Fluids

The surface whose thickness depends on the view

The surface of a glass of water looks like a perfect plane, and over most distances it nearly is. But it is at room temperature, and every ripple it could carry — from a few molecules long to a few millimetres — is shaken by heat with its own share of thermal energy. The ripples are tiny, a fraction of a nanometre high, and they add up in an odd way: every factor of ten in wavelength contributes the same amount of roughness. So the thickness of a water surface has no single value. It depends logarithmically on how large a patch is looked at, until gravity stops it at a few millimetres, and the X-ray measurements that report a third of a nanometre are reporting a property of their own resolution as much as of the water.

17 min read 5 figures Order out of the random Who is measuring

Assumes: The skin that is not a skin · The tension that vanishes where the phases meet

The skin that is not a skin found that surface tension is an energy per unit area: making more surface costs work, $\gamma$ joules per square metre, because molecules at the surface have fewer neighbours than those inside. A surface is therefore pulled flat, and the pull is strong — for water, 0.073 newtons per metre. The tension that vanishes where the phases meet followed the tension down to zero at the critical point, where the liquid and its vapour become the same substance, and found the interface spreading out as it did. It ended by noting that a real interface near the critical point is not a smooth profile but a fluctuating surface, rippled by heat.

Those ripples are not confined to the critical point. Every liquid surface carries them at every temperature, and working out how large they are for an ordinary surface, far from any transition, gives one of the stranger results in the physics of interfaces: the surface of a glass of water has no definite thickness.

A surface whose thickness depends on how much of it is looked at. The rms roughness of a water surface at 20 °C, in nanometres, against the size of the patch over which it is measured, from the thermal ripples that fit inside the patch — with gravity (solid) and without it (dashed) — for ripples down to a molecular wavelength of 0.6 nm. Over 10 nm the surface is 0.16 nm rough; over a micrometre, 0.26; over a millimetre, 0.36; and beyond a few millimetres gravity flattens the longest ripples and the roughness stops at 0.390 nm. In free fall it would keep growing as the square root of the logarithm: 0.43 nm over a metre. An X-ray reflectivity measurement sees the ripples smaller than its own coherence length, typically some micrometres, and reports a thickness of about a third of a nanometre — a number that belongs to the instrument as much as to the water. — Fig. 1 The rms roughness of a water surface at 20 °C against the size of the patch over which it is measured, from the thermal ripples that fit inside the patch, with gravity (solid) and without it (dashed), for ripples down to a molecular wavelength of 0.6 nm. Over 10 nm the surface is 0.16 nm rough; over a micrometre, 0.26; over a millimetre, 0.36; beyond a few millimetres gravity flattens the longest ripples and the roughness stops at 0.390 nm. In free fall it would keep growing: 0.43 nm over a metre.

Half a kT in every ripple

A ripple on a liquid surface stores energy two ways. Raising the surface into a wave stretches it, which costs surface energy, and lifts some liquid against gravity, which costs gravitational energy. For a ripple of wavenumber $q$ (wavelength $2\pi/q$ ) and amplitude $h$ , the energy per unit area is $\tfrac14(\gamma q^2 + \Delta\rho\, g)h^2$ , with $\Delta\rho$ the difference in density between the liquid and what lies above it. Surface tension dominates for short ripples and gravity for long ones, and they are equal at the capillary length, $\ell = \sqrt{\gamma/\Delta\rho\,g}$ , 2.7 millimetres for water — the same length that separates capillary ripples from gravity waves on a pond, which the speed below which nothing on water leaves a wake found setting the slowest speed at which anything can make waves.

In thermal equilibrium every such ripple is a mode with half a $kT$ of energy on average, the rule half a kT in a piece of wire applied to the voltage on a resistor. So the mean-square amplitude of each ripple is $kT$ divided by its stiffness: small for short, stiff ripples and larger for long, soft ones. At room temperature $kT$ is $4 \times 10^{-21}$ joules, and against a stiffness set by water’s tension the amplitudes are fractions of an ångström. No single ripple is noticeable: a ripple a micrometre long on a patch a micrometre square has a root-mean-square height of about four hundredths of a nanometre, about a seventh of the diameter of the molecules it is made of, and its neighbours in wavelength are similar. Each moves by less than an atom; only together do they make the surface a molecule thick.

What makes the result interesting is how they add.

Every decade of ripple adds the same roughness. The contribution of thermal ripples to the mean-square height of a water surface at 20 °C, per decade of their wavelength, in square nanometres, from equipartition: each ripple holds ½kT, and a stiffer surface holds it in a smaller amplitude. Below the capillary length, 2.7 mm, the surface tension alone resists the ripple and every decade contributes the same 0.0204 nm², because the number of ripples in a decade grows as the square of the wavenumber while each one's amplitude falls by the same factor. Above it gravity takes over, and the contribution dies away. So the roughness accumulates logarithmically, a little from every scale between molecules and millimetres, and no single wavelength dominates it. — Fig. 2 The contribution of thermal ripples to the mean-square height of a water surface at 20 °C, per decade of wavelength, in square nanometres, from equipartition. Below the 2.7 mm capillary length surface tension alone resists the ripple and every decade contributes the same 0.0204 nm², because the number of ripples in a decade grows as the square of the wavenumber while each one’s mean-square amplitude falls by the same factor. Above it gravity takes over and the contribution dies away.

Every decade contributes the same

The second figure breaks the roughness down by wavelength, a decade at a time. A surface of a given area holds more short ripples than long ones: the number of ripples with wavenumbers in a range grows as the square of the wavenumber, as for the modes of any two-dimensional surface, which how many ways there are to vibrate counted for a drum. Each short ripple has a smaller mean-square amplitude, because its stiffness $\gamma q^2$ is larger, by exactly the same square. The two cancel. Each decade of wavelength — from one to ten nanometres, from ten to a hundred, and so on up to millimetres — contributes the same amount to the mean-square height: for water at room temperature, 0.02 square nanometres.

So the total is a sum of equal contributions, one per decade, and its value depends on how many decades are counted. From the molecular scale up to a patch of size $L$ , the mean-square roughness is

w^2 = \frac{kT}{2\pi\gamma}\,\ln\frac{L}{2a},

with $a$ the molecular size: a logarithm. The surface’s thickness grows without limit as the patch grows, if very slowly, until the patch exceeds the capillary length; then gravity stiffens the longest ripples, the contributions per decade fall away, and the thickness stops growing. For water the limit is 0.39 nanometres, a little more than one molecular diameter. The first figure draws the growth: 0.16 nanometres over a patch ten nanometres across, 0.26 over a micrometre, 0.36 over a millimetre, and the plateau beyond.

Mathematically the logarithm is the signature of a quantity balanced on the edge between converging and diverging. In one dimension, a string’s thermal fluctuations grow with its length as the square root, without limit; in three, a solid’s are bounded. A two-dimensional surface sits exactly between, and grows logarithmically. The same marginal behaviour makes a two-dimensional crystal’s atoms wander logarithmically far from their lattice sites, so that a crystal confined to a plane has no true long-range order at any temperature above zero — the result David Mermin proved in 1968, and the reason two-dimensional order is always a delicate thing.

Equal shares from every scale

A spectrum that gives every decade the same share is familiar from a different context. The electrical noise of many devices rises at low frequency as one over the frequency, and half a kT in a piece of wire found that flicker noise is what limits how long a measurement can usefully be averaged: because each decade of frequency contributes the same noise power, averaging longer does not help as it should, and the accumulated drift grows with the logarithm of the time. A water surface’s height has a spectrum of the same form, in space rather than time. Its ripples are one-over-wavenumber-squared in the two-dimensional sense that gives equal shares per decade, and the consequence is the same: looking at a larger patch does not average the roughness away but adds to it, slowly and without end, until something — gravity, here — cuts the spectrum off.

The comparison also says why the logarithm matters so little and so much at once. Four extra decades of ripples, from a micrometre to a centimetre, add only about half as much again to the roughness, so for most purposes a water surface is a few ångströms thick and that is the end of it. But no patch is large enough for the thickness to settle, short of the capillary length, and any experiment that compares surfaces over different areas finds them different in exactly this way.

What an X-ray measures

Fig. 3 One random instant of a water surface at 20 °C along a 200 nm line, from ripples from 0.6 nm to the length of the cut, each with its equipartition amplitude and a random phase (seeded), the height exaggerated 44 times relative to the length. Its rms roughness is 0.24 nm, against 0.23 nm from the formula for a patch this size — less than a molecular diameter — rough at every scale, short ripples riding on long ones: the logarithm drawn as a picture.

The surface’s roughness can be measured, and the measurement shows the logarithm at work. X-rays reflected from a liquid surface at grazing incidence are reflected less strongly than from a perfectly sharp surface, and the shortfall measures the surface’s mean-square roughness: the reflected intensity is reduced by a factor $e^{-q_z^2 w^2}$ , where $q_z$ is the change in the X-ray’s wavevector perpendicular to the surface, so that the steeper the reflection, the more the roughness costs, and the fall of reflectivity with angle reads off $w$ . In 1985 Alan Braslau, Peter Pershan and colleagues measured water’s surface this way and found a roughness of about 0.32 nanometres, in agreement with the capillary-wave formula.

But the formula has a cut-off that belongs to the experiment. An X-ray beam does not average the surface over the whole glass. It sees the surface coherently over a lateral distance set by its own angular resolution, typically micrometres, and ripples longer than that tilt the surface slowly enough that the reflection follows them rather than being weakened by them. So the measured roughness counts only the ripples shorter than the instrument’s coherence length, and a different instrument with a different resolution measures a different thickness. Later experiments measured the faint diffuse scattering around the reflected beam, which records the ripples directly, and found the capillary-wave spectrum over several decades of wavelength — perhaps the most direct demonstration that the thickness of a liquid surface is a question of how it is looked at. The figure of a third of a nanometre is water’s roughness at the scale X-rays see, not at every scale.

The third figure draws what the X-rays are averaging: one instant of a cut through a water surface two hundred nanometres long, built from its thermal ripples with random phases. It is rough at every scale, short wiggles riding on longer swells, with no characteristic size to the bumps. Drawn at its true proportions it would be a straight line; the height has been magnified forty-four times to make it visible. The rough surface that is a mirror at a glance found the condition under which roughness stops spoiling reflection, and a water surface meets it for visible light by a factor of a thousand, which is why it makes a perfect mirror.

Six liquids and a near-critical one

How rough six liquid surfaces are. The rms thermal roughness of six liquid surfaces over a patch larger than their capillary lengths, in nanometres, from kT against surface tension: mercury, 20 °C, 0.15; water, 20 °C, 0.39; ethanol, 20 °C, 0.69; liquid argon, 85 K, 0.47; liquid helium, 2 K, 0.44; water, 370 °C, 7.23. Mercury, with nearly seven times water's tension, is the smoothest, and ethanol, with less than a third of water's, the roughest of the liquids at room temperature. Liquid argon, at 85 K but with a fifth of water's tension, is rougher than water; liquid helium, at 2 K but with a tension two hundred times smaller, is rougher than water too. And water itself, three degrees short of its critical point where its tension has nearly vanished, is many nanometres thick. The roughness goes as √(kT/γ), the ratio of the thermal energy to the energy a unit of surface costs, and only logarithmically on anything else. — Fig. 4 The rms thermal roughness of six liquid surfaces over a patch larger than their capillary lengths: mercury at 20 °C, 0.15 nm; water at 20 °C, 0.39; ethanol, 0.69; liquid argon at 85 K, 0.47; liquid helium at 2 K, 0.44; water at 370 °C, 7.23. Mercury, with nearly seven times water’s tension, is the smoothest; ethanol the roughest of the room-temperature liquids. Argon and helium, though cold, have such weak tensions that they are rougher than water; and water three degrees short of its critical point is many nanometres thick.

The roughness depends on the liquid almost entirely through one ratio: the thermal energy $kT$ against the surface tension $\gamma$ . The fourth figure compares six liquids. Mercury, whose tension is nearly seven times water’s, has the smoothest surface. Ethanol, with less than a third of water’s tension, is the roughest of the room-temperature liquids. The cryogenic liquids are instructive: liquid argon at 85 kelvin is cold, but its tension is so weak that its surface is rougher than water’s, and liquid helium at two kelvin, whose tension is two hundred times smaller than water’s, is rougher too. Temperature alone does not make a surface smooth; it is the temperature in units of what the surface costs.

The last entry closes the loop with the critical point. Water at 370 °C, a few degrees below its critical temperature, has a surface tension of less than half a millinewton per metre, and its thermal roughness is several nanometres — the fluctuating interface the earlier essay left as an open question. As the critical point is approached the tension falls to zero, and the capillary-wave roughness grows without limit. It grows alongside the interface’s own intrinsic width, and how the two combine near the critical point, which the earlier essay noted is not fully settled, is a question about exactly the ripples drawn here.

Gravity’s weak hand

Gravity flattens a surface only logarithmically. The rms thermal roughness of a large water surface at 20 °C against the strength of gravity, from a ten-thousandth of the Earth's to a hundred times it, on a logarithmic axis. Gravity enters only by setting the capillary length, beyond which it damps the longest ripples, and the roughness depends on its logarithm: 0.439 nm at a ten-thousandth of g, 0.390 nm on the Earth, 0.363 nm at a hundred g. A ten-thousandfold change in gravity changes the thickness of the surface by 13 per cent. In true weightlessness nothing stops the longest ripples, and the thickness of a free liquid surface is set by the size of its container. — Fig. 5 The rms thermal roughness of a large water surface at 20 °C against the strength of gravity, from a ten-thousandth of the Earth’s to a hundred times it, on a logarithmic axis. Gravity sets only the capillary length beyond which it damps the longest ripples, and the roughness depends on its logarithm: 0.439 nm at a ten-thousandth of g, 0.390 on the Earth, 0.363 at a hundred g. A ten-thousandfold change in gravity changes it by 13 per cent.

The last figure varies the strength of gravity. Gravity’s only role is to stiffen the ripples longer than the capillary length, cutting off the logarithm there; a weaker gravity moves the cut-off to longer wavelengths and adds a few more decades of ripples. Because each decade contributes equally, the roughness depends on gravity’s logarithm, and a change of ten thousand in gravity’s strength changes the thickness by thirteen per cent. A pond on the Moon, under a sixth of the Earth’s gravity, would have a surface about three per cent rougher than the same pond at home. In orbit, with no gravity at all, the longest ripples are limited only by the container, and a large free liquid surface in space would be thicker than the same surface on the ground by an amount that grows with the logarithm of its size — still under half a nanometre for a surface a metre across.

The weakness of the dependence is the same marginality as before. A quantity that grows as a logarithm is hardly affected by anything that changes its upper limit by any factor short of astronomical, and a surface in two dimensions is the paradigm case.

How fast the surface trembles

Equipartition gives the size of each ripple and says nothing about how fast it changes. That comes from the same two stiffnesses and the liquid’s inertia. A capillary ripple of wavenumber $q$ on deep water oscillates at $\omega = \sqrt{\gamma q^3/\rho}$ , so a ripple a tenth of a millimetre long on water rises and falls about twenty thousand times a second, and one a micrometre long a thousand times faster still, since the frequency grows as the wavelength to the power minus three-halves. Viscosity damps each ripple at a rate of about $2\nu q^2$ , with $\nu$ the kinematic viscosity, and since damping grows as $q^2$ while the frequency grows only as $q^{3/2}$ , damping wins at short enough wavelengths. On water the crossover lies somewhere around a tenth of a micrometre. Ripples longer than that ring; ripples shorter than that do not oscillate at all, but creep back towards flat as fast as viscosity lets them, and are replaced by new ones.

So the roughness the figures compute is a time average over a surface that is never still and never the same twice. At any instant the height is as random as the third figure drew it; a millisecond later the short ripples have all been replaced and only the long swells are recognisably the same.

The motion is measurable, and it turned the ripples from a nuisance into an instrument. A laser beam reflected from a liquid surface scatters weakly off the thermal ripples, and light scattered through a chosen small angle has interacted with ripples of one chosen wavelength. Each ripple that oscillates Doppler-shifts the scattered light up or down by its own frequency, so the scattered light’s spectrum shows two peaks at $\pm\omega$ , broadened by the damping. The position of the peaks gives the surface tension and their width the viscosity, measured without touching the surface — on a liquid too hot, too reactive or too small to dip a ring into, or on a monolayer of molecules spread on water whose elasticity changes the ripples’ damping. Thermal noise, which sets how thick a surface is, also reports what the surface is made of.

When a film has two surfaces

The ripples matter most where a liquid is thin. A soap film has two surfaces close together, and each is rippled; where the ripples on the two faces happen to move inward together, the film is locally thinner. The film that goes black before it bursts found that a draining film stops thinning when the pressure between its surfaces balances the suction pulling liquid out, and that the thinnest stable films are a few tens of nanometres thick. A film thinner than that can be broken by its own thermal ripples: where the attraction between its two surfaces grows faster with thinning than surface tension resists the ripple, a ripple of long enough wavelength grows instead of dying away, and the film ruptures spontaneously. The same instability, driven by the same thermal ripples, breaks up thin liquid layers spread on a solid into droplets, a process called spinodal dewetting, and it is visible under a microscope as a pattern of holes with a preferred spacing set by the fastest-growing ripple.

Where the ripple picture stops

The calculation treats the surface as a sharp sheet with a tension, displaced by ripples, and that description fails at its two ends. At the short end, ripples a few molecules long are not ripples of a continuous sheet; the surface has an intrinsic thickness of its own, the profile over which the density falls from liquid to vapour, and at the molecular scale the tension itself depends on wavelength. The molecular cut-off $a$ in the formula is a placeholder for that physics, and it enters only through the logarithm, which is why the answer is insensitive to it. At the long end, the ripples are assumed to be in thermal equilibrium with no flow and no evaporation, and a real surface has convection, air currents and contaminants that lower its tension and damp its ripples, effects larger than the thermal roughness at the scales where they act.

The equipartition rule also assumes classical statistics, which holds while $kT$ is large compared with the energy quantum of each ripple. For the shortest ripples on liquid helium, at a few kelvin, that is not true, and the ripples’ quantum zero-point motion contributes a roughness of its own that persists to absolute zero.

What a roughness number cannot convey

The figures report the roughness as a single number, an rms height, and a surface whose roughness comes equally from every scale is not described by one number. Its height has no characteristic wavelength, its slope is set by the shortest ripples and its large-scale tilt by the longest, and a measurement sensitive to one of these sees a different surface from a measurement sensitive to another. The pictures are also static: each ripple is a wave that travels, and the surface’s roughness at one instant is a snapshot of a field of capillary waves moving in every direction, with lifetimes set by viscosity — nanoseconds for the shortest, seconds for the longest. Light scattered from the surface sees that motion as a spread of Doppler shifts, and measurements of it give the surface tension and viscosity of a liquid without touching it.

Still open: what the intrinsic interface is

The capillary-wave picture separates a surface into two parts, a sharp intrinsic profile and the ripples that displace it, and the separation is not unique: where the intrinsic surface ends and the ripples begin depends on the cut-off chosen, so that the intrinsic width and the capillary roughness trade against each other. Computer simulations of liquid surfaces have tried to identify the intrinsic surface molecule by molecule, and different definitions give different answers for its width. Whether there is a natural, measurable intrinsic thickness of a liquid surface, or only a total that depends on the scale of observation, is still argued, and experiments with X-rays of different resolutions and with neutron reflection are aimed at the question.

The habit worth carrying away is to ask over what scale a quantity is measured. A liquid surface rippled by heat holds $\tfrac12 kT$ in every ripple, and because the number of short ripples grows as fast as their amplitudes shrink, every decade of wavelength adds the same 0.0204 nm² to water’s mean-square height: the thickness of the surface is $\sqrt{(kT/2\pi\gamma)\ln(L/2a)}$ , 0.16 nm over ten nanometres and 0.39 over millimetres, set as much by how far the measurement looks as by the water. Wherever a fluctuation sums equally over scales, the answer is a logarithm, and the scale of the measurement is part of the result.