Impact of Mass Transfer Limitation of Polyurethane Reactions
February 2017 www.coatingsworld.com Coatings World | 43
bonds) while the center of mass of the molecule does not
move.
• Approach 2 places parameters quantifying inter- and intra-molecular movement in the collision frequency factor of
the Arrhenius equation.
Approach 1 (1a and 1b) is an approach of modeling diffusion
as a kinetic process in series with the reaction process. Approach
2 recognizes that the frequency factor of the Arrhenius equation
accounts for the collision process (i.e., mass transfer process)
and, so, the frequency factor is expressed in terms of viscosity and both inter- and intra-molecular approaches to moiety
collision.
Approach 1a: Inter-Molecular Diffusion Rate
The diffusion rate of two reactive moieties can be modeled in a
power-law rate expression where the rate constant of diffusion
is proportional to the diffusivities of the two components [ 9] as
provided by Equation 3.
Einstein [ 10] and Smoluchowski [ 11] suggested an equation
to calculate the diffusivity of the two components. This equation
assumes that the diffusivity of a component is increasing as the
temperature increases and decreases as the viscosity of the reaction medium increases according to Equation 4.
Combining these equations gives Equation 5.
The following two assumptions were used to simplify
Equation 5:
• The molecular radius of the reactive moieties is assumed to
be equal (rA = rB = r).
• The critical distance necessary to form the encounter complex is assumed to be equal to two times the molecular
radius of the moieties (R* = 2r).
These assumptions reduce Equation 5 to Equation 6 where
A1 is a constant, T is measured in Kelvin and resin viscosity
measured in centipoise.
This equation shows that the reaction temperature and resin
viscosity are the key parameters that impact rate of diffusion
(in liquids). Backward diffusion rate constant is assumed to be
equal to the forward diffusion rate constant.
The reaction mechanism of Equation 1 is divided into
three rate expressions as shown in Table 1. For purposes of
simulation, this was assumed for all urethane reactions as pro-
posed by Ghoreishi [ 5].
Approach 1b: Inter- and Intra-Molecular Mass Transfer
Intra-molecular movement is considered to describe the local
movement of the reactive moieties. The rate of intra-molecular
diffusion is assumed to be proportional to the square root of
the reaction temperature and, more importantly, independent
of viscosity.
In Equation 7, A2 is set as one fitted parameter that leads to
a better temperature profile for the region after the gel point. By
adding inter- and intra-molecular diffusion, the total diffusion
rate becomes as noted in Equation 8.
Comparing this mechanism with that considered a one reaction step, this approach increases the number of parameters
necessary to model urethane reactions from two (the Arrhenius
parameters) to four (the Arrhenius parameters plus A1 and A2).
Approach 2: Incorporating Rate of Diffusion into
Frequency Factor
According to the kinetic theory, two conditions must be met for
reaction to occur:
• The reactive moieties must collide.
• The collision must be of sufficient energy to overcome
barriers.
For the Arrhenius equation:
The limit of no activation energy of Equation 9 yields:
The interpretation of the zero activation energy is that all
collisions of the reactive moieties lead to reaction. Hence, the
pre-exponential factor “A” is related to the collision frequency
“Z”.
Recognizing that inter-and intra-molecular mass transfers
are parallel and an additive path to collision for reaction of
Table 1. Rate expressions of the diffusion and reaction steps.
