This simulation shows how emulsion polymerization works, using styrene at 60 °C with sodium dodecyl sulfate as the surfactant and potassium persulfate as the initiator. It follows the three stages of the process, shows how the number of polymer particles depends on surfactant and initiator concentrations, shows why each particle contains an average of 0.5 radicals (Smith–Ewart Case 2), and shows why polymerization rate and molecular weight both increase with the number of particles, which is the key advantage over bulk polymerization.
Select a view with the four buttons at the top left. The surfactant and initiator sliders apply to every view except the n = 0.5 view.
Stages I–III: drag the time slider or press "play". The left side is a schematic of monomer droplets, micelles, and polymer particles. The plots show conversion, polymerization rate, and particle number versus time. Stage I (particle formation) ends when the micelles are used up. In Stage II, the rate is constant because the particle number is fixed and the particles stay saturated with monomer. Stage III begins when the monomer droplets disappear. Arrows on each plot show the time range of each stage. Check "log time axis" to expand Stage I. With high surfactant and low initiator, the droplets can disappear before the micelles are used up, and then there is no Stage II.
Particle number: particle number versus surfactant in micelles and versus initiator concentration, on log–log axes. The slopes are 0.6 and 0.4.
n = 0.5: radicals enter 64 particles at random. In Case 2, a radical that enters a particle that already contains a radical terminates both immediately, so each particle is active half the time. Cases 1 and 3 show what happens when radicals can leave the particles, or when particles are large enough to hold several radicals.
Emulsion vs. bulk: polymerization rate versus degree of polymerization. In bulk polymerization, more initiator increases the rate but decreases molecular weight. In emulsion polymerization, more surfactant increases the particle number, so rate and molecular weight both increase.
Monomer is dispersed as droplets (1–10 μm) in water that contains surfactant micelles (5–10 nm) swollen with monomer. Initiator decomposes in the water, and radicals enter micelles, which become polymer particles. Monomer diffuses from the droplets through the water to the growing particles.
Radicals formed per cm³ of water per second:
\[\rho = \frac{2 f k_d [I] N_A}{1000}\]Volume growth rate of one particle in Stages I and II:
\[\mu = \frac{k_p [M]_p\, \bar{n}\, M_0}{N_A\, \rho_p\, \phi_p}\]Radicals form new particles in proportion to the fraction of surfactant still in micelles:
\[\frac{dN}{dt} = \rho\left(1-\frac{A_p}{a_s S}\right)\] \[A_p = (36\pi)^{1/3}\sum_j \Delta N_j\,\left[\mu\,(t-t_j)\right]^{2/3}\]The simulation integrates these equations numerically. Stage I ends when the particle surface area \(A_p\) equals the area that the surfactant can cover, \(a_s S\). The result agrees with the Smith–Ewart expression:
\[N = 0.37 \left(\frac{\rho}{\mu}\right)^{2/5} (a_s S)^{3/5}\]The particle number is fixed, monomer droplets keep the particles saturated at \([M]_p\), and \(\bar n = 0.5\):
\[R_p = \frac{1000\, k_p [M]_p\, \bar{n}\, N}{N_A}\]A growing chain stops when the next radical enters its particle. Radicals enter each particle at the rate \(\rho/N\), so the degree of polymerization is:
\[X_n = \frac{k_p [M]_p\, N}{\rho}\]Both \(R_p\) and \(X_n\) are proportional to \(N\), so increasing the surfactant concentration increases both the rate and the molecular weight.
The droplets disappear at conversion \(X_c\), when the remaining monomer just saturates the particles:
\[X_c = \frac{1}{1 + \dfrac{\phi_m \rho_m}{(1-\phi_m)\,\rho_p}}\]After that, \([M]_p\) is calculated from the monomer and polymer left in the particles, and the rate decreases:
\[[M]_p = \frac{1000\,\rho_m}{M_0}\;\frac{(1-X)/\rho_m}{(1-X)/\rho_m + X/\rho_p}\]If the droplets disappear before the micelles are used up (high surfactant, low initiator), particle formation stops at \(X_c\), there is no Stage II, and the reaction goes directly from Stage I to Stage III.
The gel effect, which can raise \(\bar n\) above 0.5 late in Stage III, is not included.
With the same initiator and termination by combination:
\[R_p = k_p [M]\left(\frac{f k_d [I]}{k_t}\right)^{1/2}\] \[X_n = \frac{k_p [M]}{\left(f k_d k_t [I]\right)^{1/2}}\]The product \(R_p X_n\) does not depend on \([I]\), so any increase in rate decreases the molecular weight.
Case 1, small particles with radical desorption:
\[\bar n = \frac{\rho'}{2\rho' + k_{des}}\]Case 2, no desorption and immediate termination when a second radical enters:
\[\bar n = 0.5\]Case 3, large particles where several radicals coexist and terminate at the rate \(c\,n(n-1)\). For \(\bar n \gg 1\):
\[\bar n \approx \left(\frac{\rho'}{2c}\right)^{1/2}\]The theory value shown for Case 3 comes from the steady-state solution of the population balance for \(n\) radicals per particle, which is slightly larger than this approximation.
| \(k_p\) | 341 L/(mol s) |
| \(k_t\) | 6.0 × 10⁷ L/(mol s) |
| \(k_d\) | 3.2 × 10⁻⁶ s⁻¹ |
| \(f\) | 0.5 |
| \([M]_p\) before Stage III | 5.2 mol/L |
| \([M]\) in bulk | 8.70 mol/L |
| \(a_s\) | 0.50 nm² |
| critical micelle concentration | 8.2 mmol/L (2.36 g/L) |
| \(\rho_m\), \(\rho_p\) | 0.906 g/cm³, 1.05 g/cm³ |
| monomer charge | 30 g styrene per 100 cm³ water |
Emulsion rates are per liter of water; bulk rates are per liter of monomer. The calculated degrees of polymerization neglect chain transfer to monomer, which limits \(X_n\) for styrene to roughly 10⁴ in practice. Chain transfer lowers both the emulsion and bulk values but does not change the trends.
| \(A_p\) | total surface area of particles (cm² per cm³ water) |
| \(a_s\) | area covered by one surfactant molecule (cm²) |
| \(c\) | termination rate constant within a particle (s⁻¹) |
| \(f\) | initiator efficiency |
| \([I]\) | initiator concentration (mol/L) |
| \(k_d\) | initiator decomposition rate constant (s⁻¹) |
| \(k_{des}\) | radical desorption rate constant (s⁻¹) |
| \(k_p\) | propagation rate constant (L/(mol s)) |
| \(k_t\) | termination rate constant in bulk (L/(mol s)) |
| \([M]\) | monomer concentration in bulk polymerization (mol/L) |
| \([M]_p\) | monomer concentration in the particles (mol/L) |
| \(M_0\) | monomer molecular weight (g/mol) |
| \(N\) | number of particles per cm³ of water |
| \(\Delta N_j\) | particles formed at time \(t_j\) (per cm³ water) |
| \(N_A\) | Avogadro's number (mol⁻¹) |
| \(n\) | number of radicals in a particle |
| \(\bar n\) | average number of radicals per particle |
| \(R_p\) | polymerization rate (mol/(L s)) |
| \(S\) | surfactant in micelles, above the critical micelle concentration (molecules per cm³ water) |
| \(t\) | time (s) |
| \(X\) | monomer conversion |
| \(X_c\) | conversion at which the monomer droplets disappear |
| \(X_n\) | number-average degree of polymerization |
| \(\mu\) | volume growth rate of a particle (cm³/s) |
| \(\rho\) | rate of radical generation (radicals per cm³ water per s) |
| \(\rho'\) | rate at which radicals enter one particle, \(\rho/N\) (s⁻¹) |
| \(\rho_m\), \(\rho_p\) | monomer and polymer densities (g/cm³) |
| \(\phi_m\), \(\phi_p\) | volume fractions of monomer and polymer in a saturated particle |
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.