A reactive spherical chain-polymerization particulate system with an exothermic heat of reaction is normally expected to exhibit a parabolic temperature profile, as depicted in Figure 1. In these systems, the idea of a flat temperature profile is almost unheard of, and potentially beneficial in terms of operational and product uniformity. Also, the analysis of the model equations can become less complicated for these types of systems, because a field dynamics problem can be reduced to the relative simplicity of a lumped-parameter system.

In recent publications [1,2], it has been conceptually shown that the so-called free-radical retrograde-precipitation polymerization (FRRPP) process can result in flat temperature profiles through a combination of thermodynamic, transport, and reaction-kinetic parameters in the system. This is shown in Figure 2, wherein the profile starts at a uniform Dimensionless T (or θ in Equation (6)) of 1.0. The temperature profile ends at a steady-state value of Dimensionless T = 1.83 [3]. Note that the dimensionless r in the horizontal axis of Figure 2 is the same as h in Equation (7). This means that the reactive fluid will have h ranging from 0 to 1 only. Values of h between 1 and 2 are rescaled so that they correspond to those of the stagnant fluid boundary layer.

The prediction of the possibility of the occurrence of flat temperature profiles from FRRPP systems has been advanced from a pseudo-steady-state analysis of the energy balance of a model spherical reacting system undergoing polymerization-induced phase separation [1]. The dimensionless differential energy balance has been shown as

and the dimensionless energy source term Φ is expressed as

The dimensionless parameters are related to the following dimensional quantities in the energy balance and phase equilibria equations:

The phase behavior for the polymer-rich phase was approximated by the linear representation

where X_P is the product of the weight fractions of the monomer and polymer (or just the weight fraction of the polymer if the monomer concentration is the same for both polymer-rich and polymer-lean phases at equilibrium, as the case for the PS-S-Ether system [4,5]. Since Equation (10) provides a mesoscopic-macroscopic relationship between the temperature and monomer composition, it allows the decoupling of the mesocopic-macroscopic analysis of the time evolution behavior of the thermal aspect of the system from compositional aspects, until the details of the kinetics of microphase separation behavior is incorporated in the field equations.

In order to quantitatively characterize the relatively lower inefficiency of radical maintenance in free-radical polymerization systems, the expression for X_P in Equation (10) can be modified to include a polymer radical efficiency factor f_P; thus,

Also, a and b in Equation (10) were proposed to be obtained from experimental data points of phase equilibria experiments. It should be noted that the polymer radical efficiency factor, f_P, is related to the initiator efficiency, f, in free-radical polymerizations in a relative sense, i.e., for the same monomer, solvent, their concentrations, and operating temperatures, f_P is relatively high when an initiator with a relatively high f is used. The reason is that initiation is the starting point in time when inefficiencies of radical production take place. The survival of propagated radicals will later depend on the effectiveness of the FRRPP radical trapping mechanism, which should result in f_P < f. For example, in FRRPP of polystyrene-styrene-ether systems using azobis diisobutylonitrile (AIBN) as initiator, f_P was found to be in the order of 0.20 [1,4] even though f was known to be around 0.57 [6].

For the expectation of a flat temperature profile, θ = 1 for η = 0; thus, the dimensionless source term was symbolized as Φ_o and Equation (2) became

Then, a combined dimensionless quantity was introduced to quantitatively characterize strict FRRPP behavior, wherein the reactive polymer-rich domains attained flat temperature profiles. The dimensionless quantity was symbolized by (pronounced see-enye), and defined as

Values of from computational efforts indicated that for the FRRPP process, it should be at least below −1000 for a flat temperature profile [1].

In the unsteady-state analysis of the FRRPP system, the occurrence of the flat temperature profile from FRRPP systems was reinforced, with additional criteria for stability of steady-state behavior [2]. For a stable steady-state system, it was found that the quantity α < 0 even for an insulated reactive domain system. For a > 0, the reactive system was found to be under control with the possibility of a flat temperature profile through relatively ineffective heat removal from the fluid. With more aggressive heat removal from the fluid, the temperature profile in the reactive solid becomes more of a parabolic one.

When the monomer concentration drops to low enough values, the reactive system automatically reverts to a stable one, because the value of α automatically becomes a negative number [1]. This happens either locally (due to relatively fast reaction rate compared to diffusion rate, or polymer domain densification [2]) or in the overall reactive system, due to depletion of monomer molecules.

With assumption of a flat temperature profile, the field energy balance equation was simplified to a mixing-cup ordinary differential equation and later analyzed for its stability characteristics [2]. Results from the field partial differential equation correspond to the reacting sphere with insulated surfaces. An analytical expression for the modified dimensionless temperature (Q) vs dimensionless time (τ) was also obtained for the flat temperature profile reactive spherical domain with insulated boundary surfaces as

in which the dimensionless time was expressed as

and the modified dimensionless temperature (Q),and are obtained as

Note that T_b is the bulk fluid temperature while T_s is the reactive particle surface temperature (Figure 1).

The function, Ei or the Exponential Integral Function [7], is a special transcendental function which has been found to occur in only a few experimental systems; thus,

It can be evaluated through the following infinite series expansion for real positive arguments (x > 0):

The Euler-Mascheroni constant (also called Euler’s constant) has been cited to be equal to 0.5772156649... It should be noted from Equation (20) that the Ei function approaches −∞ at its argument (x) approaching zero from the positive side (0⁺). This is reflected in inaccuracies in results from direct evaluation Equation (14) at τ → 0⁺.

In this paper, inaccuracies of the analytical evaluation of Equation (14) are addressed, by comparing predictions of the evolution of flat temperature profiles in FRRPP systems from numerical solutions and analytical approaches. Numerical solutions used include the evaluation of the field energy equation and its lumped-parameter version for a flat temperature profile. Analytical methods employed here include the use of the exponential integral function (Equation (14)), as well the TaylorSeries-based polynomial approximations.

Since the diffusive term could be neglected for the assumption of a flat temperature profile, then the geometry of the system does not matter and a lumped-parameter analysis can be made. In dimensional form the lumpedparameter energy balance around a reactive particle can be more conveniently expressed using the convective heat transfer coefficient, h, as

where A is the heat transfer area. For insulating boundary surfaces, h = 0, and thus

Thus, the integral operation for the determination of τ at any value of Q is

Equation (23) is used as the starting material for polynomial approximation at relatively small values of dimensionless time, τ, wherein the approximation is done with the integrand.

In order to validate results of polynomial approximations of Equation (23), detailed field simulation results were also obtained. The following conditions have been deduced for the occurrence of a flat temperature profile in FRRPP systems:

1), α < 0, , , and h > 0 2), α < 0, , , and h = 0 3), α > 0, , , and h = 0 Note that only Conditions 2 and 3 are covered in this paper.

Let the function F(Q) correspond to the integrand of Equation (23); thus,

Application of the Taylor series expansion at the initial dimensionless time of zero and dimensionless temperature Q = 1 leads to

The software Mathematica^® is used to derive the various derivatives in Equation (25), and the following results were obtained up to the cubic derivative.

In order to understand the F(Q) function, it is plotted in Figure 3 at typical parameter values for both stable and unstable systems, wherein the flat temperature profile is proposed to occur.

From this figure, the area under the curve can be obtained to be equal to the dimensionless time, t. Note that for the stable system, the value of Q approaches infinity at the steady-state value of 5. With the two sets of parameters in Figure 3, the resulting plots of the Dimensionless temperature, Q, vs Dimensionless time, t, is shown in Figure 4.

Since the use of the Exponential Integral function, Ei, as defined in Equation (19), is not well understood in FRRPP systems, comparisons with straight numerical solution and Taylor Series approximations are made and reported in this paper. This exercise is very important for validation of a new conceptual phenomenon that is expressible with a rarely used transcendental function.

Figures 5(a) and 6(a) show polynomial approximation results of Q vs t for flat temperature profile histories involving typical stable and unstable parameter systems, which could be compared to numerical results in Figure 4. This stable system curve in Figure 4, in turn, has been found to compare very well with Figure 5(b), which is the numerical result of the solution of the full differential equation in Equation (22). For a representative unstable system, the dashed-curve result in Figure 4 is compared with polynomial approximation results in Figure 6(a). This representative unstable system curve in Figure 4, in turn, has been found to compare well with Figure 6(b), which is the numerical result of the solution of the full differential equation in Equation (22). In addition, exponential integral results are shown for given parameter sets in Figures 5(a) and 6(a). Figures 7(a) and 7(b) show expanded abscissa scales for Figures 5(a) and 5(b), for additional comparisons. On the other hand, Figures 8(a) and 8(b) show shorter time behavior for conditions associated with Figures 6(a) and 6(b).

These plots indicate the validity of the use of the numerical approaches, compared to the Ei approximation method using Equation (14). If one uses the polynomial approximation, for a < 0, the linear approximation would be quite accurate in the initial 60% - 70% Q-range, while the cubic approximation would be accurate in the initial 20% Q-range. For a > 0, linear and cubic approximations seem to be accurate in the small-times range, or when Q ® 1. A point of caveat here is that the polynomial approximation does not seem to be asymptotic, until the cubic terms are reached. The final obvious point to be made here is that for a < 0, the analytical-based Ei function is reasonably accurate at the above 20% Q-range.

The quantitative analysis of the occurrence of the flat temperature profile in spherical FRRPP systems indicates that numerical approaches to predict its evolution would be reasonably accurate. If an analytical approximation is to be done, for a < 0, the approach would be to use the linear Taylor series approximation up to 30% - 40% of the Q-range, and continuing to steady-state condition with the use of the Exponential Integral, Ei, function. For a > 0, the only conclusion that can be made is that both linear and cubic polynomial approximations are reasonably accurate at very small times and low Q-values, i.e., at τ → 0 and Q ® 1.

The authors are grateful for the support of the Michigan Tech Center for Environmentally Benign Functional Materials, the King Abdullah Scholarship Program, Najran and Michigan Tech Universities that allowed the continued work and publication of the quantitative investigation of the FRRPP process.