A solid, if it is not given time

Every dimensionless group this collection is organised by contains something a person can point at. A Reynolds number has a speed and a size in it; a Mach number has a speed and a stiffness; a Froude number has gravity in it and a hull. The Deborah number has none of those.

$$\mathrm{De} = \frac{\lambda}{t_{\text{obs}}}$$

Two times. A property of the material on top, a property of the experiment underneath, and nothing else — no length, no velocity, no density, not even a viscosity. It is the one group here that could be evaluated by somebody who had never been told what the substance was, provided they were told how long it had been watched.

Its claim is remarkable and nearly true: that whether something behaves as a solid or as a liquid is not a fact about the thing.

The smallest material that has a relaxation time #

To compute anything, a constitutive law is needed, and the smallest one with a λ in it is Maxwell’s — a spring of modulus $G$ in series with a dashpot of viscosity $\eta$:

$$\sigma + \lambda\,\dot\sigma = \eta\,\dot\gamma,\qquad \lambda = \eta/G.$$

Held at fixed strain it forgets its stress as $e^{-t/\lambda}$. Sheared steadily it is a Newtonian liquid of viscosity η. Struck faster than λ it is a solid of modulus $G$. It contains the whole idea and almost nothing else, which is what a model for an argument about a dimensionless group ought to be — the same reason the flat plate is the right object for an argument about lift.

Under oscillation at frequency ω the two moduli come out in closed form,

$$G' = G\frac{(\lambda\omega)^2}{1 + (\lambda\omega)^2},\qquad G'' = G\frac{\lambda\omega}{1 + (\lambda\omega)^2},$$

so the loss tangent is $\tan\delta = 1/\lambda\omega$ — exactly, at every frequency — and the two moduli cross where λω = 1. That is as clean as a dimensionless threshold ever gets. There is no coefficient to be measured, no material property left over, and nothing to argue about: the crossover is the number.

It is worth pausing on how unusual that is. Most of the thresholds this collection has measured are nothing like it: across fourteen groups the value at which two terms are equal and the value at which the observable actually moves differ by factors from one to five hundred and ninety-four, and the Rayleigh number’s threshold is an eigenvalue rather than a comparison at all. A group whose threshold is exactly one, for an exact reason, is the exception.

The crossover is found here by bisection rather than substituted, and it comes back at λω = 1 to a part in 10¹⁵. That is not a check on the arithmetic — it is a check that the routine which will shortly be pointed at a spectrum, where the answer is neither 1 nor any round number, is looking for the right thing.

Watching the same putty twice #

Before the model is stretched, it is worth seeing what the number does with materials.

The axis spans fifty orders of magnitude, and the interesting entries are the ones that appear twice. Silly putty bounced has a Deborah number of 100 and is a solid; the same lump left overnight has 3 × 10⁻⁵ and is a liquid, and there is a puddle in the morning to prove it. Nothing about the putty changed. Rock over an ice age is a liquid; rock over a second is what mountains are made of.

The relaxation times are measurements and are quoted as such. What is computed is the ratio, and the figure’s whole claim is that the ratio decides — which is a strong claim, and is about to turn out to be the wrong shape of claim rather than a wrong number.

A real material has no relaxation time #

A polymer melt does not have a λ. It has a spectrum, and the Rouse model gives one in closed form: a chain of beads and springs relaxes in modes with

$$\lambda_p = \lambda_1/p^2,\qquad p = 1,\dots,N,$$

each carrying the same modulus. The $p^{-2}$ is not a fitting choice — mode $p$ relaxes a subchain of $1/p$ of the molecule and its time goes as the square of its length — and the equal weighting is the same statement, that every mode stores $kT$.

Summing the same two formulas over that spectrum is arithmetic, and it changes the answer in kind.

One relaxation time makes the moduli separate without limit: $\tan\delta = 1/\lambda\omega$ falls for ever, so a fast enough experiment always finds a perfect solid. A Rouse spectrum makes them parallel — both rise as $\omega^{1/2}$ above the first mode — so the loss tangent goes to 1 and stays. At λ₁ω = 100 the single time gives a modulus ratio of 100 and the spectrum gives 1.04: a factor of 103.8 in the quantity a rheometer actually reports.

Forty-five degrees of loss angle, at every frequency from the first mode to the last. The material is never the solid the single time predicts, however hard it is hit.

The crossover is not a point #

The consequence for the number itself is worse than a wrong value.

For one relaxation time the moduli are within twelve per cent of each other over 0.095 of a decade — a point, for any practical purpose, and the point the Deborah number names. For a thousand-mode Rouse spectrum the same band is 2.98 decades wide, thirty-one times wider in the logarithm. The two curves do not cross so much as graze, and where they touch is set by the shortest relaxation time rather than the longest: doubling the chain length moves it.

So “the crossover frequency” of a real polymer is the midpoint of an interval three decades wide, and the property it is a midpoint of is the molecular weight rather than anything anybody would call elasticity.

This is the same failure mode this collection met in the drag on a sphere near the continuum limit and in the Womersley number’s two thresholds: a group defined by comparing two terms is a perfectly good statement about the two terms, and only sometimes a statement about where the behaviour changes.

The measurement that shows it directly #

An oscillatory experiment is one way to see this. A step-strain experiment is the other, and it is the one where the fit gets made.

On a logarithmic modulus axis a single relaxation time is a straight line. The spectrum is not, and the best single exponential through the window a rheometer would use is wrong by 49 per cent at the short-time end. It also returns a relaxation time — 0.647 λ₁ — which is a number about the fitting window rather than about the fluid, and which moves when the window does.

That is the shape of the whole problem, and it is not a failure of the fit. The fit succeeds, returns a plausible λ, and the λ is not a property of the material. It is the rheological version of what reading an exponent off a log–log plot does when the underlying law has a logarithm in it: an excellent straight line through the wrong function.

Two numbers, both called elasticity #

There is a second thing the collapse hides, and it is a confusion of names rather than of magnitudes.

De is a ratio of times. The Weissenberg number $\mathrm{Wi} = \lambda\dot\gamma$ is a relaxation time times a rate. In steady simple shear there is no observation time at all — the flow has been going for ever and will continue — so De = 0 while Wi may be a hundred, and the fluid is emphatically not Newtonian.

The upper-convected Maxwell model makes that precise and slightly alarming. In steady shear its shear stress is $\sigma_{12} = \eta\dot\gamma$, Newtonian at every rate with no elastic correction whatever, and the entire elasticity sits in the first normal-stress difference $N_1 = 2\eta\lambda\dot\gamma^2$. The ratio $N_1/\sigma_{12}$ is exactly $2\mathrm{Wi}$.

A rheometer measuring only torque is measuring a fluid it would call Newtonian. Everything that makes such a fluid climb a rod, swell out of a die and refuse to siphon in the ordinary way lives in the component that instrument does not see — which is a sharper version of the point made about an instrument that reports one viscosity for a fluid that has none.

And the one consequence that can be timed #

There is a measurable difference between the two stresses, and it is a pure number.

Switching the shear rate on gives

$$\sigma_{12}(t) = \eta\dot\gamma\left(1 - e^{-t/\lambda}\right),\qquad N_1(t) = 2\eta\lambda\dot\gamma^2\left(1 - (1 + t/\lambda)e^{-t/\lambda}\right),$$

both exact. The second carries an extra factor of $(1 + t/\lambda)$, so it arrives late: 99 per cent at $t/\lambda = 6.6384$ against the shear stress’s $4.6052 = \ln 100$. The ratio is 1.4415, and it does not depend on the fluid, the rate or the geometry.

That is a Deborah number with an experiment attached. Watch for less than about five relaxation times and the torque is settled while the normal force is not, and an instrument reporting both will report a material whose two elastic signatures disagree about what time it is. The same trap sits in every transient measurement this collection has looked at, from the lift that arrives late to the settling time of Taylor dispersion: a quantity read before its own transient has finished is a quantity about the clock.

The number rescued, by moving the material instead of the clock #

There is a way of using the Deborah number that survives the spectrum entirely, and it is the single most productive idea in polymer rheology. It works by taking Reiner’s observation seriously in the other direction: if the answer depends on the ratio of a material time to an experimental one, then changing the material time is as good as changing the experiment.

Temperature does exactly that. Warming a melt speeds up every relaxation in it, because all of the modes share one segmental friction coefficient and that is what the temperature acts on. So the whole spectrum slides bodily along the time axis — every $\lambda_p$ multiplied by the same factor $a_T$ — and the shape of the distribution is untouched.

Which means a measurement at one temperature and a measurement at another are the same curve, read at different places. Shift them horizontally by $\log a_T$ and they overlap; do it for eight or ten temperatures and the overlapping segments assemble into one master curve. That is time–temperature superposition, and its practical value is enormous: a rheometer covers perhaps four decades of frequency before inertia stops it at one end and patience at the other, and a master curve routinely spans twelve or fifteen. The behaviour at a microsecond is measured by cooling the sample and taking a minute over it.

The shift factor itself follows a form Williams, Landel and Ferry fitted in 1955,

$$\log a_T = \frac{-C_1 (T - T_{\text{ref}})}{C_2 + (T - T_{\text{ref}})},$$

with constants that are close to common across many polymers when the reference is taken at the glass transition — which is itself evidence that one mechanism is doing the shifting.

And the assumption it rests on is exactly the one this essay has been examining. Superposition works only if every mode shifts by the same factor, so that a distribution of times can be treated as one time with a scale on it. Materials for which that holds are called thermorheologically simple, and the Rouse spectrum above is simple by construction. Blends, block copolymers, filled systems and semicrystalline polymers are not: their components have different temperature dependences, the spectrum distorts rather than sliding, and the shifted segments no longer overlap.

So the failure is diagnostic rather than merely inconvenient. A master curve that will not superpose is a measurement that the material has more than one mechanism in it, and that is a stronger piece of information than any single relaxation time the fit would otherwise have returned.

Where this model stops #

Three limits, and the first two are why the figures say “the Rouse spectrum” rather than “a polymer”.

The Rouse model has no entanglements. Above a molecular weight of a few tens of thousands a melt’s chains cannot pass through one another; the relaxation is reptation rather than Rouse, the spectrum has a rubbery plateau in it, and the terminal time goes as $M^{3.4}$ rather than as $M^{2}$. Everything argued here about spectra survives that change and none of the exponents does.

Every modulus here is linear. They are defined at vanishing strain amplitude, and the interesting rheology of a polymer is what happens when the amplitude is not small. Wi is the number for that regime and nothing here computes in it beyond the closed-form normal stress.

And the chain is finite. The Rouse curves flatten above the last mode, at $\lambda_1\omega \approx N^2$, which is the segmental time and is where a real material stops being a chain and starts being a glass. The measured slopes come out at 0.532 and 0.496 rather than exactly a half for that reason, and every figure states its mode count because the answer depends on it.

Where the number came from, and what it was for #

Marcus Reiner named it in 1964, from the line in the Song of Deborah about the mountains flowing before the Lord — his point being that on a long enough view they do, so the distinction between solid and liquid is a matter of the observer’s patience rather than of the material’s nature. He wrote it as λ/t and said explicitly that the interest lay in materials for which the number is of order one, because those are the ones whose behaviour cannot be described without saying how long the experiment took.

That reading survives everything above. What does not survive is the step from “there is a time” to “there is the time”. The number is a genuine statement about an experiment and a weak statement about a material, and the reason is not that the model is crude: it is that the quantity being collapsed onto one number is a distribution.

The company it keeps #

It is worth naming the other groups on this site that are built the same way, because there are only a few and they behave alike.

A dimensionless group with no velocity in it is a statement about the material rather than about the flow, and the collection contains three of them. The Prandtl number is the ratio of two diffusivities and appears in the thermal layer’s thickness as a pure property of the substance. The Ohnesorge number, which decides where a jet stops being a jet, is built from viscosity, density, surface tension and a size, with no speed anywhere. And the Deborah number is the extreme case: it has not even a length.

Each of them is used the same way and fails the same way. A material-only group is excellent at saying which of two mechanisms is available and poor at saying where the behaviour changes, because the place the behaviour changes is decided by the flow — and a group with no flow in it cannot carry that information. The Prandtl number does not say where a boundary layer transitions; the Ohnesorge number does not say where a jet breaks; the Deborah number does not say when a material yields.

What the Deborah number has that the other two do not is an observation time, which is a property of neither the material nor the flow but of the person watching. That is what makes it the strangest group in the collection and, once its spectrum is admitted, the most honest: it says out loud that the answer depends on the experiment, which every other group here says only when pressed.

What this leaves #

The pattern is the one this collection keeps finding at the top of a ladder — a group that decides something, and a residual it cannot carry. Here the residual is a spectrum. The essays after this one meet the same shape in flows rather than in materials: a number that decides which term is the force, a number that decides where a transition sits, and in each case a second quantity that survives the collapse and changes the answer.

The place to go next is the group nearest to this one in construction and furthest from it in temperament — the frequency a wake chooses, which also has a time in it, and which collapses far better than it has any right to.