Turnbull-Fisher equation

In some cases, as in crystallization of viscous materials from the melt, the Fisher-Turnbull equation (29) is often used to describe nucleation of lipids (20, 30)... 

Fisher-Turnbull Analysis The activation free energy for nucleation, Gc, may be found from the Fisher-Turnbull equation given in Equation 1. The term in the second exponential of Equation 1 is often given as Gc/kT. Combination of Equations (1) and (3) allows development of the following equation ... 

Fig. 5. Fitting of the nucleation kinetics of palm stearin blends in sesame oil according to the Fisher-Turnbull equation (from Ref. 9).

Values of x cieaiion were used to calculate free energies of nucleation (AGc) above 20°C according to the Fisher-Turnbull equation (Strickland-Constable, 1968) ... 

Figure 6 Determination of AGc according to the Fisher-Turnbull equation. Plot of In xT versus llT(ATf showing regions of differing slopes above and below 20°C.

The activation free energy of nucleation, AG was evaluated using the Fisher-Turnbull equation [8],... 

The Fisher-Turnbull equation [Fq. (1)] was originally derived for singlecomponent systems however, it was proved to be applicable to palm oil and milkfat, which are multicomponent systems [9,10]. A fairly good linearity of the plots shows that the Fisher-Turnbull equation can be used to study such systems. 

The rate for homogeneous nucleation J) of spherical assemblies can be expressed by the classical nucleation equation (Gibbs, 1948 Volmer, 1939 Becker and Doring, 1935 Turnbull and Fisher, 1949) ... 

The rate equation predicts exponential growth once a critical supersaturation is attained, however, in practice, an optimal temperature exists below which the liquid is too viscous to nucleate and above which molecular motions prevent crystal formation. This was observed by Tamman (1925) for several organic salts. He found that the optimal nucieation temperature was lower than that required for maximal crystal growth. A similar observation was made by Mullin and Leci (1969) for the spontaneous nucieation of citric acid solutions and is shown in Figure 2.21. The viscous effects can be incorporated into the rate equation by taking into account the viscous free energy (Turnbull and Fisher 1949). 

Melts frequently demonstrate abnormal nucleation characteristics, as noted in the early work of Tamman (1925). The rate of nucleation usually follows an exponential curve (solid curve in Figure 5.2) as the supercooling is increased, but reaches a maximum and subsequently decreases (broken curve in Figure 5.2). Tamman suggested that this behaviour was caused by the sharp increase in viscosity with supercooling which restricted molecular movement and inhibited the formation of ordered crystal structures. Turnbull and Fisher (1949) quantified this behaviour with a modified form of equation 5.9 ... 

The rate of nucleation, I, has been estimated by Turnbull and Fisher [25] from the shape of AG and the influence of local viscosity, governed by a free enthalpy AG,. In Fig. 3.61 this equation is listed. The rate I applies to the case that nucleation is unhindered. The first exponent of the equation expresses retardation of nucleation due to viscosity effects with the given parameters (see Sect. 5.6). It stops nucleation as the glass transition temperature is approached. The nucleation described is homogeneous nucleation and creates a continuous stream of new crystals in the remaining melt or solution. For polymers it takes a supercooling of about 50 K to overcome the... 

In 1926, Volmer and Weber found that the nucleation rate shows a negative exponential dependence on critical free energy barrier (Volmer and Weber 1926). Becker and Doting further proposed that the activation energy for the short-distance diffusion of molecules molecules to enter the crystalline phase should be considered as well (Becker and Doring 1935). Turnbull and Fisher derived the prefactor for the rate equation of crystal nucleation (Turnbull and Fisher 1949). The rate of polymer crystal nucleation i with the change of critical free energy barrier can be expressed as... 

For monatomic systems, the steady state nucleation frequency per unit volume in a clean system (homogeneous nucleation), neglecting strain energy (valid for the crystallization of a fluid), is given by the equation (Turnbull and Fisher (1949)),... 

Homogeneous nucleation (sporadic) is often expressed by the following equation proposed by Turnbull and Fisher [60]. 

The nucleation rate is dominated by two factors. One is the critical free energy barrier of nucleation. Its exponential dependence was first proposed by Volmer and Weber (1926). The other is the diffusion energy barrier for molecules crossing over the liquid-solid interfaces. Its exponential dependence was first proposed by Becker and Doring (1935). The quantitative expression of the prefactor in the kinetic equation of the nucleation rate is given by Turnbull and Fisher (1949) as... 

The quantitative expression of the prefactor Iq in the kinetic equation of the nucleation rate is given by Turnbull and Fisher [57] as... 

In Eq. (9.164) the transport term is represented by E AG is the activation free energy required to deposit the first strip. The explicit value of AG is defined by I. The first exponential term within the bracket is thus the Turnbull-Fisher relation for the steady-state nucleation rate of monomers and nonfolded polymers. This conventional expression for the steady-state nucleation rate is modulated by the second term within the brackets. This term is present because of the basic assumption that the nuclei are composed of regularly folded chains. The extent of the modulation depends on the difference between I and T. When this difference is large there will be a signihcant effect. However, when (I — T) is small there will scarcely be any influence and the conventional expression will apply. Under these circumstances there is no indication in the equation that the chains within the nucleus are regularly folded. 

---