The pressure that stops counting molecules

The pressure that comes from counting found that osmotic pressure is the ideal gas law with the solute in place of the gas: one $kT$ for every dissolved particle per unit volume, whatever the particle is. A mole of sugar and a mole of a protein push equally hard against a membrane that holds them back. That indifference is what makes osmometry useful. Weigh out a gram of an unknown polymer, dissolve it, measure the pressure, and the number of molecules — and therefore their mass — follows from van 't Hoff’s law.

The later arguments on this subject kept the count and changed what the membrane held back: charged gels, chains with condensed counterions, plates that attract. This one keeps the membrane and the count and asks what happens to the thing being counted when the particles are long, flexible chains and the solution is concentrated enough for them to overlap. The answer is that the count survives and its object changes. Past a concentration that is surprisingly low, the osmotic pressure stops counting molecules and starts counting something smaller, with no chemical identity at all. It then no longer knows how long the chains are.

Two regimes on one plot #

A flexible polymer in a good solvent is a random coil. Its segments are strung together, but beyond a few segments each one points in a direction nearly independent of the last, so the chain wanders like the random walker whose return depends on dimension. The coil’s size is not proportional to its length $N$. For a pure random walk it would grow as $N^{1/2}$. In a good solvent segments repel one another, the walk swells to avoid itself, and the size grows as $N^{\nu}$ with $\nu$ measured and computed at 0.588. Flory’s classic estimate gives 3/5.

The figure plots the osmotic pressure against concentration for three chain lengths, each a hundred times the last. At low concentration each curve is a straight line of slope one: van 't Hoff’s law, with the pressure proportional to the number of chains. A given mass of long chains contains fewer chains than the same mass of short ones, so the longer chains sit lower, by the ratio of their lengths. This is the regime in which an osmometer weighs a polymer.

Each curve then bends upwards at a concentration marked with a dot, and the three bends happen at very different places. Beyond them all three curves merge onto a single line, of slope 2.31, and that line carries no trace of which chains are in the solution. A solution of chains a thousand segments long and a solution of chains a hundred thousand long, at the same mass concentration, push on the membrane with the same pressure.

Two things in the figure need explaining. Why do the bends happen where they do, and why does the chain length vanish from the pressure after them? One idea answers both. It was worked out by des Cloizeaux and de Gennes in the 1970s and confirmed by measurements soon after.

When the coils start to touch #

The dots mark the overlap concentration, $c^*$, at which the coils fill the solution. Each coil of $N$ segments occupies a volume of about $R^3 \propto N^{3\nu}$, so the concentration of segments inside a single coil is $N/R^3 \propto N^{1-3\nu}$. When the overall concentration reaches that value, the coils have run out of space to be separate. Since $1 - 3\nu$ is negative, $-0.76$ in a good solvent, the overlap concentration falls as the chains lengthen. A coil’s volume grows faster than its mass. Each tenfold increase in length lowers $c^*$ by a factor of 5.8, and for long chains it is small indeed: for polystyrene of a million grams per mole it is about four grams per litre, less than half a per cent by mass.

The drawing shows the two situations as random walks. On the left the coils are apart. Each is a loose tangle occupying a region much larger than its own segments fill, and the solvent between the coils is pure. The osmotic pressure counts these tangles, one $kT$ each, as van 't Hoff’s law says. A coil’s internal wriggling does not add to the pressure, because its segments are not free to wander off independently. They are tied to each other, and the chain as a whole is the particle.

On the right there are four times as many coils in the same space and they interpenetrate thoroughly. No region of the box belongs to one chain. What matters now is a length the drawing marks with a circle. Around any segment, out to a certain distance, the neighbouring segments are mostly from the same chain: the stretch of chain nearby is still a small coil of its own, swelling to avoid itself exactly as an isolated chain would. Beyond that distance segments of other chains crowd in, and the excluded-volume repulsion that swelled the coil is screened, because a segment can no longer tell its own chain’s segments from anybody else’s. That distance is the correlation length $\xi$, and the stretch of chain inside it is called a blob.

This is the key step. On scales smaller than $\xi$ the solution looks like a dilute solution of blobs: each blob is a swollen sub-coil, and blobs are packed against one another like the separate coils on the left, at their own overlap. On scales larger than $\xi$ the chains are strings of blobs, and a string of blobs is a random walk with no preferred direction. The osmotic pressure, which counts independent units, now counts blobs:

$$\Pi \approx \frac{kT}{\xi^3} .$$

A length that knows nothing about the chain #

The blob size follows from one requirement. At the overlap concentration the blobs are the whole coils, so $\xi = R$ at $c = c^*$. Above it the blobs shrink as a power of $c/c^*$, and because deep in the semidilute regime the blob is a local object — a stretch of chain surrounded by other chains — its size cannot depend on how long the whole chain is. That condition fixes the power. Writing $\xi = R\,(c/c^*)^{-m}$, with $R \propto N^{\nu}$ and $c^* \propto N^{1-3\nu}$, the length $N$ cancels only if $m = \nu/(3\nu - 1)$.

The figure plots the result. In a good solvent, with $\nu = 0.588$, the power is 0.77 (Flory’s value gives 3/4). At ten times the overlap concentration a blob is 0.17 of a coil across, and at a hundred times 0.029. In a theta solvent — a poorer solvent at the temperature where the segments’ attraction exactly cancels their excluded volume, so that the chains are ideal random walks with $\nu = 1/2$ — the power is exactly one.

Substituting into $\Pi \approx kT/\xi^3$ gives the pressure as a power of concentration alone:

$$\Pi \propto c^{3\nu/(3\nu - 1)} = c^{2.31},$$

or $c^{9/4}$ with Flory’s exponent, and $c^3$ in a theta solvent. That is the slope of the merged line in the first figure. Nothing about the chain length survives, because the blob has none: it is fixed by the concentration and by how segments repel. The unit the pressure counts is a piece of chain whose size is set by its neighbours, and it has no chemical identity at all. Its boundary moves every time the solution is diluted.

The reasoning is dimensional analysis made strict by one physical input, the demand that a local quantity not depend on a global one. The same move appeared in the counterion problem, where the counterions that never leave the chain settle at a density set by the Bjerrum length and not by the chemistry. In both cases the answer turned out to be universal for the same reason: the relevant length was set by a balance that did not involve the details.

The osmometer that stops weighing #

The practical consequence is that osmometry weighs polymers only when the solution is dilute, and the region where it works shrinks as the polymers get longer.

The figure turns the pressure back into the number an osmometer reports, the molar mass $cRT/\Pi$ that van 't Hoff’s law would infer from it, for chains of $10^5$ and $10^6$ grams per mole with coil sizes typical of polystyrene in a good solvent. At the lowest concentrations the reading is the true mass. The coils of the heavier chains overlap at 4.2 grams per litre, those of the lighter at 24, and past each overlap the reading falls away from the truth. By 200 grams per litre the two report about 6,000 and 6,300 grams per mole. The chains differ in length tenfold and the pressure has almost stopped noticing.

The reading at high concentration is not arbitrary. It is the mass of polymer in one blob: the pressure counts blobs, and an osmometer that assumes each counted unit is a molecule reports a blob’s mass as the molecular weight. At 200 grams per litre a blob of polystyrene holds about sixty segments.

This is why membrane osmometry of polymers is done by measuring at several low concentrations and extrapolating to zero, where the coils are apart. The first correction on the way there is the polymer’s second virial coefficient, the same object as the first correction to the gas law for a real gas: a measure of how much volume each coil excludes from the others. For a gas the correction stays a correction until the gas liquefies. For a polymer it becomes the whole story at a concentration of a few grams per litre, because each coil excludes a volume vastly larger than its own segments occupy.

A length that can be seen rather than inferred #

The blob would be a bookkeeping device if the only evidence for it were the osmotic exponent. It is not, because the correlation length can be measured directly. A beam of neutrons scattered from a solution in which some of the molecules carry deuterium in place of hydrogen sees the contrast between the two isotopes, and the angular spread of the scattered intensity at small angles is the Fourier transform of the correlations between segments. For a dilute solution it measures the size of a coil. For a semidilute one it measures $\xi$: the intensity falls off beyond a wavevector of about $1/\xi$, and the width of that fall-off shrinks as the concentration rises.

Daoud, Cotton, Farnoux, Jannink and their colleagues made that measurement on polystyrene in a good solvent in 1975, at the same time as de Gennes’s scaling argument, and found the correlation length falling as a power of the concentration close to three quarters, independent of the molecular weight. The osmotic measurements came from the other direction, with the pressure itself read at many concentrations and molecular weights, and every set collapsed onto one curve when plotted against $c/c^*$. Two independent instruments, one reading a force on a membrane and one reading an interference pattern, agreed on a length that neither measures directly.

The same length organises the solution’s motion. A semidilute solution flows more slowly than the solvent because its chains are tangled, and the tangles form on scales set by the blobs. The time a chain takes to wriggle out of the tube its neighbours make for it grows steeply with length, and that is the origin of the long memory in the liquid that remembers and of the extensional stiffness of a chain that cannot be stretched faster than it relaxes. The static pressure forgets the chain length; the dynamics does not, because to flow a chain must move as a whole, and a whole chain is exactly the object the pressure stopped seeing.

The membrane in an osmometer, for its part, still does what the membrane that almost holds asked of it. It must pass the solvent and hold back every chain, and for polymers that is easy: a coil is thousands of times larger than a water molecule, so the reflection coefficient is essentially one. What the membrane cannot do is decide what is counted on its far side. The chains are held back whole, and the pressure they exert is set by the blobs they are made of, which the membrane never sees.

What the mean field averages away #

The scaling result was a surprise when it was made, because a respectable theory had given a different exponent for twenty years. Flory and Huggins’s lattice model treats the solution as polymer segments and solvent molecules placed on a lattice, with the chain connectivity counted exactly for the entropy of mixing and the interactions treated on average. Every segment feels the mean concentration of segments around it.

The figure plots its prediction for chains a thousand segments long at three values of the interaction parameter $\chi$, which measures how much a segment prefers other segments to solvent. In an athermal solvent, $\chi = 0$, the pressure beyond overlap grows roughly as the square of the concentration. Its local slope between a tenth and a fifth by volume is 2.10, heading towards two, which is the virial expansion’s second term taking over. Measurements on polystyrene in toluene and on similar systems give 2.25 to 2.3. The mean field is wrong in the exponent.

The error has a clear origin. A segment in a semidilute solution does not see the mean concentration. It sees its own chain’s segments close by and a depleted region — a correlation hole — before the other chains begin, because other chains are repelled from its neighbourhood by the same excluded volume that swells it. The mean field fills in the hole and so overcounts the repulsion between the chain and its surroundings. The scaling argument keeps the hole, as the blob. Here, as in the like charges that pull together, a correlation that the average cannot contain changes a measurable answer. There it changed the sign of a force; here it changes an exponent.

The mean field gets two other things right, and the figure shows both. At $\chi = 0.5$ the square term cancels: segments attract each other just enough to offset their excluded volume, the chains are ideal random walks, and the pressure rises as the cube of the concentration. That is the theta point, and at it the scaling argument and the mean field agree, because an ideal chain has no correlation hole to average away. Past a slightly larger critical value, 0.532 for this chain length, the pressure curve develops a loop: over a range of concentrations the pressure falls as more polymer is added, which is unstable. The solution separates into a dilute phase and a concentrated one, and the solvent has stopped dissolving the polymer. The critical value approaches one half as the chains lengthen, so very long chains precipitate at the theta temperature itself.

Where the blob picture stops #

The blob picture is a scaling argument, and it says as much by what it does not compute.

The crossover is not computed. The curves in the first figure join van 't Hoff’s line to the power law with a simple interpolation, $\Pi N/ckT = 1 + (c/c^*)^{1.31}$. The two limits are theory; the shape of the bend between them is not. The true crossover function has been computed by renormalisation-group methods and measured, and it agrees with the interpolation to within some tens of per cent near overlap, which is why the figure is useful and why its numbers near the bend should not be quoted.

Prefactors are not known from the argument. $\Pi \approx kT/\xi^3$ is a scaling statement with an unknown number in front. The blob’s mass read off the osmometer figure is correct in its scaling and uncertain by a factor of order one.

Concentrated solutions leave it. When the blobs shrink to a few segments, at concentrations of tens of per cent, the swelling that defined a blob no longer has room to happen. The chains then behave as ideal random walks again, a result Flory argued for melts and neutron scattering confirmed, and the pressure is governed by local packing, which no universal argument describes.

Charged chains follow different rules. A polyelectrolyte’s blobs are stretched by electrostatic repulsion and its pressure is dominated by the counterions, which is the subject of the swelling a membrane cannot stop. The neutral-chain exponent of 2.3 applies to neutral chains only.

Still open: how the exponents change when chains are stiff or branched #

The argument above assumes flexible linear chains with segments much smaller than any length of interest. Real polymers include stiff chains such as DNA, whose persistence length of fifty nanometres is a significant fraction of the blob size over much of the semidilute range, and branched and ring polymers, whose topology changes how they fill space. For stiff chains the blob picture needs extra lengths and the crossover regions overlap. Ring polymers in concentrated solution do not interpenetrate as linear chains do, because each ring’s threading of another is limited by topology, and their size grows with length in a way that is still argued over, with simulations and experiments on synthetic rings and on chromosomes — which have been modelled as crumpled rings — disagreeing on exponents in the second decimal place. How the osmotic pressure of such solutions scales, and whether it keeps any memory of the molecule’s length, has not been settled.

The habit worth carrying away is to ask of any law that counts things what it is actually counting. Van 't Hoff’s law counts independent units, and a unit is independent only down to the length at which its surroundings stop screening it. In a dilute solution that length is the whole molecule. In a semidilute one it is a blob set by the concentration, and the pressure then carries no information about the molecules at all — the measurement weighs a length the solution chose rather than a mass the chemist made.