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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.

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.

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.

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.

In a regular solution, ideal combinatorial entropy competes with a nonzero unlike-contact energy. For equal molecular volumes and mole fraction $\varphi$ of species A, write mixing free energy per site aswhereHere $z$ is the coordination number and ${ϵ}_{ij}$ a pair-contact energy counted consistently in the lattice model. A positive $\chi$ penalizes unlike contacts; a negative value favors them. The logarithmic terms lower free energy on mixing. The sign of the enthalpy of mixing alone therefore does not determine miscibility.

At fixed temperature and composition, a homogeneous phase is globally stable only if no division into two compositions lowers its free energy. The common-tangent construction selects coexisting compositions ${\varphi }_a,{\varphi }_b$, equivalent to equality of both species' chemical potentials. Local stability to infinitesimal fluctuations instead requiresZero curvature gives the spinodal, while the coexistence boundary is the binodal. Between them a homogeneous state can be metastable and require nucleation; inside the spinodal its small composition fluctuations grow. The symmetric regular solution hasBelow this contact penalty, the free energy is convex at every composition. Above it a miscibility gap is possible around the central composition. If $\chi =A/T$ with $A>0$, heating promotes mixing, but specific interactions or temperature-dependent noncombinatorial terms can produce other trends.

For two polymers, let $N_A,N_B$ count equal-volume segments per chain, and now let $\varphi$ be the segment volume fraction of A. Connecting segments into chains reduces their independent translational entropy. The binary Flory-Huggins polymer blend hasThe energetic contact term remains extensive in the number of segments, while the favorable entropy per segment is reduced by $1/N_A$ and $1/N_B$. This is the key long-chain limit of polymer-blend miscibility: a small positive incompatibility can overcome the mixing entropy. The lattice treatment is a controlled model for this argument, rather than a claim that every real blend has composition-independent $\chi$.^[ref]

The local stability condition and critical interaction parameter becomeFor equal chain lengths, ${\chi }_c=2/N$. Similar segment chemistry and matched cohesive interactions can reduce the unlike-contact penalty; favorable specific interactions, such as complementary hydrogen bonding, can make mixing favorable even for long chains. Lower molar mass increases mixing entropy. temperature can help when it lowers the effective $\chi$, but heating is not universally sufficient. A suitable common solvent can facilitate processing or ternary mixing, and a block-copolymer compatibilizer can stabilize fine domains; neither guarantees that the final solvent-free binary material is one equilibrium molecular phase.

An oil–water emulsion illustrates emulsion stability versus molecular miscibility. A surfactant adsorbs with its water-affine head toward water and its oil-affine tail toward oil, lowering interfacial tension and the work $\gamma dA$ needed to make additional interface. Breaking a fixed oil volume $V_o$ into spherical droplets of radius $R$ gives total area $3V_o/R$, so creating small droplets still requires substantial area and usually mechanical work.

Adsorbed surfactants can oppose droplet coalescence through charged double-layer repulsion, hydrated-head interactions or steric repulsion from sufficiently extended groups. Adequate coverage, adsorption time, ionic conditions and film drainage affect the lifetime. A lower interfacial tension does not itself remove the bulk miscibility gap, and a protected emulsion is commonly kinetically stabilized. A thermodynamically stable microemulsion is a distinct situation requiring an appropriate composition and free-energy balance. The two liquids must not be called molecularly miscible simply because droplets disperse visibly.

For the capillary, balance upward contact-line force against hydrostatic column weight, neglecting meniscus volume and gas density. For inner radius $r$,The contact angle is measured through the liquid. complete wetting means ${\theta }_2=0$, so the contact-angle change of capillary rise givesThis follows from the stipulated wetting change, holding radius, bulk liquid surface tension and density fixed. One should not replace that premise with a general claim that grease always improves glass wetting; if contamination also changes the liquid surface tension, the original height and new angle alone would not determine the new height.

For two wide vertical plates separated by gap $d$, use unit width. There are two upward contact-line contributions and liquid cross-sectional area $d$, giving the capillary rise between parallel plates:Only one principal meniscus curvature contributes between wide plates, whereas a circular tube has two. For the same fluid and angle,If the plate gap equals the tube diameter, $d=2r$, the rise is half as large: the initial and completely wetting heights would be $2.50$ cm and $3.05$ cm. If instead $d=r$, the heights are equal. In either geometry, changing only the angle from $35^{\circ }$ to zero multiplies the height by $1/\cos 35^{\circ }$. Since the question specifies no plate spacing, the general gap-dependent relation is the unambiguous result; the half-height answer requires the equal-gap-to-diameter convention.

Molecular area from the adsorption slope. The supplied Gibbs adsorption equation identifies the surface excess number density $\Gamma =1/\Sigma$ throughThe logarithm is understood relative to a fixed concentration unit or reference concentration; only concentration ratios enter its differences. The pure-water point $c=0$ supplies a baseline surface tension, but cannot be included in $\ln c$.

An estimate near the later working concentration $0.055\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$ uses its two bracketing points:Taking room temperature as $298\mathrm{K}$,This corresponds to about $4.4$ adsorbed molecules per $\mathrm{n}{\mathrm{m}}^2$. The data are not perfectly linear in $\ln c$, so the inferred molecular area depends somewhat on the concentration range: a fit to all five positive-concentration points gives about $0.26\mathrm{n}{\mathrm{m}}^2$, while the higher-concentration region gives roughly $0.22\mathrm{n}{\mathrm{m}}^2$. An effective area of order $0.2$–$0.3\mathrm{n}{\mathrm{m}}^2$ is justified; a concentration-independent exact value is not.

Contact angle. Linear interpolation between $0.05$ and $0.06\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$ gives ${\gamma }_{LV}(0.055)=(48.5+45.2)/2=46.85\mathrm{mN}{\mathrm{m}}^{-1}$. For an ideal equilibrium contact angle measured through the liquid, Young's equation isThe supplied solid-vapour and solid-liquid values have difference $40\mathrm{mN}{\mathrm{m}}^{-1}$. HenceThis assumes clean homogeneous equilibrium interfaces, negligible contact-angle hysteresis, and the stated solid interfacial energies applicable to the solution.

Complete wetting. The spreading coefficient is $S={\gamma }_{SV}-{\gamma }_{SL}-{\gamma }_{LV}=40\mathrm{mN}{\mathrm{m}}^{-1}-{\gamma }_{LV}$. The onset of complete wetting occurs at $S=0$, or ${\gamma }_{LV}=40\mathrm{mN}{\mathrm{m}}^{-1}$, if the solid interfacial energies remain unchanged. The last measured value is still $42.0\mathrm{mN}{\mathrm{m}}^{-1}$, so this requires extrapolation beyond the table.

Continuing the last local slope in concentration, $(42.0-45.2)/(0.07-0.06)=-320\mathrm{mN}{\mathrm{m}}^{-1}$ per $\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$, givesContinuing a constant local Gibbs slope in $\ln c$ instead gives approximately $0.078\mathrm{mol}\mathrm{d}{\mathrm{m}}^{-3}$. Thus the data suggest a threshold near $0.08\mathrm{M}$ under continued-trend assumptions, rather than fixing one exactly. A surface-tension plateau caused by micellization, or a change in ${\gamma }_{SL}$, could prevent or shift this threshold; no such extra data are given.

Capillary rise. For a circular capillary of radius $a$, vertical surface-tension force $2\pi a{\gamma }_{LV}\cos \theta$ balances the weight $\rho g\pi a^{2}h$. Thus capillary action givesUsing $a=0.5\mathrm{mm}=5.0\times 10^{-4}\mathrm{m}$, $\rho =10^3\mathrm{kg}{\mathrm{m}}^{-3}$ and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$,This is the height above the reservoir level in the ideal static capillary approximation; meniscus-volume corrections and contact-angle hysteresis are neglected. The positive $\cos \theta$ gives a rise, rather than a depression.