Morton equations

The role of a swelling agent in the activated swelling method may be explained by considering the theoretical basis of the process. The swelling of pure polymer particles with the monomer can be described by the Morton equation ... 

If the swelling process is carried out in the presence of a swelling agent, the Morton equation may be written as follows ... 

The final form of the Morton equation for the activated swelling process may be written by expressing the equilibrium swelling radius of the seed particles in terms of initial radius by using Eq. (10). 

Figure 18 The variation of the swelling ratio of activated seed particles according to Morton equation Xmy 0.4, Xyp = 0). (a) VyIVp = 1.0, (b) VylVp = 3.0.

On the other hand, several researchers [164-167] have tried to thermodynamically describe the swelling behavior of polymer particles by one monomer. The thermodynamic approach now used is based on the so-called Morton equation given by... 

Hutchinson, H. P., D. J. Jackson, and W. Morton, Equation Oriented Flowsheet Simulation, Design and Optimization, Proc. Eur. Fed. Chem. Eng, Conf. Comput. Appl. Chem. Eng., Paris, April 1983 The Development of an Equation-Oriented Flowsheet Simulation and Optimization Package, Comput. Chem. Eng., v. 10, p. 19 (1986). 

The colloidal stability (or diffusional degradation) of monomer droplets can be further understood in terms of the Morton equation [48], in which the partial molar free energy of mixing of polymer in the monomer phase (spherical droplets) is expressed as ... 

The basic equation for swelling of a polymer particle with a monomer was given by Morton et al. who considered a system where a monomer in bulk swells polymer particles dispersed in water and where the presence of water in monomer and in polymer phase could be neglected. Denoting the monomer as compound Zp the Morton equation reads ... 

As stated by Tseng et al., the significance of a model which takes into account the absorbtion of water may be more pronounced for relatively hydrophilic systems. Also it would seem to be of importance in cases where relatively hydrophobic polymers are swollen with a mixture of a relatively hydrophilic monomer and water. This would for instance be the case when swelling a seed of polystyrene particles with methylacrylate in aqueous dispersion. In this case one would still have a relatively low value of Xf ip while the value of X p would be considerably higher than is the case for polymethylacrylate. The result will be a decrease in swelling as compared with the Morton equation because of the lowering of the activity of monomer in phase (b) and phase (C)). All in all the model of... 

The commercial polysulfldes are made from bis-chloroethylformal (formal) as shown later in equation 11. In some products trichloropropane is added as a branching agent. Table 1 shows typical properties of polysulfldes available from Morton International. These products were acquired from Thiokol Corp. in 1983. 

Morton, K. W., and D. F. Mayers. Numerical Solution ofFariial Differential Equations, Cambridge (1995). 

However, the mechanisms by which the initiation and propagation reactions occur are far more complex. Dimeric association of polystyryllithium is reported by Morton, al. ( ) and it is generally accepted that the reactions are first order with respect to monomer concentration. Unfortunately, the existence of associated complexes of initiator and polystyryllithium as well as possible cross association between the two species have negated the determination of the exact polymerization mechanisms (, 10, 11, 12, 13). It is this high degree of complexity which necessitates the use of empirical rate equations. One such empirical rate expression for the auto-catalytic initiation reaction for the anionic polymerization of styrene in benzene solvent as reported by Tanlak (14) is given by ... 

Morton, K. and Mayers, D. (1994) Numerical Solution of Partial Differential Equations. An Introduction. Cambridge University Press Cambridge. 

Hutchison, H. P., Jackson, D. J., and Morton, W., The development of an equation-oriented flowsheet simulation and optimization package—II. Examples and results, Comp, and Chem Eng. 10(1), 31 (1986). 

Researchers have examined the creep and creep recovery of textile fibers extensively (13-21). For example, Hunt and Darlington (16, 17) studied the effects of temperature, humidity, and previous thermal history on the creep properties of Nylon 6,6. They were able to explain the shift in creep curves with changes in temperature and humidity. Lead-erman (19) studied the time dependence of creep at different temperatures and humidities. Shifts in creep curves due to changes in temperature and humidity were explained with simple equations and convenient shift factors. Morton and Hearle (21) also examined the dependence of fiber creep on temperature and humidity. Meredith (20) studied many mechanical properties, including creep of several generic fiber types. Phenomenological theory of linear viscoelasticity of semicrystalline polymers has been tested with creep measurements performed on textile fibers (18). From these works one can readily appreciate that creep behavior is affected by many factors on both practical and theoretical levels. 

This process of molecular diffusion is governed by the difference in chemical potentials of the monomer in the two droplets. Morton s equation has been successfully used to describe the swelling of polymer particles with monomer [32]. According to this equation, the chemical potential of the monomer in a droplet of radius rp(p[ 0 in the presence of polymer is given by ... 

Equation (5) or (5b) is the highly important deduction of Harkins-Smith-Ewart theory. Its validity has been fully confirmed for many cases of polymerization (19). Furthermore, although it is difficult to determine the nvimber of particles, Np, accurately (19) this simple relationship has been used to determine the absolute value of the rate constant, kp, satisfactorily for the polymerization of butadiene and isoprene by Smith (20) and by Morton et al.(21). Conditions where the rate of polymerization is not proportional to the number of particles are where Trommsdorff s effect (22-24) or Gordon s unsteady state (25) principles apply. However, the existence of linear portions of the conversion-time plots proves the absence of these principles in this system. 

Morton and Corrsini used an electric analog device to simulate the Fokker-Planck equation of a linear and cubic spring. 

Morton, T. A., Myszka, D. G., Chaiken, I. M. (1995) Interpreting Complex Binding Kinetics from Optical Biosensors A Comparison of Analysis by Linearization, the Integrated Rate Equation, and Numerical Integration. Analytical Biochemistry 227 176-185. 

The fourth factor determining polymerization rate is the monomer concentration in the particles. For some monomers the ratio of monomer to polymer in the particles is about constant during part of the polymerization. Smith (57) suggested that this results from a balance between the eflFect on the monomer activity of the dissolved polymer and the eflFect of interfacial tension of the very small particles. This equilibrium was put in a quantitative form by Morton, Kaizerman, and Altier (44), who derived the following equation by combining an expression for the interfacial free energy of the particle with the Flory-Huggins equation for the activity of the solvent (monomer) in the monomer-polymer particle. 

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