Fragility and ductility of crystals

From the data provided by Mersmann [MER 01], we can deduce:

1) Fragile crystals:

Crystals with covalent bonds are typically hard and fragile, as these are directional bonds that hinder the reticular plane from slipping.

2) Ductile crystals:

Ionic bonds are not directional, which encourages slipping of reticular planes.

On collision with an agitator or pump rotor, the stress application speed is significantly greater than it would be during a Vickers hardness test, which takes place at near static speeds. Consequently, plastic deformation is hindered.

Agglomeration in suspension

Agglomeration is a result of collisions between particles. It is most significant for crystals smaller than 2 or 3 pm, becoming negligible for crystals of 30 pm.

The degree of agglomeration Z is the number of crystals present in an agglomerate. A degree of 80 has been observed. As a general rule, agglomeration only applies to crystals derived from primary nucleation. Indeed, secondary nucleation results in agitation which destroys agglomerates.

The force of interaction between the two particles is:

D: distance between the surfaces of two particles A: Hamaker constant (10-18 J) r: particle radius (m)

Debye-Huckel distance is 1/k with:

N0i: concentration of type i ions (kmol.m-3)

Г0 contains the surface potential ?o

Attractive forces exist between hydrophobic surfaces. One known example of this is the micelle formation of surfactants. On the other hand, solvated particles repel each other.

The aggregation of silica is a classic example.

In the basic medium, particles are negatively charged and do not agglomerate. However, if the concentration of electrolyte increases, surface charges are compensated and aggregation begins. However, aggregation is fast at pH < 7 due to the monosilicic acid neutralizing the OH- ions, which

were absorbed on the crystals. The concentration acts by means of the exponential of (-kD) .

According to von Smoluchowski, we can choose:

N : particle concentration

Two mechanisms apply: perikinetic and orthokinetic

1) Perikinetic:

D : diffusivity (m2.s-1)

L : particle size (m)

2) Orthokinetic:

Note that the solid volume fraction ф is:

Hence, replacing L3N by its value, the aggregation equation becomes:

The decrease in n is exponential.

Y: shear rate (s-1 (y ~ N))

n : number of particles per cubic meter

N: here, motor rotation frequency (turn.s-1)

в is inversely proportional to the suspension’s solid load (kg.m-3) due to agglomerate fracture.

Oversaturation consolidates the agglomerates.

Perikinetic aggregation becomes orthokinetic aggregation for crystal sizes from 10 to 30 pm.

Ultimately, agglomeration is favored by:

  • - high levels of diffusion;
  • - low viscosity. Indeed, diffusion is provided by the Einstein formula:

p: viscosity (Pa.s) dp: particle diameter (m) k: Boltzmann constant (1.38.10-23 J.K-1), so, high temperature;

  • - high concentration of particles (number);
  • - for particle sizes less than 2 or 3 pm, perikinetic agglomeration can be intense.

Agglomerate’s resistance to fracture is characterized by the ratio [ORO 49]:

Cc: kilomoles per cubic meter of crystal

Na: Avogadro number (6.023.1026 molecules per kilomole)

n: number of molecules in the agglomerate.

Г/K is far greater than the superficial energy, which is less than 0.1 J.m-2. The Young modulus for crystals is in the order of:

The fracture resistance of a polycrystal of 1 mm is:

The fracture resistance of polycrystals is inversely proportional to their size.

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