Diffusion equation short chains

If diffusion of short chains is prohibited, the rate equation for chain scission of sani-... 

The rate equation for polymer chain scission was presented in Chapter 3 for amorphous polymers and Chapter 4 for semi-crystaUine polymers. These equations are summarised in this section. The terms in these equations that are affected by the diffusion of short chains are highlighted. 

For amorphous polymers, the governing equations for the scission of long chains and the diffusion of short chains are... 

This may explain why short-chain surfactants with highly branched hydrophobic groups show such good wetting properties (see below). Since values of as are readily obtained (Chapter 2, Section IIIB), equation 6.23 can also be used to obtain relative diffusion coefficients from wetting times (Cohen, 1981). 

The polymerisation of PO and EO, initiated by polyfunctional starters, to make short chain polyether polyols is a reaction that is strongly dependent on diffusion. The consumption rate of PO or EO is given by two simultaneous factors the rate of the chemical reaction in the liquid phase and the efficiency of the monomer mass transfer from the gaseous phase to liquid phase (see details in section 4.1.5). The PO (or EO) consumption rate, considering the mass transfer, is described by equation 13.27 [45-50] ... 

Diffusion is the phenomenological movement of short chains from a region of high concentration to a region of low concentration. The main driving force of this movement is the concentration gradient. Pick s second law, which is also known as diffusion equation is used to describe the diffusion process ... 

In the general termination equation, Eq. (6 and 7), it is accepted that the termination rate constant, k, is diffusion controlled and, therefore, it is a strong function of chain length for short chains and a rather weak function for long chains. 

The mathematical equations that govern the heterogeneous degradation are presented in Chapter 6. A key variable in the equations is the concenlration of the short chains. These short chains diffuse from where their concentration is high to where it is low. This principle can be combined with the requirement for matter conservation and written into a mathanatical equation. Before the computer age, such equations were very difficult to solve for real devices. It is almost impossible to find an analytical function to desalbe how the concentration varies with space and time in a device of sophisticated shape. The finite element method overcomes this difficulty by using two key ideas ... 

Equations [2.26] and [2.30] have been widely used in the literature to interpret degradation data for their simplicity. The validity of these equations is often limited to the very early stage of the degradation. For amorphous polymers, the master Equation [2.15] is more generally valid. However, short chain diffusion is ignored, which is the topic of the following chapters. 

Master equation without short chain diffusion... 

At any location x, the change in the concentration of short chains comes from two parts (i) production of short chains due to chain scissions at that location and (ii) deposit or ranoval of chains from this location by diffusion, (i) can be calculated by differentiating Equation [2.9] as... 

For semi-crystalline polymers, the governing equation for chain scission rate and short chain diffusion are... 

There are three kinetic parameters in the non-dimensionalised equations l poiymer reflocts the diffusion rate of short chains in liquid and can be... 

When the two monomers are linked by a short flexible chain, intramolecular excimers can be formed. This process is still diffusion-controlled, but in contrast to the preceding case, it is not translational it requires a close approach between the two molecules via internal rotations during the excited-state lifetime. Equations (4.44), (4.45), (4.47) to (4.49) are still valid after replacing k [M] by k because intramolecular excimer formation is independent of the total concentration. Estimation of the local fluidity of a medium can be achieved by means of probes capable of forming intramolecular excimers (see Chapter 8). 

The effective diffusivity Dn decreases rapidly as carbon number increases. The readsorption rate constant kr n depends on the intrinsic chemistry of the catalytic site and on experimental conditions but not on chain size. The rest of the equation contains only structural catalyst properties pellet size (L), porosity (e), active site density (0), and pore radius (Rp). High values of the Damkohler number lead to transport-enhanced a-olefin readsorption and chain initiation. The structural parameters in the Damkohler number account for two phenomena that control the extent of an intrapellet secondary reaction the intrapellet residence time of a-olefins and the number of readsorption sites (0) that they encounter as they diffuse through a catalyst particle. For example, high site densities can compensate for low catalyst surface areas, small pellets, and large pores by increasing the probability of readsorption even at short residence times. This is the case, for example, for unsupported Ru, Co, and Fe powders. 

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