An Introduction to Rheology — Viscosity, Yield Stress, Melt

The deep dives so far have taken gelato apart element by element. They treated ice (Approfondimento 2 and 3), fat and air (Approfondimento 5 and 8), and the viscosity of the unfrozen phase (Approfondimento 4). But the smoothness, the melt and the firmness felt on the tongue are all of these at once: the whole of flow and deformation. The study that treats it is rheology (reologia), and this piece, closing the deep dives, gathers the elements back into a single sensation.

Viscosity — the resistance to flow

The most basic quantity is viscosity. Viscosity is how hard a thing is to make flow — the size of the resistance to flow. In an ideal liquid the applied shear stress runs in proportion to the speed of flow, the shear rate.

What this resistance really is, is friction. Think of a flowing liquid as many thin layers stacked one upon another. As the layers slide across one another friction arises there, and energy is thrown away as heat. The greater the friction, the more energy is thrown away, and the higher the viscosity.

The viscosity of the unfrozen phase of gelato is settled by the sugars dissolved in it and by the stabilizer holding water (Approfondimento 4). The higher the viscosity, the more firmness. Ice recrystallizes more slowly as well, though viscosity alone does not account for the mechanism (Approfondimento 3 and 4). Too high, though, and the mouthfeel turns heavy and rubbery. When Advanced Chapter 4 called stabilizers a material not to be overdone, it was this trim of viscosity that it meant.

Shear thinning — press it, and it loosens

Gelato is not a simple liquid like water, not a Newtonian fluid. Left alone it holds its shape with a high viscosity; pressed hard, it loosens and flows more readily. Raise the shear rate and the apparent viscosity falls: this property is shear thinning (pseudoplasticità). When a food dispersion departs from Newtonian behavior, this is much the commonest form it takes.

Shear thinning arises for more than one reason. Particles set in a shear field change their arrangement. Particles that are not spheres — ice crystals, fat crystals, biopolymers — line up with the flow, and once lined up the resistance falls. Flocs are drawn out and broken, releasing the liquid they held and cutting their effective volume. Solvent molecules bound to a particle can be stripped away. In the unfrozen phase of gelato, entangled stabilizer chains coming loose under shear is added to these (Approfondimento 4). Every one of them works to cut the resistance to flow.

The property earns its keep in the making and in the eating alike. In preparation the mix holds water while standing, and flows under the shear of the mantecatore. In the mouth, spread by the tongue, it loosens and melts away smoothly.

What is interesting is that the making and the eating fall into much the same band of shear rate. Stirring by machine, and chewing and swallowing — all of them overlap a range of roughly ten to a hundred per second. The same physics that loosened the mix inside the machine happens once more on the tongue. Press it, and it loosens: this is one face of melt.

How tightly packed divides liquid from solid

So far the talk has assumed flow. But gelato does not flow. It stands in the case in the shape of a mound. Why a thing that is a liquid can behave like a solid — the answer lies in how tightly the inside is packed.

Raise the volume fraction (φ) of the dispersed particles little by little, and the behavior of the system changes not smoothly but in stages. While the particles are sparse they do not act on one another, and the viscosity is settled by the continuous phase. Pack more in and the particles collide, and the viscosity climbs — and still it is a liquid that flows. Past a certain packing, though, everything alters. Each particle is ringed by neighbors and put in a cage, free to vibrate in place but unable to pass the ones beside it. Here the system turns into something that behaves as a solid under low stress, and flows only once a critical stress is passed. Pack it further and the particles lose their freedom to move at all, and it becomes an elastic solid.

The progression can be laid straight onto familiar foods. The fat globules of milk stand at about 0.04, and milk runs thin and free. Heavy cream is around 0.36, thick and slow. Mayonnaise reaches 0.80 and no longer flows at all, but holds its shape. Every one of them is oil dispersed in water, and yet the packing alone makes all that difference.

The progression shows plainly in the equation for viscosity as well.

As φ nears φ_c the bracket runs toward zero, and the power being negative, the viscosity diverges. For spheres that do not interact over long range, φ_c is about 0.6 to 0.7 and the intrinsic viscosity [η] is 2.5.

Both figures, though, belong to an ideal system, and reality departs from each. And since it departs the opposite way in each case, the two tell together.

The intrinsic viscosity rises. Let a particle be non-spherical, or swollen with the water it has taken up, or flocculated, and the intrinsic viscosity grows far larger. Ice crystals, stabilizer holding water, flocculated fat globules — all of them sit on this bulky side. So they tell far harder than their amount would suggest.

φ_c falls. Where a strong attraction or repulsion acts between particles, the packing parameter for closest packing can go considerably lower. The figure 0.6 to 0.7 holds only where the particles neither pull nor push one another from a distance.

And gelato is exactly a system of particles that attract. Partial coalescence (Approfondimento 5) is nothing but the deliberate flocculating of fat globules. The net of stabilizer, too, binds particles from a distance. Seen from the side of attraction alone, the φ_c of gelato is lower than that of ideal spheres.

Some things raise φ_c. Of these, two are treated here.

One is the spread of particle sizes. The less even the sizes, the more the small particles can slip into the gaps between the large. The same volume fraction packs more easily, so φ_c rises. In gelato the ice crystals, the fat globules and the air bubbles are none of them even in size. They mingle at sizes orders of magnitude apart. The other is the softness of the particles. A particle that can be pressed out of shape packs to volume fractions no sphere could reach. The air bubbles of gelato (Approfondimento 8) are exactly such deformable particles.

So which way does the balance move? The literature does not give the answer. That attraction lowers it, and that spread and softness raise it, are each set down on their own. So this piece too will not say what the φ_c of gelato is. One thing can be said: the textbook figure of 0.6 to 0.7 cannot be taken as it stands.

Come this far and what freezing gelato really is comes into view. To chill is to drive φ upward, and nothing else. Every time water turns to ice a particle called ice is added, and the unfrozen phase left behind shrinks (Approfondimento 1). To that are added flocculated fat globules (Approfondimento 5) and air bubbles (Approfondimento 8). Gelato is made on the precarious side of φ_c — which is why a slight difference in temperature moves its hardness so much.

Yield stress — that it can be scooped

A dispersion past closest packing behaves as a viscoelastic material, or as a plastic material. It is described by the modulus, the yield stress and the plastic viscosity. What is decisive in the rheology of gelato is this yield stress (sforzo di soglia). Under a small force gelato holds its shape like a solid. Only past a critical force, the yield stress, does it flow like a liquid.

What lies behind it is a network. Foods that show plastic behavior hold, within a liquid matrix, a network of particles bound to one another. In butter and margarine it is a network of fine fat crystals strung through the oil. While the applied force stays small the weak bonds between particles are not broken, and the material merely deforms a little. Past the yield stress the bonds break, the particles begin to slide past one another, and it flows. And take the force away and the flow stops at once.

That gelato holds the shape of a mound in the case is because gravity does not pass its yield stress. Press with a spatula and that force does pass it, and the gelato flows and is scooped out. Release, and it stops and stays in that shape. The scoopability touched on in Advanced Chapter 4 is nothing but this yield stress standing at the right height. Too high and it is stiff and cannot be scooped; too low and the shape collapses and slumps.

Then what settles that height? The strength of the attraction binding the particles together. The stronger the attraction, the harder the network is to break and the higher the yield stress. And what makes that attraction in gelato is partial coalescence (Approfondimento 5), and nothing else. Scoopability runs straight back to how far the fat globules have been flocculated — the spatula's feel is the emulsifying and the stirring made tangible.

This boundary, though, is not as sharp as the textbook figures draw it. In an ideal plastic material — a Bingham plastic — flow begins abruptly at the yield stress. But in real materials the yield point often cannot be clearly defined. Raise the stress and the network breaks down not all at once but a little at a time, over a range of applied stress. In most cases the yield stress is not a sharp point but a name given to a band of transition. Part of why an artisan cannot put just the right hardness into a single number lies here.

Viscoelasticity — stored, or spent

A plastic material switches its behavior: elastic below the threshold, viscous above it. But most foods are, more exactly, viscoelastic — they show viscosity and elasticity at the same time.

The difference lies in where the applied energy goes. In an ideal elastic solid the applied force is all stored as energy in the deformed bonds, and comes back when the force is removed. In an ideal liquid it is all spent, turned to heat by friction, and does not come back. A viscoelastic material does both. It stores part and spends part. So it does not deform fully the moment force is applied, nor return the moment force is removed — and at times it stays deformed.

The two contributions can be measured separately.

It has firmness; it is mouth-filling; it breaks cleanly. These words for flavor are, pressed far enough, the tongue reading this division between storing and spending.

Melt is the integration of everything

So rheology gathers the scattered threads into one. In the few seconds from the moment gelato enters the mouth, several physics run at once.

The heat of the mouth warms the gelato, the viscosity of the unfrozen phase falls, and the ice begins to melt. Having lost its yield stress, the gelato turns to flow on the tongue. Fine ice crystals (Approfondimento 2 and 3) are felt as smoothness. The net of fat and the fine bubbles (Approfondimento 5 and 8) are felt as creaminess. And the aroma held within opens as the temperature rises (Approfondimento 7). Melt is the name for this run of flow, melting and release, happening in concert on the tongue.

So to design melt is to bring all of these physics into mesh in the few seconds aimed at. Ice fine, fat and air in the right amount, viscosity and yield stress just so, aroma strong enough not to be lost to the cold. Every judgment in making gelato gathers at last to this one point, the few seconds in the mouth.

The numbers are the map, melt the destination

Rheology gives the means to put melt into words and numbers. Viscosity, yield stress, viscoelasticity — a sensation that could once only be called somehow firm becomes a measurable quantity. Balancing (Advanced Chapter 5), pressed far enough, is the design that brings this rheology to the values aimed at. Quality control (Advanced Chapter 12) is the management that keeps it from drifting day to day.

But one last thing. However precisely rheology can describe melt, that is no guarantee that it will taste good. A melt that can be measured is a necessary condition, not a sufficient one. Numbers are a map, not a destination. The artisan holding the map decides the last step with the tongue. Every chapter and every deep dive in this book has been there to make that step sure.