A viscous nanometre reads as slip until the gap is ten of it

A length measurable only by squeezing explained how anybody knows whether a liquid slips at a wall. No ordinary flow can see a slip length of a nanometre, because every flow divides it by a length the apparatus sets, and apparatus is large. A drainage experiment divides it by the gap — a sphere pushed towards a flat through a thin film of liquid, the force against the approach measured as the gap closes to tens of nanometres — and the drainage force for Navier slip on both surfaces has Vinogradova’s closed form, whose shortfall below the no-slip force is twice the slip length over the gap.

That essay ended on what the measurement cannot see. Molecular simulations of water against mica, and the ordering of the first few molecular layers at any smooth solid, suggest a different picture from slip: no slip at all, but a layer a nanometre or so thick in which the liquid is several times more viscous than in the bulk. Both pictures change the drainage force by an amount linear in a small length. The closing question was whether they ever differ by more than the experiment’s own scatter, and at what gap.

The flux through a stratified gap #

The drainage force is built from one quantity, the flux a pressure gradient drives through a gap of width h, and for a liquid whose viscosity varies across the gap that flux has an exact form. With the viscosity symmetric about the middle, the flow is zero at each wall and its flux is the pressure gradient times

$$G(h) = 2\int_0^{h/2} \frac{s^2}{\mu(s)}\,ds,$$

s measured from the middle. For a uniform liquid that is $h^3/12\mu$, the parallel-plate flux of every lubrication calculation. For Navier slip b on both walls, the extra flow along the walls multiplies it by exactly $1 + 6b/h$. For a layer of thickness ℓ at each wall with viscosity κμ, the integral splits at the layer’s edge and gives

$$\frac{G}{h^3/12\mu} = 1 - \left(1 - \frac1\kappa\right)\left(\frac{6\ell}{h} - \frac{12\ell^2}{h^2} + \frac{8\ell^3}{h^3}\right).$$

Compare the two. The first-order terms agree if $b = -(1 - 1/\kappa)\ell$: a viscous layer is a negative slip length, a no-slip plane moved into the liquid, and a less viscous layer is a positive one. The second-order terms do not agree: slip has none, and the layer has $+12(1 - 1/\kappa)\ell^2/h^2$. Everything that separates the two pictures is in that term and the cubic after it.

The figure draws the two walls at a 10-nanometre gap: a 1-nanometre layer at each wall three times as viscous as the liquid, and the slip length that matches it to first order, −0.667 nanometres. The layer bends the velocity profile inside itself and joins the bulk’s parabola at its edge; the slip continues the bulk’s parabola straight through the wall to a zero two-thirds of a nanometre inside the liquid. In the middle of the gap the two are within a per cent of each other.

Where they part #

The fluxes part as the gap closes. Far apart both fall below the no-slip flux as $1 - 4/h$, with h in nanometres. Closer in, the layer’s flux falls more slowly than the slip’s, because the second-order term is on its side: at 10 nanometres the layer carries 67.5 per cent of the no-slip flux and the slip 60 per cent, a difference of 7.5 per cent of the flux. The slip formula also has a pathology the layer does not. A negative slip length makes $1 + 6b/h$ vanish at $h = 6|b|$, four nanometres here, and a model that predicts no flux at all through a 4-nanometre gap full of liquid has stopped describing anything. The layer’s flux, whatever its viscosity, stays positive until the layers meet.

The immobile limit is worth writing out, because it is exact. Let κ grow without bound — a layer that does not move — and the flux factor becomes $(1 - 2\ell/h)^3$: the parallel-plate flux of a gap $h - 2\ell$. An immobile layer is exactly a wall moved into the liquid by its thickness, at every order, and nothing about the flow can distinguish it from a solid surface that is simply a nanometre further in. A drainage experiment that does not know where its surfaces are to better than a nanometre cannot tell an immobile layer from an error in its zero of distance.

Two forces that cannot be told apart by eye #

The force on a sphere of radius R approaching a flat at speed V is the flux turned inside out: continuity fixes the flux through every radius, the flux fixes the pressure gradient, and the pressure integrates to

$$F = \pi V R^2 \int_D^\infty \frac{h - D}{G(h)}\,dh,$$

with $h = D + r^2/2R$ the gap at radius r. For a uniform liquid that is Taylor’s $6\pi\mu R^2 V/D$, and for slip it is Vinogradova’s closed form; the quadrature here reproduces both to three parts in $10^{12}$. For the layer it is done by quadrature.

Both forces rise above Taylor’s, since a viscous layer is a sticky wall, as $1 + 1.33/D$ far off. At 30 nanometres they read 1.046 and 1.048, a difference of a sixth of a per cent; at 10 nanometres 1.147 and 1.169. Plotted on any scale an experiment would use, the two curves are one.

The per cent that decides it #

The figure puts the difference as a percentage against the gap, for layers two, three and ten times as viscous as the liquid and for one that does not move. It passes one per cent at 11.1, 13.1, 15.6 and 16.6 nanometres respectively: between eleven and seventeen layer thicknesses. Because the difference is a function of ℓ/D, the gap at which it reaches any level scales with the layer’s thickness; a half-nanometre layer is distinguishable only inside half those gaps. At 30 nanometres the difference is between a tenth and a quarter of a per cent.

A per cent is roughly the precision the best drainage measurements claim for their forces, and eleven to seventeen nanometres is where the question can begin to be asked. It is also close to where the continuum description begins to fail on its own terms. Below five or so nanometres water between mica surfaces shows oscillating solvation forces, one per molecular layer, and a Reynolds equation with a smooth viscosity profile has stopped being the right model. The window in which a stratified viscosity is both distinguishable from slip and still a continuum is a few nanometres wide.

The signature is a drift #

An experiment does not subtract two theories; it fits one. Fitting the slip formula to the layer’s force at each gap gives the slip length the experiment would report, and the figure plots it. Far off it is the first-order value, −0.667 nanometres for a layer three times as viscous. As the gap closes it shrinks towards zero: −0.597 nanometres at 10 and −0.525 at 5. For the immobile layer it runs from −1 nanometre down in the same way.

A slip length that depends on the gap is the layer’s signature, and it is the only one a drainage curve carries. Reported slip lengths that vary with separation have usually been attributed to something else — roughness, a rate-dependent slip, an elastic surface — and each of those produces a drift of its own. A measurement that wanted to test the layer would have to show the drift has the layer’s shape, $b(D) \to -(1 - 1/\kappa)\ell$ at large gaps and shrinking in proportion to ℓ/D, and exclude the others. The shape is computable; the exclusion is the hard part.

A slippery layer drifts too #

The same arithmetic runs for a layer less viscous than the bulk — a region depleted of solute, a film of adsorbed gas, the air layer that the squeezing essay found reading as a slip length of nearly a micron. Its first-order slip is positive, $(1/\kappa - 1)\ell$: one nanometre for a layer half as viscous, four for a fifth. And its fitted slip length shrinks as the gap closes too, to 3.47 nanometres at a 5-nanometre gap for the thinner fluid. The rule is the same in both directions: whatever the layer does, a slip length fitted to it shrinks in magnitude as the gap approaches the layer’s own thickness, because the layer’s flux saturates where slip’s keeps growing or collapsing.

What precision the test would need #

The drift is small, and its size is what a test would have to resolve. For a one-nanometre layer three times as viscous, the fitted slip length is −0.660 nanometres at a 100-nanometre gap, −0.644 at 30 and −0.597 at 10: it moves by six hundredths of a nanometre over the range where the continuum is safe and a per cent of force is measurable. An experiment that wanted to see the layer would therefore need each gap’s slip length to about two hundredths of a nanometre — a fifth of an ångström, a tenth of a water molecule’s diameter — and its surfaces’ absolute positions to the same precision, since an offset in the zero of distance is itself a first-order change in the force of exactly the immobile layer’s form. The best drainage measurements report slip lengths to a tenth of a nanometre or so, a factor of five short.

There is a second route, and it is the one the formulas point at: vary the layer rather than the gap. The first-order slip is $-(1 - 1/\kappa)\ell$, and the second-order term is $+12(1 - 1/\kappa)\ell^2/h^2$; their ratio is $-2\ell/h$, independent of κ. A set of liquids with the same bulk viscosity and different ordering at the wall, or one liquid at several temperatures, would move the first-order term and the second together if the layer picture is right, and would move only the first if the wall slips. That is a comparison of drifts across experiments rather than a measurement of one drift, and it asks for relative rather than absolute precision.

Why most of the time it does not matter #

The ambiguity is a problem for one measurement and harmless almost everywhere else, and the reason is instructive. A lubricating film’s load, the pressure a gap makes by narrowing, depends on the wall only through the flux factor G, and at the micron clearances of a bearing the difference between a nanometre of slip and a nanometre of layer is a part in a million of it. Nothing about a bearing’s performance can see the question at all.

The moving contact line of a spreading drop is the case where the microscopic length is indispensable — the stress at the edge diverges without it — and the drop a no-slip wall would never let spread found that the drop’s speed depends on that length only through its logarithm. A slip length of a nanometre, a viscous layer a nanometre thick and a precursor film of molecular thickness each supply the cutoff, and the logarithm cannot tell them apart any better than the drainage force can. Both flows need some length at the wall and are almost indifferent to which. The drainage experiment is the one place where the length itself is the quantity sought, which is why the question of what it is belongs to it.

What the wall has to be told #

A slip length is not a statement about slip; it is a boundary condition that reproduces the flow outside a thin region the calculation does not want to resolve. How many things a flow must be told counted two conditions at a viscous wall, and the Navier condition is one choice of the second. This calculation says that the same outside flow is reproduced, to first order, by any thin region with the right integral of its inverse viscosity — slip, a viscous layer, a less viscous one, a wall in a different place — and that they differ only when the gap is a few times the region’s thickness. The post array was the same kind of statement about a textured wall, where the region is a period thick; slip that is a memory of one mean free path is its gaseous counterpart, where the region is the Knudsen layer and the substitute condition fails when the gap approaches the mean free path.

For polymer melts, the wall a melt really does slip on, the question does not arise in this form. Their slip lengths are tenths of a millimetre, so large that no molecular layer could mimic them, and the slip is real sliding. The ambiguity here is peculiar to simple liquids, whose slip lengths are comparable to the thickness of the structure a wall imposes on them.

What was checked #

The force quadrature was checked against Taylor’s force with no slip, and against Vinogradova’s closed form for slip lengths of one and two nanometres at gaps of 3, 10 and 30 nanometres, to three parts in $10^{12}$. The layer’s flux factor was checked against the integral of $s^2/\mu(s)$ taken directly across the gap, split at the layer’s edge, to rounding error. The tests refuse a negative gap, a negative layer thickness and a gap of zero.

What the picture cannot show #

A step in viscosity. The layer has one viscosity and a sharp edge. Molecular simulations give a viscosity that varies smoothly over a few molecular diameters, and the second-order term depends on the profile’s shape, not only its integral; a smooth profile would move the one-per-cent gaps by an amount set by its second moment.

Rigid surfaces. At gaps of ten nanometres the drainage pressure deforms mica, glue and even silica, and an elastic surface also produces a force that falls below Taylor’s in a gap-dependent way — a drift of the fitted slip length with its own shape, which any real test would have to separate from the layer’s.

Two walls alike. The layer sits on both surfaces with the same thickness and viscosity, as between two sheets of mica. An atomic-force microscope’s sphere and flat are usually different materials, with different layers; the flux integral then splits into two unequal parts, the first-order slip is the average of the two walls’, and the second-order term depends on each wall’s layer separately, so an asymmetric pair drifts differently from a symmetric pair with the same average.

A continuum to the wall. The whole calculation is a lubrication equation with a viscosity. Below a few nanometres the liquid is layered and the flow is not a viscous continuum, which bounds the window from below.

Who worked it out #

Taylor’s drainage force is from the 1910s; Vinogradova gave the slip correction in closed form in 1995, and the dynamic surface force apparatus of Charlaix’s group measured slip lengths of water on smooth surfaces to nanometre precision in the 2000s. The idea that the first molecular layers of water at a hydrophilic surface are more viscous than the bulk comes from molecular dynamics and from force measurements of Granick’s and Klein’s groups, and the correspondence between a thin layer of different viscosity and an effective slip length is the classical derivation of the Navier condition from a layered wall.

Still open: a wall that gives #

The rival that most resembles the layer is not slip but elasticity. Mica glued to a cylindrical lens, pressed by the drainage pressure of a closing gap, flattens by a few ångströms at ten nanometres, and a flattening surface widens the gap where the pressure is highest — which lowers the force, as positive slip does, in a way that depends on the gap. The next calculation couples the Reynolds equation to the elastic deflection of a coated half-space, finds the apparent slip length that elasticity alone produces against the gap, and asks whether its drift can be told from the viscous layer’s — whether the two signatures have different shapes, or whether a surface force apparatus has been measuring the sum of both and calling it slip.