The like charges that pull together

Cement sets because microscopic sheets of calcium silicate hydrate, each carrying a strong negative charge on its surfaces, stick to one another with a few calcium ions between them. A clay soil soaked with sodium ions swells without limit when wet and turns to mud; the same clay with calcium in place of sodium swells to a fixed spacing of about a nanometre and stops. DNA in water with sodium stays an extended chain, and with spermidine or spermine it collapses into a dense toroid. Each case has two surfaces of the same charge pulling together, with no attraction between the surfaces themselves. The only thing that changed was the charge of the small ions between them.

The counterions that never leave the chain left this question open. Two DNA chains each ninety per cent neutralised by condensed ions still carry net charges of the same sign, and the Poisson–Boltzmann theory of the ions around them — the theory that located the ninety per cent — says they repel. It does not say so for particular numbers; it can be proved that within that theory like-charged objects in an electrolyte of any composition always repel. The attraction is real and measured, so the theory is wrong, and the useful thing is to find where.

The simplest system that shows it #

Take two flat parallel planes, each carrying a surface charge density $\sigma$ (in elementary charges per unit area), a distance $d$ apart. Between them put only the counterions that neutralise them, each carrying $q$ elementary charges, and no added salt. Water has a Bjerrum length $\ell_B = 0.71$ nanometres, the distance at which two unit charges interact with an energy equal to the thermal energy.

Two lengths organise everything. The first is the Gouy–Chapman length,

$$\mu = \frac{1}{2\pi q\,\ell_B\,\sigma},$$

the distance from a single charged plane at which a counterion’s electrostatic energy of attraction to the plane has risen by one $kT$. It is the thickness of the counterion layer next to an isolated plane. The second is the typical distance between neighbouring counterions on the plane, which for $q$ charges per ion and $\sigma$ per unit area is about $\sqrt{q/\sigma}$. The ratio of the two, squared up to a constant, is the coupling parameter

$$\Xi = 2\pi q^3\,\ell_B^2\,\sigma,$$

and it measures how strongly the counterions feel each other compared with how strongly they feel the plane. The pressure has a natural unit too, $2\pi\ell_B\sigma^2 kT$, which is $\sigma_e^2/2\varepsilon$ with $\sigma_e$ the surface charge in coulombs — the electrostatic pressure a surface charge exerts on a conductor, the pressure a charge puts on its own metal.

Mean field always pushes #

The Poisson–Boltzmann theory treats the counterions as a smooth density $n(z)$ that responds to the average electrostatic potential, which the density itself creates. Between two identical planes with counterions only, the equation has an exact solution: $n(z) = n_0/\cos^2(kz)$ measured from the midplane, with $k$ fixed by the condition that the ions exactly neutralise the two plates, $k\mu\tan(kd/2) = 1$.

The pressure follows from an argument that needs no solution. At the midplane the electric field vanishes by symmetry, so the only thing pushing across that plane is the counterions arriving at it, and the pressure is $kT$ times the density there. A density is never negative. The mean-field pressure is positive at every separation: 1.71 in natural units at one Gouy–Chapman length, 0.29 at four, falling as $\pi^2/d^2$ far apart and rising as the ideal-gas $2/d$ when the plates are pushed together until the counterions have nowhere to go. The figure computes the midplane density from the exact solution at each separation, checks that the counterions integrate to exactly the charge of the two plates, and checks both limits.

The argument is more general than the geometry. For any shapes and any mixture of ions, the mean-field force between like-charged bodies can be written as a stress integrated over a surface between them, and every term in that stress has the sign of a repulsion. That was proved in 1999, independently by two groups, and it is the precise sense in which the attraction seen in cement and DNA is not a subtle correction the theory underestimates. It is a force the theory cannot contain.

What the smooth cloud gets wrong #

The assumption is that each counterion responds to the average potential of all the others. That is accurate when there are many ions within the range of the potential and each interacts weakly with any one neighbour, so that a given ion sees a crowd rather than individuals. It fails when a single neighbour matters.

Put numbers on it for a moderately charged surface, half an elementary charge per square nanometre. With monovalent ions the Gouy–Chapman length is 0.45 nanometres and the ions sit about 1.4 nanometres apart on the surface: the layer is thicker than it is sparse, and the mean field is a fair description. With trivalent ions the layer thins to 0.15 nanometres, three times thinner, because each ion is pulled three times as hard, and the ions are also three times fewer, about 2.5 nanometres apart. Each ion carries three charges and repels its neighbours nine times as strongly. The layer is now a sparse, flat arrangement of highly charged ions, each much farther from its neighbours than from the plane, and each so strongly repelled by them that it clears a region of surface around itself — a correlation hole — in which no other counterion is found.

The panels show the geometry this produces. At $\Xi = 1.6$ the spacing between counterions is 3.4 Gouy–Chapman lengths, only 1.7 times the gap at which the planes would come to rest, and each ion has many neighbours within the range that matters. At $\Xi = 12.7$ the spacing is 9.6 lengths, 4.8 gaps. At $\Xi = 101$ it is 27 lengths, 13.5 gaps. When the spacing is much larger than the gap, an ion’s nearest charges are not other counterions but the two planes on either side of it, and each ion lives in a column of the gap that belongs to it alone. The hexagonal order is drawn for clarity; at room temperature the arrangement is a strongly correlated liquid rather than a crystal, but the correlation hole, which is what matters, is there in either case.

The limit where one ion decides #

Rolland Netz and André Moreira showed in 2000 that the limit of very large $\Xi$ can be taken exactly, and that it is as simple as the mean field in the opposite direction. In the mean field every ion sees all the others as a smooth cloud. In the strong-coupling limit each ion sees only the planes, because its correlation hole keeps every other ion too far away to matter at leading order.

A single ion between two equal planes of charge is in a region where the planes’ fields cancel. The field of an infinite charged plane does not fall off with distance, so between two identical planes it is equal and opposite everywhere — the same cancellation that empties the inside of a conductor of field. The ion feels no force anywhere in the gap and is equally likely to be anywhere in it.

The two profiles make the difference visible. In the mean field the counterions are drawn towards the walls whose charge attracts them, and the more room the gap gives them the more they pile up there: 2.58 times the mean density against each wall at four Gouy–Chapman lengths, 0.58 at the midplane. In the strong-coupling limit the profile is flat at every separation. The ions have not been released from the planes — they are as tightly bound as ever, since a counterion that left the gap would leave its own correlation hole uncancelled — but inside the gap nothing prefers one position to another.

The pressure follows from a theorem that holds whatever the model: the contact-value theorem. The pressure on a charged plane is $kT$ times the density of counterions touching it, minus the electrostatic pull $\sigma_e^2/2\varepsilon$ of the plane towards the layer of ions that neutralises it, which in natural units is exactly one. In the mean field the contact density is always at least that large, which is another way of reading the positive pressure. In the strong-coupling limit the ions are spread uniformly across the gap, so the contact density is the total of two plates’ worth of counterions divided by the width, $2/d$ in natural units, and

$$\tilde P = \frac{2}{d/\mu} - 1.$$

When the plates are closer than $2\mu$ the ions are crowded enough to push them apart. Beyond $2\mu$ they are too dilute at the walls, and the plates are pulled together by the electrostatic attraction between each plate and its own counterions. The free energy per unit area, which is minus the integral of the pressure, is $d/\mu - 2\ln(d/\mu)$ up to a constant, and it has a minimum at exactly $d = 2\mu$. The plates come to rest there, pressed together by nothing but their own neutralising ions.

The cube of the valence #

The coupling parameter is where the chemistry enters, and it enters steeply.

The valence appears cubed: once from the stronger pull to the plane, which thins the layer, and twice from the stronger repulsion between ions, which are also fewer. At half a charge per square nanometre, $\Xi$ is 1.6 for sodium, 12.8 for calcium, 43 for a trivalent ion and 102 for spermine’s four charges. Even a surface as highly charged as one elementary charge per square nanometre gives monovalent ions only 3.2.

Neither limit applies exactly in between, and there the answer has come from simulation. Monte Carlo simulations of explicit point ions between charged planes find the pressure following the mean field at small $\Xi$, approaching $2/d - 1$ at large $\Xi$, and becoming attractive at some separations once $\Xi$ exceeds roughly ten. The dashed line is placed there, and it separates the ions by valence rather than by surface: monovalent ions lie below it on every surface drawn, divalent ions cross it on highly charged surfaces, and trivalent and tetravalent ions lie above it on almost all of them.

That matches what is seen. Sodium never condenses DNA and never holds clay sheets or cement grains together. Calcium holds cement and clay, whose surfaces carry of the order of one charge per square nanometre or more, and does not condense DNA in water, whose effective charge after condensation is lower. Spermidine, with three charges, and spermine, with four, condense DNA readily. The observed boundary is not exactly the drawn line, because the real systems are neither flat nor made of point ions, but the ordering by valence is the ordering by $\Xi$.

How close they come #

In the strong-coupling limit the equilibrium gap is $2\mu = 1/(\pi q\ell_B\sigma)$, and it is small.

For trivalent ions the curve becomes trustworthy above 0.14 charges per square nanometre, where the predicted gap is 1.07 nanometres; for tetravalent ions above 0.06, at 1.90 nanometres; for divalent ions only above 0.47, where the gap is already down to 0.48 nanometres. Wherever the coupling is strong enough for the theory to hold, the gap it predicts is a nanometre or less, and much of the solid part of every curve lies in the band where the gap would be narrower than the ions themselves. A calcium ion with its first shell of water is about six tenths of a nanometre across.

That is not a failure of the argument; it is where the argument hands over to chemistry. The theory says the planes want to approach to within about two Gouy–Chapman lengths. What stops them in practice is the size of the ions and the water structured around them, and the observed spacings sit at about that scale: calcium clays stop swelling at a water layer or two between sheets, DNA condensed by multivalent ions packs with its surfaces about a nanometre apart, and the sheets in cement are held at a spacing comparable to a hydrated calcium ion. A correct theory at that point has to include the ions’ sizes, and the simulations that reproduce cement’s cohesion do.

A surface that takes more charge than it gives #

The same correlations predict something the mean field forbids even more plainly, and it has been seen. A charged surface in a solution of multivalent salt, rather than with only its own counterions, can bind more counterions than it needs to be neutral, and end up with a net charge of the opposite sign.

The reason is the correlation hole again. An ion arriving at a strongly correlated layer does not join a smooth cloud; it pushes the neighbouring ions aside and sits in a hole of its own, and the surface charge exposed in that hole attracts it much more strongly than the average potential would suggest. For a trivalent ion the energy gained this way is several $kT$. Once that gain exceeds the cost of adding charge to a surface that is already neutral, more ions arrive, and they stop only when the surface’s reversed charge makes the next one pay as much as it gains. Boris Shklovskii and his collaborators worked out the size of the effect around the turn of the century, and it is called charge inversion.

The consequence is visible without any theory. A colloidal particle carrying a negative charge, kept suspended in water by its own thermal jiggling, drifts towards the positive electrode in an electric field. Add a small concentration of a trivalent salt such as lanthanum chloride and the drift slows, stops, and then reverses: the particle now moves towards the negative electrode, carrying a positive charge made of more lanthanum ions than its own surface can neutralise. The same reversal happens to DNA in solutions of spermine at high enough concentration, which is one reason DNA condensed by multivalent ions can dissolve again when far more of the ion is added.

The effect of valence on clay is the agricultural version of the whole argument. A soil whose clay particles carry mostly sodium on their surfaces disperses when wet, because the platelets repel; its fine particles block the pores, water stands on the surface, and the soil sets hard when it dries. The standard remedy is to spread gypsum, calcium sulphate, which slowly dissolves and exchanges calcium for sodium on the clay. The coupling parameter on the clay’s surfaces rises eightfold, the platelets begin to attract, and they gather into crumbs with pores between them. The chemistry of the remedy is an ion exchange. What makes it work is the cube of the valence.

A correlation no density profile can draw #

The profile figure draws the density of counterions across the gap, and in the strong-coupling limit that density is flat — which looks like nothing at all happening. The attraction is not in the density. It is in the correlation between positions: given that one counterion is here, no other is within its correlation hole, on this plane or across the gap. That is a statement about pairs of ions, and a one-particle density, averaged over all positions of all the others, erases it by construction. The lattice figure puts the correlation back, but only by drawing a perfect staggered arrangement frozen in place, when at room temperature the ions form a liquid whose order extends a few neighbours at most and rearranges continually. Neither figure shows the thing that makes the pressure negative, because what makes it negative is how the ions avoid each other, and avoidance has no picture in a single frame.

Where the two limits leave off #

The planes are flat, infinite and smooth. DNA is a cylinder with charges spaced along a helix, a clay platelet is finite with charged edges, and the charges on any real surface are discrete. Discreteness matters most exactly where coupling is strong, because an ion that can sit in a pocket between surface charges is more tightly held than one on a smooth plane, and the pattern of charge on two facing surfaces can favour particular registrations of one against the other.

The ions are points and the water is a continuum. Strong coupling pushes ions to within a fraction of a nanometre of the surface, where both assumptions are weakest. Real counterions have a hydration shell that resists being stripped, the water’s permittivity near a charged surface is lower than in bulk, and the surface itself has a different permittivity from the water, so each ion is attracted or repelled by its own image. Each of these changes the numbers and none changes the sign of the effect.

No salt was added. Added salt screens the interactions beyond the Debye length, which shortens the range of both the repulsion and the correlation attraction. In a mixture of monovalent salt and multivalent counterions, the multivalent ions dominate the layer next to the surface while the salt sets the screening, and whether attraction appears depends on the balance.

And the intermediate regime has no closed theory. Between $\Xi$ of about one and about a hundred, where most real systems with divalent ions sit, neither limit is exact. Corrections to the strong-coupling limit in powers of $1/\sqrt{\Xi}$ improve it, and interpolating theories that treat each ion’s correlation hole explicitly reproduce the simulations over much of the range, but the crossover is described by fitting rather than by a single result.

Still open: whether correlation or chemistry holds DNA together #

Counting particles gave the pressure that comes from counting; a charged gel gave the swelling a membrane cannot stop, and a charged chain gave the counterions that condense onto it. Here the counting fails in a precise way. The ions are still counted at the wall, but their positions are no longer independent, and the correlation between them is the whole of the force.

The habit worth carrying away concerns what a mean field can and cannot contain. A mean-field theory can get a quantity wrong by a large factor and still give the right sign; when a force’s sign is set by correlation, the mean field gets the sign wrong, and no refinement of it within the same assumptions will fix it. The measure of when to stop trusting the average is a single dimensionless number comparing an ion’s interaction with its neighbours to its interaction with the field, and here that number rises as the cube of the valence.

What is not settled is how much of the attraction between real DNA helices is correlation of the kind drawn here and how much is specific. Measurements of the force between DNA molecules packed in arrays under osmotic stress find that the attraction depends on which multivalent ion is used, in ways a theory of point charges cannot reproduce: two ions of the same valence give different spacings, and the helical pattern of the phosphates appears to matter. Models that let ions sit in the helix’s grooves, correlating across the gap with the grooves of a neighbour, account for some of this. Whether the right description is a correlated liquid, a lattice locked to the helix, or specific binding that happens to bridge two molecules is argued over with data at the ångström scale, and the answer may be different for spermine than for cobalt hexammine.