The attraction that is the small ones' room

Entropy is a count argued that the second law is arithmetic — a system drifts towards the macrostates that can happen in the most ways — and what a system actually minimises turned that into a rule for systems at a fixed temperature: they minimise $U - TS$, so a state can win by having more arrangements as surely as by having less energy. The rubber band in that essay is the familiar case of a force that is all entropy: its chains pull back because a coiled chain has more conformations than a straight one, and heating it makes it pull harder.

The rubber band’s force acts within one object. There is a stranger case, in which entropy produces a force between two objects that do not interact with each other or with anything else in any attractive way — and it arises from a third kind of particle that is not even in contact with them most of the time. It explains why adding a pinch of polymer to a stable paint can make the pigment clump, why DNA can be pulled out of solution with polyethylene glycol, and why molecules inside a crowded cell associate far more readily than they do in a test tube. It is the depletion force, and the whole argument consists of counting room.

The shell the small ones cannot enter #

Take a liquid containing a few large particles — colloidal spheres, a micrometre across — and a great many small ones, which might be polymer coils or small micelles. Make the simplest assumptions that still leave something to compute. The large particles are hard: nothing can overlap them. The small ones cannot enter the large ones either, but among themselves they are ideal — they pass through one another freely, like the molecules of a dilute gas. No particle attracts any other.

A small particle’s centre cannot come closer to a large particle’s centre than the sum of their radii. So around every large sphere of radius $R$ there is a shell of thickness $\delta$, the small particles’ radius, that their centres are excluded from, though nothing else stops them going there. The small particles can be anywhere in the liquid except inside the large spheres and inside these shells.

When two large spheres are far apart, each excludes the small particles from its own shell, and the total excluded volume is two shells. When they come close enough that their shells overlap — that is, when the gap between their surfaces is less than one small particle’s diameter — the overlap region, shaded in the figure, is excluded only once. The small particles have gained that much volume. Nothing about the large spheres’ own energy has changed; they have neither attracted nor repelled each other. But the small particles now have more room.

More room is more arrangements. For ideal small particles at number density $n$, the number of ways of placing them grows by a factor $e^{n V_{ov}}$ when the available volume grows by $V_{ov}$, so their entropy rises by $k n V_{ov}$, and the free energy of the whole mixture falls by $kT\,nV_{ov}$. A state that can happen in more ways is more probable, by exactly the factor in the exponential that decides everything. Seen from the large spheres, the mixture behaves as though they were attracted by a potential $U(r) = -kT\,n\,V_{ov}(r)$ — an attraction that is nothing but the small particles’ room, counted.

Fumio Oosawa and Sho Asakura derived this in 1954, for colloids in a solution of polymer, and gave the mechanism a second description that is often easier to picture. The small particles exert an osmotic pressure $nkT$ on everything they touch — the pressure that comes from counting, van 't Hoff’s law for a dilute solute, the pressure a membrane that passes the solvent but holds back the solute would feel. When two large spheres are close together, the small particles are excluded from the gap between them and cannot press on the facing surfaces, while they still press on the outer surfaces. The unbalanced pressure pushes the spheres together. Integrated over the surfaces, the force is exactly the derivative of the counting potential: two descriptions of one effect, one in terms of entropy and one in terms of momentum delivered by collisions.

Finding the force without putting it in #

The argument has a step that might seem too quick: it goes from “the small particles have more room” to “the large ones attract”. The most direct test is a simulation that contains no attraction at all and asks how often the large particles are found close together.

The simulation is as stripped-down as the model. Two large discs sit on a line in a two-dimensional box with 270 small particles. A step either redraws every small particle at a random position anywhere it does not overlap a large disc, or tries to move one large disc a little; any move that does not put a small particle inside a large one is accepted. There is no energy anywhere in the program. Yet the histogram of the discs’ separation, the dots in the figure, is not flat: they are found touching about two and a half times as often as far apart, and the excess falls away smoothly to nothing at a gap of one small diameter, where the exclusion zones stop overlapping. The line is the counting prediction, the number of ways to place the small particles at each separation, and it has no adjustable part.

In a real suspension the large particles explore those separations by their own thermal wandering, the motion that the jiggle that proved atoms traces to molecular impacts; a micrometre sphere in water diffuses across its own diameter in about a second, so a pair samples its well many times a minute and the histogram builds up in an afternoon of video. That histogram is what an attraction looks like to anything that measures where particles are. The potential that would produce it, $-kT$ times the logarithm of the ratio, is about one $kT$ deep for these sizes — comparable to the thermal energy, enough to make the discs spend much more time together than apart but not enough to hold them permanently. In three dimensions, and with smaller additives, it is deeper.

The same measurement has been made on real colloids. In the late 1990s, Arjun Yodh’s group at the University of Pennsylvania held pairs of plastic spheres a micrometre across in a line of laser light — an optical tweezer that let the spheres move along it but not away from it — and recorded their separations by video, in water containing much smaller spheres or long DNA molecules. The probability distribution of the separation gave the potential directly, as in the figure, and the depletion well appeared with the depth and range the theory predicts at low concentrations. At higher ones a repulsive bump appeared beyond the well, which the simplest model does not have, and which is taken up below.

How deep and how wide #

In three dimensions the overlap of two spherical shells of radius $a = R + \delta$, whose centres are a distance $r$ apart, is $(\pi/12)(4a + r)(2a - r)^2$, and the potential follows at once.

Two features are fixed by geometry. The range of the attraction is one small particle’s diameter, because at a gap of $2\delta$ the shells just stop overlapping, so a fine bath produces a short, sharp well. And the depth at contact works out to $\phi(3/2q + 1)\,kT$, where $\phi$ is the small particles’ volume fraction and $q = \delta/R$ their size relative to the large ones. At a fixed volume fraction, halving the size of the small particles multiplies their number by eight while the overlap region shrinks only by about four, so a finer additive pulls harder: at twenty per cent by volume, particles a twentieth of the large radius give a well six kT deep, deep enough to hold large spheres together almost indefinitely.

That proportion is what makes the effect practical. Polymer coils a few tens of nanometres across, added at a few per cent to a suspension of particles a micrometre across, produce wells of several kT, and the suspension flocculates. In molecular biology the same principle is used to purify DNA and viruses: polyethylene glycol added to a solution excludes itself from the space around the large molecules, and they come out of solution together, to be collected by centrifuging. In paint and in food the effect is a hazard. A polymer thickener added to give a product body can make the pigment or the fat droplets cluster and separate — the creaming of oil-in-water emulsions thickened with polysaccharide gums such as xanthan has been traced to just this, the gum depleting the space between the droplets.

Walls and corners #

A flat wall is the limit of a large sphere whose radius goes to infinity, and it has a depletion layer of its own: small particles’ centres cannot come within $\delta$ of it. A large sphere approaching the wall overlaps its shell with that layer, and since a flat surface curves away from the sphere more slowly than another sphere would, the overlap is bigger.

For the same bath the contact depth is twice as large against a wall as against another sphere, and in a right-angled corner, where two walls both overlap the shell, it is larger again — four times, for these sizes, computed here by sampling the overlap region. Large particles in a depletion bath therefore migrate to the walls of their container and accumulate along its edges and in its corners. In 1996 Anthony Dinsmore, Yodh and David Pine used exactly this to steer colloidal spheres with no chemical treatment at all: they made a glass surface with terraces and steps, and the spheres in a bath of smaller ones gathered along the bottoms of the steps, where the most room is returned, and were pushed away from the tops of the steps, where a convex edge returns less than a flat surface would. The surface’s shape alone directed the particles, and the energies involved were a few kT.

The same geometry makes the depletion force selective about shape. Two flat surfaces brought face to face overlap their depletion layers over their whole area, so plates and rods attract far more strongly than spheres of similar size, and particles with matching surfaces — a key-shaped particle and a lock-shaped one — attract more strongly in the orientation where they fit. Colloidal “lock and key” particles assembled in this way, with a spherical particle seated in a dimple on another, were demonstrated in 2010, the dimple returning more room than any other contact.

When counting makes a suspension condense #

A single pair of particles in a well a few kT deep is a statement about probabilities. Many particles in such wells is a statement about phases: if the attraction between them is strong enough, a suspension that was evenly spread separates into a dense region and a dilute one, like a vapour condensing into a liquid. The measure that captures how strongly a pair interacts, averaged over all separations, is the second virial coefficient, the first correction to the ideal-gas pressure — the same quantity the first correction to the gas law computes for a real gas from its intermolecular potential.

For hard spheres alone the coefficient is positive, four times the spheres’ volume per particle, because they repel. The depletion attraction lowers it, and at a high enough concentration of the small particles it becomes negative — attraction has won on balance. A finding of 2000 by Gerard Vliegenthart and Henk Lekkerkerker is that for short-ranged attractions of any origin, the critical point of condensation falls where the ratio to the hard-sphere value is close to $-1.5$, almost independently of the attraction’s shape. The figure reaches that line at a small-particle volume fraction of about 0.16 for an additive a twentieth of the large radius, 0.26 for a tenth and 0.38 for a fifth: the finer the additive, the less of it is needed to make the large particles condense.

What condenses is not always a liquid. When the attraction is very short-ranged compared with the particles’ size, the dense phase that appears is a crystal or a gel rather than a colloidal liquid, because the liquid would sit at energies the crystal undercuts; for additives smaller than about a third of the large particles, a colloid–polymer mixture skips the liquid altogether. The phase diagrams of such mixtures, measured in the 1990s, follow the depletion model closely, and they have made colloids a standard model system for studying how matter condenses and crystallises — with an attraction whose strength is set by how much polymer is stirred in, and which can be switched off by diluting it.

Order that entropy builds #

The depletion force is one instance of a broader surprise: entropy, the measure usually glossed as disorder, can drive a system into a more ordered state. The cleanest example needs no second species at all. A collection of identical hard spheres, with no attraction of any kind, crystallises when it is compressed past a volume fraction of 0.494 — a result found in the first computer simulations of liquids, by Berni Alder and Thomas Wainwright and by William Wood and Jack Jacobson in 1957, and disbelieved by many at the time because nothing seemed to drive it. The resolution is the same counting as here. At high density, spheres in a disordered arrangement jam against one another and each has very little room to rattle; arranged in a lattice, each has a cell of its own to move in, and the gain in freedom of motion outweighs the loss in freedom of arrangement. The crystal has the higher entropy. Lars Onsager had found the same thing in 1949 for long thin rods, which line up into a liquid crystal at high concentration because aligned rods exclude one another from far less volume than crossed ones.

The depletion mixture combines the two. The small particles’ entropy is raised by the large ones’ order, and the large ones gather and crystallise to give the small ones room — a division in which one species becomes more ordered and the other less, and the total count of arrangements rises, as mixing what is already mixed would require of any spontaneous change. Seen as a whole, there is no paradox: the order of the large particles is paid for, many times over, by the freedom of the small ones.

The crowded cell #

The interior of a living cell is not a dilute solution. Proteins, nucleic acids and other large molecules take up a fifth to a third of its volume, and every molecule in it is surrounded by others it cannot overlap. Allen Minton and others argued from the 1980s that this crowding produces depletion-like effects on everything in the cell: reactions that join two molecules into one, or that fold a chain into a compact shape, return volume to the crowding molecules and are favoured by their entropy. Measured association constants of proteins in solutions crowded with inert polymers rise by factors of ten or more over their values in dilute buffer, and the rise follows the excluded-volume arithmetic in its broad features.

A bacterium’s single chromosome, a millimetre of DNA packed into a cell a micrometre long, is held compact partly by the same effect: the crowding proteins of the cytoplasm are excluded from the space the DNA occupies, and the DNA is pushed into a smaller volume than it would occupy on its own. The cell is full of attractions that no molecule exerts — they are the sum of everything else’s room.

Where the ideal bath stops being ideal #

The model treats the small particles as ideal, passing freely through one another. That is a good approximation for polymer coils at low concentration, which interpenetrate, and a poor one for hard small particles at high concentration. Hard small particles pack in layers against any surface, and between two large particles the layers produce a potential that oscillates, with a repulsive barrier beyond the attractive well — the bump that appeared in the optical-tweezer measurements as the concentration was raised. The simulation is two-dimensional, which changes the numbers but not the mechanism, and it uses a box small enough that the large discs’ exclusion zones are a few per cent of its area, so that the counting prediction is the exact finite-box expression rather than the infinite-bath potential. The corner and wall curves treat the walls as hard and smooth. Real colloids also carry charges, van der Waals attractions of the kind the attraction that needs no charge describes, and hydrodynamic interactions; the depletion force adds to these rather than replacing them, and experiments are designed to suppress the others in order to see it alone.

Still open: how far crowding explains the cell #

The depletion picture of macromolecular crowding predicts large effects from excluded volume alone, and the experiments with inert crowding polymers confirm them. Inside real cells the measured effects are often smaller, and sometimes of the opposite sign, because the crowding molecules are not inert: proteins carry charges and sticky patches, and weak attractions between a protein and its crowded surroundings can cancel or reverse the entropic push towards association. How much of the behaviour of molecules in a living cell is the pure counting effect, and how much is a balance between that and a large number of weak, specific interactions, is an active question, measured with labelled proteins inside living cells and with simulations of entire bacterial cytoplasms.

The habit worth carrying away is to ask who else gains when two things come together. Two large particles in a bath of small ones that cannot enter them hand back the overlap of their exclusion shells as room, so the small particles’ entropy pushes them together with a well $\phi(3/2q + 1)\,kT$ deep — 3.2 kT for an additive a tenth their size at a fifth of the volume, twice that against a wall, four times in a corner — though nothing in the mixture attracts anything. An attraction can belong to neither of the bodies it acts on, and be nothing but somebody else’s arrangements, counted.