The counterions that never leave the chain

The swelling a membrane cannot stop built the pressure of a charged gel out of two conditions — neutrality on each side of a membrane, and equal chemical potential for the salt — and one assumption it said plainly and did not test. The fixed charge was smeared through the gel as a uniform density, and every counterion was free to go wherever the two conditions sent it.

Charges on a polymer are not smeared. They sit along chains, a fixed distance apart, and near a chain the electric field is strong enough to matter more than any average. When the charges are close enough together, a large fraction of the counterions never leave the chain at all, however much solvent surrounds it. The threshold that decides “close enough” is a single length, it comes from the solvent rather than from the polymer, and in water it is seven tenths of a nanometre.

A length at which a charge and a temperature agree #

Two unit charges in a solvent of relative permittivity $\varepsilon_r$ attract or repel with an energy that falls as the inverse of their separation. The distance at which that energy equals the thermal energy $kT$ is the Bjerrum length,

$$\ell_B = \frac{e^2}{4\pi\varepsilon_0\varepsilon_r kT},$$

which in water at room temperature is 0.71 nm. Closer than that, electrostatics wins over thermal agitation; farther, agitation wins. In a vacuum the same length is 56 nm, and the factor of 78 between them is water’s permittivity — the dielectric screening the field the matter takes away describes, arriving here as the reason salts dissolve and charged molecules stay in solution at all.

A charged polymer has a charge every $b$ along its backbone, and the ratio $\xi = \ell_B/b$ is the whole of the argument. DNA carries a phosphate charge every 0.17 nm along its axis, so $\xi = 4.2$. Sodium polyacrylate, the gel in disposable nappies, has one every 0.25 nm and $\xi = 2.85$. Hyaluronan, the lubricant in joints, has one per nanometre and $\xi = 0.71$. Those three numbers turn out to put the three polymers on different sides of a threshold, and the threshold is at one.

A contest between two logarithms #

The threshold is easiest to see with a single counterion and nothing else.

The field of a long line of charge falls off as the inverse of the distance from it, and the shape of a source decides how a field falls off: a point charge’s field goes as $1/r^2$ and its potential as $1/r$, but a line’s field goes as $1/r$ and its potential grows as the logarithm of the distance. A counterion at distance $r$ from the chain therefore has a potential energy of $2\xi\,kT\ln r$ plus a constant, and its Boltzmann weight — the exponential that decides everything — is $r^{-2\xi}$.

Against that, the counterion has entropy. The volume available to it at distance between $r$ and $r + dr$ from the chain grows in proportion to $r$, and the entropy of being somewhere among that much room grows as the logarithm. So the probability of finding the counterion within some distance of the chain is a ratio of two integrals of $r \cdot r^{-2\xi}$, and everything depends on whether that integral converges at the chain.

Below $\xi = 1$ the integral converges as the rod is thinned to a line. The counterion’s energy near the chain is not low enough to beat the entropy of the far field, and it wanders off: at $\xi = 0.5$ the chance of finding it near the chain is indistinguishable from nothing. Above $\xi = 1$ the integral diverges at the chain, the weight near the line swamps everything else, and the counterion is captured: at $\xi = 1.1$ the chance is 0.996. At exactly $\xi = 1$ the two logarithms grow at the same rate and the answer depends on the logarithm of the ratio of the cell to the rod — two thirds, in the figure, and still changing.

Condensation is that divergence. Nothing chemical is involved; no bond is formed. A counterion near a sufficiently dense line of charge is trapped by the same arithmetic that makes some integrals finite and others not.

A real chain has many counterions, and they do not all collapse onto it. As the first ones condense, they cancel part of the chain’s charge, which lowers its effective $\xi$. Condensation continues exactly until the effective $\xi$ has fallen to one, at which point the next counterion sees a chain whose logarithm no longer wins. The fraction of the charge neutralised is therefore $1 - 1/\xi$. That is Gerald Manning’s result of 1969, anticipated in a two-state picture by Fumio Oosawa, and it has one striking feature: nothing about the chain enters it except the spacing of its charges.

The same contest, with a different pair of logarithms, is the transition with nothing to order: a vortex in a thin superfluid film has an energy that grows as the logarithm of the film’s size and an entropy that grows as the same logarithm, and whether vortices bind in pairs or wander free is decided by which coefficient is larger. Counterion condensation and the Kosterlitz–Thouless transition are one calculation, applied once to charges beside a line and once to vortices in a plane — which is not a coincidence, because a line charge seen end-on is a point charge in two dimensions, and a two-dimensional point charge’s potential is the logarithm.

The chain that keeps its counterions #

The single-counterion argument gives the threshold. How the many counterions actually arrange themselves requires solving for the potential they create along with the chain’s, and that is the Poisson–Boltzmann equation: the counterion density at each place is a Boltzmann factor of the potential there, and the potential is produced by the chain together with that density.

The standard way to pose the problem is to put one rod at the centre of a cylindrical cell whose radius is set by the concentration — the more dilute the solution, the wider each chain’s cell. Raymond Fuoss, Aharon Katchalsky and Shneior Lifson found the exact solution of that problem in 1951; the figure solves it numerically and checks the solution by counting its counterions, which must cancel the rod’s charge exactly.

If counterions were free, diluting the solution would spread them: the cell grows, and a counterion that can be anywhere in it would spend less and less of its time near the rod. Instead each curve develops a plateau. Close to the rod the enclosed fraction rises steeply; then it stalls; then, far out, the remaining counterions spread through the cell as a dilute cloud. Widen the cell by a factor of a hundred and the cloud moves outward while the plateau stays exactly where it was, at 0.762 of the counterions — $1 - 1/4.2$.

The number is exact in a precise sense. At the point where the curve’s slope against the logarithm of distance is steepest — its inflection — the enclosed fraction is $1 - 1/\xi$ for every cell size, a property of the equation that the figure recovers from its own numerical solution at each dilution. Three quarters of DNA’s counterions stay within a few nanometres of the chain in a solution a million times more dilute than a cell nucleus, and the quarter that leave are the only ones free to count.

The condensed counterions are not stuck in place. They move freely along the chain and exchange with the free ones constantly, which is why the word is condensation rather than binding and why the layer has no chemical specificity: sodium, potassium and caesium condense onto DNA to the same extent, because the argument knows only their charge.

What the osmometer counts #

The pressure that comes from counting establishes that an osmotic pressure counts free particles and ignores what they are. Condensation removes particles from the count without removing them from the solution.

In the cell model the pressure is simple to read: it is the counterion density at the edge of the cell times $kT$, the density of the counterions farthest from any chain. Divided by what the same number of counterions would exert as an ideal gas, it gives the osmotic coefficient, and Manning’s picture predicts its dilute limit directly. Below the threshold the counterions are nearly free and the coefficient falls only slowly from one, as $1 - \xi/2$. Above it, only the fraction $1/\xi$ is free, and those feel a chain whose effective charge is exactly at threshold, which halves their contribution again: the coefficient is $1/2\xi$.

For DNA that is 0.119, and the Poisson–Boltzmann cell reaches 0.122 at a cell of $10^8$ rod radii, approaching the limit slowly because the approach is logarithmic in the dilution. A solution of DNA carrying a mole of counterions exerts about an eighth of the osmotic pressure a mole of salt would, not because seven eighths of the counterions are attached to anything but because the chain’s field keeps them within reach of it.

Measured osmotic coefficients of real polyelectrolytes are somewhat higher than the limiting law, typically by tens of per cent, for reasons that include the finite concentration and the discreteness of the charges along the chain. But the order of magnitude, the insensitivity to which monovalent counterion is present, and the dependence on the spacing of the charges are all as the argument predicts, and they are the three things a model built on smeared-out charge gets wrong.

The charge the water sees #

The Donnan balance of the swelling a membrane cannot stop counted every fixed charge in a gel as demanding one free counterion. Above the threshold, only the uncondensed charge makes that demand.

The two regimes of the Donnan balance respond to condensation differently, which is itself a test of the picture. In a bath with little salt the pressure is simply the free counterion concentration counted as a gas, so dividing the effective charge by ξ divides the pressure by ξ: 1,235 kPa becomes 430. In a bath with plenty of salt the fixed charge is screened, the pressure goes as its square, and the same condensation divides the pressure by nearly $\xi^2$: at physiological salt, 702 kPa becomes 118.

A superabsorbent gel is sold by how much water it takes up, and it takes up hundreds of times its own weight in distilled water and a few tens of times in urine. Both numbers are well below what the fixed charge of the polymer would give if every counterion were free, and the difference between the two is the change of regime. The charge the polymer was synthesised with is not the charge the water sees, and no amount of adding charge along a chain beyond one per Bjerrum length makes the gel swell harder — the extra charge is neutralised by counterions that condense onto it as fast as it is added.

Cartilage’s charged molecules, by contrast, sit mostly below or near the threshold. Hyaluronan at $\xi = 0.71$ condenses nothing; chondroitin sulphate, with two charges per disaccharide, sits a little above one. The swelling pressure that holds a joint apart is therefore close to what the smeared-out picture predicts, which is one reason that picture worked as well as it did for tissue.

Three charges instead of one #

A counterion carrying charge $z$ feels $z$ times the chain’s field and neutralises $z$ charges when it condenses, so it condenses until the effective charge parameter has fallen to $1/z$, neutralising $1 - 1/z\xi$ of the chain.

For DNA that gives 76 per cent with sodium, 88 per cent with magnesium and 92 per cent with a trivalent ion such as spermidine. The difference between the last two looks small, and it is the difference between a DNA molecule that stays extended in solution and one that collapses into a toroid a few tens of nanometres across. Experiments with a range of multivalent ions find the collapse when roughly 89 to 90 per cent of the phosphate charge is neutralised: magnesium falls just short and does not condense DNA in water; spermidine, spermine and cobalt hexammine carry it over and do.

That collapse is how cells and viruses pack DNA. A sperm head holds its DNA folded by arginine-rich proteins carrying many positive charges; bacteriophages pack their genomes into capsids at densities approaching a crystal’s, with polyamines helping to neutralise the phosphate backbone. The threshold that decides it is set, in the mean-field picture, by the Bjerrum length of water and the spacing of phosphates on a double helix.

What the mean-field picture cannot produce is the collapse itself. Two chains each 92 per cent neutralised still carry a net charge of the same sign and, on average, repel. The attraction that pulls them together comes from correlations among the condensed counterions — a trivalent ion on one chain sitting opposite a gap between ions on the other — which the Poisson–Boltzmann equation, by replacing the counterions with their average density, cannot contain. The figure’s percentages locate the threshold; the force that acts past it is outside the model.

Where the smeared cloud stops describing the counterions #

The equation is a mean field. Poisson–Boltzmann replaces the counterions by a smooth average density, which is accurate when the ions are numerous and weakly coupled and fails as the valence rises. For monovalent ions on DNA the error is modest; for trivalent ions it is large enough to reverse the sign of the force between chains, and the numbers here are the mean-field numbers.

The chain is an infinite straight rod with smeared charge. Real charges are discrete, a chain has ends, and a flexible polyelectrolyte bends. DNA’s persistence length of about fifty nanometres makes the rod a fair description over the few nanometres where condensation happens; a flexible synthetic chain is less rod-like, and its effective spacing depends on its local conformation.

No salt has been added. The cell model here is salt-free. Added salt screens the chain beyond a Debye length, the distance past which the long-range force does not reach, and changes the free counterions’ distribution while leaving the condensed fraction roughly where Manning puts it — a result that holds approximately and is the subject of continuing refinement.

The solvent is a continuum. The Bjerrum length uses water’s bulk permittivity, and within a nanometre of a highly charged chain the water is oriented and its effective permittivity lower. The condensed layer sits exactly where that matters most.

And the limiting law is a limit. Manning’s $1 - 1/\xi$ and $1/2\xi$ are statements about infinite dilution. The cell solutions approach them logarithmically, so a real solution at a real concentration sits measurably away from both.

What the profiles cannot show #

Every figure here is a time average of counterions that never stop moving. A condensed counterion slides along the chain, leaves it, is replaced by another, and the fraction condensed is a statement about how much time counterions spend near the chain rather than about which ones are there. That motion is measurable — the condensed ions carry current along the chain, and nuclear magnetic resonance of sodium distinguishes condensed from free by how fast they tumble — and the profiles, being averages, contain none of it.

Nor do they show a chain. DNA in a cell is folded, crossed, wound round proteins and packed against its neighbours, and the cylindrical cell with one straight rod at its centre is an abstraction of that crowd. The threshold survives the abstraction because it is set within a nanometre of the chain; the pressures and the free fractions, which depend on where the free counterions go, depend on the crowd.

Still open: how like charges come to attract #

The collapse of DNA by trivalent ions, the bundling of charged filaments in cells, and the attraction between like-charged colloids in multivalent salt all require two objects of the same charge to pull together, and no mean-field theory allows it. Correlation theories — in which the condensed multivalent ions form a strongly correlated liquid, or a Wigner-crystal-like lattice, along the surfaces — produce the attraction and predict roughly the right thresholds, and simulations of explicit ions reproduce the collapse.

What is not settled is which description is right where. Near a highly charged surface in multivalent salt the counterions are strongly coupled in one regime and weakly coupled in another, and the crossover, the role of the chains’ finite size, and whether the attraction is best thought of as correlation or as a bridging of specific ions between chains are argued with experiments that measure forces between DNA arrays at ångström precision. Manning’s threshold tells where the question begins, and not how it ends.

The habit worth carrying away is to count in logarithms. When an energy and an entropy both grow as the logarithm of the same distance, the outcome is decided by one ratio of their coefficients, and nothing about the size of the system can change it — which is why a counterion near DNA, a vortex in a superfluid film and a walker in a plane that always comes home all obey rules with a sharp threshold and no dependence on scale.