A soft wall is read as slip only at second order

A viscous nanometre reads as slip until the gap is ten of it asked what a drainage measurement can tell apart. Squeeze liquid out of the gap between a sphere and a plane and the force is Taylor’s, $6\pi\eta R^2 V/D$, if the liquid sticks to both surfaces. Measured forces are often smaller, and the shortfall is read as slip through Vinogradova’s formula — a slip length of a nanometre or so. That essay showed that a thin layer of liquid at the wall with a different viscosity gives the same shortfall to first order, and that the two separate only at second order, at gaps of ten or so of the layer’s thickness.

It ended by naming a third rival that resembles both, and the one an experimenter controls least: elasticity. The surfaces in a surface-force apparatus are mica sheets glued to glass lenses, or polymer films, or soft colloids, and the drainage pressure presses on them. A surface that flattens widens the gap exactly where the pressure is highest, which lowers the force — as slip does. It asked whether the elastic signature can be told from the others, or whether a surface-force apparatus has been measuring the sum of both and calling it slip.

It can be told apart, and the reason is not the one expected. A soft wall does not imitate slip at first order at all. It imitates a spring.

The measurement and the wall #

A dynamic surface-force apparatus does not push the sphere in at a constant speed. It oscillates it, with an amplitude far smaller than the gap, and reads the force’s two parts: the part out of phase with the displacement — in phase with the velocity — is the damping, and over a rigid wall it is Taylor’s, $6\pi\eta R^2\omega/D$; the part in phase with the displacement is a stiffness, which a liquid between rigid walls does not have. Slip lengths are read from the damping.

The drainage pressure in the gap is in phase with the velocity. If the walls are elastic they deflect under it, by Boussinesq’s solution for an elastic half-space, and the deflection changes the gap the liquid drains through. Linearised in the small oscillation, the Reynolds equation and Boussinesq’s integral together become one linear problem for the complex pressure amplitude. Scaling the radius on $\ell = \sqrt{2RD}$, the width of the region that carries the pressure, leaves a single number,

$$\Lambda = \frac{48\,\eta\,\omega\,\ell^3}{\pi E^* D^3},$$

the walls’ deflection under the drainage pressure in units of the gap, with $E^*$ the combined plane-strain modulus. $\Lambda = 0$ is the rigid wall. For water against a 1 GPa coating, a 3 mm sphere at 100 Hz, Λ is 4.5 at a ten-nanometre gap; for glass it would be seventy times smaller.

A spring before a slip #

As Λ grows the damping falls and an in-phase force appears. At Λ = 10 the damping is 0.551 of Taylor’s and the in-phase part is 0.441. At larger Λ both fall, as the wall absorbs the motion the liquid would otherwise have to accommodate.

The logarithmic axes show the result the essay is about. At small Λ the in-phase force grows in proportion to Λ — a fitted slope of 1.0000 — and the damping’s deficit as its square, 2.0000. The reason is the phase. The drainage pressure is in phase with the velocity, so the wall’s deflection is too, and a deflection a quarter-cycle behind the displacement changes the gap a quarter-cycle late. Its first effect on the force is therefore a quarter-cycle out of the damping’s phase: it is a spring. Only the deflection’s effect on the deflection — the pressure the deflected gap produces, deflecting the wall again — comes back into the damping’s phase, and that is second order.

Slip behaves the other way. A slip length changes the damping at first order and adds no spring at all, because it changes how easily the liquid drains, not when. So the two are distinct in kind, and a measurement that records both parts of the force has a first-order test: if the damping is low and there is no in-phase force, the wall is not soft. If there is an in-phase force at a level a hundred times the damping’s shortfall, the wall is soft and the shortfall is its second-order consequence.

The slip it is read as #

An experiment that reads only the damping and fits a slip length will still find one. The figure gives it: $4.7\cdot10^{-5}$ of the gap at Λ = 0.1, 0.0047 at Λ = 1, 0.54 at 10 and 6.5 at 50. Below Λ of order one the reading is negligible and grows as $\Lambda^2$, the damping deficit’s own order. Above it the reading runs away, and a wall that does not slip at all is reported as slipping by several times the gap — a number nobody would believe, which is the one mercy in it.

What the soft wall’s slip looks like in nanometres #

In an experiment’s own units the signature is in the shape. For water against a 1 GPa coating — the order of the glue under a mica sheet, or of a glassy polymer film — and a 3 mm sphere, the apparent slip at 100 Hz is 0.1 nanometres at a 30 nm gap and rises steeply as the gap closes, to 9.5 nm at 3 nm, more than the gap. At 10 Hz it is a hundredth as large wherever it is small. A true slip of one nanometre reads as one nanometre at every gap and every frequency, and so does the viscous layer of the earlier essay, to first order. The elastic reading falls as the inverse square of the gap and rises as the square of the frequency, because $\Lambda \propto \omega D^{-3/2}$ and the reading goes as $\Lambda^2 D$.

That gives a second test that needs no phase information at all: change the frequency. A slip length that is the same at 10 and 100 Hz is a slip length. One that grows by a hundred is the wall.

A nanometre of slip that is not there #

Put the two cases side by side at one gap. Water drains between a 3 mm sphere and a surface at ten nanometres, oscillated at 100 Hz. If the surface is glass, with a plane-strain modulus of sixty gigapascals, Λ is 0.074: the damping is Taylor’s to one part in ten thousand, the in-phase force is seven-thousandths of it, and the slip a fit would report is three ten-thousandths of a nanometre — nothing. If the surface is a coating of one gigapascal, Λ is 4.5: the damping falls to 0.847 of Taylor’s, the in-phase force is a third of it, and a fit to the damping alone reports a slip length of 0.98 nanometres.

That is the size of slip length reported in many drainage experiments on smooth, wetting surfaces. The point is not that those reports were wrong — many were made on rigid surfaces and checked against frequency — but that a nanometre of apparent slip is exactly what a gigapascal of compliance produces at the gaps and frequencies the measurements are made at, and that the damping alone cannot tell the two apart at one gap and one frequency. At thirty nanometres the same soft surface would read as 0.10 nm, a tenth as much, which a careful experimenter would notice as a slip length that depends on where it was measured.

Why slip and a layer have no spring #

The test in the phase works because neither of the other two explanations stores anything. A slip length, as the essay that derived the drainage force with slip showed, changes how much liquid the gap passes for a given pressure gradient — a conductance — and a conductance only dissipates. A layer of different viscosity at the wall does the same. Both lower the damping at once and leave the in-phase force at zero, at any frequency, because neither the liquid nor the slip has anywhere to keep energy for a quarter-cycle. A melt that slips, whose slip length has an exponent it cannot choose, is the exception that proves the rule: a viscoelastic liquid stores energy itself, and its in-phase force would be a fourth rival with a spring of its own.

The elastic wall stores energy in its own deformation, and it returns it a quarter-cycle late. Its first-order effect is therefore a stiffness and not a conductance. Read through a slip formula it looks like a conductance only because the second-order part of its response, the part of the deflection that changes the gap in phase with the drainage, happens to lower the damping.

Three tests a measurement can make #

The calculation leaves three tests, in order of how cheaply an apparatus makes them. The first is the phase: record the in-phase force as well as the damping, and a soft wall shows an in-phase force much larger than its damping shortfall before the shortfall is resolvable at all. The second is the frequency: a slip length read at two frequencies a decade apart is the same if it is slip and a hundredfold different if it is the wall. The third is the gap: slip and a viscous layer read as constants, and a soft wall reads as a slip falling as the inverse square of the gap. The three are not independent — all follow from Λ — so a soft wall that passes one test fails the other two in the same proportion, and an apparatus that makes all three has a consistency check on its own reading of the wall. Any one of them separates the wall from slip; the earlier essay’s second-order test was needed only to separate slip from the layer, and the last of the oil is the squeeze-film arithmetic all three rest on.

How the wall moves #

The pressure field shows what the softness does to the flow. Over a rigid wall the oscillating pressure peaks on the axis at an eighth in the essay’s units, Reynolds’s value. At Λ = 10 the soft wall has given way where the pressure was largest, the peak has fallen to 0.085, and the pressure is spread over a wider region — the same flattening that a squeezed tube is held open by its own fluid found in a soft channel, where the fluid’s pressure and the wall’s compliance together decide the gap.

The deflection itself is mostly out of phase with the oscillation’s position, as the argument above requires, and largest on the axis. It is this deflection, lagging the motion by a quarter-cycle, that puts the spring into the force: the gap is narrowest a quarter-cycle late, when the sphere is already moving out again, so the liquid pushes back on the outward motion as a spring would on the displacement.

Where the spring has been seen #

A force that grows with a soft wall’s compliance and with the drainage rate is familiar from elsewhere in this collection. A ball that bounces in water and not in oil is decided by the same coupling run to large amplitude: the drainage pressure deforms the ball before it touches, the deformation stores energy, and the ball rebounds without contact if the liquid is thin enough to let the pressure build. A damper that turns into a spring found the same change of character in a squeezed gas film, where the compressibility is in the fluid rather than the wall. And nothing but the shape of the gap is the lubrication argument both rest on: a thin film’s force depends on the gap’s shape, so anything that changes the shape changes the force.

What this calculation adds is the order. In each of those the elastic or compressible effect enters the force in a phase, and the order in which it enters the two phases decides which measurement sees it first.

Checks on the coupled solve #

Boussinesq’s kernel contains the complete elliptic integral of the first kind, computed by the arithmetic–geometric mean and checked at two values to rounding. The kernel itself, applied to Hertz’s pressure distribution $\sqrt{1 - x^2}$, gives Johnson’s displacement $(\pi/4)(2 - x^2)$ inside the contact to four parts in ten thousand, which checks the treatment of its logarithmic singularity. With Λ = 0 the solve must be the rigid drainage, whose pressure is $-i/8(1 + x^2)^2$ exactly: the computed pressure agrees to $1.2\cdot10^{-4}$ and the force to $10^{-3}$, and that last part is the grid’s own error, which is divided out of every ratio reported. And the two slopes against Λ are 1.0000 and 2.0000 over a decade. The tests also refuse a negative Λ and an elliptic modulus of one.

What a half-space is not #

A coating. The wall here is a homogeneous elastic half-space. A real surface-force apparatus surface is a stiff mica sheet a few micrometres thick on a soft glue, which spreads a point load over the sheet’s own bending length before the glue feels it. The half-space is the right model when the pressure’s width $\ell$ is large against the sheet; at the smallest gaps, where $\ell$ is tens of nanometres, the sheet’s bending stiffness reduces the deflection and the effective Λ.

Viscoelasticity. A polymer’s modulus depends on frequency. Its loss part adds damping of its own, and the frequency test above then has a rival: a modulus that changes with frequency. The two still differ in shape. The elastic wall’s apparent slip rises as the square of the frequency at every gap, a line of slope two on logarithmic axes, while a polymer’s modulus away from its glass transition changes by a factor of a few across a decade and only close to the transition changes faster. Its loss part also shows in the damping rather than in the spring, so a frequency sweep over three decades that records both parts can still tell the two apart.

Small amplitude. The analysis is linear in the oscillation. Driven harder, the deflection changes the mean gap and the problem becomes the large-amplitude elastohydrodynamic one.

No slip, no layer. The calculation has neither. A real measurement may have all three, and to first order their damping effects add while only the wall adds a spring.

The convention: damping out of phase, spring in phase #

The force’s out-of-phase part, in phase with the velocity, is the damping, and is reported over Taylor’s $6\pi\eta R^2\omega/D$. Its in-phase part is reported over the same value, so the two can be compared directly. Radii are in units of $\ell = \sqrt{2RD}$, pressures in units of $12\eta\omega\ell^2/D^2$ per unit oscillation amplitude, and the apparent slip is the slip length Vinogradova’s formula, applied to the damping alone, would report.

Leroy, Charlaix, and the soft surface-force apparatus #

Elastohydrodynamic coupling in drainage was worked out for large spheres by Davis, Serayssol and Hinch in 1986, in the collision of particles in a viscous liquid. Its linearised, oscillating form, with Boussinesq’s kernel and the single parameter here, was solved and measured by Leroy and Charlaix and colleagues from 2011 on a dynamic surface-force apparatus, who used the in-phase elastic response to measure the modulus of soft coatings and thin films through the liquid without touching them. The reading of that response as an apparent slip, its second-order origin and its gap and frequency dependence are what this essay draws out.

Still open: the sheet on the glue #

The half-space is the simplest wall. A surface-force apparatus’s mica is a stiff plate on a soft foundation, and its response to a concentrated pressure has a length of its own, the plate’s bending length on the glue. The next calculation replaces Boussinesq’s kernel with the plate-on-foundation kernel for a stated mica thickness and glue modulus, and asks at what gap the plate’s stiffness takes over from the glue’s softness — whether, at the nanometre gaps where slip lengths are measured, the mica shields the glue’s elasticity enough that its spring and its apparent slip both fall below what an instrument resolves, or whether the slip lengths reported against mica carry a contribution from the glue beneath it.