Macromolecular Reaction Engineering
Volume 19, Issue 2 2400028
Research Article
Open Access

Design of Emulsion Polymerization Reactors for Monomer-Transport Limited Emulsion Polymerization

Francis Joseph Schork, 

Corresponding Author

Francis Joseph Schork

School of Chemical and Biomolecule Engineering, Georgia Institute of Technology, Atlanta, GA, 30332 USA

E-mail: [email protected]

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First published: 26 November 2024
https://doi.org/10.1002/mren.202400028

Abstract

Damkohler Number (Da) analysis can identify monomers and emulsion polymerization operating regimes where the polymerization may be monomer-transport, rather than reaction-limited. In these cases, the expected monomer concentration in the growing polymer particles will be reduced due to the transport limitation. This will reduce the expected rate of polymerization, and require the design of a larger polymerization reactor for a given production rate. In heterogenous catalysis, an effectiveness factor is used to quantify the reduction in reaction rate and necessarily increase reactor size to compensate. This paper will show that it is possible to use Da (functionally equivalent to the Thiele Modulus in heterogeneous catalysis) to estimate an effectiveness factor for emulsion polymerization. Also shown is a procedure for calculating the monomer feed ratio during binary copolymerization when one must not only take into account the reactivity ratios but also the possibility that one of the monomers is monomer-transport limited. The method provides the monomer feed ratio during the semibatch phase of a binary copolymerization. This alternative to starved-feed polymerization shall result in much faster polymerization and higher polymerization kettle utility.

1 Introduction

Schork[1] has developed a Damkohler Number (Da) analysis to aid in the determination of possible monomer-transport limitation during emulsion polymerization. Results indicate that few currently-used monomers are transport-limited; however as no biobased, biodegradable, and macromonomer are developed, more monomers with sufficiently low water solubility are likely to find commercial use. This note extends the concept of Da for monomer-transport limitation to an effectiveness factor for the design (sizing) of emulsion polymerization reactors.

The starting point for understanding monomer transport in emulsion polymerization is a monomer balance over the aqueous phase as shown below.
V a q d M a d t = F d − R p = k l a d ( M a s − M a ) N d − k p n ¯ N A M p N p = 0 $$\begin{eqnarray}\hspace*{75pt}{{V}_{aq}}\frac{{d{{M}_a}}}{{dt}} &=& {{F}_d} - {{R}_p} = {{k}_l}{{a}_d}(M_a^s - {{M}_a}){{N}_d} \nonumber\\ &&- {{k}_p}\left( {\frac{{\bar{n}}}{{{{N}_A}}}} \right){{M}_p}{{N}_p} = 0\end{eqnarray}$$ (1)
here Vaq is the volume of the aqueous phase per liter of emulsion, Fd is the rate of monomer transport out of the monomer droplets (mol sec−1-L−1), Rp is the rate of polymerization (mol sec−1-L−1), kp is the rate constant for propagation, n ¯ $\bar{n}$ is the average number of free radicals per particle, Mp is the monomer concentration in the particle, Np is the number of particles per liter of emulsion, NA is Avogadro's Number, kl is the mass transfer coefficient for monomer at the surface of the monomer droplet, ad is the surface area of a droplet, Mas is the saturation concentration of monomer in the aqueous phase, Ma is the monomer concentration in the bulk aqueous phase, and Nd is the number of monomer droplets per liter of emulsion. Equation (1), at steady state, may be written as
k l a d M a s ( 1 − f a ) N d − k p n ¯ N A f p M p s N p = 0 $$\begin{equation}{{k}_l}{{a}_d}M_a^s(1 - {{f}_a}){{N}_d} - {{k}_p}\left( {\frac{{\bar{n}}}{{{{N}_A}}}} \right){{f}_p}M_p^s{{N}_p} = 0\end{equation}$$ (2)
where
M a M a s = f a = M p M p s 0.6 = f p 0.6 $$\begin{equation}\frac{{{{M}_a}}}{{M_a^s}} = {{f}_a} = {{\left( {\frac{{{{M}_p}}}{{M_p^s}}} \right)}^{0.6}} = {{\left( {{{f}_p}} \right)}^{0.6}}\end{equation}$$ (3)
or
f p = f a 1.67 $$\begin{equation}{{f}_p} = {{\left( {{{f}_a}} \right)}^{1.67}}\end{equation}$$ (4)
where fa and fp, the fractional saturations of the aqueous (fa) and particle (fp) phases, are related by Equation (3).[2] Da is often used to estimate the relative importance of reaction rate and mass transfer, and is defined for this case as the maximum rate of polymerization divided by the maximum rate of monomer transport:
D a = R p F d = k p n ¯ / N A M p s N p k l a d ( M a s − 0 ) N d $$\begin{equation}Da = \frac{{{{R}_p}}}{{{{F}_d}}} = \frac{{{{k}_p}\left( {\bar{n}/{{N}_A}} \right)M_p^s{{N}_p}}}{{{{k}_l}{{a}_d}(M_a^s - 0){{N}_d}}}\end{equation}$$ (5)

Note that this can be gotten from Equation (2) with fp set to unity and fa set to zero. If Da is less than 0.1, a reacting system is assumed to be a reaction (not transport) limited.[1]

2 An Effectiveness Factor for Monomer-Transport Limited Batch or Semibatch Emulsion Polymerization

Rearrangement of Equation (2) gives
k l a d M a s N d k p n ¯ N A M p s N p = D a = f p ( 1 − f a ) = f p 1 − f p 0.6 $$\begin{equation}\frac{{{{k}_l}{{a}_d}M_a^s{{N}_d}}}{{{{k}_p}\left( {\frac{{\bar{n}}}{{{{N}_A}}}} \right)M_p^s{{N}_p}}} = Da = \frac{{{{f}_p}}}{{(1 - {{f}_a})}} = \frac{{{{f}_p}}}{{\left[ {1 - {{{\left( {{{f}_p}} \right)}}^{0.6}}} \right]}}\end{equation}$$ (6)
Thus the actual rate of polymerization, R p a c t $R_p^{act}$ is
R p a c t = f p R p $$\begin{equation}R_p^{act} = {{f}_p}R_p\end{equation}$$ (7)
where fp may be found by solving Equation (2) after estimating Da.

Thus, fpfunctions as an effectiveness factor. When Da is small (indicating no monomer-transport limitation), fpapproaches unity, and the actual rate of polymerization is equal to the predicted (with no monomer-transport limitation) rate. As Da increases (indicating monomer-transport limitation) fp falls, the actual rate of polymerization falls. This must be compensated for by increasing the size of the reactor in order to obtain the desired rate of polymer production.

Thus, once Da is calculated, and monomer-transport limitation is suspected, fractional saturation of the particle with monomer (the effectiveness factor, fp can be used to adjust the rate of polymerization and the necessary reactor size. (Note that the rate of polymerization may need to be reduced further to allow complete removal of the enthalpy of polymerization.) If actual conditions are not those assumed in reference,[1] the calculated Da can be adjusted from the standard conditions to conditions closer to those of the actual reactor based on its definition in Equation (4). If M p s $M_p^s$ is not known, it can be estimated from the approximation that the saturation volume fraction monomer within the particle is generally 0.63 ± 0.06.[3]

3 Copolymer Composition Control for Semibatch Emulsion Polymerization via Damkohler Analysis

For binary copolymerization, four propagation reactions must be considered:[3]
P n , m + A ⇒ k pAA P n + 1 , m P n , m + B ⇒ k pAB Q n , m + 1 Q n . m + A ⇒ k pAB P n + 1 , m Q n , m + B ⇒ k pBB Q n , m + 1 $$\begin{equation} \def\eqcellsep{&}\begin{array}{c}{P}_{n,m}+A\stackrel{{k}_{\textit{pAA}}}{\Rightarrow}{P}_{n+1,m}\\ {P}_{n,m}+B\stackrel{{k}_{\textit{pAB}}}{\Rightarrow}{Q}_{n,m+1}\\ {Q}_{n.m}+A\stackrel{{k}_{\textit{pAB}}}{\Rightarrow}{P}_{n+1,m}\\ {Q}_{n,m}+B\stackrel{{k}_{\textit{pBB}}}{\Rightarrow}{Q}_{n,m+1}\end{array} \end{equation}$$ (8)
and the reactivity ratios can be defined as
r A = k p A A k p A B ; r B = k p B B k p B A $$\begin{equation}{{r}_A} = {{{{k}_{{{p}_{AA}}}}} \over {{{k}_{{{p}_{AB}}}}}};\quad \quad {{r}_B} = {{{{k}_{{{p}_{BB}}}}} \over {{{k}_{{{p}_{BA}}}}}}\end{equation}$$ (9)
For emulsion copolymerization, the Damkohler Numbers for monomers A and B can be written as[3]
D a A c = R p A F d A = k p A ∗ n ¯ / n ¯ M p A ∗ N p k l a d ( M a A ∗ − 0 ) N d $$\begin{equation}Da_A^c = \frac{{{{R}_{pA}}}}{{{{F}_{dA}}}} = \frac{{k_{pA}^*\left( {\bar{n}/\bar{n}} \right)M_{pA}^*{{N}_p}}}{{{{k}_l}{{a}_d}(M_{aA}^* - 0){{N}_d}}}\end{equation}$$ (10)
D a B c = R p B F d B = k p B ∗ n ¯ / N A M p B ∗ N p k l a d ( M a B ∗ − 0 ) N d $$\begin{equation}Da_B^c = \frac{{{{R}_{pB}}}}{{{{F}_{dB}}}} = \frac{{k_{pB}^*\left( {\bar{n}/{{N}_A}} \right)M_{pB}^*{{N}_p}}}{{{{k}_l}{{a}_d}(M_{aB}^* - 0){{N}_d}}}\end{equation}$$ (11)
where M*aA and M*aB, are the maximum concentrations of A and B, respectively, in the aqueous phase, and M*pA and M*pB, are the concentrations of monomers A and B within the particle in equilibrium with M*aA and M*aB. The pseudo-rates constants k p A ∗ $k_{pA}^*$ and k p B ∗ $k_{pB}^*$ are defined[3] as
k p A ∗ = k p B A ( 1 − g A ) + k p A A g A $$\begin{equation}k_{pA}^* = \left[ {{{k}_{pBA}}(1 - {{g}_A}) + {{k}_{pAA}}{{g}_A}} \right]\end{equation}$$ (12)
and
k p B ∗ = k p A B g A + k p B B ( 1 − g A ) $$\begin{equation}k_{pB}^* = \left[ {{{k}_{pAB}}{{g}_A} + {{k}_{pBB}}(1 - {{g}_A})} \right]\end{equation}$$ (13)
where gA is defined as the fraction of living chains ending in an A[3]
g A = k pBA A k pAB B + k pBA A = k pBA M pA * k pAB M pB * + k pBA M pA * = M pA * r B r A k pAA k pBB M pB * + M pA * g B = 1 − g A $$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {g}_{A}=\frac{{k}_{\textit{pBA}}A}{\left({k}_{\textit{pAB}}B+{k}_{\textit{pBA}}A\right)}=\frac{{k}_{\textit{pBA}}{M}_{\textit{pA}}^{\ast}}{\left({k}_{\textit{pAB}}{M}_{\textit{pB}}^{\ast}+{k}_{\textit{pBA}}{M}_{\textit{pA}}^{\ast}\right)}=\frac{{M}_{\textit{pA}}^{\ast}}{\left[\left(\frac{{r}_{B}}{{r}_{A}}\right)\left(\frac{{k}_{\textit{pAA}}}{{k}_{\textit{pBB}}}\right){M}_{\textit{pB}}^{\ast}+{M}_{\textit{pA}}^{\ast}\right]}\\ {g}_{B}=1-{g}_{A} \end{array} \end{equation}$$ (14)
Monomer B has previously been assumed to be the monomer with potential monomer-transport limitation. That convention will be followed here. If one assumes Henry's Law holds for droplet-aqueous phase equilibrium, then
M a B ∗ = f B d M a B s $$\begin{equation}M_{aB}^* = f_B^dM_{aB}^s\end{equation}$$ (15)
where M a B s $M_{aB}^s$ is the saturation value of monomer A in the aqueous phase and f B d $f_B^d$ is the mole fraction of B in the monomer droplets.
A balance on monomer B in the aqueous phase under monomer B transport limitation yields
F d B − R p B = k p B ∗ n ¯ N A M p B a c t N p − k l a d ( M a B ∗ − M a B a c t ) N d = 0 $$\begin{equation}{{F}_{dB}} - {{R}_{pB}} = k_{pB}^*\left( {\frac{{\bar{n}}}{{{{N}_A}}}} \right)M_{pB}^{act}{{N}_p} - {{k}_l}{{a}_d}(M_{aB}^* - M_{aB}^{act})\,{{N}_d} = 0\end{equation}$$ (16)
where M a B a c t $M_{aB}^{act}$ is the actual monomer B concentration in the aqueous phase, and M p B a c t $M_{pB}^{act}$ is the actual monomer B concentration in the particle in equilibrium with M a B a c t $M_{aB}^{act}$ .
Assuming particle–aqueous phase equilibrium is governed by[2]
M p B a c t M p B ∗ 0.6 = f B p 0.6 = M a B a c t M a B ∗ = f B a $$\begin{equation}{{\left( {\frac{{M_{pB}^{act}}}{{M_{pB}^*}}} \right)}^{0.6}} = {{\left( {f_B^p} \right)}^{0.6}} = \frac{{M_{aB}^{act}}}{{M_{aB}^*}} = f_B^a\end{equation}$$ (17)
or
f B p = f B a 1.67 $$\begin{equation}f_B^p = {{\left( {f_B^a} \right)}^{1.67}}\end{equation}$$ (18)
Substituting into Equation (12) and rearranging,
k p B ∗ n ¯ N A M p B ∗ N p k l a d M a B ∗ N d = D a B c = 1 − f B a f B a 1.67 $$\begin{equation}\frac{{k_{pB}^*\left( {\frac{{\bar{n}}}{{{{N}_A}}}} \right)M_{pB}^*{{N}_p}}}{{{{k}_l}{{a}_d}M_{aB}^*{{N}_d}}} = Da_B^c = \frac{{1 - f_B^a}}{{{{{\left( {f_B^a} \right)}}^{1.67}}}}\end{equation}$$ (19)

One will observe that as f B a $f_B^a$ goes to zero (extreme monomer-transport limitation), D a B c $Da_B^c$ goes to infinity (indicating extreme monomer-transport limitation), and, as f B a $f_B^a$ approaches one (no monomer-transport limitation), D a B c $Da_B^c$ goes to zero (indicating no monomer-transport limitation),

Defining
f ′ A = f A p f A p + f B p f ′ B = f B p f A p + f B p $$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {{{f^{\prime}}}_A} = \frac{{f_A^p}}{{f_A^p \,\,+ \,\,f_B^p}}\\[10pt] {{{f^{\prime}}}_B} = \frac{{f_B^p}}{{f_A^p \,\,+ \,\,f_B^p}} \end{array} \end{equation}$$ (20)
The instantaneous copolymer compositions (from the Copolymer Equation) are
F B = r B ( f ′ B ) 2 + f ′ A f ′ B r A ( f ′ A ) 2 + 2 f ′ A f ′ B + r B ( f ′ B ) 2 $$\begin{equation}{{F}_B} = \frac{{{{r}_B}{{{({{{f^{\prime}}}_B})}}^2} + {{{f^{\prime}}}_A}{{{f^{\prime}}}_B}}}{{{{r}_A}{{{({{{f^{\prime}}}_A})}}^2} + 2{{{f^{\prime}}}_A}{{{f^{\prime}}}_B} + {{r}_B}{{{({{{f^{\prime}}}_B})}}^2}}}\end{equation}$$ (21)
A policy, then, for producing polymer of constant FB is as follows:
  • Solve Equation (21) for f′B at the desired value of FB.
  • Find f B p $f_B^p$ from Equation (20).
  • Find f B a $f_B^a$ from Equation (17).
  • Solve Equation (19) for D a B c $Da_B^c$ .
  • Solve Equation (11) for M p B ∗ $M_{pB}^*$ .
  • Solve Equation (15) for f B d $f_B^d$ .

This will provide a constant droplet concentration ( f B d $f_B^d$ ) and consequently, a constant copolymer composition during the semibatch operation. Semibatch monomer should be fed at a monomer ratio necessary to maintain f B d $f_B^d$ as shown below. Reactor sizing may be done as in Section 2 above, using the larger of D a B c $Da_B^c$ or D a A c $Da_A^c$ to calculate the effectiveness factor.

Now the rates of consumption of the two monomers can be written as
R p A = k A ∗ p n ¯ / N A M p A ∗ f A p N p $$\begin{equation}{{R}_{pA}} = k{{_A^*}_p}\left( {\bar{n}/{{N}_A}} \right)M_{pA}^*f_A^p{{N}_p}\end{equation}$$ (22)
R p B = k p B ∗ n ¯ / N A M p B ∗ f B p N p $$\begin{equation}{{R}_{pB}} = k_{pB}^*\left( {\bar{n}/{{N}_A}} \right)M_{pB}^*f_B^p{{N}_p}\end{equation}$$ (23)
R pB R pA = k pB * n ¯ / N A M pB * f B p N p k A * p n ¯ / N A M pA * f A p N p = k pB * M pB * f B p k A * p M pA * f A p = k pB * M pB * ( f B a ) 1.67 k A * p M pA * ( f A a ) 1.67 = k pB * M pB s ( f B a ) 1.67 ( f B d ) 1.67 k A * p M pA s ( f A a ) 1.67 ( f A d ) 1.67 $$\begin{eqnarray} \frac{{R}_{\textit{pB}}}{{R}_{\textit{pA}}}&=&\frac{{k}_{\textit{pB}}^{\ast}\left(\bar{n}/{N}_{A}\right){M}_{\textit{pB}}^{\ast}{f}_{B}^{p}{N}_{p}}{k{{}_{A}^{\ast}}_{p}\left(\bar{n}/{N}_{A}\right){M}_{\textit{pA}}^{\ast}{f}_{A}^{p}{N}_{p}}=\frac{{k}_{\textit{pB}}^{\ast}{M}_{\textit{pB}}^{\ast}{f}_{B}^{p}}{k{{}_{A}^{\ast}}_{p}{M}_{\textit{pA}}^{\ast}{f}_{A}^{p}}=\frac{{k}_{\textit{pB}}^{\ast}{M}_{\textit{pB}}^{\ast}{({f}_{B}^{a})}^{1.67}}{k{{}_{A}^{\ast}}_{p}{M}_{\textit{pA}}^{\ast}{({f}_{A}^{a})}^{1.67}}\nonumber\\ &=&\frac{{k}_{\textit{pB}}^{\ast}{M}_{\textit{pB}}^{s}{({f}_{B}^{a})}^{1.67}{({f}_{B}^{d})}^{1.67}}{k{{}_{A}^{\ast}}_{p}{M}_{\textit{pA}}^{s}{({f}_{A}^{a})}^{1.67}{({f}_{A}^{d})}^{1.67}} \end{eqnarray}$$ (24)
M p B ∗ M p B s 0.6 = M a B ∗ M a B s = f B d $$\begin{equation}{{\left( {\frac{{M_{pB}^*}}{{M_{pB}^s}}} \right)}^{0.6}} = \frac{{M_{aB}^*}}{{M_{aB}^s}} = f_B^d\end{equation}$$ (25)
M p B ∗ M p B s = ( f B d ) 1.67 $$\begin{equation}\left( {\frac{{M_{pB}^*}}{{M_{pB}^s}}} \right) = {{(f_B^d)}^{1.67}}\end{equation}$$ (26)

The initial (batch) monomer should be charged at the calculated ratio f B d $f_B^d$ . Monomer should be fed during semibatch operation in the ratio of RpB/RpA.

4 Summary and Conclusions

Previous work indicates that most common monomers are not monomer-transfer limited, but, as new functional monomers, macromonomers, and bio-based monomers are developed, it will be important to understand their transport limitations. Section 2, above introduces an effectiveness factor to account for the fact that the rate of polymerization may be reduced by monomer-transport limitation. This can be used to increase the size of the polymerization kettle during the design phase, or conversely, to reduce the estimated yield of an existing kettle for a new product. Section 3 describes how the monomer ratio during the semibatch phase of a semibatch (but not starved-feed) binary emulsion copolymerization should be calculated in order to account for not only the reactivity ratios but also for the extent of monomer-transport limitation of one of the monomers.

Conflict of Interest

The authors declare no conflict of interest.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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