line.

Now we inspect the behaviour of $G$ and $\unicode[STIX]{x1D6F4}$ given by (3.4)–(3.6). If the surfactant leakage in (3.1) is not too fast such that

(3.7) $$\begin{eqnarray}2V\ll 4j_{s}\ll 1,\end{eqnarray}$$

the surfactant-free zone will be small, i.e. $z^{\ast }\ll 1$ . For $z$ sufficiently far from $z^{\ast }$ , the surfactant concentration varies slowly with $z$ as $G\approx 1+4j_{s}\times \ln (z)$ (with $\unicode[STIX]{x1D6E9}\approx 1$ in (3.4)), and hence the corresponding Marangoni stress behaves as $\unicode[STIX]{x1D6F4}\approx -4j_{s}\times 1/z$ . However, for $z$ close to $z^{\ast }$ , $G\approx (1-\ln (z)/\ln (z^{\ast }))^{1/2}$ (with $\unicode[STIX]{x1D6E9}\approx \unicode[STIX]{x1D6E9}^{\ast }$ in (3.4) together with (3.5)). This yields $\unicode[STIX]{x1D6F4}\approx (1/2\ln (z^{\ast }))\times 1/z\times (1-\ln (z)/\ln (z^{\ast }))^{-1/2}$ , diverging much faster than the usual viscous stress singularity $1/z$ (Huh & Scriven Reference Huh and Scriven1971). As will be shown later, such Marangoni stress singularity will call for a new contact line structure that can account for the linear spreading law.

To confirm the features discussed above, we also solve (3.1) to determine the actual profiles of $G$ and $\unicode[STIX]{x1D6F4}$ . To better illuminate the singular nature for both $G$ and $\unicode[STIX]{x1D6F4}$ , we look at the outer region by taking $H=z$ . This admits an analytical solution:

(3.8) $$\begin{eqnarray}1-G+(2j_{s}/V)\ln [(2j_{s}-V)/(2j_{s}-VG)]=2V\ln (z).\end{eqnarray}$$

Figure 2 plots both $G$ and $\unicode[STIX]{x1D6F4}$ profiles calculated from (3.8) at a fixed value of $V$ with various values of $j_{s}$ under the constraint (3.7). We also plot the asymptotic results (3.4) and (3.6) (with $\unicode[STIX]{x1D6E9}=1$ ), showing an excellent agreement with those given by (3.8). As shown in figure 2(a), for a given value of $j_{s}$ , $G$ declines as the distance $z$ to the contact line decreases. Notice that $G$ vanishes at some particular point, which exactly corresponds to the complete depletion point $z^{\ast }$ – the lower $j_{s}$ the smaller $z^{\ast }$ in accordance with (3.5). On approaching $z^{\ast }$ , such a decline in $G$ gets sharper. For a very little surfactant leakage having $j_{s}=$ 0.01 or lower, $G$ shows merely a small decrease from the equilibrium value (i.e. $G=1$ ) away from the contact line. But near the contact line, because the influence of the leak becomes perceivable, a small depletion region has to form in response to the leak, showing a very sharp decrease in $G$ . Such a boundary-layer-like surfactant concentration profile also resembles the simulation result obtained by Karapetsas et al. (Reference Karapetsas, Craster and Matar2011) for an insoluble surfactant (see their figure 3a). As for $\unicode[STIX]{x1D6F4}$ , its magnitude is increased as $z$ is decreased, and rises more sharply on approaching $z^{\ast }$ , as displayed in figure 2(b). The value of $\unicode[STIX]{x1D6F4}$ blows up as $z\rightarrow z^{\ast }$ , signifying a Marangoni stress singularity. All these features confirm those given by (3.4)–(3.6).

Figure 2.

(a) Calculated surfactant distribution $G$ for insoluble superspreader with various values of surfactant leakage flux $j_{s}$ at $V=0.001$ . Lines: exact solution equation (3.8). Symbols: approximate solution equation (3.4). (b) Shows the corresponding Marangoni stress $\unicode[STIX]{x1D6F4}=-G_{z}$ whose exact and approximate values are evaluated using (3.1) and (3.6) respectively. All the results are calculated at $H=z$ . $G$  vanishes at the depletion point $z^{\ast }$ near which rapid decline in $G$ and divergence in $\!\unicode[STIX]{x1D6F4}$ are evident.

Figure 3.

Schematic illustration of the interface profile and flow field near the contact line. Owing to surfactant leakage near the contact line, there exists a complete depletion point $z^{\ast }$ below which there is no surfactant on the interface. The Marangoni shearing toward the contact line can lead to a pressure build-up, bending the interface into a microscopic capillary nose with dynamic contact angle $\unicode[STIX]{x1D703}^{\ast }$ greater than the apparent dynamic contact angle $\unicode[STIX]{x1D703}_{0}$ .

3.2 Emergence of a new contact line structure

As indicated by (3.6), for small $j_{s}$ , a Marangoni stress singularity can exist on approaching the contact line. Because it diverges at a much faster rate than the usual viscous stress singularity $1/z$ , there is no way to dissipate it by the viscous force exerted by the substrate through the movement of the contact line (as it is derived under $j_{s}\gg V/2$ due to (3.7)). It thus follows that the dissipation of the Marangoni forcing has to be sought at the microscopic level at the scale comparable to or below $z^{\ast }$ , implying that a new contact line structure must emerge, which is explained as follows.

According to (3.6), an intensified Marangoni stress $\unicode[STIX]{x1D6F4}=-G_{z}<0$ is shearing the interface in the direction toward the contact line. From (2.10), this shearing will lead pressure $P=-\unicode[STIX]{x1D6FE}H_{zz}$ to build up in the direction toward the contact line, i.e. $P_{z}>0$ . The mounting pressure near the contact line will in turn bend the air–liquid interface into a capillary nose (see figure 3), making the actual contact angle $\unicode[STIX]{x1D703}_{actual}$ greater than $\unicode[STIX]{x1D703}_{0}$ . Such bending of the interface with an increased dynamic contact angle is consistent with the small ridge at the moving front observed in the simulation study by Karapetsas et al. (Reference Karapetsas, Craster and Matar2011) (see their figure 3a). Since it is the surfactant leakage responsible for $\unicode[STIX]{x1D6F4}<0$ that bends the interface, $\unicode[STIX]{x1D703}_{actual}$ should mainly vary with $j_{s}$ , regardless of the contact line speed $U$ as long as (3.7) is satisfied. Apparently, the behaviour of $\unicode[STIX]{x1D703}_{actual}$ cannot be described by the classical clean-interface de Gennes–Cox–Voinov law that links the apparent dynamic contact angle to $U$ (Voniov Reference Voniov1976; de Gennes Reference de Gennes1985; Cox Reference Cox1986a ). Instead, a new wetting law must emerge to govern how $\unicode[STIX]{x1D703}_{actual}$ varies with $j_{s}$ . Because of surfactant leakage and Marangoni stress, the physics involved in determining this new law will be very different from those in deriving the de Gennes–Cox–Voinov law. So we anticipate that the corresponding contact line structure will also be distinct from the latter’s.

In fact, because the interface becomes clean below $x^{\ast }(=z^{\ast }L\ll L)$ , the strong Marangoni shearing will transform into an enormous capillary pressure and this pressure will act as an additional wetting force to drive the contact line. The situation is quite distinct from the usual de Gennes–Tanner capillary wetting that occurs in a somewhat macroscopic sense over the outer length scale $L$ (de Gennes Reference de Gennes1985). In contrast, such capillary wetting takes place in a microscopic sense at the length scale $x^{\ast }$ . More importantly, it is not purely driven by surface tension but by the Marangoni shearing set-up by the surfactant leakage near the contact line. It is this local Marangoni-driven capillary wetting responsible for constant wetting speed and hence the linear spreading law seen in superspreading.

We should stress that the peculiar features mentioned above are the propositions that closely follow the two analytical results derived from our theory under the ‘little but fast’ leak condition (3.7): (i) the complete depletion point $z^{\ast }$ given by (3.5), and (ii) the Marangoni stress singularity at $z^{\ast }$ in (3.6). Although somewhat similar views can be drawn from the simulations by Karapetsas et al. (Reference Karapetsas, Craster and Matar2011), here we will provide a rigorous proof for these propositions, which is given next.

3.3 Marangoni-driven capillary wetting: a new dynamic contact angle relationship

To determine the actual contact line speed caused by the Marangoni-driven capillary wetting described above, we need to establish the corresponding dynamic contact angle relationship. Hence we consider (2.10) by neglecting the $V$ term under (3.7). Further combining (3.2), we arrive at

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}H^{2}H_{zzz}=6j_{s}/G.\end{eqnarray}$$

Note here that the surface tension parameter $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D703}_{0}^{2}\unicode[STIX]{x1D70E}_{clean}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}_{0}$ is taken to correspond to the clean interface’s tension $\unicode[STIX]{x1D70E}_{clean}$ . This is because we want to determine the microscopic contact angle as $z$ approaches the complete depletion point $z^{\ast }$ , the pressure has to match that at $z^{\ast }$ . Substituting (3.4) into (3.9) and writing $H$ in terms of $\unicode[STIX]{x1D6E9}$ using (3.3), (3.9) becomes

(3.10) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}^{2}\unicode[STIX]{x1D6E9}_{zz}=6\unicode[STIX]{x1D6FE}^{-1}j_{s}\times z^{-2}[1+(8j_{s}/\unicode[STIX]{x1D6E9})\times \ln (z)]^{-1/2}.\end{eqnarray}$$

For the left-hand side of (3.10), we have used Voinov’s approximation $H_{zzz}\approx \unicode[STIX]{x1D6E9}_{zz}$ from $H_{z}=\unicode[STIX]{x1D6E9}+z\unicode[STIX]{x1D6E9}_{z}\approx \unicode[STIX]{x1D6E9}$ .

Equation (3.10) will be used to determine the actual contact angle $\unicode[STIX]{x1D703}_{actual}$ . Because $z^{\ast }$ is small, $\unicode[STIX]{x1D703}_{actual}$ can be roughly represented by $\unicode[STIX]{x1D703}(z^{\ast })$ . Therefore, the purpose of solving (3.10) is to find $\unicode[STIX]{x1D6E9}(z^{\ast })$ . Following the approach to solving the Bretherton equation for deriving the de Gennes–Cox–Voinov wetting law (see appendix A, § A.1), equation (3.10) can be solved approximately using Voinov’s method (see appendix A, § A.2). Integrating both the left- and right-hand sides of (3.10) respectively and making use of the fact that $\int _{1}^{z}\unicode[STIX]{x1D6E9}^{2}\unicode[STIX]{x1D6E9}_{zz}\,\text{d}z\approx \unicode[STIX]{x1D6E9}^{2}\unicode[STIX]{x1D6E9}_{z}$ (see (A 3) in appendix A), equation (3.10) becomes

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}^{2}\unicode[STIX]{x1D6E9}_{z}=6\unicode[STIX]{x1D6FE}^{-1}j_{s}\times \unicode[STIX]{x1D719}(z),\end{eqnarray}$$

where $\unicode[STIX]{x1D719}(z)$ is the integral below (obtained by (A 11) in appendix A):

(3.12) $$\begin{eqnarray}\unicode[STIX]{x1D719}(z)=\int _{1}^{z}z^{-2}[1+(8j_{s}/\unicode[STIX]{x1D6E9})\times \ln (z)]^{-1/2}\,\text{d}z\approx \ln (z).\end{eqnarray}$$

Integrating (3.11) from $z=1$ from $z=z^{\ast }$ and recognizing $\unicode[STIX]{x1D6E9}(z=1)=1$ and $\int _{1}^{z^{\ast }}\ln (z)\,\text{d}z\approx 1$ for small $z^{\ast }$ , we obtain

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}^{3}(z^{\ast })=1+18\unicode[STIX]{x1D6FE}^{-1}j_{s}.\end{eqnarray}$$

With $\unicode[STIX]{x1D6E9}(z^{\ast })=\unicode[STIX]{x1D703}^{\ast }/\unicode[STIX]{x1D703}_{0}$ , $\unicode[STIX]{x1D6FE}=(\unicode[STIX]{x1D70E}_{clean}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}_{0})\unicode[STIX]{x1D703}_{0}^{2}$ and $j_{s}=J_{s}\unicode[STIX]{x1D702}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}_{0}\unicode[STIX]{x1D703}_{0}\unicode[STIX]{x1D6E4}_{0}$ , the slope $\unicode[STIX]{x1D703}^{\ast }$ at $z^{\ast }(\ll 1)$ , which can represent the actual contact angle $\unicode[STIX]{x1D703}_{actual}$ , can be determined as

(3.14) $$\begin{eqnarray}\unicode[STIX]{x1D703}^{\ast 3}=\unicode[STIX]{x1D703}_{0}^{3}+18Ca_{1}.\end{eqnarray}$$

Here, instead of the usual capillary number $Ca=\unicode[STIX]{x1D702}U/\unicode[STIX]{x1D70E}_{clean}$ , equation (3.14) is characterized by the modified capillary number based on the speed of surfactant leakage $J_{s}/\unicode[STIX]{x1D6E4}_{0}$ ,

(3.15) $$\begin{eqnarray}Ca_{1}\equiv \unicode[STIX]{x1D703}_{0}^{3}\unicode[STIX]{x1D6FE}^{-1}j_{s}=(J_{s}/\unicode[STIX]{x1D6E4}_{0})\unicode[STIX]{x1D702}/\unicode[STIX]{x1D70E}_{clean}.\end{eqnarray}$$

Somewhat surprisingly, equation (3.14) looks quite similar to the well-known de Gennes–Cox–Voinov law (see (A 9) in appendix A). As $\unicode[STIX]{x1D703}^{\ast }$ in fact measures the slope of the ‘capillary nose’ which is surfactant free, it should match the clean-interface contact angle described by the de Gennes–Cox–Voinov law (see (A 9) in appendix A):

(3.16) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{clean}^{3}\approx 9Ca\ln (z^{\ast }L/\ell ).\end{eqnarray}$$

Here the length scale ratio in the logarithmic term is taken as the length $z^{\ast }L$ of the surfactant-free zone (with $z^{\ast }$ given by (3.5)) to a microscopic cutoff length $\ell (\ll z^{\ast }L)$ or an inner length scale (due, for instance, to the precursor film ahead of the capillary nose or to wall slip). The associated microscopic contact angle $\unicode[STIX]{x1D703}(\ell )$ is assumed negligible here. Matching $\unicode[STIX]{x1D703}^{\ast }$ given by (3.14) to $\unicode[STIX]{x1D703}_{clean}$ given by (3.16), we can determine the contact line speed as

(3.17) $$\begin{eqnarray}U=\frac{1}{9}\frac{\unicode[STIX]{x1D70E}_{clean}}{\unicode[STIX]{x1D702}}\unicode[STIX]{x1D703}_{0}^{3}(1+18\unicode[STIX]{x1D6FE}^{-1}j_{s})[\ln (z^{\ast }L/\ell )]^{-1},\end{eqnarray}$$

wherein $\ln (z^{\ast })=-(1+18\unicode[STIX]{x1D6FE}^{-1}j_{s})^{1/3}/8j_{s}$ is a result of combining (3.5) and (3.13). For $18\unicode[STIX]{x1D6FE}^{-1}j_{s}\sim O(1)$ or smaller (i.e. $18Ca_{1}\sim O(\unicode[STIX]{x1D703}_{0}^{3})$ or smaller), $U$ scales as the capillary velocity $\unicode[STIX]{x1D703}_{0}^{3}\unicode[STIX]{x1D70E}_{clean}/\unicode[STIX]{x1D702}$ , just like that of capillary wetting. Notice that during the spreading $\unicode[STIX]{x1D703}_{0}$ is gradually decreasing whereas the surfactant leakage contribution $18Ca_{1}$ in (3.14) is kept constant. Because the ratio of the latter to $\unicode[STIX]{x1D703}_{0}^{3}$ in (3.14) varies as $\unicode[STIX]{x1D703}_{0}^{-3}$ , the contribution from surfactant leakage $18Ca_{1}$ to $\unicode[STIX]{x1D703}^{\ast }$ would become increasingly important during the spreading. In other words, as long as $18\unicode[STIX]{x1D6FE}^{-1}j_{s}$ is not too small compared to unity, $\unicode[STIX]{x1D703}^{\ast }$ might still be dominated by $18Ca_{1}$ from the surfactant leakage and hence might not change significantly during spreading. In fact, if $18\unicode[STIX]{x1D6FE}^{-1}j_{s}$ is too small, the situation will reduce to the usual capillary wetting and there will be no superspreading at all, making (3.17) no longer applicable.

On the other hand, if surfactant leakage is sufficiently strong such that $18\unicode[STIX]{x1D6FE}^{-1}j_{s}\gg 1$ (i.e. $18Ca_{1}\gg \unicode[STIX]{x1D703}_{0}^{3}$ ), not only does the contact angle $\unicode[STIX]{x1D703}^{\ast }\approx (8Ca_{1})^{1/3}$ become independent of $\unicode[STIX]{x1D703}_{0}$ in (3.14), but also the contact line speed (3.17) reduces to

(3.18) $$\begin{eqnarray}U=2\left(\frac{J_{s}}{\unicode[STIX]{x1D6E4}_{0}}\right)[\ln (z^{\ast }L/\ell )]^{-1}.\end{eqnarray}$$

It is essentially the speed of surfactant leakage $J_{s}/\unicode[STIX]{x1D6E4}_{0}$ which is constant and independent of $\unicode[STIX]{x1D703}_{0}$ . This speed comes from the surface flow set-up by the surfactant leakage near the contact line through $u_{s}\unicode[STIX]{x1D6E4}=-J_{s}$ which gives $u_{s}\sim -J_{s}/\unicode[STIX]{x1D6E4}_{0}$ . This surface flow is sustained by the corresponding Marangoni shearing working with surface tension to push the contact line forward, making the contact line advance at a constant speed given by (3.18). Together with the fact that $U$ with not too small $18\unicode[STIX]{x1D6FE}^{-1}j_{s}$ also does not change significantly during spreading, $U$ given by (3.17) can be deemed as a constant, thereby explaining the linear spreading law seen in superspreading.

4.1 Surfactant leakage due to corner diffusion and corresponding Marangoni shearing

Having analysed superspreading with an insoluble surfactant, we now move on to soluble surfactant which is more relevant to superspreading. A similar contact line structure can also form due to surfactant leakage around the contact line. But unlike direct surfactant transfer from the interface onto the substrate for insoluble surfactant, the surfactant transport (2.5) is driven by a bulk surfactant concentration gradient resulting from desorption of surfactant from the interface to the substrate (see figure 1 b). This process establishes a corner diffusion flux $J_{D}={\mathcal{D}}(C_{1}-C_{2})/h$ from the surfactant-rich interface at concentration $C_{1}$ toward the surfactant-poor substrate at concentration $C_{2}({<}C_{1})$ , where ${\mathcal{D}}$ is the bulk diffusivity of surfactant. In the dimensionless form, this flux can be re-written as $j_{D}^{\prime }=j_{D}/H$ , where $j_{D}=({\mathcal{D}}\unicode[STIX]{x0394}C/\unicode[STIX]{x1D703}_{0}\unicode[STIX]{x1D6E4}_{0})/u_{M}=\unicode[STIX]{x1D702}{\mathcal{D}}(\unicode[STIX]{x0394}C/\unicode[STIX]{x1D6E4}_{0})/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}_{0}\unicode[STIX]{x1D703}_{0}^{2}$ measures the strength of the corner diffusion flux relative to Marangoni convection and $\unicode[STIX]{x0394}C=C_{1}$ – $C_{2}$ . Hence (2.9) becomes

(4.1) $$\begin{eqnarray}[(G_{z}H+2V)G]_{z}=-4j_{D}/H.\end{eqnarray}$$

We again focus on the little but fast leak scenario:

(4.2) $$\begin{eqnarray}2V\ll 4j_{D}\ll 1,\end{eqnarray}$$

where Marangoni shearing toward the contact line can be established by a weak surfactant leakage without being mitigated by the contact line’s sweeping. Under this condition, equation (4.1) reduces to

(4.3) $$\begin{eqnarray}[G_{z}HG]_{z}=-4j_{D}/H.\end{eqnarray}$$

Using Voinov’s approximation (3.3): $H=z\unicode[STIX]{x1D6E9}$ modulated by a slowly varying function $\unicode[STIX]{x1D6E9}(z)$ , $G$ can be determined below by integrating (4.3) with $G(z\rightarrow 1)=1$ :

(4.4) $$\begin{eqnarray}G=[1-(4j_{D}/\unicode[STIX]{x1D6E9}^{2})\times (\ln (z))^{2}]^{1/2},\end{eqnarray}$$

with the corresponding compete depletion point

(4.5) $$\begin{eqnarray}z^{\ast }=\exp (-\unicode[STIX]{x1D6E9}^{\ast }/2j_{D}^{1/2}).\end{eqnarray}$$

The Marangoni stress is then

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D6F4}=-G_{z}=(4j_{D}/\unicode[STIX]{x1D6E9}^{2})\times (\ln (z)/z)\times [1-(4j_{D}/\unicode[STIX]{x1D6E9}^{2})\times (\ln (z))^{2}]^{-1/2}.\end{eqnarray}$$

For $z$ close to $z^{\ast }$ , since $\unicode[STIX]{x1D6E9}\approx \unicode[STIX]{x1D6E9}^{\ast }$ , equation (4.4) behaves as $G\approx [1-(\ln (z)/\ln (z^{\ast }))^{2}]^{1/2}$ , exhibiting a very sharp decay near $z^{\ast }$ . The corresponding Marangoni stress is $\unicode[STIX]{x1D6F4}\approx \ln (z^{\ast })\times (\ln (z)/z)\times (1-(\ln (z)/\ln (z^{\ast }))^{2})^{-1/2}$ , again diverging much faster than the viscous stress singularity $1/z$ .

Figure 3 plots both $G$ and $\unicode[STIX]{x1D6F4}$ profiles by numerically solving (4.1) with $H=z$ under (4.2). Similar to figure 2 for insoluble surfactant, the larger $j_{D}$ is the sharper the decrease for $G$ and hence the greater build-up of $\unicode[STIX]{x1D6F4}$ . The results also show an excellent agreement with the approximation solution (4.4) and (4.6) with $\unicode[STIX]{x1D6E9}=1$ , confirming the sharp decline in $G$ and the divergence in $\unicode[STIX]{x1D6F4}$ near $z^{\ast }$ due to weak surfactant transfer shown above. The boundary-layer-like profile for $G$ also resembles the simulation result reported by Karapetsas et al. (Reference Karapetsas, Craster and Matar2011) for soluble surfactant (see their figure 9a).

4.2 New contact line structure and dynamic contact angle relationship

To determine the dynamic contact angle relationship based on (4.6), we need (2.10) by neglecting the $V$ term under (4.2):

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}HH_{zzz}=(3/2)G_{z}.\end{eqnarray}$$

Substituting (4.6) into (4.7) and re-writing it in terms of $\unicode[STIX]{x1D6E9}$ using $H=z\unicode[STIX]{x1D6E9}$ and $H_{zzz}\approx \unicode[STIX]{x1D6E9}_{zz}$ , we arrive at

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}^{3}\unicode[STIX]{x1D6E9}_{zz}=-(6\unicode[STIX]{x1D6FE}^{-1}j_{D})\times (\ln (z)/z^{2})\times [1-(4j_{D}/\unicode[STIX]{x1D6E9}^{2})\times (\ln (z))^{2}]^{-1/2}.\end{eqnarray}$$

Following the procedures given by appendix A (see § A.3), to solve (4.8) we first integrate its left- and right-hand sides to yield

(4.9) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}^{3}\unicode[STIX]{x1D6E9}_{z}=-(6\unicode[STIX]{x1D6FE}^{-1}j_{D})\times \unicode[STIX]{x1D713}(z),\end{eqnarray}$$

where $\unicode[STIX]{x1D713}(z)$ is the integral below (obtained by (A 13) in appendix A):

(4.10) $$\begin{eqnarray}\unicode[STIX]{x1D713}(z)=\int _{1}^{z}(\ln (z)/z^{2})\times [1-(4j_{D}/\unicode[STIX]{x1D6E9}^{2})\times (\ln (z))^{2}]^{-1/2}\,\text{d}z\approx (\ln (z))^{2}/2.\end{eqnarray}$$

Integrating (4.9) from $z=1$ to $z=z^{\ast }$ and knowing $\unicode[STIX]{x1D6E9}(z=1)=1$ and $\int _{1}^{z^{\ast }}[\ln (z)]^{2}\,\text{d}z\approx -2$ for small $z^{\ast }$ , we arrive at

(4.11) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}^{4}(z^{\ast })=1+24\unicode[STIX]{x1D6FE}^{-1}j_{D}.\end{eqnarray}$$

Compared to (3.13) for an insoluble superspreader, the last term due to surfactant leakage has the same dependence on surface tension and surfactant leakage flux. However, the power of $\unicode[STIX]{x1D6E9}$ is different. The additional power comes from the corner diffusion term $-4j_{D}/H$ in (4.3). Re-writing (4.11) in terms of actual physical quantities, we obtain the following dynamic contact angle relationship for a soluble superspreader:

(4.12) $$\begin{eqnarray}\unicode[STIX]{x1D703}^{\ast 4}=\unicode[STIX]{x1D703}_{0}^{4}+24Ca_{2},\end{eqnarray}$$

characterized by the modified capillary number

(4.13) $$\begin{eqnarray}Ca_{2}\equiv \unicode[STIX]{x1D703}_{0}^{4}\unicode[STIX]{x1D6FE}^{-1}j_{D}=(D\unicode[STIX]{x0394}C/\unicode[STIX]{x1D6E4}_{0})\unicode[STIX]{x1D702}/\unicode[STIX]{x1D70E}_{clean}.\end{eqnarray}$$

So the velocity here $D\unicode[STIX]{x0394}C/\unicode[STIX]{x1D6E4}_{0}$ can be interpreted as the velocity of the surface flow induced by the corner diffusion flux, as indicated by (4.3). Matching $\unicode[STIX]{x1D703}^{\ast }$ given by (4.12) to $\unicode[STIX]{x1D703}_{clean}$ with (3.16), the contact line speed is found to be

(4.14) $$\begin{eqnarray}U=\frac{1}{9}\frac{\unicode[STIX]{x1D70E}_{clean}}{\unicode[STIX]{x1D702}}\unicode[STIX]{x1D703}_{0}^{3}(1+24\unicode[STIX]{x1D6FE}^{-1}j_{D})^{3/4}[\ln (z^{\ast }L/\ell )]^{-1},\end{eqnarray}$$

wherein $\ln (z^{\ast })=-(1+24\unicode[STIX]{x1D6FE}^{-1}\text{ }j_{D})^{1/4}/2j_{D}^{1/2}$ is a result of combining (4.5) and (4.11). Similar to (3.17) for an insoluble superspreader, $U$ again scales as the capillary wetting velocity $\unicode[STIX]{x1D703}_{0}^{3}\unicode[STIX]{x1D70E}_{clean}/\unicode[STIX]{x1D702}$ . Impacts of surfactant leakage are mainly reflected in $(1+24\unicode[STIX]{x1D6FE}^{-1}j_{D})^{3/4}=(1+(D\unicode[STIX]{x0394}C/\unicode[STIX]{x1D6E4}_{0})\unicode[STIX]{x1D702}/\unicode[STIX]{x1D70E}_{clean}\unicode[STIX]{x1D703}_{0}^{4})^{3/4}$ . So for $24\unicode[STIX]{x1D6FE}^{-1}j_{D}\gg 1$ , equation (4.14) is reduced to

(4.15) $$\begin{eqnarray}U=\frac{1}{9}\frac{\unicode[STIX]{x1D70E}_{clean}}{\unicode[STIX]{x1D702}}\left(\frac{\unicode[STIX]{x1D702}D\unicode[STIX]{x0394}C/\unicode[STIX]{x1D6E4}_{0}}{\unicode[STIX]{x1D70E}_{clean}}\right)^{3/4}[\ln (z^{\ast }L/\ell )]^{\text{ }-1}.\end{eqnarray}$$

Again, $U$ is independent of $\unicode[STIX]{x1D703}_{0}$ . But $U$ will vary as the $3/4$ power of the driving surface flow $D\unicode[STIX]{x0394}C/\unicode[STIX]{x1D6E4}_{0}$ , in contrast to the insoluble surfactant result (3.18) in which $U$ is linearly proportional to $J_{s}/\unicode[STIX]{x1D6E4}_{0}$ . The reason for this difference is that the power of $\unicode[STIX]{x1D703}^{\ast }$ in (4.12) does not match that of $\unicode[STIX]{x1D703}_{clean}$ in (3.16) because of the additional corner diffusion term $-4j_{D}/H$ in (4.1).

A remark is worth making below concerning the spreading speed given by (4.14). Recall that the concentrated Marangoni stress near the contact line is a result of a tiny surfactant leakage imposed by our model to account for the effects of the surfactant transfer from the interface to the substrate. This feature also agrees with what Karapetsas et al. (Reference Karapetsas, Craster and Matar2011) observed in their simulation study. And yet, because of the leak, the contact line is actually surfactant free. So this Marangoni stress does not drive the contact line directly. Instead, it transforms into an enormous Laplace pressure, which in turn drives the contact line in a clean-interface manner. It is this Marangoni-driven capillary wetting that drives superspreading, leading to a constant spreading speed given by (4.14) and why the resulting speed scales as the capillary velocity. It is also this reason why the spreading speed $U$ does not directly depend on the Marangoni stress but on the leakage flux $j_{D}$ .

4.3 Criterion of superspreading

Because only a certain class of surfactants can exhibit superspreading, this implies that, to be a superspreader, certain requirements have to be met. While the present analysis is made under the constraint (4.2), a superspreader must at least satisfy

(4.16) $$\begin{eqnarray}2V<4j_{D}<1,\end{eqnarray}$$

where $V=U/u_{M}$ and $j_{D}=({\mathcal{D}}\unicode[STIX]{x0394}C/\unicode[STIX]{x1D703}_{0}\unicode[STIX]{x1D6E4}_{0})/u_{M}$ with $u_{M}=\unicode[STIX]{x1D703}_{0}\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}_{0}/\unicode[STIX]{x1D702}$ . The criterion $4j_{D}<1$ to ensure that surfactant leakage is weak so that the surfactant depletion zone can be confined in a small region near the contact line. On the other hand, the leak has to be sufficiently strong to establish the surfactant concentration gradient needed to push the contact line without being mitigated by the contact line’s sweeping. This demands $2V<4j_{D}$ .

In addition to (4.16), we also need the following condition from (4.11) to ensure that the Marangoni flow is strong enough to bend the interface by making $\unicode[STIX]{x1D703}^{\ast }$ significantly deviated from $\unicode[STIX]{x1D703}_{0}$ according to (4.12):

(4.17) $$\begin{eqnarray}24\unicode[STIX]{x1D6FE}^{-1}j_{D}>1,\end{eqnarray}$$

with $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D703}_{0}^{2}(\unicode[STIX]{x1D70E}_{clean}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}_{0})$ . Hence, to satisfy both (4.16) and (4.17), $j_{D}$ has to be within the following range under which superspreading might occur:

(4.18) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}/24<j_{D}<1/4.\end{eqnarray}$$

It is worth pointing out that to make (4.18) hold it is necessary to have $\unicode[STIX]{x1D6FE}/24>V/2$ so that $V/2<j_{D}$ in (4.16) can be replaced by $\unicode[STIX]{x1D6FE}/24<j_{D}$ . In fact, $\unicode[STIX]{x1D6FE}/24>V/2$ is equivalent to $V/\unicode[STIX]{x1D6FE}=Ca_{clean}\unicode[STIX]{x1D703}_{0}^{-3}<1/12$ , meaning that surface tension force has to be sufficiently strong compared to the viscous force. We further notice that $j_{D}=({\mathcal{D}}\unicode[STIX]{x1D702}\unicode[STIX]{x0394}C/\unicode[STIX]{x1D6E4}_{0})/\unicode[STIX]{x1D703}_{0}^{2}\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}_{0}$ is approximately proportional to the surfactant concentration $C_{0}$ because superspreading generally occurs in the high concentration regime above the CMC where the surface coverage $\unicode[STIX]{x1D6E4}_{0}$ is nearly constant. As will be illustrated later in § 5, the actual value of $C_{0}$ is close to the lower bound from $\unicode[STIX]{x1D6FE}/24<j_{D}$ in (4.18). This underpins the need for a sufficiently strong surfactant leakage in driving the contact line with Marangoni shearing. Also because $j_{D}>\unicode[STIX]{x1D6FE}/24>V/2$ here, not only $j_{D}>V/2$ guarantees that the as-established surfactant concentration gradient is not diminished by the contact line’s sweeping, but also $\unicode[STIX]{x1D6FE}/24>V/2$ ensures that the Marangoni shearing can turn into a capillary wetting force to drive the contact line.

On the other hand, if the surfactant leakage is too strong, Marangoni shearing would no longer be concentrated near the contact line to make the Marangoni–capillary mechanism work. In fact, in this case the entire interface would be short of surfactant, which reduces to the standard capillary wetting and hence does not lead to superspreading. This demands $1/4>j_{D}$ to give the upper bound of $C_{0}$ . All these requirements result in a range for $C_{0}$ , restricting the surfactant concentration and properties for seeing superspreading.

To better see how the condition for seeing superspreading is determined by surfactant concentration, we re-write $j_{D}$ in terms of the adsorption length $\unicode[STIX]{x1D706}\equiv \unicode[STIX]{x1D6E4}_{0}/\unicode[STIX]{x0394}C$ . This allows us to transform (4.18) to the following range for the dimensionless adsorption length $\unicode[STIX]{x1D6EC}\equiv \unicode[STIX]{x1D706}\unicode[STIX]{x1D70E}_{clean}/\unicode[STIX]{x1D702}{\mathcal{D}}$ :

(4.19) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D70E}_{clean}}{\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}_{0}}\right)\unicode[STIX]{x1D703}_{0}^{-1}<\frac{1}{4}\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D703}_{0}<24\unicode[STIX]{x1D703}_{0}^{-4}.\end{eqnarray}$$

In (4.19), the lower bound comes from $j_{D}<1/4$ and the upper bound corresponds to $\unicode[STIX]{x1D6FE}/24<j_{D}$ in (4.18). In $\unicode[STIX]{x1D6EC}$ , the diffusivity of surfactant molecule ${\mathcal{D}}$ can be evaluated using the Stokes–Einstein equation ${\mathcal{D}}=k_{B}T/6\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}a$ with $k_{B}T$ being the thermal energy and $a$ the size of surfactant molecule. Hence $\unicode[STIX]{x1D6EC}$ can be re-written as $\unicode[STIX]{x1D6EC}=6\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}a\unicode[STIX]{x1D70E}_{clean}/k_{B}T$ , measuring the amount of the work needed to spread a surfactant monolayer of thickness $a$ over distance $\unicode[STIX]{x1D706}$ under the actions of surface tension $\unicode[STIX]{x1D70E}_{clean}$ .

As indicated by (4.19), to be a superspreader, the adsorption length $\unicode[STIX]{x1D706}\equiv \unicode[STIX]{x1D6E4}_{0}/\unicode[STIX]{x0394}C\sim \unicode[STIX]{x1D6E4}_{0}/C_{0}$ , has to be within a certain range. Apparently, not every surfactant can meet this criterion. As $\unicode[STIX]{x1D706}$ essentially represents the adsorption isotherm of a surfactant, for a given surfactant and a given surface coverage $\unicode[STIX]{x1D6E4}_{0}$ , equation (4.19) will fix the range of surfactant concentration $C_{0}$ in which superspreading will be observed. In addition, the range strongly depends on the apparent dynamic contact angle $\unicode[STIX]{x1D703}_{0}$ , suggesting that surfactant wettability and surface chemistry also have a part to play. As a potential candidate of superspreader has to fulfil all of the requirements listed above, this restrains the types of surfactants that become superspreaders.

Recall that (4.19) comes from the need for a local surfactant leakage under the flux constraint (4.18). It seems that an arbitrary surfactant can produce such a leakage effect, which actually is not true. The reason is that to meet (4.18), the surfactant might require a specific affinity and a special kinetic pathway to adsorb onto the substrate, so that the surfactant transfer from the interface to the substrate can occur at the desired level. And to acquire such affinity and sorption kinetics, it is also necessary for the surfactant to possess certain chemical properties or structure that favour such a very restrictive adsorption process. In other words, the special physical and chemical properties needed for being a superspreader are actually hidden in the local surfactant leakage flux imposed by the present hydrodynamic model. As superspreading can also be influenced by other factors such as dynamic surface tension (Rafai & Bonn Reference Rafai and Bonn2005), evaporation (Semenov et al. Reference Semenov, Trybala, Agogo, Kovalchuk, Ortega, Rubio, Starov and Velarde2013), wetting and surface chemistry (Ivanova, Zhantenova & Starov Reference Ivanova, Zhantenova and Starov2012), surfactant assembly (Stoebe et al. Reference Stoebe, Lin, Hill, Wards and Davis1997; Kumar et al. Reference Kumar, Varanasi, Tilton and Garoff2003a ), humidity (Ivanova et al. Reference Ivanova, Zhantenova and Starov2012), etc., the criterion (4.18) or (4.19) should be viewed as a part of the requirements for superspreading. Hence, it is a sufficient condition for superspreading. That is, if this criterion is satisfied, superspreading might occur; but if superspreading occurs, then it must be fulfilled. Because we do not take into account all of these effects, to see superspreading, the range of surfactant concentration predicted by (4.19) would be wider than the actual concentration range occurring in experiments, which is indeed found in the case considered in the next section. Nevertheless, we at least verify that local surfactant leakage is one of the necessary elements for triggering superspreading.