During urethane reactions, large changes in viscosity lead to large changes in mass transfer rates. After the gel point, the mass transfer of blowing agents from the gel to cells essentially ceases; however, mass transfer for reactions continues, albeit at a slower rate. This paper discusses fundamental approaches to account for the impact of these changes in mass transfer on reaction rates.
Three different approaches were studied to describe the rate of molecular movement of the reactive moieties during polyurethane reactions. Inter- and intra-molecular moiety movements were studied as separate diffusion steps. Inter-molecular movement stops at the gel point and intra-molecular movement has been added to describe moiety movement after the gel point. The collision frequency factor in the Arrhenius equation was modified based on inter- and intra-molecular approaches. Simulation results for reaction temperature and viscosity profiles show good agreement with the experimental data.
Diffusion-limited reactions are reactions that occur quickly and the rate of diffusion of the reacting components in the reaction medium is the limiting step. Smoluchowski , Collins and Kimball , and Noyes  were the first to discuss and develop this phenomena. When the reacting components are mixed in a reacting media, diffusion rate is proportional to the time it takes to bring these components (or the reactive part in these components) into a reacting distance to allow a reaction to occur.
For a liquid phase reaction, the viscosity of the reaction mixture is the key physical property to which rates of diffusion correlate. In a thermoset polymerization reaction, the viscosity of the resin increases due to the formation of long chain polymers and polymer crosslinking. This leads to increasing viscosity and decreasing the rate of diffusion of the reacting components. For thermoset reactions, viscosity ultimately approaches infinity and, so, diffusion becomes the controlling step at some point during polymerization.
For a simple irreversible reaction of two reacting components, A and B, the diffusion and the reaction steps can be represented by the following steps:
• The diffusion of the two components from a large separate distance in the reaction medium to a proximity at which the chemical forces become dominant and the pseudo species (AB)* is forming.
• The dissociation of the pseudo species as the reacting components diffuse apart.
• The reaction of the pseudo species to form a product.
The pseudo species (AB)* is considered as an active complex if it is at the saddle point of the potential energy surface of A and B, or as a collision complex if there is no interaction between A and B. The term “encounter complex” has been used to cover cases of either the active or collision complex .
Polyurethane (PU) forming reactions were considered for studying and discussing the impact of diffusion of the reacting components on the reaction rates during the reaction. A polyurethane is defined as a long chain of organic units joined by urethane links. Alcohol moieties in the polyol and isocyanate moieties react to produce urethanes according to Equation 2.
The properties of the polyol and the isocyanate (molecular structure and molecular weight) impact the properties of the polyurethane produced . Catalysts, crosslinkers, and light stabilizers also impact the polymerization reaction [6-8].
This paper discuss two approaches to provide an insight and effective method to more accurate simulation of urethane forming reactions. These approaches are as follows.
• Approach 1a assumes that the reactive moieties diffuse together in the mixture to form an encounter complex which may disassociate or react to form a product. This type of diffusion is considered as the inter-molecular movement of the reactive alcohol and isocyanate moieties.
• Approach 1b is intra-molecular movement of the reactive moieties which is considered in parallel to the inter-molecular movement. Intra-molecular movement is the movement of segments of the molecule (no breaking of chemical 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  as provided by Equation 3.
Einstein  and Smoluchowski  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 proposed by Ghoreishi .
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 polymer systems, the frequency factor should include a driving force for collision according to Equation 11.
To quantify this driving force, these rates were multiplied by the probability of successful reaction as a result of collision which is referred to as the steric hindrance factor “ρ”. This mechanism results in Equation 12.
Substituting this result into Equation 8 and Equation 9 yields Equation 13.
This approach requires two fitted kinetic parameters as compared to modeling diffusion separately which requires a total of four parameters.
Several researchers proposed different mechanisms, approaches, and equations to identify the impact of resin viscosity and mass transfer limitation during the reaction [12-15]. These works included assumptions such as:
• Constant rate of reaction.
• Neglecting the impact of temperature changes.
• Using empirical or semi-empirical equations for rates of diffusion and reaction.
The results of these studies are highly specific to the system studied. The research presented here differs from previous works as follows:
• The impact of mass transfer on polymer reactions was successfully evaluated using two fundamental approaches.
• Modeling is performed through the gel point where there is a transition from the mass transfer from the inter- to intra-molecular movement.
• Modeling is performed with over 50 ordinary differential equations to account for the elementary reactions and processes.
The fundamental nature of the research presented in this paper, including approaches to account for mass transfer in the frequency factor of reaction rates, has the prospect of being widely applicable to a number of polymerization reactions.
Simulation has been built on the simulation code of Zhao, Ghoreishi, and Al-Moameri [5, 6, 16-19] for the reaction kinetics and blowing agents. Group contribution simulation presented by Fu  has been used for viscosity calculations.
Table 2 shows the gel formulation of the urethane forming reaction. The experiments were performed by the following steps.
1. The B-side components in Table 2 were mixed together in a closed beaker.
2. The A- and B-side materials were poured in a plastic cup and mixed using a 2000 rpm mixer blade attached to a floor-model drill press for 10 seconds.
3. LabView software with type-K thermocouple was used to measure reaction temperature.
4. The viscosity profiles of the polyurethane gel were measured using a Cole-Parmer basic viscometer.
All experiments were carried out at ambient temperature to avoid deviations in reaction kinetics .
Mixing rate was kept the same for all gel experiments as it may affect the viscosity profiles . All experiments were carried out at ambient temperature in order to avoid deviations in reaction kinetics and molecular weights of the polymer . Foam forming data from a previous publication  was used to evaluate the simulation approaches of this paper in foam-forming reactions. Table 3 shows the specification of the polyol and isocyanate.
Results and Discussion
Approach 1a: Inter-Molecular Diffusion Rate
The experimental and simulation results of reaction temperature and viscosity profile for the formulas in Table 2 are shown in Figure 1. For the diffusion of the reacting moieties considered as only inter-molecular diffusion, the simulation shows that the reactions stop at the gel point. Resin viscosity increases to infinity at the gel point and the rate of diffusion of the reacting moieties decreases to zero according to Equation 6. After the gel point, the temperature and height cease to increase and unreacted moieties cease to further react. The good fit of the simulation results of reaction temperature and viscosity profiles in the region before the gel point supports using the second order reaction kinetic parameter and fraction alcohol moieties that were reported in previous studies .
Figure 2 shows the ratio of the diffusion rate to the reaction rate. The results show that the ratio is decreasing rapidly when the viscosity of the resin increases rapidly at the gel point. In comparison, the rate of reaction increases according to the Arrhenius equation. According to this ratio profile, the reaction rate is the limiting step in the region before the gel point, and the diffusion rate is the limiting step after the gel point.
An example of a concentration of one encounter complex (AB)* is shown in Figure 3. The higher rate of diffusion at the beginning of the reaction leads to instant increase in complex concentration, and then the concentration decreases as the driving force for formation of the encounter complex decreases.
Approach 1b: Inter- and Intra-Molecular Mass Transfer
The simulation results of Figure 4 show that the reactions continue after the gel point. The movement of sections of a polymer allow the reaction without a net movement of polymer molecules relative to each other. This result supports the addition of the intra-molecular movement.
The A1 and A2 parameters in the rate of diffusion were optimized to provide a fit to the data.
Figure 5 shows the ratio of the rate of diffusion to the rate of reaction. The ratio decreases throughout the reaction time in the region before the gel point in which many monomers that diffuse together diffuse apart before reacting. In the region after the gel point, the rate of diffusion decreases to zero as the viscosity of the polymer increases to high values. The rate of reaction proceeds through the intra-molecular mechanisms of polymer segments.
Figure 6 shows an example of the concentration profile of one encounter complex. The trend is similar to the profile of the inter-molecular mechanism except it shows a low but significant rate of reaction after the gel point. Other encounter complexes decrease to zero at the same time when the maximum reaction temperature is attained.
Approach 2: Incorporating Rate of Diffusion into Frequency Factor
Equation 13 is considered as a reaction rate expression for both catalytic and homogeneous reactions. Good agreements were obtained with the experimental result profiles. One value of A1 and A2 (shown in Table 4) was used for the rate of all reactions including homogeneous and catalytic reactions. A good fit to the data demonstrates that these frequency factor values are independent of the actual encounter complex where the impact of catalysis is reflected in lower activation energies and catalyst count complexes.
Table 4 lists the kinetic parameters used in the simulation approaches for inter-and intra-molecular movements and the frequency factor.
For a chemical reaction to occur, the reacting molecules need to diffuse to proximity and a sufficient energy state to overcome the activation energy. The impact of the rate of mass transfer on reaction rate was simulated for rigid urethane polymerization reactions using two approaches.
The first approach simulates the mechanism of the rate of diffusion. This approach shows that the reaction is reaction limited in the region before the gel point, and diffusion limited after the gel point. When only the inter-molecular movement is considered, the simulation results of the temperate profile stops increasing at the gel point as the rate of diffusion of the moieties decreases to zero due to the increase in viscosity.
Temperatures continue to increase after the gel point (infinite viscosity) in urethane polymerization systems and, so, it is clear that reaction mechanisms exist that do not rely on movement of entire molecules relative to the polymer matrix. Reactions based on moiety collisions from intra-molecular movement must be responsible for reactions after the gel point and this mechanism is in parallel with inter-molecular.
The second approach incorporates the sum of the inter- and intra-molecular movements as a pre-exponential factor in the Arrhenius equation. This approach uses fewer parameters and solving fewer differential equations.
The two approaches provide a good fit to the experimental data of reaction temperature and resin viscosity; however, the use of diffusion mechanisms is critical to simulate polymerization and incorporating the diffusion mechanisms in the reaction rate expression is more efficient.
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