The sand that darkens when it is wet

Why the sky is blue and the sunset is not began with scattering by particles much smaller than light’s wavelength, when the particle is the size of the wave with particles comparable to it, and why a litre of water is not blue for the reason the sky is with the one liquid whose colour is absorption. Everything a scatterer removes found what a single scatterer takes from a beam, the cloud light has to walk through followed light through a thick cloud by a random walk, the walk that interference can stop found when that walk halts, and the full moon that is too bright found a powdered surface reflecting light straight back.

The random walk is the key to a familiar observation that is usually left unexplained. White things are white because they scatter light many times and absorb very little of it; paper, snow, salt and dry sand all return most of the light that falls on them, after it has bounced among their grains or fibres. Wet them, and they darken. Water is clear — a millimetre of it absorbs a negligible fraction of visible light — so it is not adding darkness of its own. This essay follows the two ways water changes where the light goes, both of which lengthen its path through the material, and finds that the darkening is the material’s own absorption, given more chances.

Light that the film sends back #

The first mechanism needs only a film of water on top. A surface that scatters light diffusely sends it back up in every direction. Without water, all of it leaves into the air. With a film of water over it, light going up meets the water’s surface from inside, and only the part within the critical angle can leave: the window a diamond is cut inside found that light inside a denser medium escapes only through a cone, and the rest is totally reflected back down.

For light scattered evenly in every direction, the escape cone of water passes a fraction $1/n^2$ of it, reduced a little further by ordinary reflection at angles inside the cone. The figure computes it: under water, 53 per cent escapes on the first attempt and 47 per cent is sent back down to the surface. Under oil, with a higher index, more is trapped: 58 per cent. The cone light has to find to get out met the same trap inside a light-emitting diode, where most of the light generated in a high-index crystal never escapes; here the light was generated nowhere, but it is trapped the same way.

The light sent back down is not lost. It returns to the surface, which reflects a fraction of it — the surface’s albedo — and absorbs the rest. The reflected part comes up again, and again only half escapes. Each round trip costs the light a fraction of itself to absorption at the surface, and the number of round trips a typical photon makes is set by how strongly the film traps it.

How much darker, and why the dark get darker #

Summing the round trips gives the reflectance of a surface under a water film, as a function of its reflectance dry. It was worked out in this form by John Lekner and Michael Dorf in 1988, building on earlier measurements of wet soils, and the figure plots it. A bright surface of reflectance 0.9 falls only to 0.77 when wet: it absorbs little on each bounce, so extra bounces cost it little. A mid-grey surface of 0.5 falls to 0.32, and a dark one of 0.2 to 0.11 — just over half its dry value.

That asymmetry explains something about colour that is otherwise puzzling. A wet coloured surface — a red brick, a painted wall, a pebble — looks not only darker but richer, its colour more saturated. A red surface reflects red strongly and absorbs green and blue. Under water, the red, which is reflected well, survives its extra trips nearly intact; the green and blue, already poorly reflected, lose a larger fraction of what is left on every return. The ratio of red to blue in the returning light grows, and the colour deepens. Pebbles on a beach look brightest in colour when the sea has just washed over them, and fade to dusty pastels as they dry, for this reason.

How many times the light comes back #

The film’s effect can be put as a count. A photon scattered up from the surface escapes with probability a little over a half; otherwise it goes back down, is reflected with probability equal to the albedo, and tries again. The average number of times light meets the surface before it finally leaves, or is absorbed, is $1/(1 - aR_{\text{int}})$. For a white surface with albedo 0.95 under water that is about 1.8 encounters instead of one; for a mid-grey surface, 1.3. Each extra encounter multiplies the surviving light by the albedo, and that is where the extra darkness comes from.

The count also says what makes the effect strong or weak. It grows with the film’s index, which sets how much is trapped, and with the surface’s albedo, which sets how much trapped light survives to be trapped again. It does not depend on the film’s thickness at all, provided the film is thin enough not to absorb and thick enough to have a flat top: a film of water a tenth of a millimetre deep traps exactly as much as a puddle. That is why a surface merely damp enough to glisten already looks as dark as it will get from the film alone.

Two explanations, sixty years apart #

The first mechanism was proposed by the Swedish meteorologist Anders Ångström in 1925, in a study of how much sunlight different kinds of ground reflect. He measured wet and dry soils and sands, found the wet ones reflecting markedly less, and attributed it to total internal reflection in the water film sending light back to be absorbed. The explanation stood for sixty years, until Sean Twomey, Craig Bohren and John Mergenthaler pointed out in 1986 that it could not be the whole story: materials wetted so that no continuous film formed on top still darkened, and some darkened more than the film mechanism allowed. They proposed the second mechanism — index matching in the pores, which makes each grain scatter more weakly and more nearly forward, so that light penetrates further before turning back. Lekner and Dorf then combined the two and showed that both are needed to account for measured darkening.

The history is a small example of a common pattern: an explanation that is correct as far as it goes, accepted because it is correct, and incomplete in a way nobody tested until someone asked what would happen if its essential ingredient, the film, were removed. The effect survived the removal, which meant there was a second cause, and for many materials the second cause turned out to be the larger.

Why not just a reflection #

It might seem that the film should make a surface brighter, not darker, since the top surface of the water adds a mirror-like reflection. It does, but only in one direction. The water’s surface reflects about seven per cent of diffuse light from outside, and all of it goes into the mirror direction, producing the glint off a wet road. Seen from any other angle, that reflection is absent, and the surface looks darker; seen in exactly the mirror direction, the glint can make it brighter than dry. A wet pavement lit by a street lamp shows both at once: bright streaks of glare towards the lamp, and dark, saturated patches everywhere else. The light has been rearranged, not added.

Water in the pores #

The second mechanism needs no film on top. Sand, soil, paper and cloth are porous: light enters the gaps between their grains or fibres and is scattered at every surface where it passes from grain to air and back. The strength of each scattering event depends on the contrast in refractive index between grain and gap.

The fraction of light reflected where a quartz grain meets air is 4.7 per cent; where it meets water it is 0.57 per cent, eight times less; where it meets oil, whose index is closer still, 0.07 per cent. The reflection at a single surface is only a crude measure of how strongly a grain turns light aside — a large grain also refracts light, bending it at each surface by an amount that depends on the same contrast — but both weaken as the index of the surroundings approaches the grain’s. At an exact match the grain disappears optically: a jar of glass beads covered with a liquid of the glass’s own index looks like a jar of liquid, and a glass rod dipped in it vanishes.

Weaker scattering means that light entering a wet bed goes further before being turned back. The cloud light has to walk through found that light in a thick scattering medium random-walks, with a step set by how far it goes before its direction is randomised. Lengthen the step and the walk penetrates deeper, and a photon that eventually re-emerges has travelled a longer path through the grains — and every grain absorbs a little. More path through an absorbing material means more absorption, even though no individual grain has changed.

The thick bed #

The simplest model of a thick scattering bed, developed by Paul Kubelka and Franz Munk in 1931 for paints, describes it by two numbers: an absorption coefficient $K$ and a scattering coefficient $S$. Its reflectance depends only on their ratio, and falls steeply as absorption gains on scattering. Dry sand reflecting 0.4 sits at $K/S = 0.45$. Water in the pores leaves $K$ unchanged — the grains absorb what they absorbed before — and reduces $S$. If it reduces it fourfold, $K/S$ becomes 1.8 and the reflectance falls to 0.18, less than half its dry value; a water film on top of the saturated sand takes it to about 0.10.

The factor of four is an assumption chosen to show the size of the effect, not a measurement: how much index matching reduces the effective scattering of a real bed depends on the grains’ size, shape and packing, and is found by experiment. Measurements on wet sands and soils give darkening of roughly this size, with the film and the pores both contributing, and remote-sensing instruments that estimate soil moisture from satellites use exactly this darkening as their signal.

The shirt that becomes see-through #

A thin layer behaves differently from a thick bed, and the difference is visible every time a white garment gets wet.

A dry white shirt is a thin layer of transparent fibres with air between them. It scatters strongly enough to reflect about half the light falling on it and transmit about as much, diffusely. Skin behind it is seen only by light that passes through the cloth twice, in and out, and that is a small fraction: here eight per cent, swamped by the cloth’s own fifty-three per cent. The shirt looks white and opaque. Wet, the fibres are surrounded by water, each scatters much less, and the layer’s scattering thickness falls. It now reflects only about a quarter of the light and transmits three-quarters, and the skin behind it returns as much light as the cloth does. The shirt looks grey where it lies against nothing and skin-coloured where it lies against skin. Nothing has become transparent that was not transparent before; the fibres have stopped turning light round.

Paper does the same thing: a drop of oil on paper makes a translucent spot, which is why greaseproof paper is made by matting fibres so tightly that little air remains between them, and why a grease stain on a paper bag lets light through. Frosted glass wetted with water becomes almost clear, for the same reason: the rough surface scatters because of the contrast between glass and air, and water fills the roughness with something nearly glass.

Snow, clouds and paint #

The same physics works in reverse wherever whiteness matters. Snow is white because ice crystals and air make a high-contrast scattering medium, and it greys as it melts because meltwater fills the gaps and reduces the contrast, and because larger, rounded grains scatter less per unit mass. Fresh snow reflects about ninety per cent of sunlight; wet, old snow reflects sixty or seventy, and the difference, spread over snowfields, is a significant term in the energy budget of melting glaciers. White paint depends on particles of titanium dioxide, whose refractive index of about 2.6 makes them scatter strongly even embedded in a binder of index 1.5; paints made with chalk, whose index nearly matches the binder, are white when dry and powdery — with air between the chalk particles — but turn translucent when the binder soaks in, which is why chalk makes a good filler and a poor pigment.

Where the model stops #

The film model assumes a flat, diffusely reflecting surface under a flat film, with the light inside the film distributed evenly in direction. Real wet surfaces are rough, films are uneven, and puddles are not films. The Kubelka–Munk model treats the scattering bed as uniform and the light within it as two diffuse streams, one up and one down, which is crude for layers only a few scattering lengths thick and for light arriving at a steep angle; more accurate calculations solve the full equation of radiative transfer. And the index-matching reduction in scattering depends on details the models here do not include: the size of grains compared with the wavelength, their shapes, and how much water they hold — a partly wet bed, with water only at the grain contacts, darkens less than a saturated one.

A bed does not go from dry to saturated in one step, either. As water is added it first coats the grains and gathers in rings at their points of contact, held there by surface tension; the pores then fill progressively, and only near saturation does a film form on top. The darkening therefore grows with water content, steeply at first as the contacts fill and the grains’ surfaces are wetted, then more slowly, and it stops changing once every pore is full. A beach shows all the stages in a few metres, from the pale dry sand above the tide line through the darker damp band to the darkest, glistening sand the last wave has just left.

What the pictures cannot show #

The figures show reflectances, averaged over direction and colour, and cannot show what makes wet surfaces look wet: the combination of darkening, saturation and a mirror-like glint, whose balance changes with the angle of view and the position of the light. The eye reads that combination very reliably as wetness, which is why painters render a wet street with dark, saturated colours and a few bright highlights, and why a surface that is merely darker does not look wet. And the figures cannot show the paths themselves — the long, random, looping routes photons take through a wet bed, each one a random walk made longer by every grain that failed to turn it round.

Still open: how to read a soil’s water from the light it returns #

Satellites estimate soil moisture over whole continents, and the darkening of wet soil is one of the signals they use, alongside the microwave emission of the soil, which depends on water through its dielectric constant. The optical signal is complicated by everything else that changes a soil’s reflectance — its minerals, organic matter, roughness and vegetation cover — and the relation between water content and darkening differs between soils and saturates when the pores are full. Models of radiative transfer through wet, rough, particulate surfaces that predict the darkening from a soil’s measured properties, without calibration against the same soil, are still being developed, and how precisely the optical signal alone can measure moisture over varied terrain is an open question.

The habit worth carrying away is to ask where the light went before asking what absorbed it. Water in and on a porous surface absorbs almost nothing itself: a film returns 47 per cent of the upwelling light for another chance at absorption, and water in the pores cuts each grain’s scattering eightfold so light wanders deeper — and on the longer path the material’s own absorption does the darkening, most of all for surfaces that were dark already. A wet shirt goes see-through for the same reason, with transmission taking the place of absorption.