A mica sheet hides its glue only at the smallest gaps, and changes how the false slip falls

A soft wall is read as slip only at second order put a dynamic drainage measurement against a wall that gives. A sphere oscillates over a plane in a liquid, the drainage pressure presses on the wall, and the wall’s deflection changes the gap the liquid drains through. Solved together, the flow and the elastic wall gave a force whose first change is a spring, in phase with the displacement, and whose damping — the part slip lengths are read from — falls only at second order. Read through a slip formula, that second-order fall is a slip length that rises as the square of the frequency and falls as the square of the gap. Against a coating of one gigapascal, water at a three-nanometre gap reads as slipping by 9.5 nanometres.

The wall there was an elastic half-space: the same material all the way down. That essay named the obvious objection itself. The surfaces in a surface-force apparatus are mica sheets a few micrometres thick, cleaved molecularly smooth and glued to glass lenses, and mica is stiff — some hundred and seventy gigapascals in the plane of the sheet — while the glue under it is a polymer a hundred times softer. A stiff sheet on a soft layer does not respond to a concentrated pressure the way a half-space does. It bends, and a bent plate spreads its load over a distance of its own before passing it to the foundation. The question was whether, at the gaps where slip lengths are measured, the sheet shields the glue enough that its spring and its false slip both disappear, or whether the slip reported against mica carries something of the glue beneath it.

It shields only at the smallest gaps, and only by a modest factor, and where it shields it changes the one signature that would have given the glue away. The false slip survives at nanometres where the sheet is three micrometres thick, and the gap test that separated it from true slip is the test the sheet weakens.

A plate that refuses short wavelengths #

The cleanest way to see what a sheet does is in wavenumbers. A pressure that varies across the surface as a ripple of wavenumber $q$ pushes an elastic half-space down by an amount in proportion to $2/(E^* q)$, with $E^*$ the plane-strain modulus: long ripples push deeply, short ones shallowly, which is what Boussinesq’s integral says in the radius. A thin plate of bending stiffness $B$ on top adds a resistance $Bq^4$ to the same ripple, so the two together deflect by

$$\hat u(q) = \frac{\hat p(q)}{Bq^4 + E^* q/2} = \frac{2}{E^* q} \cdot \frac{1}{1 + (q\lambda)^3}, \qquad \lambda = \left(\frac{2B}{E^*}\right)^{1/3}.$$

At wavelengths long against $\lambda$ the sheet does nothing and the half-space answers. At wavelengths short against $\lambda$ the sheet refuses to bend and the deflection collapses as $q^{-4}$. The length $\lambda$ is the sheet’s bending length on the glue. With $B = E_m t^3/12(1 - \nu^2)$ for a mica sheet of thickness $t$, modulus 175 gigapascals and Poisson’s ratio a quarter, on a glue of plane-strain modulus one gigapascal — the earlier essay’s coating — it is 9.4 micrometres for a sheet three micrometres thick, 3.1 for one micrometre and 15.7 for five. It grows in direct proportion to the thickness.

The drainage pressure has a width of its own. Between a sphere of radius $R$ and a plane at gap $D$ it is spread over $\ell = \sqrt{2RD}$, the radius at which the gap has doubled. For a 3 mm sphere that is 3.5 micrometres at a two-nanometre gap, 4.2 at three, 7.7 at ten, 13.4 at thirty and 24.5 at a hundred. It is tempting to picture the pressed region as the size of the gap, nanometres; it is micrometres, set by the geometric mean of the sphere’s radius and the gap rather than by the gap, and that is the whole reason the sheet matters. A pressed region of nanometres would sit far inside any mica sheet’s bending length; a pressed region of micrometres sits right across it.

So the problem keeps the half-space’s single parameter, $\Lambda = 48\eta\omega\ell^3/\pi E^* D^3$, now built on the glue’s modulus, and gains one more: $\tilde\lambda = \lambda/\ell$, the bending length in pressed widths. For three micrometres of mica it is 0.38 at a hundred nanometres, 0.70 at thirty, 1.2 at ten and 2.2 at three. The crossover is in the middle of the range drainage measurements use.

The dent spreads and shallows #

The first figure shows the mechanism in the radius. On bare glue the drainage pressure dents the wall most on the axis, where the pressure peaks, and the dent falls away over a couple of pressed widths. Under a sheet whose bending length is a third of the pressed width the dent is nearly the same: 0.89 of the bare glue’s on the axis. At a bending length equal to the pressed width it is about half as deep on the axis, 0.54, and slightly wider — 0.34 of the bare axis value two pressed widths out, against 0.33 for the glue. At three pressed widths the axis sinks 0.23 and the dent is almost flat across the region the liquid drains through; at ten, 0.076, and flat.

That flatness is the important part. A dent that is deep on the axis and shallow outside changes the gap’s shape where the pressure is highest, and the pressure feels it most. A flat dent changes the gap by the same amount everywhere, which is equivalent to moving the whole sphere: the sheet has stopped being a soft surface and become a stiff spring between the plane and its glass support.

The shielding goes as a square #

The second figure gives the size of the effect on the damping, where the slip is read. At a small glue softness, $\Lambda = 0.1$, it divides the sheet’s damping deficit by the bare glue’s. The ratio is 0.92 at a bending length of 0.3 pressed widths, 0.49 at 1, 0.12 at 3 and 0.014 at 10.

The dashed line is the limit the flat dent suggests. A sheet much stiffer than the pressed region sinks uniformly, and it is the solution for a point load on a plate on a half-space that says by how much: $2F/(3\sqrt3\,E^*\lambda)$ for a net force $F$, a spring of stiffness $3\sqrt3\,E^*\lambda/2$. In series with the rigid drainage that spring gives a force $F_0/(1 + i\varepsilon)$ with

$$\varepsilon = \frac{\pi^2 \Lambda}{48\sqrt3\,\tilde\lambda},$$

a damping $1/(1 + \varepsilon^2)$ and an in-phase part $\varepsilon/(1 + \varepsilon^2)$. Its damping deficit, $\varepsilon^2$, falls as $\tilde\lambda^{-2}$, and at ten pressed widths the full solution is within a tenth of it. The sheet divides the glue’s false slip by about the square of its bending length over the pressed width, once that ratio is a few; below one it divides it by almost nothing.

For three micrometres of mica on a gigapascal of glue the spring is stiff in everyday terms — about twenty-five newtons per millimetre of travel. What matters is the liquid in series with it. The drainage damping of water in a three-nanometre gap under a 3 mm sphere, at 100 hertz, amounts to an impedance of thirty-five newtons per millimetre: the same size as the sheet’s spring. Two impedances of the same size in series share the motion, and that is why the sheet at three nanometres still lets the glue be seen.

The orders survive the sheet #

The earlier essay’s main result was an ordering: a soft wall adds a spring at first order in its softness and lowers the damping only at second, while a true slip lowers the damping at first order and adds no spring at all. The third figure asks whether the sheet disturbs that. Under a sheet three pressed widths in bending length, the in-phase force rises with the glue’s softness at a fitted slope of 1.0000 and the damping deficit at 2.0000. The sheet makes both smaller, by the factors of the second figure, and leaves their orders exactly where they were.

The reason is the same as before, and it does not care what is underneath. The drainage pressure is in phase with the velocity, so the wall’s deflection is too, whether that deflection is a deep dent or a flat sink; its first effect on the force is a quarter-cycle out of the damping’s phase, a spring. Only the deflection’s effect on itself comes back into the damping. A sheet is still something that stores energy and returns it late, and a slip length still is not.

Two of the earlier essay’s three tests therefore survive the sheet intact. The phase test — record the in-phase force, and a soft wall announces itself with an in-phase part far larger than its damping shortfall — works under mica exactly as on bare glue, because the ratio of the two is set by the order and not by the factor. At three nanometres and 100 hertz under three micrometres of mica the in-phase force is 0.48 of Taylor’s damping while the damping has fallen to 0.45; at thirty nanometres the in-phase force is still 0.064 when the damping is 0.996, fourteen times the shortfall. The frequency test survives too: the false slip is second order in Λ, Λ is proportional to the frequency, so it still grows as its square.

The false slip in nanometres #

The fourth figure puts it in the units an experiment reports. Water, a 3 mm sphere, 100 hertz, a glue of one gigapascal, and the slip a fit to the damping alone would return from a wall at which nothing slips. On bare glue it is the earlier essay’s curve: 9.5 nanometres at a three-nanometre gap, 0.98 at ten, 0.10 at thirty.

Under a micrometre of mica — the thinnest sheets in common use — almost nothing changes: 8.8 nanometres at three, 0.099 at thirty. The sheet’s bending length, 3.1 micrometres, is below the pressed width at every gap drawn. Under three micrometres the reading falls to 2.7 nanometres at three, 0.38 at ten and 0.067 at thirty; under five, to 1.1, and 0.043 at thirty. At a hundred nanometres every curve has nearly rejoined the bare glue’s, because the pressed width there is twenty-five micrometres and longer than any sheet’s bending length.

Set beside the slip lengths reported in drainage experiments on smooth wetting surfaces — from nothing to a few nanometres — the figure does not say the sheet hides the glue. A three-micrometre sheet leaves a false slip of nearly three nanometres at a three-nanometre gap and 100 hertz, and a thin sheet leaves almost all of it. The sheet shields by factors of three to ten at the gaps where slip is most often claimed, and by less than a factor of two from ten nanometres out.

For the larger lenses of a classical surface-force apparatus, radius a centimetre rather than a few millimetres, the pressed width is nearly twice as large at the same gap and the sheet does even less: at thirty nanometres the false slip under three micrometres of mica is 3.6 nanometres against 4.1 on the bare glue.

The gap test is the one the sheet takes away #

The third of the earlier essay’s tests was the gap. A slip length or a viscous layer reads as a constant at every gap; the soft wall’s false slip falls as the inverse square of the gap, because $\Lambda \propto D^{-3/2}$ and the reading goes as $\Lambda^2 D$. An experimenter who saw a slip length triple when the gap halved, and then triple again, would suspect the wall.

The fifth figure shows that under a sheet that signature is no longer there to be seen. It plots the local exponent of the false slip in the gap, at 10 hertz where every reading is small and the second-order picture is clean. Bare glue holds −2 throughout. The sheets begin at −2 at large gaps, where they do nothing, and drift towards −1 as the gap closes: near three nanometres the exponent is −1.68 under a micrometre of mica, −1.29 under three and −1.17 under five.

The −1 is the series spring’s. A sheet sinking as one piece is a spring whose stiffness does not depend on the gap at all; the drainage’s impedance rises as $1/D$, so the damping deficit, the square of their ratio, rises as $1/D^2$, and the slip it is read as — the deficit times the gap — as $1/D$. Under thick mica the false slip falls only as the inverse of the gap, and over the narrow range of gaps a measurement spans that is far harder to tell from a constant than the inverse square was. A slip length that rose by two-thirds as the gap closed from five nanometres to three would sit within the scatter of most fits.

So the sheet is not a shield. It is a filter that removes a factor of a few from the false slip and, at the same time, the shape that made the false slip recognisable. Of the three tests, the one an apparatus makes most easily — measure at several gaps — is the one it weakens, and the two it leaves are the phase, which needs the in-phase force recorded rather than discarded, and the frequency, which needs a second frequency.

Why the half-space was not wrong #

None of this makes the half-space a mistake. It is the right model for a soft polymer film thick against the pressed width, and it is what the sheet becomes at large gaps. Nor is the sheet a refinement that only matters at the margins. If the pressed region were nanometres wide, as the gap is, every gap would sit far inside a mica sheet’s bending length and the sheet would be a perfect shield; if it were millimetres, the sheet would be invisible. At micrometres the sheet’s length and the liquid’s are the same size, and that coincidence — between a few micrometres of mica on glue and the geometric mean of a millimetre sphere and a nanometre gap — is why neither model alone describes the apparatus.

The drainage arithmetic underneath is unchanged: the essay that derived the drainage damping with slip gives the formula every false slip here is read through, and a viscous nanometre reads as slip until the gap is ten of it showed that a thin layer of different viscosity is indistinguishable from it at first order, and the last of the oil is the squeeze-film arithmetic all of them rest on. Where a wall genuinely slips, as a polymer melt really does, the slip length is microns to millimetres and none of these impostors could mimic it. The sheet adds a fourth candidate to that list that is again indistinguishable at one gap and one frequency, now with a gap dependence closer to the other three.

Where the spring meets the sheet elsewhere #

A spring in series with a squeezed film is familiar from other scales. A ball that bounces in water and not in oil is decided by the same coupling run to large amplitude, with the ball’s own elasticity as the spring. A damper that turns into a spring found the same change of character in a gas film, where the compressibility is the fluid’s. And a squeezed tube is held open by its own fluid is a soft channel in which the wall’s compliance and the fluid’s pressure set the gap together, as they do here. Nothing but the shape of the gap is the lubrication argument under all of them; the sheet changes the shape less, and the gap more uniformly, than a bare soft wall does.

Checks on the plate kernel #

The kernel is computed as a double sum over the radial grid and a uniform grid of wavenumbers, with Bessel functions of the first kind tabulated once. It is checked independently on the rigid drainage pressure $1/(1 + x^2)^2$, whose Hankel transform is $(q/2)K_1(q)$, so that its displacement under the sheet is a single integral: the matrix agrees with that integral to two parts in ten thousand at four radii, at bending lengths of one and a fifth of a pressed width.

The two limits are checked against independent solutions. At a bending length of a twentieth of a pressed width the damping deficit and in-phase force agree with the earlier essay’s half-space solve — Boussinesq’s elliptic-integral kernel, a different discretisation entirely — to under a tenth of a per cent. At twenty pressed widths and a large softness the damping agrees with the series spring to a tenth of a per cent and the in-phase force to one and a half, the residue of the pressed region’s finite width. The solve refuses a negative softness, a bending length shorter than its wavenumber grid can resolve, a sheet of no thickness and an incompressible one.

What the sheet model leaves out #

The glue is a half-space. It is taken as thick against both the pressed width and the bending length. Glue layers under mica are often no thicker than the pressed widths at the larger gaps here; a glue bonded to glass at that depth is stiffer than a half-space, so every number for the bare glue is an upper bound and the sheet’s relative effect is somewhat larger at large gaps than drawn.

The sheet is thin and perfectly bonded. Classical plate theory holds while the bending length is several sheet thicknesses — three times here, at the edge of comfort — and a sheet that has partly delaminated from its glue is a different, softer object.

One wall is soft. The sphere is rigid. With mica on glue on both surfaces, as in a classical apparatus, the two compliances add and Λ doubles, with the same bending length.

The liquid is plain. No slip and no layer are included; to first order their damping effects add to the wall’s, which is exactly why the false part cannot be subtracted without the tests above.

Who measured the glue through the mica #

That the glue under mica is compliant enough to matter was known to the surface-force community from contact mechanics: Sridhar, Johnson and Fleck modelled the apparatus’s glued mica as a layered elastic solid under adhesive loads in 1997, and showed that the glue’s compliance shapes the contact radii the apparatus measures. In drainage, the elastohydrodynamic coupling of a sphere and a soft surface was worked out by Davis, Serayssol and Hinch, and its linearised, oscillating form was used by Leroy, Charlaix and colleagues to measure the moduli of thin films through the liquid. What this calculation adds is the crossover between the two descriptions — the bending length against the pressed width — and its consequence for the one test that most slip measurements make, the dependence on the gap.

Still open: glue of a finite thickness #

The glue here is bottomless, and the next step is to give it a floor. A layer of thickness $g$ bonded to rigid glass responds like a half-space to ripples shorter than $g$ and like a stiff mattress — a Winkler foundation of stiffness set by $g$ — to longer ones, so the bare glue’s false slip should saturate once the pressed width passes $g$, and the sheet’s bending length on such a layer changes from $(2B/E^*)^{1/3}$ to $(Bg/E^*)^{1/4}$. The next calculation would carry the exact kernel of a bonded layer under a plate, and ask for stated glue thicknesses whether there is a range of gaps where the glue is felt through the sheet at all — a window bounded below by the sheet’s bending length and above by the glue’s thickness — and whether it closes for the thin glue layers a careful apparatus builder uses.