2. Non-Marangoni, visco-capillary spreading

We consider the axisymmetric spreading of a nonvolatile, nonpolar droplet of constant volume $V$ and dynamic viscosity $\mu$ (figure 1). In a nonpolar solvent, the liquid–vapour surface tension remains constant, so that no Marangoni flow arises (Tadmor et al. Reference Tadmor, Multanen, Stern and Yakir2021). The surfactants carried by the droplet can either remain in the bulk flow or be adsorbed onto the solid substrate. Owing to its spreading, the droplet base radius $r_B$ , contact angle $\vartheta$ and thickness $h$ (including top thickness, $h_T$ , figure 1 b) evolve in time $t$ . We assume that the effect of gravity is negligible compared with surface tension (Starov et al. Reference Starov, Kalinin and Chen1994), and are interested in finding the time evolution of $r_B(t)$ .

Figure 1.

(a) Image sequence of a typical spreading experiment of a $2 \ \unicode{x03BC} \mathrm{l}$ , $6 \ \mathrm{mM}$ octadecylamine-laden tetradecane droplet on mica (see movie 1). (b) Geometry of the axisymmetrically spreading droplet. The inset reveals the droplet shape near the propagating front $r_B$ , where the force at the triple line leads to a curvature gradient (solid, red), in contrast to the approximate sketch (dash, black).

In the case of fully wetting, surfactant-free droplets, spreading is driven by the motion of the contact line, via a force per unit length of $\gamma _{\textit{SV}}-\gamma _{\textit{SL}}-\gamma _{\textit{LV}} \cos \vartheta$ , where $\gamma _{\textit{SV}}$ , $\gamma _{\textit{SL}}$ and $\gamma _{\textit{LV}}$ are the solid–vapour, solid–liquid and liquid–vapour interfacial energies, respectively. This motion modifies the fluid thickness, leading to a gradient in the curvature of the liquid–vapour interface (figure 1 b, inset) and resulting in a pressure gradient that governs the flow (Hocking & Rivers Reference Hocking and Rivers1982; Starov et al. Reference Starov, Kalinin and Chen1994). Modelling the flow via lubrication theory yields a nonlinear partial differential equation for $h(r,t)$

(2.1a) \begin{equation} \frac {\partial h}{\partial t} + \frac {\gamma _{\textit{LV}}}{3\mu r}\frac {\partial }{\partial r}\left (\textit{rh}^3\frac {\partial }{\partial r}\left [\frac {1}{r}\frac {\partial }{\partial r}\left \{r\frac {\partial h}{\partial r}\right \}\right ]\right ) = 0, \end{equation}

along with an integral equation representing global mass conservation

(2.1b) \begin{equation} 2\pi \int _0^{r_B} \textit{rh} \,\mathrm{d}r = V . \end{equation}

To close the problem, (2.1a ) requires boundary conditions (BCs). As opposed to the case of gravity currents, where the BCs do not affect the bulk scaling (Hocking & Rivers Reference Hocking and Rivers1982; Huppert Reference Huppert1982; Starov et al. Reference Starov, Kalinin and Chen1994; Sayag & Worster Reference Sayag and Worster2013), one of these BCs is a contact-line condition that is required to regularise the moving-contact-line singularity (Voinov Reference Voinov1976; Cox Reference Cox1986; King & Bowen Reference King and Bowen2001) and may thus introduce logarithmic corrections that involve a microscopic length scale. While these logarithmic corrections affect the scaling imposed by the bulk flow, they do not significantly change over the course of an experiment (§ S1 of the Supplementary Material (SM) available at https://doi.org/10.1017/jfm.2026.11419, Winkels et al. (Reference Winkels, Weijs, Eddi and Snoeijer2012)). Therefore, these corrections can be absorbed into the pre-factors of the scaling relations implied by the bulk equations (2.1)

(2.2) \begin{equation} h \sim \left (\frac {\gamma _{\textit{LV}}}{\mu V^2} t \right )^{-1/5}\!; \qquad \qquad r_B \sim \left (\frac {\gamma _{\textit{LV}} V^3}{\mu } t \right )^{1/10} \!, \end{equation}

where the pre-factors of order $1$ depend on the BCs and can be found experimentally (Chen Reference Chen1988) or estimated theoretically (Hocking & Rivers Reference Hocking and Rivers1982; Starov et al. Reference Starov, Kalinin and Chen1994). Equation (2.2) is often referred to as Tanner’s law (Tanner Reference Tanner1979; Starov et al. Reference Starov, Kalinin and Chen1994).

3. Faster spreading of surfactant-laden droplets

To explore the impact of surfactants on the propagation of nonpolar droplets in the absence of Marangoni stresses, we conduct laboratory experiments on nonvolatile, nonpolar solvents laden with surfactants. Specifically, we test octadecylamine-laden tetradecane droplets spreading on mica substrates (ODA/TD on mica) in a wide range of concentrations $c_0$ (between 0.6 and 148 mM) and volumes (between 1 $\unicode{x03BC}$ l and 4 $\unicode{x03BC}$ l). In each case, we mixed pure TD, which has $\gamma _{\textit{LV}}\approx 26.56 \ \mathrm{mJ\,m^{-2}}$ , with ODA, and measured the evolution of the droplet radius from a top view (figure 1 a). Combining our measurements together with available experimental data of the same system (Tadmor et al. Reference Tadmor, Baksi, Gulec, Jadhav, N’guessan, Sen, Somasi, Tadmor, Wasnik and Yadav2019, Reference Tadmor, Baksi, Gulec, Jadhav, N’guessan, Somasi, Tadmor, Tang, Wasnik and Yadav2020, Reference Tadmor, Multanen, Stern and Yakir2021), we find that the spreading is consistent with Tanner’s law at low surfactant concentrations ( $c_0\lesssim 12 \ \mathrm{mM}$ ) (figure 2). In particular, the spreading in this regime reduces to a single master curve when normalised by $V^{3/10}$ . However, at higher surfactant concentrations ( $c_0\gtrsim 12 \ \mathrm{mM}$ ), we consistently obtain a power-law spreading of $t^{1/7}$ . At the ‘crossover’ concentration of $12 \ \mathrm{mM}$ the spreading evolution transitions during the experiment from $t^{1/10}$ at early times to $t^{1/7}$ at later times (figure 2 b,c). In addition to the change in the scaling with respect to time for a certain concentration, we deduce a dependence of the spreading rate on the surfactant concentration for $c_0\gtrsim 12 \ \mathrm{mM}$ , as evident by the parallel but not colinear trends in figure 2. To quantitatively assess these exponents, we performed global log–log regression fits over all data points in each regime. The resulting best-fit exponents and $95\,\%$ confidence bounds (§ 2 of the SM) confirm consistency with $t^{1/10}$ in the Tanner regime and with $t^{1/7}$ in the surfactant-dominated regime.

Figure 2.

(a) Results of spreading experiments for ODA/TD on mica performed at a range of concentrations and volumes, showing $r_B/V^{3/10}$ against $t$ on a log–log scale. Dashed lines follow trends corresponding to $t^{1/10}$ (black) and $t^{1/7}$ (blue), and the transition between them is revealed at the ‘crossover’ concentration ( $12 \ \mathrm{mM}$ for this particular system): (b) a zoomed-in view of the crossover on a log–log scale; (c) the spreading exponent, obtained through $\mathrm{d}\ln {r_B}/\mathrm{d}\ln {t}$ , plotted against time to validate the crossover between two distinct power-law trends.

Intriguingly, the 1/7 exponent that we measure does not correspond to the broad variety of Marangoni scaling laws in the literature (Rafaï et al. Reference Rafaï, Sarker, Bergeron, Meunier and Bonn2002; Wei Reference Wei2018); particularly, while this 1/7 exponent is faster than Tanner’s law, it is slower than the Marangoni exponents ranging between $1/6$ and 1 (superspreading). We also note that across all concentrations, the surfactants did not alter $\gamma _{\textit{LV}}$ , which was measured before and after each experiment (Tadmor et al. Reference Tadmor, Multanen, Stern and Yakir2021). Additionally, we distinguish our 1/7 exponent from the 1/7 exponent in two-dimensional droplets (Wei Reference Wei2018) and molecular-kinetic theory models (de Gennes et al. Reference de Gennes, Brochard-Wyart and Quéré2004), which do not account for surfactant effects in contrast with our results that vary with $c_0$ . Furthermore, the nonpolar solvent does not dissociate the substrate (Gee et al. Reference Gee, McGuiggan, Israelachvili and Homola1990), and micelles cannot form; only inverted micelles, which do not impact surface tension or adsorption, can form (Tadmor et al. Reference Tadmor, Multanen, Stern and Yakir2021). Therefore, our experimental results imply the existence of a surfactant-governed mechanism distinct from the known Marangoni laws, in which the main impact of the surfactants on the flow is through a non-uniform adsorption and subsequent modification of the solid–liquid interfacial energy (Tadmor et al. Reference Tadmor, Baksi, Gulec, Jadhav, N’guessan, Sen, Somasi, Tadmor, Wasnik and Yadav2019, Reference Tadmor, Baksi, Gulec, Jadhav, N’guessan, Somasi, Tadmor, Tang, Wasnik and Yadav2020, Reference Tadmor, Multanen, Stern and Yakir2021).

4. Evidence of a $\gamma _{\textit{SL}}$ gradient and adsorption mechanism

4.1. Experimental evidence of non-uniform adsorption and a $\gamma _{\textit{SL}}$ gradient

To explore the hypothesised non-uniform surfactant adsorption and $\gamma _{\textit{SL}}$ posited in figure 3(a), we performed three independent experimental tests. First, we evaluated the presence of a surfactant monolayer on substrates using spectroscopic ellipsometry, finding a non-homogeneous surfactant monolayer whose density increased radially toward the droplet centre across all samples (§ S3 of the SM, figure 3 a). In the second test, we evaluated the non-uniformity of $\gamma _{\textit{SL}}$ by allowing a large ( $16 \ \unicode{x03BC} \mathrm{l}$ ) surfactant-laden droplet (6 mM ODA/TD) to spread over a mica substrate for 10 s, removing it completely from the surface using a wipe (Kimwipe), then blowing compressed nitrogen on the substrate to remove excess microdroplets without altering it (de Gennes et al. Reference de Gennes, Brochard-Wyart and Quéré2004; Tadmor et al. Reference Tadmor, Baksi, Gulec, Jadhav, N’guessan, Sen, Somasi, Tadmor, Wasnik and Yadav2019). Following this, we placed smaller droplets of pure TD at different radial locations relative to the original larger droplets (figure 3 b, inset), and measured their equilibrium contact angles, $\vartheta _{eq}$ . This process was repeated three times for reproducibility. We observed that $\vartheta _{eq}$ grew larger closer to the centre of the original larger droplets (figure 3 b). This implies a decreasing local $\gamma _{\textit{SL}}$ towards the centre of the larger droplets, so that $\partial \gamma _{\textit{SL}}/\partial r \gt 0$ (§ S4 of the SM). In the third test, we separated surfactant-laden droplets from the substrate, while applying a uniform force, by attaching a Kimwipe, which was much larger than the droplet, to a rectangular slide, and observing that the separation was instantaneous. As the Kimwipe is much larger than the droplet and separation is rapid, the applied normal stress is effectively uniform; meanwhile, due to the instantaneous detachment, viscous relaxation during detachment is negligible, so that the measured force reflects the total adhesive resistance associated with variations in $\gamma _{\textit{SL}}$ . We observed that the core of the droplet detached, while a rim of fluid remained attached to the substrate (figure 3 c). Since the applied force was uniform, this indicates weaker adhesion of the droplet centre to the substrate, implying lower $\gamma _{\textit{SL}}$ in that region, and stronger adhesion of the droplet front, implying a higher $\gamma _{\textit{SL}}$ .

Figure 3.

(a) Schematic description of the adsorption process and resulting $\gamma _{\textit{SL}}$ gradient. (b) Results of the big-droplet–small-droplet experiments (second $\gamma _{\textit{SL}}$ -gradient test), showing the radial decrease in the contact angle. Error bars indicate standard deviation over three trials. The inset shows schematically (not to scale) how smaller droplets were placed relative to the original larger droplet. (c) Separation of a droplet from a substrate by a uniform force (third $\gamma _{\textit{SL}}$ -gradient test), where the rim remaining attached demonstrates the interfacial energy gradient.

Each of these three tests provides independent evidence of a positive radial gradient of $\gamma _{\textit{SL}}$ . Furthermore, the absence of any other notable difference with nonpolar, surfactant-free droplets leads to the ansatz that $\partial \gamma _{\textit{SL}}/\partial r$ has dynamical consequences on spreading. We emphasise that a gradient seems crucial, as supported by evidence that a uniform $\gamma _{\textit{SL}}$ had no dynamical effect on the spreading of visco-gravity drops (Kumar et al. Reference Kumar, Zuri, Kogan, Gottlieb and Sayag2021).

5. Modelling the dynamical impact of $\gamma _{\textit{SL}}$ gradient on spreading

To model the effect of non-uniform $\gamma _{\textit{SL}}$ in our system, we consider the variation of the free energy of the solid–liquid interface, $\delta E_{\textit{SL}}$ , owing to a unit area displacement of the droplet $\mathrm{d}A$ and an interfacial energy increment $\Delta \gamma _{\textit{SL}}$

(5.1) \begin{equation} \delta E_{\textit{SL}} = \Delta \gamma _{\textit{SL}} \mathrm{d}A. \end{equation}

As the change in the shear force $\mathrm{d}F$ is $-\partial E_{\textit{SL}}/\partial r$ , and the shear stress is $\mathrm{d}F/\mathrm{d}A$ , a spatial gradient of $\gamma _{\textit{SL}}$ produces a tangential traction acting on the liquid through the interfacial layer of adsorbed surfactants adjacent to the solid. While the substrate itself is rigid and does not deform, adsorption-induced spatial variations in $\gamma _{\textit{SL}}$ modify the interfacial free energy and thereby result in a shear stress reduction on the liquid, analogous in form (but not in physical origin) to Marangoni stresses at liquid–vapour interfaces. Additionally, since the size of a surfactant molecule (at most a few nm) is negligible relative to the droplet thickness, we consider this shear stress reduction to be effectively on the solid substrate. Accordingly, the total tangential stress acting on the liquid at $z=0$ is

(5.2) \begin{equation} \left .\tau _{rz}\right |_{z=0} = \left .\mu \frac {\partial u}{\partial z}\right |_{z=0}-\frac {\partial \gamma _{\textit{SL}}}{\partial r} . \end{equation}

Physically, a positive gradient of the solid–liquid interfacial energy reduces the friction experienced by the liquid at the substrate (figure 3), thereby promoting motion along the solid surface. Importantly, since the adsorption gradient extends over a finite region along the substrate behind the contact line, this effect modifies the basal stress over a macroscopic portion of the wetted area rather than acting solely as a localised contact-line forcing (as is the case in the aqueous superspreading systems mentioned in § 1). Therefore, we assume that a positive slip velocity, $u_s \triangleq u|_{z=0}$ exists, and is associated with an effective slip length $\beta _{\textit{SL}}$ , consistent with observed effects of a wettability gradient on the motion of droplets (de Gennes et al. Reference de Gennes, Brochard-Wyart and Quéré2004). We model the slip via a Navier-type condition (Hocking & Rivers Reference Hocking and Rivers1982)

(5.3) \begin{equation} \left .\mu \frac {\partial u}{\partial z}\right |_{z=0}\approx \mu \frac {u_s}{\beta _{\textit{SL}}}. \end{equation}

Evaluating the Stokes equations under the lubrication approximation with a no-shear BC at the free surface, and combining (5.2) and (5.3) at the solid–liquid interface, yields

(5.4) \begin{equation} u_s = \frac {\beta _{\textit{SL}}}{\mu }\left (h \frac {\partial }{\partial r}\left [\frac {\gamma _{\textit{LV}}}{r}\frac {\partial }{\partial r} \left \{r\frac {\partial h}{\partial r}\right \}\right ]+\frac {\partial \gamma _{\textit{SL}}}{\partial r}\right ) \! , \end{equation}

which transforms the dynamic BC of (5.2) into the required kinematic BC for the Stokes equations (Shikhmurzaev Reference Shikhmurzaev1997). Vertical integration of the local mass balance together with the radial flux leads to a new equation for $h(r,t)$

(5.5) \begin{equation} \frac {\partial h}{\partial t} + \frac {1}{\mu r}\frac {\partial }{\partial r}\left (\textit{rh}\left [\left \{\frac {h}{3}+\beta _{\textit{SL}}\right \}h\frac {\partial }{\partial r}\left \{\frac {\gamma _{\textit{LV}}}{r}\frac {\partial }{\partial r}\left (r\frac {\partial h}{\partial r}\right )\right \}+\beta _{\textit{SL}}\frac {\partial \gamma _{\textit{SL}}}{\partial r}\right ]\right ) = 0 , \end{equation}

which encodes a competition between capillarity and the $\gamma _{\textit{SL}}$ gradient, and also competition between $h$ and $\beta _{\textit{SL}}$ within the capillary regime. To explore these competitions, we require a deeper understanding of $\beta _{\textit{SL}}$ , which we expect to be of the order of nanometres (hence $\ll h_T$ ) (Barrat & Bocquet Reference Barrat and Bocquet1999; Bocquet & Charlaix Reference Bocquet and Charlaix2010). Even when $\partial \gamma _{\textit{SL}}/\partial r$ dominates capillarity, spreading will decay if the droplet approaches Young’s equilibrium, obtained at the equilibrium contact angle, $\vartheta _{eq}$ . In our experiments, $\vartheta _{eq}\ll \vartheta \ll 1$ , so the force per unit length, $\gamma _{\textit{LV}}(\cos {\vartheta _{eq}}-\cos {\vartheta })$ , representing the departure from Young’s equilibrium, is approximated by $\gamma _{\textit{LV}}\vartheta ^2/2$ . Therefore, $\beta _{\textit{SL}}\propto \vartheta ^2$ , as has also been found experimentally in hydrophilic conditions (Barrat & Bocquet Reference Barrat and Bocquet1999; Bocquet & Charlaix Reference Bocquet and Charlaix2010). Additionally, theory (Wörner et al. Reference Wörner, Cai, Alla and Yue2018) and experiments (Bocquet & Charlaix Reference Bocquet and Charlaix2010) have shown that $\beta _{\textit{SL}}\sim r_B$ for fixed $\vartheta$ . Consequently, $\beta _{\textit{SL}}\sim r_B\vartheta ^2$ , implying that

(5.6a) \begin{equation} \frac {\beta _{\textit{SL}}}{r_B}\approx k_{\textit{SL}}\left (\frac {h_T}{r_B}\right )^2\! , \end{equation}

where $k_{\textit{SL}}$ is a dimensionless proportionality constant. The Langmuir adsorption mechanism implies the scaling $\Delta \gamma _{\textit{SL}}\sim \bar {k}_{\textit{ads}} c_0 \gamma _{\textit{SL},0}$ (Diamant et al. Reference Diamant, Ariel and Andelman2001; Tabor et al. Reference Tabor, Eastoe and Dowding2010), where $\gamma _{\textit{SL},0}$ is the initial solid–liquid interfacial energy. Therefore,

(5.6b) \begin{equation} \beta _{\textit{SL}}\frac {\partial \gamma _{\textit{SL}}}{\partial r} \sim k_{\textit{SL}} \left (\frac {h_T}{r_B}\right )^2 \bar {k}_{\textit{ads}} c_0 \gamma _{\textit{SL},0}. \end{equation}

Considering the $h\ll \beta _{\textit{SL}}$ limit, and scaling (5.5) in the capillarity-dominated case we reveal using (5.6a ) that the spreading evolves like $r_B\propto t^{1/13}$ , which we did not observe experimentally. This is consistent with our expectation that $h\gg \beta _{\textit{SL}}$ holds for our experiments. In that limit, comparing the competing contributions of the solid–liquid interfacial energy and capillarity using (5.5) yields a new dimensionless number (§ S6 of the SM for further details)

(5.7) \begin{equation} {I_{\!J}} \triangleq \frac {r_B}{h_T} \frac {\gamma _{\textit{SL},0}}{\gamma _{\textit{LV}}} k_{\textit{SL}}\bar {k}_{\textit{ads}}c_0, \end{equation}

which we term the Israelachvili number. This number is similar to the Marangoni number as it also compares an interfacial energy gradient with capillarity (Wei Reference Wei2018). Here, $I_{\!J}$ is the product of three factors: the first, $r_B/h_T\gg 1$ , grows with time; the second, $\gamma _{\textit{SL},0}/\gamma _{\textit{LV}}$ , can span two to three orders of magnitude (ignoring liquid metals, which either do not wet substrates or form interfacial bonds with them, both beyond the scope of this work), and varies between $\approx 1.5$ and $\approx 40$ in our experiments (see § 6, § S2 of the SM). The third factor, $k_{\textit{SL}} \bar {k}_{\textit{ads}} c_0$ , depends on $k_{\textit{SL}}$ , whose magnitude, while constant, is not yet known. The physical significance of this factor is the efficiency with which bulk surfactant concentration modifies the solid–liquid interfacial energy gradient thus reducing basal stress at the solid–liquid interface.

When ${I_{\!J}}\ll 1$ , capillarity dominates, (5.5) simplifies to the visco-capillary model (2.1a ), and $r_B$ evolves via Tanner’s law, as expected for low $c_0$ ; below a threshold concentration (e.g., $c_{0,T}\approx 6 \ \mathrm{mM}$ for ODA/TD on mica), $I_{\!J}$ remains lower than $1$ throughout spreading, and the spreading rate is independent of the initial surfactant concentration (figures 2, 4). Meanwhile, in the case of ${I_{\!J}}\gg 1$ , where the $\partial \gamma _{\textit{SL}}/\partial r$ dominates, our analysis yields a similarity scaling of the form

(5.8) \begin{equation} h\sim \left (\frac {k_{\textit{SL}}\bar {k}_{\textit{ads}}c_0 \gamma _{\textit{SL},0}}{\mu V^{\tfrac{3}{2}}} t\right )^{-2/7}; \ \enspace r_B\sim \left (\frac {k_{\textit{SL}}\bar {k}_{\textit{ads}}c_0 \gamma _{\textit{SL},0}V^2}{\mu } t\right )^{1/7} \!, \end{equation}

which implies faster spreading at high surfactant concentrations relative to Tanner’s law; above a higher threshold concentration (e.g. $c_{0,{G}}\approx 18 \ \mathrm{mM}$ for ODA/TD on mica), for which $I_{\!J}$ is greater than $1$ from very early times, the spreading evolves with the new 1/7 exponent throughout. We note that the spreading is proportional to $V^{3/10}$ and $V^{2/7}$ at low and high $I_{\!J}$ , respectively, and as $3/10\approx 2/7$ , spreading is practically insensitive to the effect of volume relative to the effect of $c_0$ .

Because $I_{\!J}$ increases with time through the ratio $r_B/h_T$ , we expect that for any nonpolar, surfactant-laden, nonvolatile droplet there is a range of crossover concentrations around a concentration $c_{0,{I_{\!J}}}$ for which ${I_{\!J}}=1$ at some time during spreading (provided Young’s equilibrium is not reached beforehand). Spreading under such concentrations will initially follow Tanner’s law ( ${I_{\!J}}\ll 1$ ) and will later transition to a 1/7 exponent ( ${I_{\!J}}\gg 1$ ). We observed such a path in our preliminary experiments, where $c_{0,{I_{\!J}}}=12 \ \mathrm{mM}$ for ODA/TD on mica (figure 2 b,c). Imposing ${I_{\!J}}=1$ at $c_{0,{I_{\!J}}}\sim 10 \ \mathrm{mM}$ for ODA/TD on mica, and taking $r_B/h_T\gtrsim 10$ , $\gamma _{\textit{SL},0}/\gamma _{\textit{LV}}\approx 10$ , $\bar {k}_{\textit{ads}}\sim 10 \ \mathrm{mM}^{-1}$ , we obtain that $k_{\textit{SL}}\sim 10^{-4}$ . Consequently, (5.6a ) and $r_B\gg h_T$ imply that $\beta _{\textit{SL}}$ is of the order of 1–10 nm, reaffirming that the $t^{1/13}$ scaling is not dominant in our system.

6. Universality of the new $\boldsymbol{t^{1/7}}$ spreading law

To explore the universality of the impact of surfactants on the spreading of nonpolar droplets, we conducted experiments on a total of twelve different combinations of solvents (TD, dodecane (DD), hexadecane (HD)), substrates (mica, Au, Si) and surfactants (ODA, dodecylamine (DDA), sodium dodecyl sulphate (SDS), sodium dodecyl benzene sulphonate (SDBS)). The properties of all systems are provided in § S2 of the SM. For each experimental setting we performed experiments in a range of surfactant concentrations and volumes, measuring the evolution of the droplet radius as before. Additionally, $\gamma _{\textit{LV}}$ was measured before and after each experiment and no variations were recorded.

Figure 4.

Similar measurements to those in figure 2 across 12 systems: (a) ODA/TD on mica, Au and Si (including all results presented in figure 2, e.g. the $c_0\geqslant 18$ mM marker contains all concentrations 18–148 mM from figure 2), (b) DDA/TD on mica, Au and Si, (c) SDS/TD on mica and Si and ODA/DD on mica, (d) SDBS/TD on mica and Si and ODA/HD on mica. In each panel, the data are presented on two vertical axes; Right: data for ${I_{\!J}}\lesssim 1$ , normalising $r_B$ by $(\gamma _{\textit{LV}}V^3/\mu )^{1/10}$ . Left: data for ${I_{\!J}}\gtrsim 1$ , normalising $r_B$ by $(\bar {k}_{\textit{ads}}c_0 \gamma _{\textit{SL},0}V^2/\mu )^{1/7}$ . Dashed lines correspond to trends of $t^{1/10}$ (black) and $t^{1/7}$ (blue). We mark transitions at ${I_{\!J}}\approx 1$ via arrows colour coded by substrate: mica (green), Si (grey), Au (gold).

As shown in figure 4, we observe that all twelve experimental settings reduce to the same trends; below a range of ‘transitional’ concentrations, Tanner’s law is followed, while above that range, the new 1/7 exponent arises. Equation (2.2) predicts that all data should collapse when $r_B$ is normalised by $(\gamma _{\textit{LV}}V^3/\mu )^{1/10}$ for $I_{\!J}\ll 1$ ; meanwhile, (5.8) predicts that all data should collapse when $r_B$ is normalised by $(\bar {k}_{\textit{ads}}c_0\gamma _{\textit{SL},0}V^2/\mu )^{1/7}$ for $I_{\!J}\gg 1$ . These master-curve collapses are indeed shown in figure 4. Within the transitional range, arrows indicate the time when the spreading regime transitions from Tanner’s law to the new 1/7 law. Moreover, we do not observe any other spreading evolution, and the value of $k_{\textit{SL}}\approx 10^{-4}$ is confirmed across all the systems for which the rest of the parameters in $I_{\!J}$ are known (Hołysz et al. Reference Hołysz, Mirosław, Terpilowski and Szcześ2008; Patil et al. Reference Patil, Santhanam, Biswas and Ayappa2010; You et al. Reference You, Liu, Liu and Fang2016; Hartje et al. Reference Hartje, Munsky, Ni, Ackerson and Snow2017; Holec et al. Reference Holec, Dumitraschkewitz, Vollath and Fischer2020; Fry et al. Reference Fry, Moody, Spikes and Wong2020). This suggests that $k_{\textit{SL}}$ is approximately constant for the wide class of substrates and liquids considered here. Once this parameter is fixed, the model quantitatively predicts the crossover via the criterion $I_{\!J}\sim 1$ . This allowed us to determine the values of $\bar {k}_{\textit{ads}}$ through $c_{0,{I_{\!J}}}$ for systems in which it was unknown (§ S2 of the SM), and could be used to determine any unknown parameter in $I_{\!J}$ , provided all other parameters are known.

Finally, although $h(r,t)$ was not measured explicitly, all droplets appeared parabolic throughout spreading, suggesting bulk self-similarity while also confirming that (5.8) is in agreement with our experiments through the global mass balance (2.1b ). Since the substrate at the contact line is freshly wetted, adsorption behind the front does not directly modify the local interfacial energy and consequently the Young angle; thus, as in classical Tanner spreading, contact-line physics regularises the motion but does not determine the dominant bulk scaling.