Equation Turnbull

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)),... 

The frequency I of heterogeneous nucleation is represented by the following equation (Turnbull and Vonnegut 1952) ... 

Cohen and Turnbull [87] generalized somewhat the theoretical concepts of the relationship between diffusion and self-diffusion of liquids modelled by assemblies of rigid spheres and obtained on the basis of the theories of Frenkel and Eyring, Fox and Flory [88] and Williams, Landell and Ferry [89] the equation ... 

In a later work Cohen and Turnbull [90] defined the free volume of liquid as a part of the thermal expansion being freely redistributed throughout the volume without any change in energy. Macedo and Litovitz [91] pointed out that Cohen-TurnbulTs equation (74) ensues from an already known equation obtained independently by Fulcher [92] and Tamman [93] ... 

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 ... 

Figure 4.12 The points give the measured viscosity-temperature relationship for e-terphenyl, while the shaded regions are the viscosities predicted by the Adam-Gibbs equation (4-10) using A5(7 ) measured for o-terphenyl. The two shaded regions represent alternative fits of the Adam-Gibbs parameters, one fit to the high-temperature, and the other to the low-temperature, data. (From Greet and Turnbull, reprinted with permission, from J. Chem. Phys. 47 2185, Copyright 1967, American Institute of Physics.)...

The method is based on classical nucleation and growth equations for amorphous materials and a derived expression for AG based on the expressions of Turnbull,Hoffman, and Thompson and Spaeten. Using the calculated AG value, published material property data such as modulus and surface energy, and the measured crystallization or glass transition temperature (T ) obtained from differential scanning calorimetry (DSC), analytical expressions for nucleation rate and growth rate can be written. These expressions are then used as the basis for a pixel-by-pixel modeling approach for visualization of the microstructural evolution of the cross-section of a thin... 

Turnbull and Vatts ( ) determined the equilibrium pressures for the disproportionation of ZrClg(cr) in the temperature range from 613 K to 723 K by a modified dew point technique. 2nd and 3rd law analyses of their vapor pressure-temperature equation give the same enthalpy of reaction A H (298.15 K) - 30.7 kcal mol" for 2ZrClg(cr) + 2 The... 

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

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) ... 

In equation (3.01), A and B are empirical constants, Vocc is the volume occupied by the constituent particles and v/ is the free volume. In equation (3.02), r]a, C and To are constants. VTF equation implies that viscosities of glass forming supercooled liquids are non-Arrhenius and To is the temperature which linearizes the data of the non-Arrhenius plot. Cohen and Turnbull (Cohen and Turnbull, 1959 Turnbull and Cohen, 1961,... 

The Cohen and Turnbull free-volume model [30] assumes a liquid composed of hard-sphere molecules and voids in which diffusion occurs whenever a void larger than some minimum volume V forms in the body of the liquid and a molecule jumps into it. The equation for the diffusion coefficient is... 

Turnbull and the Doolittle equations, both of which are essentially free-... 

To modify the Cohen-Turnbull equation for use at elevated pressures we put it in the form... 

Figure 4-19. Variation of (P) with pressure. The points o represent experimental values of (P) for cis-decahydronaphthalene. Points are experimental values for n-pentadecane. Solid curves are calculated from the modified Doolittle equation, dashed curves from the modified Cohen-Turnbull equation. Data by Hogenboom, Webb and Dixon [151.

The original work on the free-volume model by Cohen and Turnbull - showed that the fluidity obeyed the Doolittle equation (2.8). We show that the percolation ideas developed in this paper give rise to the same equation for the fluidity. 

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). 

Time-temperature superpositioning was originally derived from free volume models, which assume that the rates of molecular motions are governed by the available unoccupied space. Early studies of molecular liquids led to the Doolittle equation, relating the viscosity to the fractional free volume, f =V /(V - Vo), where V is the specific volume and Vo is the occupied volume normalized by the mass) (Doolittle and Doolittle, 1957 Cohen and Turnbull,... 

The constants correspond to coo = C, B = yVflEf, and Tq = T o (7b is the Vogel temperature). With Cohen and Turnbull delivered this free-volume model, a theoretical justification of the empirical VFTH equation and the equivalent Williams-Landel-Ferry (WLF) [Williams et al., 1955] equation as well as of the empirical free-volume models of viscosity [Fox and Flory, 1950 Doolittle, 1951]. 

ISM parameter (VDW analogue) constant in Tait equation 0.0894 (Ch. 6), or Cohen-Turnbull the pre-exponential factor... 

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 ... 

Cohen and Turnbull (1959) clarified the physical significance of the Doolittle equation by demonstrating that the probability P v ) of finding a hole of size v or larger in a field of free volume can be expressed as... 

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... 

It is known that glassy polymer membranes can have a considerable size-sieving character, reflected mainly in the diffusive term of the transport equation. Many studies have therefore attempted to correlate the diffusion coefficient and the membrane permeability with the size of the penetrant molecules, for instance expressed in terms of the kinetic diameter, Lennard-Jones diameter or critical volume [40]. Since the transport takes place through the available free volume in the material, a correlation between the free volume fraction and transport properties should also exist. Through the years, authors have proposed different equations to correlate transport and FFV, starting with the historical model of Cohen and Turnbull for self diffusion [41], later adapted by Fujita for polymer systans [42]. Park and Paul adopted a somewhat simpler form of this equation to correlate the permeability coefficient with fractional free volume [43] ... 

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