Angle of contact

Drop a bead of water on a freshly cleaned glass slide and it spreads into a thin, wide film, hugging the surface as if it wants to cover as much glass as possible. Drop a bead of mercury on the very same slide and it does the opposite — it draws itself into a tight, shiny sphere that barely touches the glass at all, and if you tilt the slide it goes rolling off. Same solid, same size of drop, two completely opposite behaviours. The single number that captures this difference, precisely enough to predict it in advance, is the angle of contact — the angle at which a liquid's surface meets a solid wall. This is not a cosmetic detail. It is the angle of contact that decides whether water climbs up inside a thin glass capillary tube (as in the xylem of a tree, a wick, blotting paper, the soil around a plant's roots) or whether mercury gets pushed down inside the very same tube, refusing to rise. It decides why a raincoat's threads keep water out while a paper towel pulls it in, and why a drop of detergent added to greasy water suddenly lets the water spread and clean, instead of beading up uselessly. Underneath every one of these moments is the same tug-of-war: does a liquid molecule at the edge of a drop feel a stronger pull from its own kind (cohesion) or from the solid it is touching (adhesion)? Get that one contest right, and the angle of contact — and everything that follows from it — falls into place. This article defines the angle of contact precisely, connects it to cohesive and adhesive forces, reads off the resulting meniscus shape, and shows how this one angle plugs into the capillary rise formula used numerically below.

Key takeaways

The angle of contact $\theta$ is the angle, measured inside the liquid, between the tangent to the liquid surface and the tangent to the solid surface at the point where solid, liquid and gas all meet.
$\theta$ is decided by a tug-of-war between cohesive force $F_c$ (liquid–liquid) and adhesive force $F_a$ (liquid–solid): adhesion winning gives an acute angle (wetting), cohesion winning gives an obtuse angle (non-wetting).
Water on clean glass wets it, $\theta \approx 0°$ , with a concave meniscus; mercury on glass does not wet it, $\theta \approx 138°$ , with a convex meniscus.
The angle of contact is not a fixed property of a liquid alone — it depends on the solid surface, the surrounding gas, surface cleanliness, and temperature.
The vertical pull of surface tension at the wall is $S\cos\theta$ per unit length of contact line; this single projection is what drives capillary rise or depression.
The sign of $\cos\theta$ alone tells you the outcome: $\theta < 90°$ gives rise, $\theta = 90°$ gives no effect, $\theta > 90°$ gives depression, exactly as seen in $h = \dfrac{2S\cos\theta}{\rho g r}$ .
Key idea: the angle of contact θ, set by whether adhesion or cohesion wins at the solid–liquid edge, fixes both the meniscus shape (concave for acute θ, convex for obtuse θ) and the sign of capillary rise through the vertical surface-tension component S cos θ in h = 2S cos θ / (ρ g r).

The core insight

The angle of contact is the tell-tale angle, measured inside the liquid at a solid wall, whose acute or obtuse value — set by whether adhesion or cohesion dominates — fixes the meniscus shape and, through the vertical component $S\cos\theta$ of surface tension, the sign and size of capillary rise.

Where it fits in the syllabus

This topic sits inside the CBSE Class 11 Physics unit Mechanical Properties of Fluids, in the surface tension section, right after surface energy and the surface-tension–surface-energy relation and just before drops, bubbles and capillary rise are treated numerically. You are expected to define the angle of contact, explain qualitatively why it depends on the relative strengths of cohesive and adhesive forces, know the standard contrasting examples — water–glass (wetting, acute) and mercury–glass (non-wetting, obtuse) — and connect the angle of contact to the meniscus shape and to the capillary rise formula $h = \dfrac{2S\cos\theta}{\rho g r}$ . A full force-resolution derivation of why a particular θ results from particular molecular forces is beyond Class 11 scope; what is examined is the qualitative reasoning and the correct use of $\theta$ inside the capillary rise formula.

The physical picture

Look closely at the edge where a liquid surface meets a solid wall — say, water climbing the inside of a glass tube. The liquid surface does not stay flat right up to the glass; it curves, forming a meniscus, and the curve meets the wall at a definite angle. That angle, $\theta$ , measured inside the liquid, between the tangent to the liquid surface and the tangent to the solid surface at the three-phase contact line (solid, liquid, gas all meeting at one line), is the angle of contact.

\theta = \angle(\text{liquid surface tangent},\ \text{solid surface tangent})\ \text{measured inside the liquid}

where: the tangent to the liquid surface is drawn at the exact point of contact, curving away from the wall into the bulk liquid, and the tangent to the solid surface runs along the wall itself. The physical reading is that $\theta$ is not something you choose or that a container "has" on its own — it is the outcome of a competition at that contact line. A liquid molecule sitting right at the edge feels two pulls: the cohesive force $F_c$ , the attraction toward other liquid molecules, and the adhesive force $F_a$ , the attraction toward the solid's molecules. If $F_a > F_c$ , the molecule is pulled toward the solid more than it is pulled back into the liquid, the liquid spreads and climbs the wall, and the result is an acute angle — the liquid wets the solid. If $F_c > F_a$ , the molecule is pulled back into the liquid bulk more strongly than toward the solid, the liquid pulls away from the wall, and the result is an obtuse angle — the liquid does not wet the solid. When the two forces are roughly matched, $\theta \approx 90°$ and the surface meets the wall almost flat, with negligible curving either way.

Figure

Two glass tubes side by side, one showing water with an acute contact angle and a concave meniscus curving upward at the walls, the other showing mercury with an obtuse contact angle and a convex meniscus curving downward at the walls

Because the meniscus shape follows directly from $\theta$ , you can read off wetting behaviour just by looking at the curve: acute $\theta$ (water–glass, $\theta \approx 0°$ ) always goes with a concave meniscus, while obtuse $\theta$ (mercury–glass, $\theta \approx 138°$ ) always goes with a convex meniscus. The angle of contact for a given liquid–solid pair is not fixed for all time — it also depends on the surrounding gas, the cleanliness of the liquid (a trace of detergent or oil sharply lowers water's contact angle, which is exactly why detergent lets water "wet" and lift grease off a greasy plate that plain water beads off of), and on temperature (contact angle generally falls as temperature rises). The reason $\theta$ matters beyond a qualitative picture is that it enters directly, through a cosine, into the capillary rise formula:

h = \frac{2S\cos\theta}{\rho g r}

where: $S$ is the surface tension of the liquid (N/m), $\theta$ is the angle of contact, $\rho$ is the liquid's density (kg/m³), $g = 9.8\ \text{m/s}^2$ , and $r$ is the inner radius of the capillary tube (m). The physical reading: when $\theta < 90°$ , $\cos\theta > 0$ , so $h > 0$ and the liquid rises above the level in a wide open vessel — this is capillary rise. When $\theta > 90°$ , $\cos\theta < 0$ , so $h < 0$ and the liquid is depressed below the outside level, exactly as mercury is pushed down inside a glass capillary. When $\theta = 90°$ , $\cos\theta = 0$ and $h = 0$ : the level inside the tube sits at the same height as outside, with no rise or depression at all.

Derivation AWhy the vertical pull of surface tension gives capillary rise or depression, and how θ enters

The goal here is to see, by balancing forces on the column of liquid that has been drawn up (or pushed down) inside a thin tube, exactly how the angle of contact and the surface tension together decide the height $h$ that appears in $h = \dfrac{2S\cos\theta}{\rho g r}$ . The key physical fact that makes the derivation work is that surface tension acts along the liquid surface, tangent to it at the contact line, tilted at the angle of contact $\theta$ to the vertical wall of the tube — so only the vertical component of that tension force actually does the job of lifting (or holding down) the liquid column, and it is precisely this projection that brings $\cos\theta$ into the formula.

Step 1. Picture a thin capillary tube of inner radius $r$ dipped vertically into a wide dish of the liquid. At the contact line where the meniscus meets the glass, the liquid surface pulls on the wall with a surface tension force per unit length equal to $S$ , directed along the liquid surface, tangent to the meniscus at the wall. Because that tangent makes angle $\theta$ with the vertical tube wall, this pull has a vertical component of magnitude $S\cos\theta$ per unit length of the contact line, and a horizontal component $S\sin\theta$ that is balanced all the way around the tube's circular rim, so it contributes no net lift.

Step 2. Find the total vertical pulling force. The contact line runs around the inner circumference of the tube, a length $2\pi r$ . So the total upward (or downward, if $\theta > 90°$ ) pulling force supplied by surface tension is the force per unit length times this length: $F_{\text{vertical}} = S\cos\theta \times 2\pi r = 2\pi r S\cos\theta$ .

Step 3. This vertical pull must support the weight of the extra liquid column of height $h$ drawn up above (or depressed below) the outside free-surface level, inside the tube of cross-sectional area $\pi r^2$ . That column has volume $\pi r^2 h$ , mass $\rho \pi r^2 h$ , and weight $\rho \pi r^2 h g$ . At equilibrium, the upward pull from surface tension exactly balances this weight: $2\pi r S\cos\theta = \rho \pi r^2 h g$ .

Step 4. Solve for $h$ . Cancel one factor of $\pi r$ from both sides, leaving $2S\cos\theta = \rho r h g$ , and divide through by $\rho g r$ to isolate $h$ : $h = \dfrac{2S\cos\theta}{\rho g r}$ , exactly the capillary rise formula. The derivation makes the role of $\theta$ transparent: it converts the full surface-tension force $S$ (always positive, tangent to the meniscus) into the vertical component $S\cos\theta$ that alone does the lifting or holding-down. A narrower tube (smaller $r$ ) gives a larger $h$ because the same tension force, acting around a shorter circumference, lifts a proportionally smaller cross-section of liquid — why capillary effects are dramatic in fine tubes and negligible in wide ones.

Worked examples

Board level - reading the sign of cos θ for water and mercury

Water in a clean glass capillary tube has an angle of contact $\theta \approx 0°$ (complete wetting), while mercury in a similar tube has $\theta = 138°$ . For both liquids placed in identical thin glass capillary tubes, use the sign of $\cos\theta$ to predict qualitatively what happens to each liquid's level compared to the level in a wide open vessel, and state the resulting meniscus shape in each case.

Step 1. For water, $\cos 0° = 1$ , which is greater than zero, so in $h = \dfrac{2S\cos\theta}{\rho g r}$ the value of $h$ comes out positive: water rises above the level in the open vessel. Because $\theta$ is acute (wetting), the meniscus is concave.

Step 2. For mercury, $\cos 138° = -\cos 42° \approx -0.743$ , which is less than zero, so $h$ comes out negative: mercury is depressed below the outside level. Because $\theta$ is obtuse (non-wetting), the meniscus is convex.

Step 3. Putting the two together: the sign of $\cos\theta$ alone — without computing any numerical value of $h$ — tells you whether a capillary tube shows rise (wetting, concave meniscus, $\theta < 90°$ ) or depression (non-wetting, convex meniscus, $\theta > 90°$ ). Water rises with a concave meniscus; mercury is depressed with a convex meniscus.

Competitive level - numerical capillary rise and depression from θ

A glass capillary tube of inner radius $r = 0.20\ \text{mm} = 2.0\times10^{-4}\ \text{m}$ is dipped first into water ( $S = 0.072\ \text{N/m}$ , $\rho = 1000\ \text{kg/m}^3$ , $\theta \approx 0°$ ) and then into mercury ( $S = 0.465\ \text{N/m}$ , $\rho = 13{,}600\ \text{kg/m}^3$ , $\theta = 138°$ ). Take $g = 9.8\ \text{m/s}^2$ . Find the capillary rise for water and the capillary depression for mercury.

Step 1. For water, substitute directly into $h = \dfrac{2S\cos\theta}{\rho g r}$ with $\cos 0° = 1$ : $h = \dfrac{2 \times 0.072 \times 1}{1000 \times 9.8 \times 2.0\times10^{-4}} = \dfrac{0.144}{1.96} \approx 7.35\times10^{-2}\ \text{m}$ .

Step 2. So $h \approx 7.35\ \text{cm}$ of rise for water — this is the kind of order-of-magnitude capillary rise seen in a fine glass tube, and matches the qualitative sign found in the board-level example.

Step 3. For mercury, use $\cos 138° \approx -0.743$ : $h = \dfrac{2 \times 0.465 \times (-0.743)}{13{,}600 \times 9.8 \times 2.0\times10^{-4}} = \dfrac{-0.691}{26.66} \approx -2.59\times10^{-2}\ \text{m}$ .

Step 4. The negative sign means depression, not rise: mercury's level inside the tube sits about $2.59\ \text{cm}$ below the free surface outside — noticeably smaller in magnitude than water's rise, mainly because mercury's much larger density $\rho$ sits in the denominator and pulls $|h|$ down even though mercury's surface tension $S$ is over six times water's. Water rises about 7.35 cm; mercury is depressed about 2.59 cm, both driven by the same formula through the different signs and sizes of cos θ, S and ρ. Timing note: on a competitive exam, don't re-derive $h = \dfrac{2S\cos\theta}{\rho g r}$ from scratch — have it memorised, substitute carefully keeping every unit in SI (radius in metres, not mm), and double-check the sign of $\cos\theta$ before reporting rise vs. depression; this whole calculation should take under ninety seconds per liquid.

Common traps

Measuring the angle of contact on the wrong side — it must always be measured inside the liquid, never the angle on the gas side.
Assuming all liquids wet all solids — wetting/non-wetting is a property of the specific liquid–solid–gas combination (water wets clean glass but not wax or grease; mercury does not wet glass at all).
Mixing up which meniscus goes with which angle: concave goes with acute/wetting (water–glass), convex goes with obtuse/non-wetting (mercury–glass) — reversing this is a common exam slip.
Thinking the angle of contact is a fixed constant for a liquid on its own — it depends on the solid surface, the surrounding gas, surface cleanliness/impurities, and temperature.
Forgetting to convert the tube radius to metres before substituting into $h = \dfrac{2S\cos\theta}{\rho g r}$ , which throws the answer off by a factor of a thousand.
Reporting a negative $h$ as "no rise" instead of correctly stating it as depression — the sign is informative, not an error to drop.

Common trapAdhesion versus cohesion decides the angle, not the density or weight of the liquid

Do not reach for density, weight or "heaviness" to explain why mercury does not climb a glass tube while water does — that instinct is wrong and will cost marks. Density enters the capillary rise formula $h = \dfrac{2S\cos\theta}{\rho g r}$ only as the thing being lifted, sitting in the denominator; it plays no role in deciding whether $\theta$ is acute or obtuse in the first place. The angle of contact itself is set entirely by the tug-of-war between cohesive force (liquid–liquid attraction) and adhesive force (liquid–solid attraction) at the contact line. Mercury atoms are far more strongly attracted to each other than to glass, so cohesion wins, the surface pulls away from the wall, and $\theta$ is obtuse — this is a molecular-force fact, true even in the (hypothetical) absence of gravity. Once $\theta$ is obtuse, the formula then uses mercury's large density to compute how much the level is depressed, but the depression's existence and sign come from $\cos\theta$ being negative, not from mercury being heavy.

You should be able to…

Define the angle of contact precisely, stating exactly where and on which side it is measured.
Explain, in terms of cohesive and adhesive forces, why some liquids wet a solid (acute $\theta$ ) and others don't (obtuse $\theta$ ).
State and justify the meniscus shape (concave/convex/flat) that goes with a given angle of contact.
Derive, from a force balance at the contact line, how the vertical component $S\cos\theta$ of surface tension leads to $h = \dfrac{2S\cos\theta}{\rho g r}$ .
Use the sign of $\cos\theta$ to predict capillary rise vs. depression, and compute $h$ numerically when $S$ , $\rho$ and $r$ are given.
Give water–glass and mercury–glass as standard contrasting examples, with approximate angle values and the everyday reasons (like detergent action) that follow from them.

Ask the tutor: curious how detergent molecules physically lower water's angle of contact enough to lift grease off a plate — ask the AI tutor to connect that soap-and-grease example to the cohesion-versus-adhesion picture here.

Why does water climb up a thin straw but not a thick pipe by any noticeable amount?

The upward pull from surface tension acts only along the contact line, a distance $2\pi r$ around the rim, while the weight it has to lift grows with the cross-sectional area $\pi r^2$ of the raised column. In the formula $h = \dfrac{2S\cos\theta}{\rho g r}$ , that ratio of "pull around a line" to "weight over an area" leaves $h$ inversely proportional to $r$ : halve the radius and the rise doubles. A thin straw has a small enough $r$ that the rise is many centimetres — easily visible. A thick pipe has such a large $r$ that the same formula predicts a rise of a fraction of a millimetre, far too small to notice against the flat surface. The physics — adhesion pulling water up the wall — is identical in both; only the geometry changes the visible outcome.

Does the angle of contact change if I use a plastic tube instead of glass, or add soap to the water?

Yes to both, and this is exactly why the angle of contact is described as depending on the solid surface and on liquid purity, not as a fixed number for "water." Water wets clean glass almost perfectly ( $\theta \approx 0°$ ) because glass has a strong adhesive attraction for water molecules, but on a waxy or greasy plastic surface the adhesive pull is much weaker, so water's contact angle on plastic can be noticeably larger — sometimes large enough that water beads up rather than spreading. Adding detergent to water lowers its effective surface tension and its contact angle on most surfaces, which is precisely why soapy water spreads into grease and dirt that plain water would simply bead away from — the everyday "wetting power" of soap is this same angle-of-contact physics at work.

If the angle of contact is exactly 90 degrees, does that mean the liquid and solid don't interact at all?

No — it means the two competing forces, cohesion within the liquid and adhesion toward the solid, happen to be closely matched, not that either force is absent. At $\theta = 90°$ the liquid surface meets the wall without curving noticeably up or down, so the meniscus looks essentially flat at the edge, and $\cos 90° = 0$ makes the capillary rise formula give $h = 0$ : the level inside a thin tube of that material sits at (very nearly) the same height as the free surface outside. Both forces are still fully present and still pulling; they simply produce no net vertical curvature at the wall when they are balanced this way.

Is angle of contact the same thing as the shape of a floating drop, like the water-strider effect?

Related but not identical. The angle of contact is specifically about where a liquid meets a solid surface — a container wall, a tube, a leaf. The rounded shape of a free water drop, or a water-strider's legs dimpling the surface without sinking, is a separate consequence of surface tension acting on a free liquid surface, not by itself a solid contact angle. That said, when the strider's leg does touch the water and resists wetting its waxy hairs, that resistance is an angle-of-contact effect — a very obtuse, non-wetting angle, similar in kind to mercury on glass.