A thousandth of a monolayer holds a bubble still

The surface that moves with the flow worked out the drag on a clean bubble rising slowly through a liquid. Its surface is free to move, so the liquid slides past it instead of being held at rest, and the drag is two-thirds of what a rigid sphere of the same size would feel: Hadamard and Rybczynski’s $4\pi\mu U a$ against Stokes’ $6\pi\mu U a$. Bubbles in ordinary water do not rise at the clean speed, and the essay gave the reason: surfactant, swept to the rear by the moving surface, piles up there into a stagnant cap whose gradient of surface tension holds the interface still. The drag then depends on how big the cap is, and runs from the clean value with no cap to Stokes’ with the cap over everything.

That essay named the answer — Sadhal and Johnson’s exact solution of 1983 — without computing it, and left two questions. How fast does the drag climb as the cap grows? And how much surfactant does a cap of a given size need? The second is the one that decides whether real bubbles are mobile or rigid, and the answer is that they need very little, because what the surfactant has to resist is the viscous stress of a slow flow, which is tiny.

A surface with two boundary conditions #

The calculation is Stokes flow past a sphere, with the boundary condition changing type part-way round it. Over a cap centred on the rear stagnation point, from the rear out to a half-angle $\theta_c$, the surface is held still: no slip, as on a solid. Over the rest of the surface the liquid exerts no shear on the gas, and the surface moves freely, as on a clean bubble. Everywhere the surface is impermeable. How many things a flow must be told counted how many conditions a viscous flow needs at a boundary; here the count is the same everywhere, but which condition applies switches at one circle of latitude.

The flow is written as Stokes’ uniform stream plus a series of decaying axisymmetric modes, each of which vanishes on the sphere so that impermeability holds automatically. Each mode has one unknown coefficient, and the coefficients are fitted by least squares at sixteen hundred points round the sphere: zero surface velocity at the points on the cap, zero shear at the points off it. The drag is read from the coefficient of the one mode that carries momentum to infinity, the Stokeslet.

The first figure is the drag against the cap angle. With no cap the series gives the clean bubble’s drag exactly, and with the cap over the whole sphere it gives Stokes’ exactly: both to fifteen digits. In between it rises smoothly, slowly at first — a cap of thirty degrees adds two and a half per cent — then steeply through the middle angles, and it levels off before the cap has covered the bubble. Half the climb from the clean drag to the rigid drag is done by a cap of 73 degrees, which covers 35 per cent of the surface. Nine-tenths of it is done by 112 degrees, and the last tenth is spread over the front third of the bubble.

The edge the series cannot hold #

The figure also shows something about the method worth knowing. The series truncated at ninety modes falls short of the exact drag by up to 0.7 per cent, and adding modes closes the gap only slowly: at 45, 90, 180 and 300 modes the drag at a 60° cap is short by 1.3, 0.71, 0.38 and 0.25 per cent. The error falls as the inverse of the number of modes, which is the signature of a singularity. At the cap’s edge the surface goes from still to sliding in no distance at all, and the shear on the cap side rises as the inverse square root of the distance to the edge. A sum of smooth modes cannot represent that, and each extra mode buys only a little more of it.

Knowing the rate is enough to fix it. If the error is proportional to the inverse of the number of modes, two truncations, at N and 2N, combine into an estimate with the error removed, and the extrapolated drags in the figure lie on Sadhal and Johnson’s closed form, $1 + (2\theta_c + \sin\theta_c - \sin 2\theta_c - \tfrac13\sin 3\theta_c)/4\pi$, to 0.06 per cent at every angle tried. The closed form was found by a method — a mixed boundary-value problem solved with a dual series of Legendre functions — that builds the singularity in from the start, which is why it is exact.

What the cap does to the rest of the surface #

The second figure follows the surface’s own speed from the rear stagnation point to the front. On a clean bubble the surface flows from front to rear at half the stream’s speed at the equator, the motion that makes the liquid’s job easier. A cap stops the surface over its own extent, as it must, and it also slows the free surface ahead of it: the free part has less room to accelerate before it meets the still part, and near the cap’s edge it must come to rest. With a cap of 90 degrees the free front half moves at about seven-tenths of the clean bubble’s speed at 120 degrees from the rear, and the slowing reaches all the way to the front stagnation point. That is why the drag climbs faster than the area covered. Holding the rear still drags down the motion of the front as well, and the drag counts the whole surface’s motion, not only the cap’s.

The third figure is the shear the outer flow puts on the cap — the stress the surfactant has to resist. It is zero at the rear stagnation point, by symmetry, and rises towards the cap’s edge, steeply at the end, where it carries the inverse-square-root singularity. The stress is in units of $\mu U/a$, the viscous stress of the flow, and that is the whole point of the next step: it is small. For a bubble a tenth of a millimetre across rising at eight millimetres a second in water, $\mu U/a$ is about a sixth of a pascal.

The surfactant that holds the cap #

The cap is held still by a gradient of surface tension. Where surfactant is concentrated the tension is lower, so a surface with more surfactant at the rear than near the cap’s edge pulls towards the edge — the Marangoni stress — and on a stagnant cap that pull exactly balances the shear of the flow at every point. For an insoluble surfactant dilute enough that each molecule lowers the tension by the same amount, the surface pressure $RT\Gamma$ it makes, with $\Gamma$ the number of moles per area, rises from nothing at the cap’s edge to a peak at the rear, and its gradient at every point is the shear of the previous figure.

The fourth figure integrates that balance from the cap’s edge back to the rear for three caps. The concentration is zero at the edge — the cap’s edge is defined as the place where the surfactant runs out — and rises steeply there, where the shear is singular but integrable, then more gently towards the rear. Its peak, at the rear stagnation point, is a surface pressure of 0.99, 1.94 and 2.70 times $\mu U$ for caps of 45, 90 and 135 degrees. So the surface pressure a stagnant cap needs is of the order of $\mu U$, the viscous stress times the bubble’s radius — about $10^{-5}$ newtons per metre for the small bubble above, ten thousand times smaller than the 0.07 newtons per metre of water’s surface tension.

The drag follows the load, not the coverage #

The fifth figure puts the two halves together. It plots how far the drag has climbed from the clean bubble’s to the rigid sphere’s against the bubble’s total surfactant load, averaged over its whole surface and expressed as a surface pressure in units of $\mu U$. A load of 0.2 makes a 60-degree cap and a third of the climb. A load of 0.56 makes a 90-degree cap and 70 per cent of it. A load of one makes a 120-degree cap and 94 per cent. Beyond that, a great deal more surfactant buys very little, because the front of the bubble, where the cap would have to extend, is where the shear the surfactant must hold is largest relative to what the cap gains.

The dashed line is the cap’s share of the surface for the same loads, and the drag runs ahead of it at every load. With a cap over a third of the surface the drag is half-way to rigid; with half the surface held it is three-quarters of the way. A bubble does not have to be coated to behave as a solid; it has to have its rear held.

How little a real bubble needs #

The sixth figure turns the load into something measurable. An air bubble 50 micrometres in radius rising in water at room temperature has a Reynolds number of about one, at the edge of where Stokes flow holds, and rises clean at 8.16 millimetres a second. A typical soluble surfactant packs a full monolayer at about 3.2 micromoles per square metre. Expressed as a share of that monolayer, the 90-degree cap needs a mean coverage of 4.3 ten-thousandths and slows the bubble to 6.03 millimetres a second. The 120-degree cap needs 7.1 ten-thousandths and slows it to 5.55, within two per cent of a rigid sphere’s 5.44.

That is the answer to the question the earlier essay left. A bubble a tenth of a millimetre across is effectively rigid once it carries a thousandth of a monolayer, gathered at its rear. The number of molecules in the 120-degree cap is about forty million, seven hundred-thousandths of a picomole. Water with a nanomolar impurity holds that much in a sphere a quarter of a millimetre in radius around the bubble, and the bubble sweeps through far more water than that in the first second of its rise. The bubbles of above their resonance, bubbles make water faster and the nuclei of a crevice keeps the nucleus a free bubble loses are smaller still, and since the load a cap needs per area falls as the square of the radius, on this arithmetic they are held still by even less. No laboratory water is clean enough to keep it out.

The scale also explains the size dependence the earlier essay described. The load a cap needs is proportional to $\mu U$, and for a bubble in Stokes flow $U$ grows as the square of the radius, so a larger bubble needs more surfactant per area to hold the same cap, while it also outruns the adsorption that would supply it. Small bubbles in ordinary water are rigid; large ones may keep a mobile front, and the crossover is a comparison of how fast surfactant arrives with how much the flow’s stress demands.

What the calculation was checked against #

The two ends of the curve are exact: no cap gives the clean bubble’s drag and a full cap gives Stokes’, both to fifteen digits, because at those ends the boundary condition is the same everywhere and one mode carries the whole solution. In between, the extrapolated drag matches Sadhal and Johnson’s closed form to 0.06 per cent at six cap angles, while the raw ninety-mode truncation is off by up to 0.7 per cent, which is the edge’s singularity measured. Away from the edge the boundary conditions themselves are satisfied: on a 90-degree cap the surface speed is zero to four thousandths of the stream’s, and on the free surface the shear is zero to the same. The surfactant integral uses a substitution that concentrates its points at the edge, where the stress is singular, and the total load moves by under a per cent between 180 and 300 modes. The checks refuse a cap angle outside the sphere and a negative load.

What the calculation leaves out #

Solubility and kinetics. The surfactant here is insoluble and the cap is in equilibrium. A real soluble surfactant exchanges with the liquid, adsorbing over the front and desorbing at the rear, and the cap’s size is then set by that balance and by how long the bubble has been rising — which is why bubbles slow down as they rise. The load computed here is what the cap holds; how long it takes to collect it is a separate calculation.

A dilute surfactant. The surface pressure is taken as proportional to the concentration. At a thousandth of a monolayer that is accurate; near a full monolayer the relation curves and the cap’s concentration profile changes shape.

Surface diffusion and a sharp edge. Surfactant diffuses along the surface, which smooths the cap’s edge over a length set by the ratio of diffusion to the surface’s speed. The singular stress at the edge is an artefact of the sharp edge, and a diffusing cap has none; the drag, which is an integral, hardly notices.

Inertia. At a Reynolds number of one the Stokes solution is at its edge. Larger bubbles carry wakes, and the cap’s effect on a wake — a rigid rear separates, a mobile one does not — is a different and larger effect than this.

The boundary condition is a field #

The earlier essay ended on the observation that the honest boundary condition at a bubble’s surface is neither slip nor no slip: the interface carries a field of its own, and where that field is strong enough the fluid sees a wall. This calculation puts a number on “strong enough”. The field has to supply a surface pressure of order $\mu U$, which is the viscous stress of the flow times the size of the body, and in slow flows of water that is so small that almost any contamination meets it. A viscosity made of particles found that a suspension’s viscosity depends on whether its particles’ surfaces move, and a swimmer that cannot go backwards that a surface which moves on purpose can propel a body through a world with no inertia; both questions turn on the same boundary condition, and in real water both are answered, for small enough bodies, by the trace chemistry of the surface rather than by the fluid.

The same smallness of viscous stress against surface tension turns up wherever slow flows meet interfaces. The threshold the walls decide found the Marangoni effect driving a flow on its own; here it stops one. In both, a gradient of surface tension that would be invisible in a static measurement — a part in ten thousand of the tension — is a match for everything the viscosity can do.

Who worked it out #

Hadamard and Rybczynski found the clean bubble’s drag independently in 1911. Frumkin and Levich explained in 1947 why bubbles in real water rise as solids, by the surfactant swept to the rear, and Savic gave the stagnant-cap picture in 1953. Davis and Acrivos solved the cap problem numerically in 1966, and Sadhal and Johnson found its exact solution in 1983, by a dual series of Legendre functions that holds the edge’s singularity. The relation between cap angle, surfactant and bulk concentration was taken up by Cuenot, Magnaudet and Spennato in 1997 and many since, as the clean-versus-contaminated question became central to bubble columns and flotation.

Still open: the cap a soluble surfactant builds while the bubble rises #

Every cap here is in equilibrium with a fixed amount of insoluble surfactant. A bubble released into water with a soluble surfactant starts clean, collects surfactant over its front by adsorption from the liquid, sweeps it to the rear and grows a cap as it rises, and its speed falls as the cap grows. The next calculation couples the cap’s load to adsorption from a dilute bulk concentration through a diffusion layer over the bubble’s front, follows the cap’s growth and the bubble’s speed along its rise, and asks over what distance a bubble of a given size reaches its contaminated speed — and whether, for small bubbles in ordinary tap water, that distance is shorter than the bubble itself.