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Condensation in a wetting confined geometry at a vapor pressure below planar-interface saturation. A concave liquid meniscus has lower equilibrium vapor pressure in the capillarity approximation. Gap size, contact angle and surface adsorption affect the actual onset.

Two wide vertical plates separated by gap $d$ provide two contact lines per unit width. Upward surface-tension force $2\gamma \cos \theta$ balances column weight $\Delta \rho gdh$. Compared with a circular tube of radius $r$, the height ratio is $r/d$ for the same fluid and contact angle. Equal plate gap and tube diameter give half the tube rise.

A liquid spreads macroscopically over a solid with zero macroscopic equilibrium contact angle. In the simple interfacial-energy model the threshold is ${\gamma }_{SV}-{\gamma }_{SL}={\gamma }_{LV}$, or zero spreading coefficient; positive coefficient favors a spreading film. Extrapolating a concentration-dependent liquid-vapour surface tension to this threshold assumes the other interfacial energies remain fixed and no intervening adsorption or micellization behavior changes the trend.

For unchanged fluid, gravity and capillary geometry, equilibrium rise is proportional to the cosine of the contact angle measured through the liquid. Changing from angle $\theta$ to complete wetting therefore multiplies the height by $1/\cos \theta$. If contamination also changes liquid surface tension or density, the contact-angle ratio alone does not fix the new height.

A contact line can remain pinned while the observed contact angle changes between receding and advancing limits. Surface roughness and chemical heterogeneity can produce this hysteresis. Slowly adding or removing liquid from a sessile drop measures advancing and receding angles separately. An observed pinned angle therefore need not be the single equilibrium angle predicted by Young's equation for a perfectly homogeneous surface.

For a circular capillary of inner radius $r$, balance the vertical force $2\pi r\gamma \cos \theta$ against the hydrostatic column weight $\rho g\pi r^{2}h$. The displayed signed height follows, neglecting meniscus-volume corrections. A wetting liquid with $\theta <\pi /2$ rises, whereas a nonwetting liquid with $\theta >\pi /2$ is depressed. The contact angle is measured through the liquid. The expression uses radius rather than diameter and presumes equilibrium at the specified wall chemistry.

The gravity-capillary drop with a meridional curvature approximation uses the stipulated $\kappa =-h^{\prime \prime }$ for this calculation. At equilibrium the thin-film equation givesRegularity at $r=0$ forces $C=0$. In the wetted interior $h>0$, the equilibrium pressure gradient therefore vanishes, givingThis is the capillary length. Integrating the constant-coefficient ordinary differential equation givesThe central boundary condition $h^{\prime }(0)=0$ sets $C_2=0$, and contact at $r=R$ gives $C_0=-C_1\cosh (R/l)$. Write $C_1=-\alpha l$. The prescribed contact angle fixesHere $\alpha$ is the dimensionless coefficient of this drop solution, unrelated to the hinge angle in question 1.

The axisymmetric volume integral is $V=2\pi {\int }_0^{R}rh(r)dr$. Substituting $\xi =r/l$ givesFor numerical evaluation, with $s=R/l$, the integral isTogether with $\alpha =\tan {\theta }_c/\sinh s$, this determines $R$ for fixed $V$ and ${\theta }_c$.

The Bond number $R^2/l^2=\rho g{R}^2/\gamma$ distinguishes the dominant balances. For $R\ll l$, surface tension dominates gravity and the surface is nearly a small-slope constant-curvature cap:For $R\gg l$, gravity flattens the interior, while surface tension bends the edge over the capillary length. The stipulated model givesup to relative corrections of order $l/R$. The contact angle fixes the edge slope and therefore the height scale; conservation of mass then fixes the lateral extent. Consistency of lubrication theory requires small slopes, in particular $\tan {\theta }_c\ll 1$.

The stated approximation differs from axisymmetric free-surface curvature. For an axisymmetric graph with the same sign convention, the sum of principal curvatures isThe omitted $h^{\prime }/r$ term is generally of the same order as $h^{\prime \prime }$. For example, $h=H(1-r^2/R^2)$ gives stipulated $\kappa =2H/R^2$ but axisymmetric small-slope $\kappa =4H/R^2$. Thus the required hyperbolic-cosine profile solves the explicitly prescribed reduced model, not the full axisymmetric capillary equilibrium. Keeping both small-slope terms instead giveswhere $I_0,I_1$ are modified Bessel functions. The qualitative capillary length and Bond number balances remain valid.

For the motionless uniform layer, the heat equation reduces to $T_{zz}=0$. The lower temperature and upper heat-transfer condition determine the quasistatic temperature of a conductively cooled film:When $\alpha h/\kappa \ll 1$, $T_s=T_0-\Delta T\alpha h/\kappa +O(\Delta T(\alpha h/\kappa {)}^2)$. For a varying film let its vertical velocity scale as $Uh/L$, as required by incompressible flow. Relative to vertical thermal diffusion, horizontal thermal diffusion is smaller by $(h/L{)}^2$, and either horizontal or vertical thermal advection is smaller by $U{h}^2/(\kappa L)$. Time variation on the film's transport timescale $L/U$ has the same small factor. Thus the leading heat equation is again $T_{zz}=0$ at each $x,t$ when both specified dimensionless parameters are small. This argument retains the small $\alpha h/\kappa$ assumption. If thickness were externally varied on a faster timescale $t_h$, one would additionally need $h^2/(\kappa t_h)\ll 1$; the spatial conditions alone do not control arbitrary rapid time dependence.

Define $b={\gamma }^{\prime }\Delta T\alpha /\kappa >0$. The surface tension at the surface is ${\gamma }_s={\gamma }_0+bh$, so the Marangoni stress is ${\gamma }_{s,x}=b{h}_x$. With capillary pressure $p=-{\gamma }_0{h}_{xx}$ to leading order, horizontal Stokes flow in the film solves $\mu u_{zz}=p_x$, $u(0)=0$, $\mu u_z(h)={\gamma }_{s,x}$. HenceThe continuity equation gives the thermocapillary thin-film equationTo justify using constant surface tension in the capillary pressure, compare the two volume fluxes on horizontal scale $L$: their magnitudes are ${\gamma }_0{h}^4/(3\mu L^3)$ and $b{h}^3/(2\mu L)$. When both mechanisms are retained at leading order, their balance gives $bh/{\gamma }_0=O((h/L{)}^2)\ll 1$. Thus replacing ${\gamma }_s$ by ${\gamma }_0$ in the curvature term incurs only a relative small correction. Equivalently, small $\alpha h/\kappa$ ensures this if ${\gamma }^{\prime }\Delta T/{\gamma }_0$ is bounded; if it were parametrically large, the extra small surface-tension-variation condition would have to be imposed separately.

The capillary pressure is higher beneath a crest than beneath a trough, so pressure-driven volume flux drains the crest and smooths thickness. The Marangoni effect acts oppositely: a thicker region has a colder surface and therefore larger surface tension. Surface velocity is pulled towards that region, supplying more liquid and amplifying the thickness variation. The arrows below represent these two mechanisms separately.

Put $a_c={\gamma }_0/(3\mu )$ and $a_m=b/(2\mu )$. Substitution of $h=\hat{h}H$, $x=\hat{h}X/ϵ$, $t=\hat{t}\tau$ gives coefficients $a_m{ϵ}^2\hat{t}$ and $a_c{ϵ}^4\hat{t}/\hat{h}$. Making both equal to one yieldsIn these scales the fractional surface tension change is $b\hat{h}/{\gamma }_0=2{ϵ}^2/3$, making the previous approximation explicit.

The linear stability analysis about $H=1$ gives ${\eta }_{\tau }+{\eta }_{XX}+{\eta }_{XXXX}=0$. For a normal mode $e^{s\tau +ikX}$,Long waves with $0<|k|<1$ grow, $|k|>1$ decay, and $k=0,\pm 1$ are neutral. The neutral $k=0$ mode changes the mean film thickness. The dispersion relation is even in $k$, with its two maxima at $\pm 1/\sqrt{2}$.

For a steady positive film with zero volume flux, divide $H^2{H}_X+H^3{H}_{XXX}=0$ by $H^3$ and integrate to obtain $H_{XX}+\ln H=C$. Choosing $\hat{h}$ as specified fixes $C=0$. Multiplication by $H_X$ and another integration then give the zero-flux thermocapillary film profiles first integralFor $H>0$, $V^{\prime }=\ln H$ and $V^{\prime \prime }=1/H>0$: the unique minimum is $V(1)=-1$, while $V(0^{+})=0$ and $V(e)=0$. This potential energy interpretation classifies the profiles without assuming a sinusoidal shape at finite amplitude.

For $E=-1$, the only profile is the uniform film $H=1$. For $-1<E<0$, there are two positive turning heights $H_{-}<1<H_{+}<e$ and a smooth periodic film oscillates between them. Its period isFor $E=0$, the maximum is $e$ and the lower endpoint is zero. A drop reaches zero thickness in finite distance, with zero limiting slope but unbounded curvature since $H_{XX}=-\ln H$. Its half-width isThus the formal limiting profile consists of drops of peak height $e$ and footprint width $2\sqrt{\pi e}$, with zero limiting contact angle. Identical drops can touch, or a zero-thickness region can separate them in the degenerate zero-flux model. These are limiting wet-region solutions, not everywhere positive twice differentiable films; the singular contact region is not resolved by lubrication theory. A useful parametric drop profile isFinally let $E=-1+\delta E$ with small positive $\delta E$. Writing $H=1+\eta$ gives $V=-1+{\eta }^2/2+O({\eta }^3)$ and $H_{XX}=-\eta +O({\eta }^2)$. The small-amplitude steady profile is $H=1+\sqrt{2\delta E}\cos (X-X_c)+O(\delta E)$, with period tending to $2\pi$. Its wavenumber tends to the neutral boundary $|k|=1$ of the unstable band, not to the fastest-growing wavenumber.

The Young–Laplace equation is $p_l-p_v=\gamma (1/R_1+1/R_2)$ with signed principal curvatures positive for a convex liquid droplet. A wetting bridge between plates has a strongly concave meridional curvature. If the contact angle through the liquid is $\theta$, the plate separation is $d$, and the bridge radius is approximately $a\gg d$, thenFor complete wetting, the leading suction is $p_v-p_l\simeq 2\gamma /d$. The neglected azimuthal curvature is of order $1/a$. A numerical pressure cannot be specified without the separation, surface tension and wetting condition. The full signed-curvature expression remains valid when the narrow-gap approximation is unsuitable.

Let $P_{\mathrm{sat}}$ denote the flat-interface saturation pressure at the same temperature, $P_V$ the surrounding equilibrium vapor pressure, and ${\overline{V}}_l$ the incompressible liquid molar volume. Equating liquid and ideal-vapor chemical potentials to their common reference at $P_{\mathrm{sat}}$ givesThus the pressure contribution to the attractive normal force on a wetted circular area isFor capillary pressures large compared with the vapor-pressure difference, and a bridge wetting essentially the plate area,Here $F$ is the magnitude of the external separating force balancing attraction, and $P_V<P_{\mathrm{sat}}$. A finite contact angle adds a contact-line force $2\pi a\gamma \sin \theta$ at each plate's rim, counted once in the force on that plate, while the exact pressure also retains both curvatures. The rim term is normally smaller than the narrow-gap pressure term by order $d/a$ for appreciable wetting; it cannot be neglected uniformly near $\theta =90^{\circ }$. For complete wetting the leading equivalent expression is $F\simeq 2\pi a^2\gamma /d$. The corresponding equilibrium humidity is $P_V/P_{\mathrm{sat}}\simeq \exp [-2\gamma {\overline{V}}_l/(RTd)]$.

These are quasistatic, equilibrium expressions for an existing bridge, neglecting viscous suction, contact-angle hysteresis, disjoining forces and deformation of the plates. The wetted radius must replace $a$ if the liquid does not reach the plate edge. During actual separation a bridge can shrink, depin or break, so this local force balance is not a promise of a constant force for arbitrary travel.

Use a self-assembled monolayer of an alkanethiol with a polar terminal group, such as $\mathrm{HS}(\mathrm{C}{\mathrm{H}}_2{)}_{\mathrm{n}}\mathrm{OH}$. Sulfur anchors to gold while the exposed hydroxyl groups interact favorably with water. A methyl-terminated monolayer would instead present a low-polarity outer surface and usually retain hydrophobic character.

Prepare a clean gold surface, allow adsorption from a compatible clean solution, and rinse away unbound material. Dense coverage and molecular ordering help make the terminal chemistry reproducible. Mixed chain lengths or incomplete coverage can change the wetting response, so the presence of an OH group in the bottle alone is not sufficient. An original experiment demonstrated hydrophilic hydroxyl-terminated films and showed that mixing chain lengths changed surface order and wettability.^[ref]

This specifies the resulting interface rather than treating every freshly prepared gold surface as intrinsically identical: contamination, preparation and adsorbates influence the starting contact angle.

The contact angle is measured through the liquid between the solid surface and the tangent to the liquid–vapor interface at the contact line. The right panel of the wetting diagram labels that angle and the three interfacial tensions.

At $0^{\circ }$, the ideal macroscopic limit is complete wetting: a film spreads rather than remaining a finite-angle cap. Angles between $0^{\circ }$ and $90^{\circ }$ are conventionally called hydrophilic for water; the drop wets the surface relatively well. At $90^{\circ }$ the tangent is vertical. Between $90^{\circ }$ and $180^{\circ }$ the water drop is conventionally hydrophobic, with increasingly weak wetting. The exact $180^{\circ }$ limit represents vanishing macroscopic contact in an ideal model, not a routine finite-area sessile drop on a real solid.

The categories describe water under specified conditions. Roughness, contamination and contact-line pinning can make advancing and receding angles differ from each other and from the ideal equilibrium angle.

This is the first printed “(vi)”, concerning adhesion. The reversible work per unit area to separate the solid–liquid interface into solid–vapor and liquid–vapor interfaces isSubstitute ${\gamma }_{SV}-{\gamma }_{SL}={\gamma }_{LV}\cos {\theta }_C$ from Young's equation:This is the Young-Dupré equation. It has units $\mathrm{J}{\mathrm{m}}^{-2}$, equivalent to $\mathrm{N}{\mathrm{m}}^{-1}$. The liquid–vapor tension is the multiplier; the abbreviated printed subscript does not denote a new interface. Within this equilibrium model, smaller water contact angle means larger reversible work of adhesion for fixed liquid–vapor tension.

Taking the printed reference to part(iii) literally, measure an equilibrium water contact angle with a calibrated sessile-drop goniometer on a smooth homogeneous surface. With independently constrained interfacial tensions, test ${\gamma }_{SV}-{\gamma }_{SL}={\gamma }_{LV}\cos {\theta }_C$. Advancing and receding measurements reveal pinning and hysteresis; contact angle alone cannot independently determine both unknown solid-interface energies.

If the intended reference was instead the polymer-size prediction in the preceding part, use static light scattering or small-angle neutron scattering at both temperatures on appropriately dilute solutions. At small scattering wave number $q$, the Guinier approximationgives $R_g$ from the slope of $\ln I$ against $q^2$. Control concentration, aggregation and scattering contrast, and compare the extracted radii. Dynamic light scattering normally measures a hydrodynamic size rather than directly the same radius of gyration. Answering both readings avoids silently correcting the printed cross-reference.

A Pickering emulsion is stabilized by particles adsorbed at the liquid–liquid interface. A partially wetting sphere removes a patch of that interface. The smaller detachment barrier into either bulk phase is approximatelywhere $\theta$ is the three-phase contact angle. At approximately neutral wetting, $\theta =90^{\circ }$, the stated particle radius givesat $25^{\circ }\mathrm{C}$. Such a large attachment energy makes thermal detachment unlikely. A sufficiently covered interface resists drop coalescence, and a water-wet particle preference supports an oil-in-water dispersion. However, radius alone does not guarantee stabilization: completely water-wet or oil-wet particles have a vanishing attachment barrier in this model, and sufficient particle coverage is necessary. Sedimentation and droplet ripening are separate failure modes. The classical particle-stabilized-emulsion experiments provide the historical basis of the terminology.^[ref]

At a straight contact line, take forces per unit transverse length and measure the contact angle $\theta$ through the liquid. Solid–vapor surface tension pulls outward along the solid, solid–liquid surface tension pulls inward, and the liquid–vapor force has horizontal component ${\gamma }_{LV}\cos \theta$. Tangential equilibrium givesThis is Young's equation. The substrate supplies the normal reaction balancing the vertical capillary component. The original force sketch labels directions at a right-hand contact line. The result assumes smooth, chemically uniform, rigid surfaces in partial-wetting equilibrium, with negligible pinning and line tension.

Wetting concerns how a liquid covers a solid in contact with another phase, such as its vapor. For an ideal flat homogeneous surface, balance of the three interfacial tensions gives Young's equationwhere the contact angle $\theta$ is measured through the liquid. Small angle means favorable contact; partial wetting gives a finite angle. The spreading coefficient $S={\gamma }_{SV}-{\gamma }_{SL}-{\gamma }_{LV}$ compares a covered surface with a dry one. Positive $S$ favors complete wetting, while negative $S$ permits a finite-angle droplet in the partial-wetting regime.

A sessile-drop experiment images the tangent at the contact line on a clean level substrate. Varying drop volume slowly distinguishes advancing and receding angles; their difference is contact-angle hysteresis from pinning and heterogeneity, so a single observed angle need not equal the ideal equilibrium value. A capillary-rise experiment gives a second consequence: the Young–Laplace equation and hydrostatic balance predict $h=2{\gamma }_{LV}\cos \theta /(\rho gr)$ in a narrow circular tube when meniscus-volume corrections are negligible. A wetting liquid rises, whereas a sufficiently nonwetting one is depressed.

For a smooth homogeneous solid and a partially wetting liquid at equilibrium, tangential balance at the contact line gives Young's equationThe angle is measured through the liquid. Replacing solid-vapour area by solid-liquid plus liquid-vapour area changes free energy by $-S$ per wetted area, where the spreading coefficient isFor partial wetting, $S={\gamma }_{LV}(\cos \theta -1)<0$. Positive $S$ favours a spreading film and zero macroscopic equilibrium contact angle; the finite-angle Young construction cannot be extrapolated to $\cos \theta >1$.

The intended chemically matched surface approximation is ${\gamma }_{SL}\simeq 0$. It gives $S\simeq 23-21.3=1.7\mathrm{mN}{\mathrm{m}}^{-1}>0$, so increasing the wetted area lowers free energy. The initially steep drop spreads and flattens toward complete wetting:The initial $90^{\circ }$ angle describes a nonequilibrium shape and must not be inserted into equilibrium Young's equation to fix ${\gamma }_{SL}$. More precisely, complete wetting requires ${\gamma }_{SL}\le 1.7\mathrm{mN}{\mathrm{m}}^{-1}$ in this surface-energy model; chemical similarity alone is not a measurement of that interfacial energy.

Water on such a low-energy, oil-like surface is expected to bead into a comparatively rounded, high-angle drop because the substrate does not compensate its cohesive interactions. Its numerical contact angle needs the water-solid interfacial energy, which is not supplied.

For a clean liquid interface at fixed temperature and composition, surface tension is the reversible free-energy cost per added area, $\gamma =(\partial F/\partial A{)}_{T,V,N}$, so $dF=\gamma dA$. Its units are both $\mathrm{J}{\mathrm{m}}^{-2}$ and $\mathrm{N}{\mathrm{m}}^{-1}$. Molecules at an interface have a different environment from those in the bulk; creating more interface changes their interactions and entropy. For constant $\gamma$, the interfacial free energy is $\gamma A$ up to a reference constant. This is not generally identical to the internal surface energy: the interfacial entropy density is $-d\gamma /dT$, giving internal-energy density $\gamma -Td\gamma /dT$.

A line of length $\ell$ in one interface transmits a tangential force $\gamma \ell$, perpendicular to the line. A soap film has two interfaces, giving force $2\gamma \ell$ on a sliding rim. Quasistatic extension by $dx$ creates total area $2\ell dx$, recovering that force from $dF=2\gamma \ell dx$. Uniform positive surface tension favors minimum area at fixed volume, hence an isolated spherical droplet when gravity is negligible. curvature gives the Young–Laplace equation, with outward normal chosen consistently:A spherical liquid drop has $\Delta p=2\gamma /R$; a thin soap bubble with two similar interfaces has approximately $4\gamma /R$. The capillary length $\sqrt{\gamma /(\Delta \rho g)}$ compares gravity with capillarity. surface tension also restores capillary waves, rather than providing a bulk static shear modulus.

Interfacial contamination and added surfactants can change $\gamma$ substantially; a nonuniform value produces tangential Marangoni stress. A rim-force measurement should therefore control temperature, cleanliness and the number of interfaces. Capillary-rise methods additionally need the contact angle, because the vertical component is proportional to $\gamma \cos \theta$. An independent frame-force measurement can distinguish a contact-angle change from a true change of surface tension.