# Dilatancy (fluid)

Dilatancy (from Latin dilatus retarding precedent, delaying, depressed, Part. Perf. Of differre ), also shear thickening is in the rheology of the property of a non-Newtonian fluid , at high temporal changes of the shear (i. E. At high shear rate) has a higher viscosity to demonstrate. In English, a dilatant fluid is also called shear-thickening , ie " shear-thickening " or " shear-hardening ".

The increase in viscosity is caused by a structural change in the fluid , which ensures that the individual fluid particles interact more strongly with one another (e.g. get caught) and so slide past one another more poorly. For concentrated suspensions , dilatancy can be described as a shear-induced phase transition .

The viscosity (tenacity) of a dilatant fluid thus increases with the shear rate , but does not depend on time at a constant shear rate . If, on the other hand, the viscosity does not drop immediately after the shear force has been reduced, but rather as a function of time , it is called rheopexy .

The dilatance of granular materials is dealt with in Dilatance (granular matter) .

## Mathematical-physical modeling Figure 3: Shear stress-shear rate diagram:
1: Shear-thickening (dilatant) fluid
2: Newtonian fluid
3: Shear-thinning (pseudoplastic) fluid
4: Bingham-plastic fluid
5: Casson-plastic fluid
To the right is the shear rate and upwards the resulting one Shear stress applied.

If the shear stress is plotted as a function of the shear rate (shear rate) , liquids with dilatancy are typically characterized by a flow law of the form ${\ displaystyle \, \ tau}$ ${\ displaystyle \, {\ frac {du} {dy}}}$ ${\ displaystyle \, \ tau = K \ left ({\ frac {du} {dy}} \ right) ^ {n}}$ With

• the consistency factor  K
• the flow index n > 1 (curve 1 in the diagram).

The opposite behavior, the decrease in viscosity with the shear rate or shear stress, corresponding to an index n <1, is called structural viscosity (also shear thinning) and can be found e.g. B. in highly polymeric solutions (curve 3 in the diagram).

For Newtonian liquids such as water, n = 1, i.e. the viscosity is independent of shear stress or shear rate (curve 2 in the diagram).

## Examples

• The behavior of a starch pulp can be easily observed in the experiment: for this, starch is mixed with water, so that a watery pulp is created. If you slowly pull a spoon through the pulp, it appears liquid, at higher speeds the pulp becomes so tough that it no longer flows but rather crumbles away. However, the chunks become liquid again after a very short time and merge with the rest of the pulp. This behavior is an example of dilatance, but not rheopexy .
• Even cooked cheese behaves dilatant: it can move slowly, at higher speed but firm and tears.
• Zinc pastes with a high solid content also show dilatance. This can cause the ointment mill used to make this paste to seize and must be observed accordingly.
• Suspensions of fine particles in concrete (concrete suspensions) also have the properties of a dilatant liquid.

## use

The US manufacturer Dow Corning made from silicone - polymer to dilatant bouncing putty , which was primarily as children's toys on the market to date. In addition to its normal kneadability, this substance behaves completely differently in the event of sudden mechanical stress: if you throw a ball from the material to the ground, it springs back like a rubber ball ; If you hit a piece with a hammer very quickly, it will break into many small, sharp-edged pieces, almost like ceramic . Sharp edges and smooth fracture surfaces are also formed when torn . Technical applications are not known.

A material with similar properties has recently been used as the Active Protection System ( APS ), for example in motorcycle clothing : specially shaped pads that contain a dilatant composite material allow the wearer to move freely. In the event of an abrupt blow as a result of a fall, however, the material "hardens" to a hard rubber-like consistency, distributing the forces acting on a larger part of the body and thus preventing injuries.

Dilatant liquids are currently being tested in conjunction with Kevlar fabrics in the manufacture of bulletproof and puncture-proof protective vests . Due to the impregnation with the liquid, the tissue receives such a high resistance to penetration that even metallic arrowheads , shot from heavy arrow bows , are unable to penetrate a tissue a few millimeters thick.

## Explanation as a shear-induced phase transition

There are different explanations for dilatancy in concentrated suspensions of colloidal (Brownian) particles . Here the approach is explained in more detail to consider dilatancy as a shear-induced phase transition .

The starting point for the explanation of dilatancy in concentrated suspensions are the intermolecular forces of colloidal particles. The interaction of electrically stabilized colloidal particles is described by the DLVO theory . It is characterized by two opposing forces. On the one hand, the particles are attracted by van der Waals forces . On the other hand, there are charges on and around the particles ( electrical double layer ), which cause colloidal particles to repel each other. The graphic shows the two-particle potential U (h) of electrically stabilized particles. It consists of a primary and secondary minimum, caused by the van der Waals attraction, and a potential wall due to the electrostatic repulsion, the maximum of which is at . ${\ displaystyle h _ {\ mathrm {max}}}$ For example, if the concentration of the particles in a suspension is increased, the mutual distance becomes smaller and smaller. From a certain critical concentration onwards, thermal fluctuations ( Brownian movement ) are sufficient to overcome the potential wall. The colloidal suspension becomes unstable, i.e. H. the Brownian particles coagulate and adhere to one another due to the Van der Waals attraction. Associated with this is the separation into two phases, a concentrated (coagulated) phase and a phase with almost no particles. The same can be achieved by reducing the electrostatic repulsion, for example by adding suitable ions (salt). In both cases the suspension shows an equilibrium phase transition from a so-called colloidal fluid to a colloidal solid. ${\ displaystyle h_ {0}}$ The increase in the particle concentration is tantamount to a decrease in the volume per particle. From a macroscopic point of view, a shear that maintains the volume of a liquid element should not produce any effect. However, the macroscopic view neglects the granular structure of a suspension consisting of solid particles. In order to get an understanding of dilation in suspensions, one has to switch to the mesoscopic level . For this purpose, consider a spherical density fluctuation from a large number of colloidal particles, as shown schematically in the graphic.

During its service life, this is deformed by continuous shear at the shear rate by the value : ${\ displaystyle \ tau}$ ${\ displaystyle \ gamma}$ ${\ displaystyle u}$ {\ displaystyle {\ begin {aligned} u & = \ tau \ gamma \ end {aligned}}} With increasing volume fractions, the lifetime τ of a density fluctuation grows and finally becomes infinite near the so-called closest packing of spheres ( lubrication theory ). In other words, a suspension behaves like a solid as the concentration of colloidal particles increases. In the case of continuous shear deformation, a density fluctuation is elongated (elongation axis) and at the same time compressed perpendicular to it (compression axis) during its service life as shown in the graphic. Colloidal particles approach each other along the compression axis. If the shear deformation is large enough, the particles can come so close that they overcome the potential barrier. When that happens, they stick together, increasing the shear viscosity. ${\ displaystyle u}$ If you consider this effect as an activation process , you can approximate both the equilibrium and the shear-induced phase transition with a simple formula. If we denote the number of coagulation events per unit of time as, then it is given by the Boltzmann statistics : ${\ displaystyle f}$ ${\ displaystyle f = f_ {0} \ exp \ left ({\ frac {U_ {B} - \ sigma V} {k _ {\ mathrm {B}} T}} \ right)}$ where is the applied shear stress, activation volume and thermal energy. The frequency has the maximum value when the exponent just disappears. For the equilibrium phase transition is determined by the maximum of the potential barrier . For the critical shear stress for the occurrence of the dilatance can be determined by: ${\ displaystyle \ sigma}$ ${\ displaystyle V}$ ${\ displaystyle k _ {\ mathrm {B}} T}$ ${\ displaystyle f_ {0}}$ ${\ displaystyle \ sigma = 0}$ ${\ displaystyle U_ {B} = U (h _ {\ mathrm {max}})}$ ${\ displaystyle \ sigma> 0}$ ${\ displaystyle \ sigma _ {C}}$ ${\ displaystyle \ sigma _ {C} = {\ frac {U_ {B}} {V}}}$ Considering the dilatance as a shear-induced phase transition allows the calculation and explanation of experimental results:

1. Dilatancy disappears at low volume fractions. Since the lifetime of density fluctuations is very short at low volume fractions, these disappear before they can be sufficiently deformed.${\ displaystyle \ tau}$ 2. There are two forms of dilatation: reversible and irreversible dilatancy. In the case of irreversible dilatancy, once formed particle clusters cannot separate from one another because they are trapped in the primary potential minimum. In the reversible case, the colloidal particles, caused by the surface roughness, do not come close enough to not be separated again by thermal excitations. However, this process takes a certain amount of time and leads to rheopexy .
3. Dilatation can be influenced by a suitable choice of the particles and their interaction (volume fraction, salt concentration, suspending medium, etc.). The equation for critical stress suggests that this becomes lower the lower the potential maximum , i.e. the closer a suspension is to an equilibrium phase transition. If, for example, the volume concentration or the salt content of an electrically stabilized suspension is increased, the shear rates or the shear stress decrease in order to trigger dilation.${\ displaystyle U_ {B}}$ 4. Suspensions that have already been coagulated show no dilatation, but only shear thinning.