The liquid that remembers

Silly putty bounces off a table and, left on it, spreads into a puddle overnight. A glacier looks like rock and moves like a river. Pitch in the University of Queensland’s funnel has produced nine drops since 1930, and a lump of the same pitch struck with a hammer shatters into fragments with sharp edges.

Each of those is usually presented as a curiosity about a peculiar material. None of them is: each is the ordinary behaviour of a substance with a relaxation time, watched on two sides of it.

The ratio of the two — the material’s relaxation time divided by the time of the observation — is the Deborah number, named after the prophetess who sang that the mountains flowed before the Lord. The point of the name is exactly the point of this essay: on a long enough view, they do.

Two experiments, and what they separate #

Rheology has two elementary experiments and they are complementary. In a creep test a stress is applied suddenly and held, and the strain is watched. In a relaxation test a strain is applied suddenly and held, and the stress is watched. A purely elastic solid gives a step in both and nothing else; a purely viscous liquid gives a ramp in the first and an immediate collapse in the second.

Each of those three curves is the response of an arrangement of two elements: a spring, which stores energy and responds instantly, and a dashpot, which dissipates it and responds at a rate. In series they make a Maxwell material, which flows. In parallel they make a Kelvin–Voigt material, which does not. A spring across a Maxwell arm makes the standard linear solid, which is the simplest arrangement that behaves like anything real.

That is the definition worth having. A liquid is a material whose stress relaxes to zero at fixed strain; a solid is one whose stress does not. It is a statement about the long-time limit of a function, so answering it requires waiting, and how long is exactly what nobody can know in advance.

Where the memory comes from #

A relaxation time is a physical thing rather than a fitting parameter, and its origin differs between materials in a way that explains the spread of thirty-seven decades in the first figure.

In a simple liquid it is the time for a molecule to move past its neighbours — about as long as it takes a molecule to travel its own free path, a picosecond in water, set by the same molecular motions that carry momentum sideways. In a polymer melt it is the time for a chain to escape the tube its neighbours make around it, which scales as the cube of the chain’s length and so runs from milliseconds to hours across a modest range of molecular weights. In a glass it is the time for a cooperative rearrangement of a region, and it rises so steeply on cooling that it passes from seconds to millennia over a few tens of degrees.

Viscosity over modulus #

The relaxation time has been described and not computed, and it has a two-symbol expression that makes the thirty-seven decades legible.

A Maxwell material is a spring of modulus $G$ in series with a dashpot of viscosity $\eta$. Hold it at a fixed strain and the stress decays exponentially with

$$\tau = \frac{\eta}{G},$$

which is the only combination of the two with units of time. So a material’s memory is not its viscosity and not its stiffness but the ratio, and that is worth stating because the two are constantly conflated.

The conflation is easy to expose. Glycerol is over a thousand times more viscous than water and pours noticeably slowly, and its elastic modulus at short times is much the same as any liquid’s — so its relaxation time comes out at around a nanosecond. It is a slow liquid with no memory whatever, and nothing about it is viscoelastic on any timescale an experiment can reach. Being difficult to pour and being elastic are unrelated properties.

Run the ratio the other way and the same point appears. A polymer melt and a small-molecule liquid can have similar viscosities and relaxation times differing by nine decades, because the melt’s modulus at short times is enormously lower — it is entropic, coming from the chains’ resistance to being straightened, rather than from stretching bonds. A small modulus in the denominator is what buys a long memory, and it is why elasticity in a liquid is a phenomenon of long molecules.

So the axis in the opening figure is a ratio, and the materials are spread across it by their moduli quite as much as by their viscosities. A substance that flows slowly is not thereby a substance that remembers, and the pitch in the funnel is doing two separate things — flowing very slowly, and having a memory — that its usual telling runs together.

What it looks like when the two regimes meet #

The interesting behaviour is never deep in either limit; it is where the Deborah number is of order one, and the material does both things at once.

A velocity gradient sustained by a stress, with momentum diffusing across it, is the definition of a Newtonian fluid — and it is what the memory-free case looks like. The stress depends on the rate now and on nothing that happened before, so the material has no state beyond its current motion. Everything in this essay is what appears when that is false.

Three consequences of the middle regime are worth naming because they are visible without instruments.

A polymer solution climbs a rotating rod instead of being flung outward — the opposite of what happens to an ordinary liquid in a spinning bucket, whose surface is pushed outward into a paraboloid. The rotation stretches the chains along the circular streamlines, they pull back along those lines like stretched elastic, and the resulting inward hoop stress squeezes the fluid toward the axis and up the rod. A Newtonian liquid has no such stress and forms a dip.

A jet of it can be cut with scissors and the free end retracts. That is elasticity in something that is unmistakably pouring.

And a filament of it resists breaking. A Newtonian thread cannot stay a thread — surface tension pinches it into drops at a wavelength the instability picks out — and an elastic stress along the filament opposes exactly the thinning that the instability requires, so a polymer solution spins into fibres where water splashes.

The fastest-growing wavelength of the instability that breaks a thread sets the spacing of the drops, and in a viscoelastic liquid it is quite different: the elastic stress resists the pinching, so the thread thins into a long filament instead of breaking cleanly. That is the everyday signature — a fluid that strings rather than drips — and it is the memory acting on a timescale shorter than the relaxation time.

The glass, which is the extreme case #

The most consequential material in this essay is the one whose relaxation time is longest, and it is ordinary window glass.

Cooling a liquid that fails to crystallise makes its relaxation time rise steeply — for silicate glasses by fourteen decades over a few hundred degrees. The conventional definition of the glass transition temperature is the point at which that time reaches a hundred seconds, which is a definition about an experiment rather than about a substance: it is the temperature at which the material stops flowing within the patience of the person measuring it.

Below that temperature the material is not in equilibrium and never reaches it, which makes it one of the few everyday substances whose state depends on the direction time ran while it was being made. It continues to relax, at an ever-decreasing rate, toward a state it will not attain — a process called physical ageing, and one that changes the mechanical properties of a polymer component slowly over its whole service life.

A simple liquid’s relaxation follows a straight line on a plot of log-rate against reciprocal temperature, with the slope reading off a barrier. A glass-forming one does not: its line curves, the apparent barrier grows as it cools, and there is no single activation energy to quote. That is the extreme case of memory — a relaxation time that lengthens without limit while nothing about the structure changes discontinuously.

The often-repeated claim that medieval window panes are thicker at the bottom because the glass has flowed is false, and the arithmetic here is why: at room temperature the relaxation time of window glass is of order 10²⁵ seconds, which is fifteen decades longer than the age of the universe. The panes are uneven because that is how crown glass was made.

Measuring a spectrum rather than a number #

A single relaxation time is a model, and no real material has one. What a real material has is a distribution — chains of different lengths, regions of different sizes, structures that rearrange on many scales — and the response is a superposition.

A material with one relaxation time decays as a single exponential, straight on a log plot. Almost no real material does. What is measured instead is a spectrum of relaxation times, and quoting a single one is a summary of a distribution rather than a property — which is why two materials with the same nominal relaxation time can behave quite differently under a given deformation.

The standard way to measure the distribution is to stop using steps and use oscillations. Drive the material sinusoidally at a frequency and separate the response into the part in phase with the strain — the storage modulus, which is elastic — and the part in phase with the rate, the loss modulus, which is dissipative. Sweep the frequency and the two curves are the spectrum. Their crossover — the frequency at which storage and loss are equal — is the reciprocal of the dominant relaxation time, and it is the single number most rheological data sheets are actually reporting when they quote one.

Tracking a small bead embedded in a material gives a mean squared displacement whose slope is the answer: linear in time means viscous, flat means elastic, and anything between reports the mixture. That is microrheology, and its advantage is that it needs a microlitre of material and no rheometer — which is why it is the technique for anything biological.

Two numbers that do the same work in other subjects #

The Deborah number is one of a family, and recognising the family is most of what makes it useful.

A Weissenberg number is the same ratio with the observation time replaced by the reciprocal of a shear rate — how much elastic memory survives one unit of deformation, rather than one unit of clock time. The two coincide for a steady shear and differ for a start-up or a stop, which is exactly where the interesting transients are.

A Knudsen number is the ratio of a molecular free path to the apparatus, and it decides whether a gas is a continuum at all; the viscosity of a gas stops being independent of density precisely where that ratio reaches one. A Péclet number is the ratio of an advection time to a diffusion time. A Fourier number is a heat-conduction time against an observation.

Every one of them is a duration divided by a duration, and every one of them separates a regime in which a description applies from one in which it does not. The pattern is worth carrying because it converts a qualitative question — is this material a solid? — into an arithmetic one, and because the arithmetic can be done before the experiment rather than after it.

Decades from hours #

The claim that a spectrum is measured over many decades of frequency raises an obvious objection: an instrument can sweep four or five decades, and the interesting behaviour covers ten or twelve. The resolution is one of the most useful tricks in the subject and it is worth setting out.

Raising the temperature of a polymer speeds up every relaxation process in it, and — crucially — speeds them all up by very nearly the same factor. So a measurement made at a higher temperature is equivalent to one made at the reference temperature at a shorter time, and the whole response curve simply slides along the logarithmic time axis without changing shape.

That makes an experiment out of it. Measure the response over the four decades available at each of a dozen temperatures, then slide each curve horizontally until it overlaps its neighbours, and the result is a single master curve covering fifteen decades. The horizontal shifts required are the shift factors, and their dependence on temperature is itself a measurement: for polymers above their glass transition it follows a characteristic form with two parameters, and the parameters carry information about the free volume available for rearrangement.

Two things make it worth naming rather than treating as a data-reduction step. It is what lets a manufacturer state the creep of a component over thirty years from a week in a laboratory, which is a prediction of a kind very little of materials science can make. And it fails informatively: if the curves refuse to overlap, more than one mechanism with different temperature dependences is present, and the failure of the superposition is the discovery.

What it does on the way out of a die #

The regime where the two behaviours meet is not only a laboratory curiosity; it is the constraint on every process that pushes a polymer through a hole.

Extrude a melt through a die and it emerges wider than the hole — sometimes twice as wide. The chains were stretched and aligned by the converging flow into the die, and on leaving they relax and pull the material back into a shorter, fatter shape. The amount depends on how far the material got through its own relaxation while it was still in the die, which is the ratio of the residence time to $\tau$: a long die at a low rate lets the chains forget and the swell is small, a short die at a high rate does not. Nothing about the geometry of the hole predicts the shape of what comes out.

Push harder and the same elasticity produces defects rather than a dimension. Above a threshold the extrudate’s surface roughens into a regular ripple, and above a further one the whole flow becomes unsteady and the product emerges in irregular lumps. Both are instabilities driven by the stored elastic stress finding a way to release itself, and both put a hard ceiling on how fast a line can be run.

The practical consequence is that a polymer process is designed against a time rather than against a force. The quantity to compute is how long the material spends in each part of the tool against its own relaxation time, and the ordinary engineering instinct — that going faster costs more pressure and nothing else — is the one that fails.

Where the model stops #

Everything above is linear. Doubling the stress is assumed to double the strain, and for a polymer that holds only at strains of a few per cent. Beyond that the chains align, the relaxation time itself changes with the deformation, and the superposition that produced every curve in this essay is invalid. Large-amplitude behaviour is a separate subject with no comparable general framework.

Springs and dashpots are a language, not a mechanism. Any linear response can be fitted by enough of them in some arrangement, and the fit says nothing about what is happening molecularly. Two materials with identical measured spectra can be structurally unrelated, in the way that two systems obeying one equation need have nothing else in common.

Temperature is not a parameter here and dominates in practice. Most polymer relaxation times follow a steep law in temperature, so a measurement at one temperature and a long time is equivalent to one at a higher temperature and a short time. That equivalence — time–temperature superposition — is how decades of behaviour are inferred from hours of measurement, and it fails whenever more than one mechanism is present with different temperature dependences.

And flow past a body is not this subject. Everything here is a material’s constitutive response, measured in a gap or in a filament with no obstacle in it. What a viscoelastic fluid does when made to flow around something — where the elastic stresses can drive instabilities at vanishing Reynolds number — belongs to the collection that owns flows.

What the pictures cannot show #

The Deborah figure draws seven materials as seven points, and a real material is not a point. It is a smear across several decades, because it has a distribution of relaxation times, and drawing it as a single number is exactly the simplification the section on spectra says is wrong. The figure is honest about the ordering and dishonest about the width.

Nor can any of these figures show what a material is doing while it relaxes. The curves are macroscopic averages; what is happening is that some large number of local rearrangements are occurring at a rate, and the exponential is a consequence of their being independent. A drawing of the microstructure would have to be a drawing of an assumption.

Where this ladder goes next #

The rung below established that the stress in a fluid need not be proportional to the rate of strain, which broke the idea that a fluid has a viscosity. This rung breaks something older: the idea that a substance is a solid or a liquid at all. What replaces both is a response function — a stress that depends on the entire history of the strain rather than on its present value — and the two rungs are the two axes of that function.

The habit worth carrying away is a question. Before asking what a material is, ask how long the question is being asked over. Nothing about pitch changes between the hammer blow and the decade in the funnel; what changes is the comparison being made, and a classification that depends on the observer’s patience is not a classification of the material. The same discipline applies to a river’s bed, a mountain range, a soldered joint under load, and the mantle of the Earth — all of them solids on the timescale of the question usually asked and fluids on the timescale of the interesting one.

What is left on this ladder is the non-linear regime the model stops at, and the one measurement that reaches it: the response to a large oscillation, where the storage and loss moduli stop being numbers and become functions of the amplitude as well.