Crystallisation Flashcards
Crystal Habit
Describes general external shape of crystal
Gibbs crystal habit theory
Crystal growth attempts to minimise total free energy of the surface
High surface energy = low morphological importance
Wulff crystal habit theory
Distance of facet from its centre is proportional to surface energy
BFDH crystal habit theory
facet growth is inversely proportional to the interplanar spacing, d_hkl
Hartman & Perdock crystal habit theory
Growth rate is proportional to the attachment energies of crystal facets
Miller Indices
Integers that are inversely proportional to the intercepts of the crystal face
h = a/x, k = b/y, l = c/z
Miller Indices methodology
- Determine the intercepts of the face along the crystallographic axes
- Take reciprocals
- Clear fractions
- Reduce to the lowest terms
Polymorphs
Molecularly identical crystals with different lattice structure
Dihedral angle
Angle between 2 facets
consistent irrespective of particle size
Supersaturation
Solution containing more solute than thermodynamically stable in solutions (and able to overcome energy barrier to form crystal surface)
Saturation
Crystal is at thermodynamic equilibrium with solution (maximum concentration that is thermodynamically stable)
Degree of Supersaturation
Delta_C = C - C*
Supersaturation Ratio
S = C/C*
Relative Supersaturation
Phi = Delta_C/C* = S - 1
Moyers & Rousseau (van’t Hoff Relationship)
ln(x) = Hf/RT * (T/Tm - 1)
Hf = Latent heat of fusion Tm = melting point
Predicition of solubility for very small particles
ln (cr/c) = 2Msigma/(niRTro_sr)
Supersaturation Equation (Sigma)
sigma = Delta(mu)/RT
Primary Nucleation
In the absence of seeded crystals
Homogeneous nucleation
formation of crystals from supersaturation only
Heterogeneous nucleation
formation of crystals from the presence of insoluble material
Secondary nucleation
Occurs in the presence of seeded crystals
Empirical equation for secondary nucleation
dN/dt = k_n(c - c)^i = B
Classical Nucleation Theory
Surface energy of small crystals provides a thermodynamic barrier to forming new surface
Delta Gr = - 4/3 pir^3 * Delta Gv + 4pir^2 *sigma
(Assumes Nucleus to be spherical and interacial tension to isotropic - non directionally dependent)
Critical Radius
Minimum size of crystal for it to be more energy beneficial to grow than redissolve
Method of solving for r_crit and Delta Gr_crit
Set d(delta Gr)/dr = 0 Solve for r_crit put equation for r_crit into delta Gr equation to solve for delta Gr_crit
Homogeneuous rate of nucleation
J = Fexp((-16pi*sigma^3 vm^2)/(3k^3 T^3(lnS)^2)
Heterogeneous rate of nucleation
J = K_n * delta(c)^n
Derivation of homogeneous nucleation rate
- Start with Arrhenius:
J = F*exp(-delta G/kT) - Insert delta G_crit
- Rearrange ln(cr/c*) for r and insert
- Use molar volume to remove terms and insert some constants in pre-exponential factors
Define induction period
Delay between supersaturation & first crystal formation
ti is proportional to 1/J
2 Stage process of crystallisation
- Birth = Nucleation
- Growth (Volume Diffusion & Surface diffusion)
3 Crystal growth regimes
- Continuous Growth (lots of kink sites)
- Surface Nucleation (several nuclei on suface)
- Spiral growth (single nucleus on surface)
Crystal habit modification
- growth rates are not constant along facets
- Specific adsorption of species on particular facets modifies crystal growth
Crystal Yield Equation
y = Rw1 (c1 - c2(1 - E))/(1 - c2(R - 1))
R = ratio of hydrate: anhydrous MW E = ratio of solvent evaporated to initial solvent
Crystal yield derivation
- Solute balance
w1c1 = w2c2 + y/R - Solvent Balance
w1 = w2 + y(R - 1)/R + w1E - Rearrange solvent balance for w2
- Insert w2 into solute balance
- rearrange for y
Derivation of mass of seeds required equation:
- Take the mass of crystal assuming spherical
m_s = ro*pi/6 * d_s^3 - Take equation for yield based on volume difference:
y = ropi/6(d_p^3 - d_s^3) - Write out m_s/y
- Then multiply both sides by y
Optimum cooling curve
- Growth rate will increase as crystal grows
- dC/dT of solubility curve decreases as C decreases, therefore need to increase (-dT/dt) to accommodate
- Need to cool faster to remain in metastable region
- Slow initial cooling but increasing over time
Practice sketching crystallisation curves
slides 50 and 51
