Spreading of rotating liquid annular ring with hysteresis effects

A known volume of liquid (l) is placed onto a rotating horizontal solid surface (s) and is surrounded by a passive gas (g), as sketched in Fig. 1. The location of the free interface is denoted by h and the location, where all three phases (solid, liquid, gas) are in contact, is termed contact line and denoted by a. The angle within the liquid at the contact line is termed (apparent) contact angle \(\theta \).

We concentrate on a droplet which has experienced a central liquid break up. Such a break up may occur if the interface h gets very close to the solid surface and an attractive disjoining pressure causes a breakup, or it may be forced by a non-wetting particle placed in the center. In both cases there are two contact lines, at an inner radius b and at an outer radius a. The apparent contact angles are then \(\theta _b\) and \(\theta _a\), respectively. Both contact lines may move in time t with their velocities \(b_t\) and \(a_t\), respectively. Here the subscript t indicates a partial derivative with respect to time t. This notation for partial derivatives is present also in the further equations, to allow for a compact formulation.

In this article two cases are considered: (f) a liquid layer or film, unbounded in the direction normal to the sketch, and (d) a rotationally symmetric droplet. The liquid film is treated in Cartesian \((x,y,z)^T\) coordinates and specific equations are denoted by (f). The droplet is treated in cylindrical \((r,\varphi ,z)^T\) coordinates and all equations specifically valid for this case are denoted by (d). In both cases, gravity acts parallel to the z-axis, i.e. the gravitational acceleration is \({\textbf{g}}=(0,0,-g)^T\). For the liquid film a volumetric force linearly increasing with the distance from the z-axis is considered. This force has a certain analogy to a centrifugal force and may be realized by an air jet acting onto the free interface, resulting in a pressure variance (cf. Moriarty et al. [26]). For the case of the droplet, the system rotates around the z-axis, with a certain speed of rotation \({\varvec{\Omega }}(t)=(0,0,\Omega )^T\), leading to true centrifugal forces. The spreading of the rotationally symmetric droplet (d) is in the focus of the mathematical formulation and all results are discussed for this rotationally symmetric droplet. The equations for the spreading symmetric film (f) are given in the “Appendix” and no results are presented.

The system of three phases has to be characterized by certain properties. The liquid (l) is characterized by its dynamic viscosity \(\mu \) and its density \(\varrho \). The free liquid–gas (l–g) interface is characterized by its surface tension \(\sigma \), and dynamic wetting (i.e. the dynamic contact lines) can be characterized by a static advancing contact angle \(\theta _A\) and a static receding contact angle \(\theta _R\). The Hamaker constants for a generalized disjoining pressure are \({\mathcal {H}}_i\). Further, the variables of the liquid flow are the velocity \({\textbf{w}}=(u,v,w)^T\) and the pressure p. Further, we have the (constant) pressure \(p_g\) in the surrounding gas, which can be set to zero. Since both, density and viscosity of the surrounding gas are much smaller than density and viscosity of the spreading liquid, a one-sided model formally derived by Burelbach et al. [15] can be employed, while these authors even included evaporation (not considered in the present article). This passive-gas approximation removes the need to solve for the gas flow.

We shall develop all equations valid for the rotationally symmetric spreading droplet in this section, while the reader may find the equations specifically valid for the plane spreading of a film in the “Appendix”. The equationsensure conservation of mass and momentum for a Newtonian liquid within the frame of continuum mechanics. The appropriate boundary conditions at the l–s interface arewith the normal and tangential vectors at the solid surface \({\textbf {n}}_s, {\textbf {t}}_{s}\), a slip coefficient \(\gamma '\), and the deformation-rate tensor \(\overline{\textbf{D}}\). Equation (3) represents a Navier slip condition, as used by Hocking [27] or Dussan [28], for a spreading liquid. As shown by Huh and Scriven [29], this relaxes the shear-stress singularity at the contact line, formerly found by Moffatt [30]. Equation (4) reflects the impermeability of the solid. At the free l–g interface, the boundary conditions arewith the stress tensor \(\overline{\textbf{T}}\), and the normal and tangential vectors at the free interface \({\textbf{n}}\) and \({\textbf{t}}\). Boundary condition (5) is of kinematic nature and ensures, that no fluid passes through the interface. As evaporation and condensation are neglected, the flow remains tangential to the free interface. Boundary condition (6) reflects the normal-stress balance and includes the pressure jump across a curved interface of mean curvature 2K. Boundary condition (7) reflects the tangential-stress balance, whereas the passive-gas approximation results in a shear-free interface.

The speed of the contact line depends on the profile (i.e. on the slope) of the free interface near the contact line. From a macroscopic point of view, the liquid spreads normal to the contact line and a macroscopic correlation, the so-called Tanner law Tanner [4], couples the speed of the contact line \(a_t\) to the (macroscopic) contact angle \(\theta \). Its generalized form, as e.g. introduced by Ehrhard and Davis [31], can be used if the spreading is determined by hydrodynamics [11]. This form iswith a mobility coefficient \(\kappa _A\) and a mobility exponent \(q \ge 1\). Theoretical considerations for a perfectly wetting liquid (\(\theta _A=0\)) by [32] and molecular simulations by He and Hadjiconstantinou [33] suggest \(q=3\). Similarly, from the data of Hoffman [34], de Gennes [1] for small speeds of the contact line concludes \(q=3 \pm 0.5\). Also, Rose and Heins [35], Friz [36], and Schwartz and Tajeda [37] propose from their experimental data and from physical reasoning that \(\tan {\theta } \propto a_t^{1/3}\). For \(\theta \ll 1\) this dependency recovers \(q=3\) for cases of perfect wetting. It should be kept in mind, though, that the spreading law (8) remains an approximation, reflecting the integral physics in a correct manner. A generalization of this law, including contact angle hysteresis, leads to the spreading lawThe mobility coefficients and static contact angles for a receding contact line (\(\kappa _R, \theta _R\)) and for an advancing contact line (\(\kappa _A, \theta _A\)) may be different. If \(\theta _R\ne \theta _A\) holds, a contact-angle hysteresis is present in the spreading law (9). The behavior of the contact lines can be qualitatively presented in a diagram, featuring \(\theta = f(a_t)\), see Fig. 2.

To take advantage of the geometric disparity within the shallow layer of liquid, we engage characteristic values \(f_0\) for the scalingDimensionless variables are denoted by \(f^*\). Thereby, \(a_0\) is the initial spreading length or radius of the liquid, \(\theta _0\) is the initial contact angle, \(\kappa _A \theta _0^q\) is an estimate of the initial speed of the contact line (for perfect wetting), and \(a_0 \theta _0\) is an estimate of the initial height of the liquid. The vertical velocity scale is dictated by conservation of mass. The time scale is inferred from the initial horizontal extent of the liquid and the initial speed of the contact line. The pressure scale is determined by a balance of pressure forces and viscous forces in the horizontal direction. The resulting dimensionless groups areHere, the capillary number C reflects the ratio of viscous and capillary forces, the Bond number G the ratio of gravitational and capillary forces, the Reynolds number \(\textrm{Re}\) the ratio of inertial and viscous forces, the Ekman number \(\textrm{Ek}\) the ratio of viscous and Coriolis forces, and the centrifugal number \(\Phi \) the ratio of centrifugal and capillary forces. The dimensionless slip length \(\gamma \) reflects the ratio of the slip length and the characteristic vertical height and the dimensionless Hamaker constants \(H_i\) reflect the ratio of intermolecular and capillary force.

For simplicity all asterisks, marking scaled variables, will be dropped from here. We further apply the lubrication approximation \(\theta _0 \rightarrow 0\) with the constraintsto the Eqs. (1–7), to obtain a simplified (linearized) system of equations. After an integration of the simplified conservation equations over the liquid layer of thickness h and the application of all simplified boundary conditions at \(z=0\) and \(z=h\), an evolution equation for the position of the free interface h can be derived. This is given byThe truncation error within the above evolution equation is of the order \({{\mathcal {O}}}(\theta _0^2)\). If we neglect the disjoining pressure corrections, this evolution equation can be solved analytically in the limit \(C\rightarrow 0\), which holds for most liquids. This quasi-steady solution iswith the integration constants \(C_1, C_2, C_3\), and the modified Bessel functions of first and second kind \(\textrm{I}_i\) and \(\textrm{K}_i\), both of order i. The quasi-steady solution (14d) appears to be valid as long as \(C \ll 1\) holds, i.e. as long as capillary forces dominate over viscous forces.

If a rotational-symmetric liquid droplet with a dry central patch is present, the boundary conditions are obviouslyHence, we expect a contact condition both at the inner contact line b (cf. Eq. (15d)) and at the outer contact line a (cf. Eq. (16d)). Moreover, an integral volume conservation must be fulfilled (cf. Eq. (17d)), as long as evaporation and condensation are not considered. The integration constants in Eq. (14d) can be determined and, for the case of the spreading droplet, they arewithThe contact angles at both contact lines can be determined from the local slopes \(h_r\) of the droplet interface h at the contact lines. Hence, the velocities of both contact lines \(a_t\) and \(b_t\) can be linked to the respective contact angles by means of Tanner’s law and we obtainObviously, in all cases the contact-line speed can be determined by solving an initial-value problem, though a coupling via the solution for the free interface h in Eq. (14d) persists. These initial-value problems are solved numerically by engaging a Adams–Bashforth–Moulten predictor corrector scheme.

To understand the effect of volumetric forces of gravitational or centrifugal nature, it appears sufficient to discuss droplet profiles h(r) for varied strength of these forces and for a given set a, b. In dimensionless form, the gravitational forces are characterized by the Bond number G, the centrifugal forces by the centrifugal number \(\Phi \). For the droplet profiles in Fig. 3, the inner contact line is taken very close to the center, i.e. at \(b=10^{-3}\), while the outer contact line is at \(a=1.5\). Such a situation is typical immediately after the central break up of the liquid droplet, potentially forming a central dry patch. Please note, that all profiles given in Fig. 3 physically represent thin rotationally symmetric droplets with finite contact angles \(\theta _a,\theta _b\) and a small-radius dry patch at the droplet center. The vertical extend of the droplets appears strongly magnified, as separate scaling and different scales on both axes in Fig. 3 are engaged.

In Fig. 3a the centrifugal number is varied in the range \(\Phi \; \epsilon \; [0,5]\) and the Bond number is constant and very small, i.e. \(G = 10^{-3}\). Hence, we have a situation where gravity has essentially no effect, while the effect of the centrifugal forces can be discussed separately. At small centrifugal forces (\(\Phi \le 2\)) the droplet profile h leans towards the center. Hence, the inner contact angle \(\theta _b\) is greater than the outer contact angle \(\theta _a\). This can be understood from the droplet profile for \(\Phi = 0\), for which no centrifugal and hardly any gravitational forces are in effect. Hence, the sole effect of capillary forces leads to a droplet profile h with constant mean curvature. The increase of the inner contact angle to values greater than the static advancing contact angle, i.e. for \(\theta _b > \theta _A\), would subsequently lead to spreading at the inner contact line, i.e. the dry patch would close. For greater centrifugal numbers (for \(\Phi > 2\)), centrifugal forces increase and the center of the liquid mass moves towards the outer contact line a. Linked to this observation, the visible curvature of the droplet profile h changes from a simple convex shape to a convex–concave–convex shape. This also leads to a decrease of the inner contact angle \(\theta _b\) and an increase of the outer contact angle \(\theta _a\). Again, if the outer contact angle would exceed the static advancing contact angle, i.e. for \(\theta _a > \theta _A\), the liquid would subsequently spread at the outer contact line, leading to a further decrease of the inner contact angle \(\theta _b\). As soon as the inner contact angle falls below the static receding contact angle, i.e. for \(\theta _b < \theta _R\), the liquid would subsequently recede at the inner contact line, leading to a further opening of the central dry patch. Droplet profiles with a dry central patch as given in Fig. 3 for \(G \simeq 0\), i.e. without gravitational forces, are likewise inferred by McKinley et al. [20]. Their droplet profile agrees reasonably well with our profile for \(\Phi = 1\), particularly the steep decent of the droplet profile as \(r \rightarrow 0\) is remarkable. As these authors use a different dimensionless group for the centrifugal forces and do not vary the strength, a quantitative comparison is not made.

In Fig. 3b the sole effect of gravitational forces can be inspected, as centrifugal forces are switched off (\(\Phi =0\)) and gravitational forces are varied by setting the Bond number to values in the range \(G \; \epsilon \; [10^{-3},100]\). For \(G=10^{-3}\) we recover the droplet profile from Fig. 3a (for \(\Phi =0\)) which is characterized by the sole effect of capillary forces. Increasing the Bond number, more and more, leads to a leveling of the liquid and the droplet profile h appears more and more flat between the contact lines, though it remains convex in all cases. This leveling also causes an outward shift of the center of mass, which in the limit \(G \rightarrow \infty \) can be found at the radius \(r \rightarrow (a+b)/2 \simeq 0.75\). In parallel, the increasing Bond number G increases the outer contact angle, and for \(\theta _a > \theta _A\) would cause spreading, i.e. an outward movement of the outer contact line. McKinley et al. [20] have not included gravitational forces in their investigations. Hence, the present results broaden the knowledge on the effect of such vertical volumetric forces onto the spreading.

Even though the droplet profiles in Fig. 3 have been obtained for a specific choice of the contact-line positions, this general behavior as consequence of varied volumetric forces can be found for all other combinations of contact-line positions, both in static and spreading situations.

For all what follows, we choose \(G=1\) and \(\Phi =1\) as basis, which physically means that both gravitational and centrifugal forces have a considerable effect. Moreover, the contact-line positions are chosen as \(b=10^{-3}\) and \(a=2.011\). These contact-line positions are selected because the central height \(h(r=0)\) of the liquid layer in this case is close to zero, eventually leading to a central break up of the droplet.

For the perfectly wetting case, the static advancing and receding contact angles are \(\theta _A=\theta _R=0\). Physically, perfect wetting occurs if strong attractive molecular interactions between liquid and solid molecules are present, while the interaction between gas and solid molecules remains weak. Hence, in any case the liquid would spread along the solid starting from both contact lines. The macroscopic model of Tanner reflects this physical behavior correctly, as it predicts spreading for any contact angle \(\theta _a>0\) or \(\theta _b>0\), i.e. the inner contact line would move inward to close the dry patch, while the outer contact line would move outward.

In Fig. 4a droplet profiles h(r) are given for a slightly varied outer contact-line position a. We recognize for \(a = 2.011\) a positive contact angle \(\theta _b>0\) at the inner contact line (cf. Fig. 4b), confirming the above arguments for a closing of the dry patch. By moving the outer contact line outward, i.e. for \(a > 2.011\), the inner contact angle \(\theta _b\) decreases, leading to negative inner contact angles and even to a negative height of the liquid layer in the central region, i.e. \(h(r)<0\). This is obviously a non-physical result, as e.g. the conservation of the liquid volume (cf. Eq. (17d)) is no longer meaningful with negative volume contributions. Physically, as soon as the l–g interface gets close to the liquid–solid (l–s) interface, molecular interactions between both interfaces become important and the disjoining-pressure corrections become essential. Indeed, the droplet profiles in Fig. 4 are obtained from the analytical solution (14d), for which the disjoining-pressure corrections have been neglected. The effect of the disjoining-pressure corrections is discussed in more detail in Sect. 3.5.

In Fig. 5a the profile of the droplet shortly after the beginning of the spreading is plotted for the static contact angles \(\theta _A=\theta _R=0.5\). The initial position of the inner contact line is \(b=10^{-3}\), i.e. a small dry patch is present. The early droplet profiles show that the outer contact line a appears almost immobile, while the inner contact line recedes. This steepens the annular ring on both sides, i.e. both the inner and the outer contact angles \(\theta _b\) and \(\theta _a\) increase. Hence, the speed of the outer contact line a also increases, leading to an outward movement of both contact lines. In other words, the entire annular ring moves outward, whereas its height decreases due to its increasing (mean) radius. In Fig. 5b the profiles of the annular ring are plotted for later times. The outward movement of the annular ring persists. The lubrication approximation remains valid because, even scaled, since the radial wetted distance is eight times greater than the liquid height. Also note the different scales on the ordinates in Fig. 5a, b. McKinley et al. [20] also report similar behavior of droplet spreading as discussed in Fig. 5a. In detail, an increasing central dry patch followed by a steepening of the annular droplet and an outward movement. As their results are obtained without gravitational forces, a detailed comparison to our present results for \(G=1\) appears to be difficult.

In the upper right of Fig. 5b, droplet profiles for static contact angles \(\theta _A=\theta _R \; \epsilon \; [0.35,0.7]\) and for \(a=20\) are depicted. Great static contact angles lead to a steep droplet profile near the outer contact line a, as the outer spreading appears restrained. This occurs since the inner contact line b has to approach the outer contact line a to steepen the contact angle \(\theta _a\). If the static contact angles are small, the receding of the inner contact line b and its influence onto the profile of the droplet can be recognized.

The complete spreading is summarized in Fig. 6a by plotting the position of the contact lines b and a as function of time t. The curves are obtained for varied static contact angels \(\theta _A = \theta _R \; \epsilon \; [0.35,0.7]\). At the beginning (for small t) the inner contact line appears to be mobile, while the outer contact line is hardly mobile. Only for the small static contact angle \(\theta _A=\theta _R=0.35\), both contact lines start to move outward immediately. For greater static contact angles, the inner contact line b first has to approach the outer contact line a to steepen the outer contact angle \(\theta _a\). According to Tanner’s law, the speed of the (inner) receding contact line b rises for smaller contact angles \(\theta _b\). Also, from Fig. 6a, the receding contact line b moves slower for smaller static contact angles, as the outer advancing contact line a obviously controls the spreading. The behavior of the contact lines in time, as discussed in Fig. 6a, can similarly be found in the work of McKinley et al. [20]. Most of the observed features of these authors are qualitatively in accordance with our present results. As these authors did not account for gravitational forces, again a detailed comparison to our present results for \(G=1\) appears to be difficult. Moreover, these authors analyzed their results for short times only, such that they could not detect the asymptotic power-law behavior.

In Fig. 6b the above spreading is plotted in the form of a log–log plot. At the beginning (for small t) the mobile inner contact line b, for all static contact angles, approaches the same asymptotic behavior, i.e. all curves approach an identical slope. Later, at \(t \simeq 100\), the outer contact line a starts to move and also approaches, for all static contact angles, the same asymptotic behavior. A closer inspection of all the curves reveals that for both contact-line positions the same asymptotic spreading law occurs. Particularly, for \(t \rightarrow \infty \), the slopes of all curves appear to be identical. The offset from the first to the second asymptotic behavior occurs, because the advancing contact line starts to move and to control the spreading.

The asymptotic behavior of the contact lines, as obvious from straight lines in the log–log diagram of Fig. 6b, suggests a power-law ansatz of the forms \(a \propto t^\alpha \) and \(b\propto t^\beta \) for the temporal behavior of both contact lines. In Table 1 the exponents of these power laws are summarized for all computed static contact angles. The first exponent \(\beta \) is determined from the (mean) slope at the first inflection point of the b curves in the log–log plot. The second exponents \(\alpha ,\beta \) are determined from the (mean) slope of all curves at the last computed time step \(t=10^4\). The slopes of the asymptotes, and hence the exponents, scatter less than 3% around the mean values. To summarize, the general spreading behavior according to \(a,b \propto t^1\) appears suitable during these asymptotic phases.

If hysteresis is present, several combinations of static contact angles, fulfilling the constraint \(\theta _A>\theta _R\), are possible. For simplicity, the mobility coefficients are chosen to be equal, i.e. \(\kappa _R = \kappa _A\). In Fig. 7a the profile of the droplet is plotted for \(a=2.5, \theta _A=0.55\), and for varied static receding contact angles \(\theta _R \; \epsilon \; [0.3,0.55]\). The droplet can spread as an annular ring, as in the case of spreading without hysteresis. With decreasing \(\theta _R\), hence increasing hysteresis \((\theta _A-\theta _R)\), the receding of the inner contact line b appears more pronounced and the contact angle \(\theta _b\) increases. In Fig. 7b the profile of a droplet with \(\theta _A=1, \theta _R=0.9\) is shown for several times. The initial droplet profile features \(a \simeq 2\) and \(b \simeq 0\). After the initial break up, both the outer a and the inner contact line b recede (solid lines). Hence, as both contact lines approach each other, the height of the droplet increases. Further, capillary forces lead to a greater contact angle at the inner contact line b. As soon as \(\theta _b > \theta _A\) is fulfilled, the inner contact line starts to advance and the droplet closes in the center (dotted lines). Initial states with desired contact-line positions and angles, in general, can be obtained e.g. by varying the rotational speed of the solid, i.e. by varying \(\Phi \) in the simulations.

The spreading history after break up of the central liquid layer is summarized in Fig. 8a in the form of a log–log diagram for the case plotted in Fig. 7a. If the static receding contact angle is in the range \(\theta _R < 0.3\), disjoining-pressure corrections would have to be taken into account to prevent a negative liquid height. For \(\theta _R = 0.3\) the droplet obviously spreads into a static equilibrium, as the contact-angle hysteresis represses the mobility of the inner contact line b at a later stage in time. Hence, the outer contact angle \(\theta _a\) cannot steepen enough to exceed the static advancing contact angle \(\theta _A\) and, thus, remains immobile. Though, for a static receding contact angle of \(\theta _R \ge 0.35\), the outer contact angle exceeds the static advancing contact angle, allowing the outer contact line to advance outwards. In other words, the droplet spreads outwards as an annular ring.

The movement of both contact lines can be divided into four stages:During the spreading, the mobility exponent q hardly has an influence onto the asymptotic slopes in Fig. 8a. This has been explicitly confirmed by comparing the asymptotic slopes for \(q = 1, 2, 3\). Therefore, the asymptotic behavior is a result of the coupling of a and b, respectively of \(\theta _A\) and \(\theta _R\). The exponents of the power laws for the asymptotic behavior are summarized in Table 2. Once more, with the exception of the static equilibrium (for \(\theta _R = 0.3\)), the laws \(a,b \propto t^1\) appear suitable for a spreading with hysteresis during these asymptotic phases.

In Fig. 8b the movement of the contact lines is summarized in a log–log plot, for the situation depicted in Fig. 7b, i.e. for \(\theta _A=1\) and varied \(\theta _R\) in the range \(\theta _R \; \epsilon \; [10^{-3},1]\). If there is only a small contact-angle hysteresis with \(\theta _R > 0.8\), the outer contact line starts to recede and the inner contact line finally advances, leading to an (inner) closing of the droplet. In all other cases, i.e. for \(\theta _R \le 0.8\), a static equilibrium is approached.

Hence, if contact-angle hysteresis is present in the system, there are four possible states which may be obtained. In Fig. 9 these states are plotted in a map, as function of the static advancing contact angle \(\theta _A\) an the static receding contact angle \(\theta _R\).It appears worthwhile to mention that the combination \(\theta _A = \theta _R\) on the bisectrix of Fig. 9 recovers the behavior of spreading without hysteresis. According to the map, a stable static equilibrium without an annular ring cannot be expected, since small disturbances of the contact angles would lead either to an annular situation or to a closing of the central dry patch. This suggests that a static equilibrium of an annular droplet without hysteresis cannot be stable, a result which was already discussed by McKinley and Wilson [21].

Figure 9a provides the flow map for \(G=1\) and \(\Phi \; \epsilon \; [0.5,2]\). Increasing the centrifugal number \(\Phi \) shifts the transition of the states to greater static advancing contact angles \(\theta _A\). This occurs since only greater static advancing contact angles \(\theta _A\) can slow down the outer contact line movement, such that the inner contact line can follow. In Fig. 9b the parameters are \(\Phi =1\) and \(G \; \epsilon \; [10^{-3},10]\). The influence of the Bond number G appears to be relatively weak, if compared to the influence of the centrifugal number. Though, for greater Bond numbers G, the transitions are similarly shifted to greater static advancing contact angles \(\theta _A\). This appears plausible since gravity tends to propel the spreading. Nevertheless, typical Bond numbers as \(G=10\), taken from the experiments of Ehrhard [6], show a considerable influence. If the mobility coefficients follow \(\kappa _A > \kappa _R\), the advancing contact line is more mobile than the receding contact line. Once more, in this case all transitions are shifted to greater \(\theta _A\). If \(\kappa _A < \kappa _R\) holds, the effects are vice versa.

Quantitative experimental results for the spreading of rotating droplets with a central dry patch are not known to the authors. However, Mukhopadhyay and Behringer [25] report that a partially wetting droplet breaks up in the center as the height tends to zero. In contrast, a perfectly wetting droplet does not break up. This qualitative behavior appears to be consistent with the results of our present work.

The spreading of perfectly wetting rotating droplets without a dry patch at the center has been frequently analyzed, without engaging a dynamic contact-angle law as e.g. that of Tanner [4]. Instead, a disjoining-pressure term has been introduced, responsible for the spreading. The disjoining-pressure term is present in the entire liquid domain, though it has a significant effect only in regions with \(h \ll 1\). In the present article, we engage a hybrid model by coupling both, a disjoining-pressure correction and the spreading law of Tanner. This enables us to use a quasi-stationary solution, decoupled from the spreading itself for weak capillary forces, i.e. for \(C \ll 1\). Furthermore, it is possible to include a contact-angle hysteresis by generalizing Tanner’s law. Finally, this opens the possibility to investigate the contact-line instability for a rotating droplet, spreading beyond the radius at which a negative liquid height would occur without a disjoining-pressure correction (cf. Boettcher [23] or Boettcher and Ehrhard [24]).

The disjoining pressure prevents a break up of the droplet at the center and, therefore, the boundary condition for the inner contact line (15d) has to be replaced by a symmetry condition and the integral volume conservation (17d) has to be modified. Hence, we havewhich leads with neglected disjoining-pressure correction and \(\delta \rightarrow 0\) to analytical solutions for h, namelyIf the disjoining-pressure correction are taken into account, the quasi-steady versions of the evolution equations need to be solved numerically by using a collocation method for singular boundary-value problems, as given e.g. by Auzinger et al. [38]. As there is a singularity due to the disjoining pressure for \(h=0\), a small deviation \(\delta =0.002\) from \(h=0\) at the contact line is used for relaxation. In Fig. 10a the droplet profiles for different times are plotted for a disjoining pressure with the exponents \(i=3,4\) and the corresponding attractive and repelling Hamaker constants of \(H_3=10^{-13}\) and \(H_4=10^{-10}\) (full lines). A comparison between these results and those of the analytical solutions (23d) (dashed lines) shows, that only the two curves with \(a=2.5, 3.0\), i.e. the curves with a dry central patch, differ substantially from the solid curves. The other three dashed curves with \(a=1.0, 1.5, 2.0\) are sitting on top of the corresponding solid curves and can hardly be recognized. The mean relative error for h of those three curves is only 0.051%. The disjoining pressure makes an appearance as \(h|_{x,r=0}\rightarrow 0\). The spreading of a droplet with the prevention of a break up at the center, therefore, can successfully be modeled. This is true, since Tanner’s law provides the dynamics of the contact line and the disjoining pressure prevents the break up of the liquid, without noteworthy effects elsewhere. The error for sufficiently small Hamaker constants is negligible, because the influence of the disjoining pressure on the liquid thickness is restricted to the remaining thin liquid layer at the center. This situation is depicted in Fig. 10b. The dashed line again gives the droplet profile from the analytical solution (23d), the solid lines are obtained for varied \(H_4\). Obviously, the profiles in the droplet center, with decreasing \(H_4\), converge to a thin horizontal liquid layer, while in the outer region (for greater r) the droplet profiles can hardly be distinguished. For comparison, Table 3 lists the relative errors between the outer contact angles \(\theta _a\), derived numerically with disjoining pressure, and derived analytically. For e.g. the contact-line position \(a=2\), the contact angles \(\theta _a\) are almost identical, provided \(H_4<10^{-8}\) holds. Because the outer contact angle \(\theta _a\) for small \(H_4\) remains to be a good approximation even beyond the validity of the analytical solution, it seems as if the analytical solution may be even used after the height at the center falls below zero, as long as only the speed of the spreading is of interest.