The best a tube can do is its own radius

Taylor’s dispersion gives a tube an effective axial diffusivity $D(1 + \mathrm{Pe}^2/48)$, and the essay that computes it draws two conclusions from the form: the spreading rises as the square of the velocity, and it rises as the molecular diffusivity falls. Both are surprising and both are about a coefficient.

A coefficient that rises monotonically with velocity offers no advice. Run the tube faster and the dye spreads faster; run it slower and it spreads more slowly; there is no velocity at which anything is best, and the essay does not claim there is. That is a complete description of the physics and it is the wrong question for anybody who has a tube and wants to use it.

Per metre, not per second #

The variance of a slug grows as $\sigma^2 = 2 D_{\text{eff}} t$. A user of the tube does not care how long the slug has been in it; they care how wide it is when it comes out, and what it has to be compared against is how far apart two different substances have drifted — which also accumulates with distance. So the quantity to minimise is the variance per unit length,

$$H = \frac{\sigma^2}{L} = \frac{2D_{\text{eff}}}{u} = \underbrace{\frac{2D}{u}}_{\text{molecular}} + \underbrace{\frac{a^2}{24D}\,u}_{\text{Taylor}},$$

and dividing by $u$ is the whole of the difference. The first term is molecular diffusion along the tube, which the essay below computes and then discards as negligible — correctly, as a contribution to the coefficient, and not at all as a contribution to $H$, because it is the only term that matters when the flow is slow.

A sum of a term falling as $1/u$ and one rising as $u$ has a minimum. The optimum exists because two mechanisms with opposite dependence on velocity are being added, and the essay below has only one of them in view.

Chromatography calls $H$ the plate height — the height of one theoretical equilibrium stage — and $L/H$ the number of theoretical plates, which is the figure of merit for a column. The name is inherited from distillation and is a historical accident; the quantity is a variance per unit length and nothing else.

It is worth being explicit about why a variance per length rather than a width per length, because the distinction carries the whole argument. Variances of independent contributions add; widths do not. A column’s spreading, an injector’s, a detector’s and a connecting tube’s combine by adding their variances, so a quantity defined as variance over length can be added up along a train of components and the total read off. A width over length cannot, and a chemist quoting “peak width per metre” would find the numbers refusing to combine.

That additivity is also why the two terms in the expression above may simply be summed. Molecular diffusion along the tube and Taylor’s dispersion are not independent in any deep sense — they are the same molecules — but their contributions to the variance are, because one comes from a displacement along the tube and the other from a displacement across it followed by advection, and the two are uncorrelated. The additivity is an assumption and it is the one every plate-height equation rests on, including the term that is added to it in packed columns and disputed at the end of this essay.

A minimum with nothing in it #

The minimum of $B/u + Cu$ is at $u^* = \sqrt{B/C}$ and takes the value $2\sqrt{BC}$. Putting Golay’s coefficients in gives two results, and the second of them is startling:

$$u^* = \frac{\sqrt{48}\,D}{a}, \qquad H_{\min} = 2\sqrt{2D\cdot\frac{a^2}{24D}} = \frac{a}{\sqrt3}.$$

The best a bare tube can do is its own radius divided by the square root of three. The diffusivity has cancelled. The velocity has cancelled. Whatever is flowing, whatever it is dissolved in, and however fast it is being pushed, a tube’s smallest attainable spreading per metre is a property of its bore alone.

The cancellation is not a coincidence and it is worth seeing why, because it is the same kind of statement as a duct that forgets everything but one number: a property of the geometry survives when the fluid’s own constants do not. One term goes as $D$ and the other as $1/D$, so their geometric mean — which is what $2\sqrt{BC}$ is — has no $D$ in it. The same argument applied to the velocity is what makes the optimum velocity carry the $D$ instead: the diffusivity has to go somewhere, and it goes into where the minimum is rather than into how deep it is.

The physical reading of $a/\sqrt3$ is worth having as well, because the algebra alone leaves it looking like an accident. At the optimum the two mechanisms contribute equally, by construction. The molecular term contributes $\sqrt{BC}$, which is the variance a slug picks up by diffusing along the tube during the time it takes to travel a unit length — and at the optimum that time is exactly the time it takes to diffuse across the bore. So the minimum plate height is the distance a molecule diffuses along the tube while it crosses it, which is the radius, times a number of order one.

A tube’s best performance is therefore a statement about a race between two diffusions in perpendicular directions, and the fluid’s diffusivity cancels because it is the same diffusivity in both. That is also why the result is so robust: it survives any change to the fluid, and it would only be changed by something that made diffusion anisotropic.

The practical form is a sentence a chemist can act on. To halve a column’s plate height, halve its bore — and nothing else will do it. Not the carrier gas, not the temperature, not the pressure, and not the flow rate, all of which move the optimum velocity around and leave the depth of the minimum where it was.

What the minimum is made of, on each side of it #

Reading the two branches separately says what a column is doing wrong when it is run badly, and the two answers are opposites.

To the left of the minimum the flow is too slow. The slug is in the tube long enough for molecular diffusion to spread it appreciably along the tube, and that spreading has nothing to do with the flow at all — it would happen in a sealed capillary with no flow whatever. A column run there is being ruined by waiting.

To the right the flow is too fast. Taylor’s mechanism dominates: fluid on the axis outruns fluid near the wall, the solute cannot diffuse across the bore quickly enough to average the two, and the slug is sheared out. A column run there is being ruined by hurrying.

The two failures are distinguishable in an experiment and they look nothing alike. A column run too slowly gives broad symmetric peaks that get broader if the sample is left in longer; one run too fast gives broad peaks whose width is proportional to the flow rate. A chemist reading a bad chromatogram can tell which side of the minimum the column is on without knowing any of the arithmetic, simply by halving the flow and seeing which way the peaks go.

There is a third reading of the same curve, which is about time rather than quality. The minimum is broad — the plate height is within twenty per cent of its best over a factor of about two in velocity either side — so a column run at twice its optimum loses very little resolution and takes half as long. That is why real chromatography is routinely done to the right of the optimum, and why the “optimum practical gas velocity” quoted in the trade is well above the one drawn here.

Where the diffusivity went, and what it costs #

Fifty microns a second, on a thirty-metre column, is a run lasting seventeen days. That single number is why gas chromatography is done in open tubes and liquid chromatography is not.

The difficulty is not that the liquid column is bad; the figure above says it reaches the same plate height as the gas column, and would give about the same number of plates. It is that it reaches it at a velocity nobody can wait for, and running it any faster puts it on the rising branch where Taylor’s dispersion dominates and the resolution collapses. Liquid chromatography escapes by abandoning the open tube altogether and packing the column with particles — which replaces the single wide streamline bundle with a tortuous path whose transverse diffusion distance is the particle size rather than the tube’s radius, and buys back three or four decades in the only length that matters.

That is the same substitution Darcy’s law makes for a porous medium, seen from the other side: a packed bed’s pore is its own length scale, and here it is being chosen deliberately to be small.

The price of that substitution is pressure, and it is steep. Flow through a packed bed goes as the square of the particle size for a given gradient, so reducing the particle diameter by a factor of ten to gain a factor of ten in plate height costs a factor of a hundred in the pressure needed to drive it. That is the entire history of liquid chromatography in one sentence: particles have fallen from tens of microns to under two, plate heights have fallen with them, and the pumps have gone from tens of atmospheres to a thousand. The limit on a modern liquid column is not fluid mechanics but the burst pressure of a steel fitting, which is an unusual place for a transport problem to end up.

What holding the solute does, and where it stops #

A chromatographic column does not merely transport; it retains. Coat the wall with something the solute dissolves in, and the solute spends a fraction of its time stationary — which is what makes two substances come out at different times in the first place.

The reason there is a cost at all is not the delay. A solute that merely waited, uniformly, would emerge later and no wider. The cost comes from the radial structure: a wall that adsorbs is a sink at the edge of the tube, so the concentration profile across the bore is no longer the one Taylor computed, and the mismatch between where the solute is and how fast the fluid is moving there is larger. Golay’s mobile-phase coefficient carries that, and it rises from $1/24$ at no retention to $11/24$ as the retention becomes complete.

Eleven over one, so the plate height rises by $\sqrt{11}$ and no more. That the penalty is bounded is the useful part: a chemist choosing a stronger stationary phase to improve a separation pays a factor of at most 3.3 in plate height, against a gain in selectivity that can be a factor of many, and the trade is therefore almost always worth making.

The shape of the approach matters as much as the bound. Half the penalty is paid by a retention factor of one and three-quarters of it by a factor of three, so the expensive part of retention is the first bit of it — and a column operating at a retention factor of five is already within a few per cent of the saturated value. The bound is an asymptote rather than a value the curve reaches: expanding the coefficient for large $k$ gives a penalty of $\sqrt{11}\,(1 - 8/11k)$, so what is left to pay falls as $8/\sqrt{11}k$ and the curve is still two parts in a thousand short of $\sqrt{11}$ at a retention factor of a thousand. Increasing retention beyond that is free in plate height and costs only time, which is a much easier bargain to describe than a penalty that kept rising.

The stationary-phase film adds a term of its own that this bound does not cover, and it behaves the other way. Its coefficient goes as the square of the film thickness divided by the solute’s diffusivity in the film, which for a polymer is three or four decades below its diffusivity in a gas — so a film of a quarter of a micron contributes about as much as the whole mobile-phase term, and one of two microns dominates everything. A thick film is the one way of making an open tubular column genuinely bad, and it is the reason columns are specified by film thickness at all.

Four real columns #

A 250-micron capillary carrying helium, with a quarter-micron film and a retention factor of three, comes out at an optimum velocity of 0.19 metres a second, a plate height of 211 microns, and 142,000 theoretical plates in thirty metres. Those are the numbers a working gas chromatograph is specified at, and none of them was fitted: they follow from a bore, a film thickness and two diffusivities.

Widening the bore to 530 microns — which is done, because a wider column takes more sample — costs a factor of 2.5 in plate height and a factor of 2.5 in optimum velocity, so the column both separates less well and takes longer. The figure says why in one line: the mobile-phase term goes as $a^2$ and the optimum velocity as $1/a$.

The bare tube in the same table is the useful control. With nothing on the wall it reaches 72 microns of plate height and 416,000 plates — three times the coated column’s — and it separates nothing at all, because without retention every solute comes out at the same time. That is the trade the whole technique is built on: plates are cheap and selectivity is not, and a column is coated in the full knowledge that the coating costs two-thirds of the theoretical performance the empty tube had.

What the picture cannot show #

The flow is laminar and fully developed. A capillary column at its optimum has a Reynolds number of a few, so the first assumption is safe; the second is not entirely, since the first few centimetres of any column are an entry region in which the profile is still forming.

The column is straight. Coiling a thirty-metre capillary into a twenty-centimetre oven introduces a secondary circulation in the cross-section — the same Dean flow assumed absent above — which reduces the plate height by stirring the bore, and is one of the reasons a real column sometimes beats this arithmetic.

Adsorption is taken as instantaneous. The stationary-phase term assumes the solute equilibrates with the film faster than it moves along, and for a thick film or a slow phase it does not; that adds a third $C$ term which this expression contains and which is set to zero here for the uncoated cases.

And the slug is taken as a point, rather than a distribution with a residence time of its own. A real injection has a width, the detector has another, and a column’s measured plate count is degraded by both — which is why a manufacturer’s figure and a user’s differ by a factor that has nothing to do with the column.

Who found it, and when #

Golay derived the plate-height equation for an open tubular column in 1958, and presented it with the proposal that columns be open capillaries rather than packed tubes — a suggestion that took a decade to be accepted and then took over the subject entirely. Van Deemter’s equation for packed columns, of 1956, is the same structure with a third term for the irregular flow paths between particles.

The surprising connection is with an entirely different way of using a tube. A differential-pressure flow meter is judged by how much permanent pressure it costs for how much signal it gives, and the answer is a ratio in which the signal rises as the square of the velocity and the loss rises as the square as well, so the trade has no optimum. Here two mechanisms with opposite powers of velocity are added and one appears immediately. The presence or absence of an optimum is decided by whether the competing quantities scale with the same power, and neither problem’s physics announces which case it is until the quantities have been written down.

Still open: what the packed column’s third term really is #

Van Deemter’s equation has a term Golay’s does not — a constant, independent of velocity, called eddy diffusion — and after sixty-five years it is the least well founded quantity in the subject.

Its origin is that a packed bed offers a solute many paths of different length and different speed, so a slug spreads by taking them. Written as a constant times the particle diameter it is a plausible guess, and measured plate-height curves fit it well enough to be useful. What it does not do is explain why the constant is what it is, and the most careful treatments — Giddings’ coupling theory — argue that the term should not be constant at all, but should couple to the mobile-phase term in a way that makes it velocity-dependent at low flow.

The calculation that would settle it is a straightforward one: a random packing generated once, the Stokes flow through it solved, and a cloud of tracer released and followed, with the variance measured against distance at a sweep of velocities. The plate-height curve that came out would be a measurement rather than a fit, and the question it would answer is whether the constant term survives at all when the geometry is solved rather than parameterised — which is the difference between a correlation with three fitted numbers and a theory of a packed column.