This simulation shows the number fraction Nj (blue) and weight fraction Wj (green) of polymer chains as functions of the number of monomer units for polymerization in a batch reactor. Select "step growth" or "free radical" polymerization with the buttons, and use the sliders to set reaction rate constants, initial concentrations, and reaction time. The number average chain length N̄j, the weight average chain length W̄j, and the polydispersity are shown. Distributions are calculated using the Flory–Schulz model, and the equations used for the calculations are displayed when the Details button is selected. The reaction steps for each type of polymerization are displayed when the Reaction steps button is selected. The x and y axis ranges adjust automatically as the parameters change.
The number fraction \(N_j\) and weight fraction \(W_j\) of polymer chains for a polymer with \(j\) monomer units are:
\[ N_j = \frac{y_j}{\sum y_j}, \qquad W_j = \frac{j\,y_j}{\sum j\,y_j}, \]
where \(y_j\) is the mole fraction and \(p\) is the fraction of functional groups that have reacted:
\[ y_j = (1-p)\,p^{\,j-1}. \]
For step-growth polymerization:
\[ p = 1 - \left(M_0 k_p t + 1\right)^{-1}, \qquad M = M_0(1-p). \]
For free-radical polymerization:
\[ p = \frac{k_p M}{k_p M + k_t \tau + \sqrt{2 k_t k_i f I_0 e^{-k_i t}}}, \]
\[ M = M_0 \exp\!\left[\, 2 k_p \sqrt{\frac{2 f I_0}{k_i k_t}}\,\left(e^{-k_i t/2} - 1\right) \right], \]
where \(M_0\) and \(I_0\) are the initial monomer and initiator concentrations; \(M\) is monomer concentration; \(k_p\), \(k_i\) and \(k_t\) are the rate constants for the reaction, initiation and termination; \(t\) is time; \(\tau\) is the terminator concentration and \(f\) is fraction of chains initiated by initiator.
The number \(\bar{N}_j\) and weight \(\bar{W}_j\) average chain lengths are:
\[ \bar{N}_j = \frac{\sum j\,P_j}{\sum P_j}, \qquad \bar{W}_j = \frac{\sum j^{2} P_j}{\sum j\,P_j}, \]
where \(P_j = y_j M\) is the polymer concentration.
The polydispersity \(D\) is:
\[ D = \frac{\bar{W}_j}{\bar{N}_j}. \]
Symbols
- Nj = number fraction of polymer chains with j monomer units
- Wj = weight fraction of polymer chains with j monomer units
- N̄j = number average chain length
- W̄j = weight average chain length
- D = polydispersity
- j = number of monomer units in a chain
- p = fraction of functional groups that have reacted
- M0, M = initial and current monomer concentrations
- I0 = initial initiator concentration
- kp = reaction (propagation) rate constant
- ki = initiation rate constant
- kt = termination rate constant
- τ = terminator concentration
- f = fraction of chains initiated by initiator (0.5)
- t = time
In step-growth polymerization, chains grow by reaction between functional groups on any two molecules: monomers, oligomers, or polymers. A typical example is a condensation polymerization, such as a diol reacting with a diacid to form a polyester and water:
\[ \mathrm{HO{-}R{-}OH} + \mathrm{HOOC{-}R'{-}COOH} \rightarrow \mathrm{HO{-}R{-}OOC{-}R'{-}COOH} + \mathrm{H_2O} \]
The product still has a reactive group at each end, so it continues to react. Typical steps are:
- monomer + monomer → dimer: \(\;M_1 + M_1 \xrightarrow{k_p} P_2\)
- dimer + monomer → trimer: \(\;P_2 + M_1 \xrightarrow{k_p} P_3\)
- dimer + dimer → tetramer: \(\;P_2 + P_2 \xrightarrow{k_p} P_4\)
- in general, any two chains combine: \(\;P_i + P_j \xrightarrow{k_p} P_{i+j}\)
Each step also releases a small molecule (H2O in this example). Assuming the rate constant \(k_p\) is the same for every step (equal reactivity), the concentration of unreacted functional groups \(M\) decreases as
\[ -\frac{dM}{dt} = k_p M^2 \quad\Rightarrow\quad M = \frac{M_0}{1 + k_p M_0 t}, \]
which gives the conversion \(p = 1 - M/M_0\) used in Details. Monomer is consumed quickly, but long chains form only at very high conversion: the number average chain length is \(\bar{N}_j = 1/(1-p)\), so \(p = 0.99\) is needed for \(\bar{N}_j = 100\).
Free-radical polymerization is a chain-growth process: a radical adds monomer units one at a time, rapidly building a long chain until the radical is destroyed. Typical steps are:
- Initiation. The initiator (for example, a peroxide or AIBN) decomposes into two radicals, and a fraction \(f\) of these radicals starts chains by adding to a monomer:
\[ I \xrightarrow{k_i} 2R^{\bullet}, \qquad R^{\bullet} + M \rightarrow P_1^{\bullet} \]
- Propagation. The growing radical chain adds monomer units, one at a time. This step repeats hundreds to thousands of times for each chain:
\[ P_j^{\bullet} + M \xrightarrow{k_p} P_{j+1}^{\bullet} \]
- Termination. Two growing radicals react with each other (by combination or disproportionation), or a growing radical reacts with a terminator \(T\), producing "dead" polymer that no longer grows:
\[ P_j^{\bullet} + P_k^{\bullet} \xrightarrow{k_t} \text{dead polymer}, \qquad P_j^{\bullet} + T \rightarrow P_j \]
The radical concentration is small and nearly constant (quasi-steady state), because radicals are formed and destroyed at about the same rate:
\[ [P^{\bullet}] = \sqrt{\frac{2 f k_i I}{k_t}}, \qquad I = I_0\, e^{-k_i t}. \]
Monomer is consumed by propagation, \(-dM/dt = k_p M [P^{\bullet}]\), which integrates to the equation for \(M\) in Details. The probability \(p\) that a growing chain adds another monomer instead of terminating is the propagation rate divided by the sum of the propagation and termination rates, which gives the equation for \(p\) in Details. Unlike step-growth polymerization, long chains form right away, even at low monomer conversion.
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 is a JavaScript/HTML5 implementation of a Mathematica simulation by Rachael L. Baumann and Nathan S. Nelson. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.