Wetting and spreading

Wetting phenomena are ubiquitous in nature and technology. A solid substrate exposed to the environment is almost invariably covered by a layer of fluid material. In this review, the surface forces that lead to wetting are considered, and the equilibrium surface coverage of a substrate in contact with a drop of liquid. Depending on the nature of the surface forces involved, different scenarios for wetting phase transitions are possible; recent progress allows us to relate the critical exponents directly to the nature of the surface forces which lead to the different wetting scenarios. Thermal fluctuation effects, which can be greatly enhanced for wetting of geometrically or chemically structured substrates, and are much stronger in colloidal suspensions, modify the adsorption singularities. Macroscopic descriptions and microscopic theories have been developed to understand and predict wetting behavior relevant to microfluidics and nanofluidics applications. Then the dynamics of wetting is examined. A drop, placed on a substrate which it wets, spreads out to form a film. Conversely, a nonwetted substrate previously covered by a film dewets upon an appropriate change of system parameters. The hydrodynamics of both wetting and dewetting is influenced by the presence of the three-phase contact line separating “wet” regions from those that are either dry or covered by a microscopic film only. Recent theoretical, experimental, and numerical progress in the description of moving contact line dynamics are reviewed, and its relation to the thermodynamics of wetting is explored. In addition, recent progress on rough surfaces is surveyed. The anchoring of contact lines and contact angle hysteresis are explored resulting from surface inhomogeneities. Further, new ways to mold wetting characteristics according to technological constraints are discussed, for example, the use of patterned surfaces, surfactants, or complex fluids.

Figure 1

(Color online) Plant leaves after rain.

Figure 2

(Color online) Young’s equation can also be interpreted as a mechanical force balance on the three-phase contact line; the surface tension is an energy per unit area, equivalent to a force per unit length acting on the contact line.

Figure 3

(Color online) The three different possible wetting states according to Young’s equation.

Figure 4

(Color online) Zisman plot. Liquids of different liquid-vapor surface tension are deposited on a Terphene surface, and the variation of the contact angle of the liquid with the solid is measured, yielding the critical surface tension of the Terphene surface. A liquid with this surface tension just wets the surface completely. The liquids used here are aqueous solutions of the soluble surfactant cetyl trimethyl ammonium bromide (CTAB) (70. To within the experimental uncertainty, the straight line goes through the origin.

Figure 5

Two stages of the spreading of a silicone drop on a glass substrate, which it wets completely: (a) $t=3.25\mathrm{s}$ after deposition, (b) $t=18.25\mathrm{s}$. Below the drop’s surface one sees its reflection. The drop volume is $V=1.7\times {10}^{-4}{\mathrm{cm}}^3$, and $\gamma ∕\eta =10.6\mathrm{cm}∕\mathrm{s}$. Although the equilibrium contact angle is zero, the “apparent” angle is finite, and the drop is well fitted by a spherical cap. From 122.

Figure 6

The radius $R$ of the drop of Fig. 5 as a function of time, described by Tanner’s law (10).

Figure 7

A cartoon of an advancing contact line. In a frame of reference in which the contact line is stationary the solid moves with velocity $U$ to the right.

Figure 8

A spreading drop that partially wets the solid surface, whose shape is determined by the apparent contact angle. The interface near the corner is highly curved, so ${\theta }_{\mathrm{ap}}$ is larger than the angle ${\theta }_m$ seen on a scale of nanometers. On this scale the interface fluctuates, and the corrugation of the solid surface is seen. The size of the crossover region between microscopic and macroscopic, the “intermediate” region, is estimated based on Eq. (58).

Figure 9

The edge of a water drop receding on a disordered surface. The defects (size $10\mu \mathrm{m}$) appear as white dots.

Figure 10

Force per unit length exerted on a plate that is immersed in a liquid and then withdrawn (buoyancy force has been subtracted). From 395.

Figure 11

Advancing and receding contact angles on a composite substrate as a function of the surface fraction of defects. (a) Experimental measurements for liquid tin on $\mathrm{Si}{\mathrm{O}}_2$ partially covered by a regular array of Si defects. Adapted from 160. (b) Prediction adapted from the calculation by 503.

Figure 12

(Color online) The wetting phase diagram calculated by 407, where $t=(T-T_c)∕T_c$ is the reduced temperature, $h_1$ is the surface field, and $h$ is the bulk field. The wetting line $W$ (which is a parabola in the mean field approximation) has the critical temperature $T_c$ at its apex. At $T_w^1$ there is a tricritical point which separates first-order from second-order wetting transitions. The surface of prewetting transitions is bounded by a line of critical points $C_{\mathrm{pre}}^1$, called the critical prewetting line, which merges into the wetting line at $T_w^1$.

Figure 13

(Color online) Schematic effective interface potential for a first-order wetting transition (a) and for a (short-range) critical wetting transition (b).

Figure 14

(Color online) Surface phase diagram measured by 483 for the wetting of alkanes and mixtures of alkanes of different (effective) chain lengths, corresponding to different differences $\Delta \gamma$ between surface tensions of the pure alkanes and pure methanol.

Figure 15

The continuous divergence of the wetting layer thickness observed for the wetting of methanol on nonane $\left({\mathrm{C}}_9\right)$, compared to its behavior on undecane $\left({\mathrm{C}}_{11}\right)$ for which the transition is discontinuous (first order).

Figure 16

Singular part of the surface free energy, as determined directly from contact angle measurements, giving the surface specific heat exponent ${\alpha }_s$. The wetting of methanol on nonane $\left({\mathrm{C}}_9\right)$ is continuous, and so the slope of the surface free energy vanishes in a continuous fashion. For undecane $\left({\mathrm{C}}_{11}\right)$, the transition is discontinuous (first order), and a discontinuity in slope is indeed apparent.

Figure 17

(Color online) Sequence of two wetting transitions for the wetting of hexane on (salt) water, salinity $1.5\mathrm{mole}∕\mathrm{l}$ NaCl. From 517.

Figure 18

Singular part of the surface free energy, $1-\cos {\theta }_{\mathrm{eq}}$ vs temperature, for hexane on brine $\left(2.5M\right)$. The filled circles are obtained upon increasing the temperature, and the open circles upon decreasing $T$: clearly a hysteresis around the first-order transition is observed. The open symbols likely correspond to the metastable extension (undercooling) of the frustrated complete wetting state. From 454.

Figure 19

Film thickness $l$ of the mesoscopically thick film of liquid pentane on water as a function of temperature. This film is present in the frustrated complete wetting state between the first-order and the continuous transition, observed for pentane on water. The circles represent ellipsometry data of 459, while the solid curve is the result of a calculation based on Lifshitz theory (187; 581). With this approach, the film thickness is obtained from the amplitudes of the two leading terms of the long-range interactions between adsorbate and substrate. This type of calculation requires only the macroscopic dielectric properties (static permittivities and refractive indices) of the isolated media (47; 582).

Figure 20

Generic wetting phase diagram: alkane chain length vs glucose concentration. Filled symbols show the first-order transition (dashed line). Open symbols represent the continuous transition (solid line). The two transition lines produce three phase regions: regular partial wetting (PW), complete wetting (CW), and frustrated complete wetting (FCW). The critical end point (CEP) is the crossover between first order and critical wetting.

Figure 21

(Color online) Wetting of a wall by a colloid-polymer mixture. (a) Phase-separated mixture of fluorescently labeled polymethylmethacrylate (PMMA) colloids and polystyrene polymer in decalin taken under uv light. (b) A blowup of the encircled region of (a) by means of laser scanning confocal microscopy (dimensions $350\times 350\mu {\mathrm{m}}^2$). The interfacial profile is accurately described by the balance between the Laplace and hydrostatic pressures. However, the wetting layer is unexpectedly thick in this picture (3; 1).

Figure 22

(Color online) Interface width as a function of the wetting layer thickness. The dashed line is ${\xi }_{\perp }^2=0.29\mu \mathrm{m}$ $\mathcal{l}$, while ${\xi }_{\perp }^2=0.23\mu \mathrm{m}$ $\mathcal{l}$ is the theoretical prediction obtained for the wetting parameter $\omega =0.8$ using the independently measured surface tension (269).

Figure 23

Superhydrophobicity by surface texture. Left: Patterned surface. Right: Water droplet posed on the patterned surface. Superhydrophobic behavior results from the fact that due to the patterning the droplet is mainly in contact with air (77).

Figure 24

The apparent contact angle as a function of the root mean squared amplitude of the applied voltage (ac field) for a drop of oil. The filled squares are for increasing field (advancing contact line), the open circles for decreasing field (receding contact line). The solid line is the theoretical curve (48). From 397.

Figure 25

Parabolic flow in a wedge near an advancing contact angle. The substrate moves with speed $U$ to the right.

Figure 26

The dependence of the interface slope on the distance from the contact line (378). The solid curve is based on Eqs. (52, 53) with $L$ treated as an adjustable parameter (186). Also plotted in the lower two frames are deviations $\Delta \theta$ from theory, the solid curve showing a running average over $5\mu \mathrm{m}$.

Figure 27

Schematic of the procedure for continuum modeling.

Figure 28

Apparent dynamic contact angles of perfectly wetting fluids measured in a capillary (285) and for a plunging plate (543). Each symbol corresponds to a different fluid and/or substrate. The solid line is Eqs. (52, 53) with ${\theta }_m=0$ and $x∕L={10}^4$.

Figure 29

Schematic of a moving contact line for $S_i>0$, $A>0$. Ahead of the apparent contact line one finds a precursor “foot” driven by van der Waals forces.

Figure 30

Apparent dynamic contact angle data from 285 and 543 for partially wetting fluids. The value of $g\left({\theta }_{\mathrm{eq}}\right)∕\ln (x∕L)$, with $x∕L={10}^4$, is added to the abscissa to make the theoretical curve the same as in Fig. 28. Each symbol corresponds to a different experiment, the equilibrium contact angle is taken from experiment. Those data sets from 543 that showed evident contact angle hysteresis were not included.

Figure 31

Contact line velocity plotted as function of the applied force per unit length for various temperatures (445). Solid lines are exponential fits to the experimental data.

Figure 32

Schematic of the theory by 64, associating contact line motion with the hopping of molecules between the potential wells provided by the substrate.

Figure 33

(Color online) MD simulation of a Couette flow with a moving contact line. The dots are the instantaneous molecular positions of the two fluids, projected on the $x\text{−}z$ plane. The black (gray) circles denote the wall molecules. The upper panel shows the symmetric case, ${\theta }_{\mathrm{eq}}^s=90°$; the lower the asymmetric case, ${\theta }_{\mathrm{eq}}^a=64°$. From 447.

Figure 34

Comparison of the MD (symbols) and continuum (lines) interface profiles for the symmetric and asymmetric potentials shown in Fig. 33. The MD results were obtained by averaging over at least ${10}^5$ atomic time scales, the continuum calculation is based on the diffuse interface theory, with parameters determined from the MD simulation. From 449.

Figure 35

A plate withdrawn from a bath of nonwetting liquid at speed $U_{\mathrm{pl}}$, which forms a contact line at height $z_{\mathrm{cl}}$. The capillary numbers based on the plate speed $U_{\mathrm{pl}}$ are ${\mathrm{Ca}}_{\mathrm{pl}}=9.7\times {10}^{-3}(△)$, ${\mathrm{Ca}}_{\mathrm{pl}}=10.2\times {10}^{-3}(◼)$, ${\mathrm{Ca}}_{\mathrm{pl}}=10.7\times {10}^{-3}(◻)$, ${\mathrm{Ca}}_{\mathrm{pl}}=11.2\times {10}^{-3}(●)$, and ${\mathrm{Ca}}_{\mathrm{pl}}=11.5\times {10}^{-3}(○)$. These values are all above the critical capillary number ${\mathrm{Ca}}_{\mathrm{cr}}$, so the contact line moves up very slowly, and each curve is traced out in a time-dependent fashion for each experiment. The full line is found from the thin-film equations (55, 56), including gravity. From 164.

Figure 36

Dewetting patterns as observed with the AFM. The height scale ranges from black $\left(0\mathrm{nm}\right)$ to white $\left(20\mathrm{nm}\right)$. (a) Spinodal dewetting, (b) homogeneous (thermal) nucleation, and (c) heterogeneous nucleation. From 511.

Figure 37

(Color online) Reconstructed surface potentials $V\left(h\right)$ for PS films on a silicone wafer coated with a layer of SiO of varying thickness $d_{\mathrm{Si}\mathrm{O}}$. The fits are to the curve $V\left(h\right)=c_i∕h^8+A_{\mathrm{Si}\mathrm{O}}∕12\pi h^2+(A_{\mathrm{Si}}-A_{\mathrm{Si}\mathrm{O}})∕12\pi {(h+d_{\mathrm{Si}\mathrm{O}})}^2$, with experimentally determined Hamaker constants. For the particular kind of wafer used, the strength of the short-range interaction is $c_i=5.1\times {10}^{-77}\mathrm{J}{\mathrm{m}}^6$. The arrow marks the thickness $h^{*}$ of the microscopic film left behind after dewetting. From 511.

Figure 38

(Color online) The driven spreading of a thin film on a horizontal substrate. A small sinusoidal perturbation of wavelength $\lambda =1.4\mathrm{mm}$ is imposed optically, which grows into a sequence of fingers of the same wavelength. By applying perturbations of different wave numbers in a series of experiments, the dispersion relation for the instability is measured and compared with linear stability theory (with no adjustable parameters in the stability analysis). From 229.

Figure 39

(Color online) A plate being withdrawn from a bath of viscous liquid, with the contact line pinned at the edge of the plate. (a) Below the rivulet transition the contact line forms a corner. (b) Above the transition a rivulet comes out of the tip of the corner, and eventually decays into droplets. From 163.

Figure 40

Drops of silicone oil running down planes at increasing inclination. As the velocity increases, a corner first forms, which becomes unstable to the ejection of drops at even higher speeds. From 435.

Figure 41

(Color online) The normalized speed $\mathcal{C}$ of the drop as function of the normalized opening angle $\Phi$. Symbols represent rescaled experimental data for silicone oils with viscosities 10, 104, and 1040 cP. Data were rescaled using the receding static contact angle ${\theta }_r$.

Figure 42

The splash produced by a sphere impacting on water is caused by the contact line of the solid-air-water interface becoming unstable, so a sheet of water detaches from the solid. On the left, no instability occurs for a static contact angle of ${\theta }_{\mathrm{eq}}=15°$, while for ${\theta }_{\mathrm{eq}}=100°$ (right) a splash is produced. From 180.

Figure 43

Photograph of a spreading trisiloxane solution droplet at high surfactant concentration for which the radius increases linearly in time; clearly the droplet is no longer a spherical cap.

Figure 44

(Color online) Deposit from an evaporating droplet of two different salt solutions, top, ${\mathrm{Na}}_2\mathrm{S}{\mathrm{O}}_4$, bottom, NaCl. This shows that the wetting properties of the salt crystals are important for determining where crystals form. From 518.

Figure 45

A contact line pinned on a single defect.

Figure 46

Receding meniscus pinned on a defect. The experimental shape of the contact line (dots) is well fitted by Eq. (103) with $L=2.4\mathrm{mm}$. From 405.

Figure 47

Distortion of the contact line at equilibrium for a strong defect. Top: For a given $y_{\infty }$ there may be three equilibrium positions of the contact line, among which one $\left(y_2\right)$ is unstable. Middle: For an advancing line far enough from the defect, $y_M\simeq y_{\infty }$. Bottom: When $y_{\infty }=y_a$, $y_M$ jumps forward; the energy dissipated in the jump is the hatched area.

Figure 48

Hysteresis as a function of the fraction $X$ of the surface covered by defects. From 461.

Figure 49

Successive positions of the contact line. Almost nothing happens for $t<0\mathrm{ms}$; then an avalanche occurs which is nearly finished at $t=500\mathrm{ms}$.

Figure 50

Schematic of the velocity $U$ as a function of $f$. The dashed curve corresponds to zero disorder.

Figure 51

(Color online) Roughness $W\left(L\right)$ of the contact line as a function of the scale $L$. Dots: Measurements for a weakly disordered substrate for several values of $L_D$ (477). Lines: Calculated scaling function (262).

Figure 52

Width distribution of a contact line $\varphi \left(z\right)$, where $z=w^2∕W^2$. Symbols: Measured distribution for $L=500\mu \mathrm{m}$. Full (dashed) line: Scaling function for $\zeta =0.50\left(0.39\right)$.

Figure 53

Schematic of the velocity variation. At $T=0$ (line), one expects a sharp depinning transition at $f=f_c$, which is blurred by thermal fluctuations at $T\ne 0$ (dashed line).

Figure 54

Activation energy as a function of the hysteresis for liquid hydrogen on various disordered cesium substrates.

Figure 55

Advancing thin film/thick film boundary. Helium film of thickness $50\mathrm{nm}$ on a cesium substrate at $1.87\mathrm{K}$. From 401.