The number that cannot break a drop

A drop of one liquid in another, sheared, is pulled out by the viscous stress and pulled back by its own skin. The ratio of the two is the capillary number,

$$\mathrm{Ca} = \frac{\mu_c \dot\gamma a}{\sigma},$$

and it decides how fine an emulsion gets, how thick a coating is, how small the droplets in a spray are, and whether a polymer blend has any strength. It is the group behind the two-thirds law for a withdrawn plate and behind the size a pendant drop is allowed, and in both of those places it does what a dimensionless group is supposed to do.

Here it does half of it. The deformation collapses onto Ca almost perfectly. The breakup does not collapse onto it at all, and the gap between those two facts is a piece of physics rather than a missing correction.

Taylor’s answer, which is exact and nearly indifferent #

The small-deformation theory is Taylor’s, from 1932 and 1934, and its result is one line:

$$D = \frac{L - B}{L + B} = \mathrm{Ca}\,\frac{19\lambda + 16}{16\lambda + 16}, \qquad \lambda = \mu_d/\mu_c.$$

Linear in Ca — the drop deforms in proportion to the stress applied, which is the whole content of a first-order theory — and then the part worth staring at. The prefactor runs from exactly 1 at $\lambda = 0$ to exactly 19/16 as $\lambda\to\infty$. That is an eighteen per cent spread over the entire physically possible range of the viscosity ratio: from a gas bubble to a solid particle, and back.

Read at face value, the theory says the viscosity ratio does not matter.

Two other exact numbers come out of the same calculation and both are used below. The drop’s relaxation time after the flow is switched off,

$$\tau = \frac{(2\lambda + 3)(19\lambda + 16)}{40(\lambda + 1)}\cdot\frac{\mu_c a}{\sigma},$$

and the fact that the deformation is a shape, so it can be written as a traceless symmetric tensor and given an equation of motion.

A model with one line in it #

Nothing above says what happens when Ca is not small, and the measurements say something abrupt happens. To get there a model is needed that reduces to Taylor’s at small Ca and is defined at large Ca, and the smallest one is a linear relaxation law for the shape:

$$\frac{\mathrm{d}\mathsf A}{\mathrm{d}t} = \mathsf\Omega\!\cdot\!\mathsf A - \mathsf A\!\cdot\!\mathsf\Omega + c_1 \mathsf E - \frac{\mathsf A}{\tau}.$$

Three things act on the drop’s shape: the rate of strain drives it, the vorticity rotates it, and interfacial tension relaxes it. That is stated rather than derived, and its two coefficients are then fixed by requiring it to reproduce Taylor’s answer where Taylor’s answer holds. The relaxation time is Taylor’s τ, and matching the small-Ca deformation forces

$$c_1 = \frac{5}{2\lambda + 3}$$

exactly — which is the coefficient Maffettone and Minale write down for the same tensor, arrived at here from Taylor’s two formulas and nothing else. The calibration is checked to machine precision at seven viscosity ratios, and it is worth checking: both of Taylor’s formulas are ratios of polynomials that look alike, and a transposed coefficient would produce a model that behaved plausibly and was not calibrated to anything.

What comes out, and it saturates #

In steady simple shear the equation has a closed solution. Writing $s$ for the strain rate and $w$ for the vorticity,

$$b = \frac{\tau c_1 s}{1 + w^2\tau^2},\qquad a = -w\tau b,\qquad |\mathsf A| = \frac{c_1\,\dot\gamma\tau}{2\sqrt{1 + \dot\gamma^2\tau^2}},$$

and the last of those saturates. However hard the drop is sheared, the deformation cannot exceed $c_1/2 = 5/2(2\lambda+3)$.

At $\lambda = 1$ the ceiling is 0.5; at $\lambda = 4$ it is 0.227; at $\lambda = 10$, 0.109. And a drop that cannot reach the deformation at which drops break cannot break.

That is a qualitative prediction from a model with no fitted breakup criterion in it: simple shear has a ceiling and the ceiling falls with the viscosity ratio. It is the thing Taylor’s first-order result cannot express, because a straight line has no ceiling.

Why the rotation is the whole of it #

The mechanism is visible in the orientation.

At vanishing shear rate the drop’s major axis sits at forty-five degrees, which is the direction of maximum stretching for a simple shear. As the rate rises the axis turns into the flow direction — 45.0° at Ca = 10⁻⁴ and 0.033° at Ca = 400 — because the vorticity carries the drop round faster than the strain can stretch it. Aligned with the flow there is nothing left pulling the ends apart.

And the reason simple shear is so bad at this is exact. Its velocity gradient has one non-zero entry, which splits evenly:

$$\nabla\mathbf u = \dot\gamma\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix} = \underbrace{\frac{\dot\gamma}{2}\begin{pmatrix}0&1\\1&0\end{pmatrix}}_{\text{strain}} + \underbrace{\frac{\dot\gamma}{2}\begin{pmatrix}0&1\\-1&0\end{pmatrix}}_{\text{rotation}}.$$

The stretching rate is $\dot\gamma/2$ and the rotation rate is $\dot\gamma/2$: exactly equal, which is the same identity that makes $Q$ vanish everywhere in a parallel shear layer. So the capillary number, built from $\dot\gamma$, is twice the stretching the drop actually feels, and the other half of it is spent turning the drop away from the stretching.

The axis the number has no room for #

If the rotation is what saturates the deformation, then a flow with less rotation should not saturate — and it does not.

At $\alpha = 0$ the flow is a simple shear and sits exactly at the point where strain and rotation balance. To the left it is rotation-dominated and the drop barely deforms at all; to the right the strain wins, and at $\alpha = 1$ — pure planar extension, with no vorticity — the deformation is linear in Ca for ever. At Ca = 1000 the extensional deformation is 4,375 times the shear one.

So a drop in an extensional flow breaks at any viscosity ratio, and a drop in simple shear may not break at all, at the same capillary number. The number has no term for the flow type, and the flow type is the difference between an emulsion and a suspension of unbreakable blobs.

This is the same axis the Q-criterion is built on and the same one the frames essay shows is not objective — the split of a velocity gradient into strain and rotation — arriving here as a piece of process engineering.

What the measurements say, and what the theory says #

Put the two side by side.

Taylor’s own criterion — break when the first-order deformation reaches a half — gives a critical capillary number of $0.5\cdot 16(\lambda+1)/(19\lambda+16)$, which runs from 0.500 at $\lambda = 0$ to 0.421 as $\lambda\to\infty$. A span of 1.14. On a logarithmic axis that is a horizontal line.

The measurements span 18.1 over the same range and then go to infinity. At $\lambda = 10^{-3}$ the critical number is 8.7; it falls to a minimum of about 0.48 near $\lambda = 1$; it rises again to 2.2 at $\lambda = 3$; and above $\lambda = 4.08$ there is no critical number, because there is no breakup.

The theory is not a rough version of that curve. It is the wrong shape, and it is wrong in the one place the number is used. That is worth saying plainly because Taylor’s result is correct: it is an exact first-order statement about the deformation, checked here to be linear in Ca at every viscosity ratio, and it says nothing about breakup because a linear theory has no threshold in it.

What the number is actually used for #

It is worth being concrete about where this bites, because the failure is not academic.

Emulsification. A mixer is a device for making the capillary number large, and the standard design rule is to raise the shear rate until Ca exceeds the critical value. For an oil-in-water emulsion with a viscosity ratio near one that rule works, and the drop size that comes out of it is $a \sim \sigma\,\mathrm{Ca}_{\rm crit}/\mu_c\dot\gamma$ — smaller drops for faster mixing, which is the behaviour everybody expects. For a silicone oil in water, with a ratio of 30, no amount of shearing produces drops at all, and the process engineer’s response is to introduce an extensional element: a contraction, an orifice, a static mixer with a change of section. That is a change of flow type rather than of flow strength, and the capillary number cannot express it.

Polymer blending. The morphology of a two-phase blend — droplets, or co-continuous strands — is set by the same competition, at viscosity ratios that are routinely between 0.1 and 10 because both phases are melts. Twin-screw extruders are built with kneading blocks specifically because the shear zones between them do not break the dispersed phase and the elongational zones do.

And coating, where the capillary number does exactly what it is supposed to. In the withdrawn plate there is no drop and no rotation, the film thickness goes as $\mathrm{Ca}^{2/3}$, and the collapse is clean over four decades. The difference is not that one problem is harder; it is that one of them has an object with an orientation in it and the other does not.

The two ends of the measured curve #

The Grace curve has two branches and they fail the theory in opposite directions, which is worth separating because they are different physics.

At high viscosity ratio the drop is stiffer than what surrounds it, it rotates as a nearly rigid body, and the strain has less and less purchase. That is the branch this essay’s model explains: the ceiling falls as $1/(2\lambda+3)$ and crosses the breakup deformation at a finite λ.

At low viscosity ratio the drop is far less viscous than the matrix and deforms into a long thin thread rather than a fat ellipse. A thread does not break by being stretched; it breaks by the Rayleigh–Plateau instability, which needs the stretching to stop — a thread held in a steady extension is stabilised by it and breaks only when the flow is switched off or when it leaves the strong region. The critical capillary number rises at that end because the mechanism has changed, not because the drop is harder to deform.

So a single curve labelled “critical capillary number” is two mechanisms glued together at a minimum near λ = 1, and the shape model has no representation of the second one. That is the honest reading of the figure and it is not visible in it.

Where the boundary comes from #

The shape model does have one, and it needs exactly one number that measurement supplies.

The ceiling is $5/2(2\lambda+3)$, and it falls through a critical deformation $D^*$ at

$$\lambda^* = \frac{1}{2}\left(\frac{5}{2D^*} - 3\right).$$

Setting $D^* = 0.236$ puts the boundary at $\lambda^* = 3.797$, against the measured 4.08. That is one fitted number for one number predicted, so it is not a prediction of where the boundary is. What is predicted, with nothing fitted, is that there is a boundary in simple shear and is not one in extension — and that it comes from the rotation rather than from anything about surface tension or about the drop.

The low-viscosity-ratio end of the measured curve is not explained here at all. A very low-viscosity drop breaks by a different mechanism — it deforms into a long thread and the thread breaks by a capillary instability, which is the mechanism behind a jet’s break-up length rather than anything in this model. The shape model has one degree of freedom and cannot represent a thread.

The interface that is not free #

Everything above treats the drop’s surface as a boundary with one property: a tension $\sigma$, the same everywhere on it. Almost no drop anybody makes on purpose has such a surface, because an emulsion that is meant to survive being poured has a surfactant in it, and a surfactant does something to the interface that lowering $\sigma$ does not describe.

The surface of a deforming drop is itself in motion — the outer fluid drags it, and it circulates. That circulation sweeps adsorbed surfactant towards the rear, where it accumulates, so the concentration is no longer uniform and neither is the tension. A gradient of tension is a tangential stress, and it points forward, against the motion that produced it. The interface therefore resists being dragged, and in the limit of a strong enough gradient it stops moving altogether: the surface goes rigid, not because the drop’s contents changed but because the skin acquired a way of pushing back.

The cleanest evidence is a measurement that has nothing to do with breakup. The drag on a clean fluid sphere is Hadamard and Rybczynski’s,

$$F = 2\pi\mu_c a U\,\frac{2 + 3\lambda}{1 + \lambda},$$

which for a bubble at $\lambda \to 0$ is $4\pi\mu_c a U$ — two thirds of Stokes’ law, so a clean bubble should rise half again as fast as a rigid sphere of the same size. Small bubbles in ordinary water rise at the Stokes rate. Trace contamination, at concentrations far below anything that measurably alters the surface tension itself, is enough to immobilise the interface, and the bubble falls back to behaving like a solid particle — which is the whole subject of another essay in this collection, and the same mechanism read as a drag rather than as a deformation.

For this essay the consequence is that $\lambda$ as measured in a viscometer is not the $\lambda$ the drop is behaving at. A contaminated low-viscosity drop reads as a much stiffer one, which moves it to the right along every curve above — towards the lower ceiling and, if it goes far enough, past the boundary into the region where simple shear cannot break it at all.

And surfactants add a breakup mechanism the model has no room for. A drop with surfactant swept to its ends develops a locally very low tension there, the ends become pointed, and a fine thread of droplets is emitted from the tip while the parent remains intact — tip streaming, at capillary numbers below the critical one, producing daughters one or two orders of magnitude smaller than anything the ellipsoid picture predicts. Taylor saw it in the 1934 paper and reported it as an oddity. It is now one of the routine ways of making a fine emulsion, and it is a mechanism whose existence depends on the surface being unequal to itself.

What the model does not contain #

Four limits, and the first is the reason the word “model” appears on every figure.

The shape equation is stated, not derived. Its two coefficients are calibrated to Taylor’s exact result and its structure — drive, rotate, relax — is an assumption. That the calibration lands on $5/(2\lambda+3)$, which is the published coefficient, is evidence that the structure is the right one; it is not a derivation.

The drop stays an ellipsoid. A real drop near breakup develops pointed ends, then a neck, then two daughters and a string of satellites. None of that is in a second-order tensor.

Inertia is absent everywhere. These are creeping flows, which is right for the emulsions and blends the number is used for and wrong for a spray.

And the flow is steady. Grace’s measurements are steady-shear measurements, and a drop broken by a transient — spun up quickly, or passed through a contraction — breaks at capillary numbers well below the steady critical one, because it has not had time to rotate into alignment. That is the same distinction between an instantaneous and a settled answer that the startup of a viscoelastic shear turns on.

Where the work came from #

G. I. Taylor did the drop in 1932 and 1934, in the same run of work that produced most of what this collection’s creeping-flow essays rest on. The four-roll mill in the 1934 paper exists precisely to make a flow whose type can be varied, which is the axis this essay is about — he built an apparatus for the residual before anybody had a name for it.

Grace’s measurements are from 1971 and are the ones everybody plots; de Bruijn’s correlation to them, with its pole at 4.08, is from 1989. Bentley and Leal mapped the flow-type axis experimentally in the 1980s with a computer-controlled four-roll mill, and Maffettone and Minale wrote the ellipsoid model in 1998.

The order matters. The exact first-order theory came first and is still exactly right. The measurements that showed it could not predict breakup came forty years later, and the model that explains why came sixty years after that.

What this leaves #

Two numbers survive the collapse: the viscosity ratio, which the deformation hardly sees and the breakup depends on entirely, and the flow type, which the capillary number has no room for at all.

The next essay takes a group with a time in it rather than a rate, and finds that the residual there is the phase — which a peak force cannot measure and a whole record can: whether an oscillating flow has time to make a wake.