Classical Nucleation Theory (CNT)

The process of precipitation of particles using the solvent shifting method involves a concept known as ‘nucleation.’, the nucleation process is described based on classical nucleation theory(101). However, the process may include some ‘non-classical’ ways of nucleation.

‘Nucleation’ is a process that takes place when a system is brought into a non-equilibrium metastable state. This metastable state is described as a supersaturated condition that represents a local minimum of the free energy. Due to this local minimum free energy, the system is stable towards small fluctuations (102). However, with an increase in microscopic fluctuations, the random collisions of the dissolved molecules of the solute in the solution lead to the formation of the clusters of a new phase which are calle ‘Nuclei’. These nuclei represent the new state that represents the new minimum of the free energy possessed by the bulk system, and the transformation of the phase (from clusters to nuclei) takes place through the energy barrier which represents the local maximum free energy state(103,104).

19 Figure 1 Free energy diagram of Classical Nucleation Theory (CNT).

The process of nuclei formation in a supersaturated solution consisting of homogeneously distributed molecules can be explained through random collisions that lead to the aggregation of these molecules towards the formation of clusters or ‘embryos’. As the cluster or embryo reaches a particular size known as ‘critical’ size, it can be regarded as the ‘critical nucleus.’

Assuming it to be spherical, the radius of this critical nucleus can be predicted on the basis of CNT (Figure 1). It is important to note that, according to CNT assumptions, the collisions between any two pre-existing clusters, the break-off of pre-existing clusters into two or more smaller clusters or the collisions between two particles are ignored (105,106).

By considering the nature of nucleus as ‘spherical,’ the CNT assumes that the interior of the nucleus is in the bulk new-phase state, and the interfacial tension (𝜸) is the same as for a planar interface of the coexisting new and mother phases. With these assumptions, the free energy of homogeneous nucleation (Δ𝑮homo) for a spherical particle with radius 𝒓, the surface energy 𝜸 and the bulk free energy of volume ΔGv, is quantitatively represented as

Equation 1 𝚫𝐆_{𝐡𝐨𝐦𝐨} = 𝟒 𝝅𝒓^𝟐𝛄 + ^𝟒

𝟑 𝝅 𝒓^𝟑𝚫𝐆_𝐯

20 The free energy of the bulk nucleus is expressed as the difference between the free energy of the bulk particle and the solution as

Equation 2 Δ𝑮𝒗= − ^{𝑲𝑩 𝑻 𝐥𝐧(𝑺)}

𝑽𝒎

Where kB is Boltzmann's constant, T is temperature (K), S is the degree of supersaturation and Vm is the molecular volume

In the Equation 3, as the surface term is always positive while the bulk free energy is always negative, the maximum free energy through which a cluster will pass to form a stable nucleus can be obtained by differentiating ΔG with respect to r and setting it to zero (dΔG/dr = 0).

This gives critical free energy ΔGcrit: Equation 3 Δ𝑮Crit = ^𝟒

𝟑

𝛑 𝐫

The critical radius of the nucleus is given by:

Equation 4 𝑹Crit = − ^𝟐𝛄

𝚫𝐆𝐯

Combining Equation 3 and Equation 5 then give:

Equation 5 R Crit = ^{𝟐𝛄𝑽𝒎}

𝑲𝑩 𝑻 𝐥𝐧(𝑺)

Thermodynamically, the critical size reflects a metastable state as any radii smaller than rcrit

will be unstable and dissolve back in the solution while any radii larger than rcrit will lead to the unlimited growth, this is explained in Figure 1.

The green curve represents surface free energy, and blue curve represents the bulk free energy.

Combining these two terms allows the prediction of the critical radius at which a cluster of molecules become stable and start growing as a stable nucleate (red curve at rcrit).

From the Figure 1, it can be said that the positive surface term offers a barrier to form the critical nucleus. Once it is formed, the negative volume term then allows the nucleus to grow into a particle and attain the bulk properties.

Considering the kinetic barriers, the rate of nucleation (J) can be expressed as;

21 Equation 6

𝐉 = 𝐀 𝐞

−∆𝐆 𝐊𝐁𝐓)

Where the first exponent (^𝐄𝐚

𝐊𝐁𝐓) represents the kinetic barriers with an overall activation energy Ea, while the second exponent (^−∆𝐆

𝐊𝐁𝐓) represents the thermodynamic barrier. The parameter A is a pre-exponential factor which is determined by the properties of the nucleating solute. The theoretical value of the pre-exponential factor has been reported to be ~1030 cm⁻³ s⁻¹; however, it is very difficult to measure in practice.

The kinetic barriers are generally neglected due to difficulty in their quantification. On the other hand, the thermodynamic barrier can be calculated based on the assumptions of CNT, which mainly involve consideration of the similar behaviour of the nanoscopic nuclei, and macroscopic phase, i.e., the nuclei possess same structure and corresponding interfacial energies as the bulk. This is the background for nucleation based on CNT.

However, there have been discussions about the assumptions of the CNT and their lack of applicability to the extremely small clusters of few (~ 10) molecules, where the radius of curvature is so high that the centre of the curve is not in the thermodynamic limit and the interface is sharply curved, thus changing its free energy. Erdemir et al.(107) have discussed such shortcomings of CNT in detail, thus taking the discussion towards the other non-classical theories too.