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# Copolymerization-Reaction-Modeling — Analysis Notes
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**Source:** [gtancev/Copolymerization-Reaction-Modeling](https://github.com/gtancev/Copolymerization-Reaction-Modeling)
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**Author:** Georgi Tancev, PhD
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**Relevance to project:** ⭐⭐⭐⭐⭐ — Core model for radical copolymerization simulations
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---
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## What This Code Does
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Simulates **batch bulk radical copolymerization** of two monomers (A and B) in a 15 L reactor. The system of ODEs tracks concentrations of initiator, both monomers, and both radical species over time (~2 h).
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### Inferred Monomer System (from hardcoded parameters)
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| Parameter | Monomer A | Monomer B |
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|-----------|-----------|-----------|
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| Molar mass | 104 g/mol | 100 g/mol |
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| Density | 0.90 kg/L | 0.94 kg/L |
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| kp₀ | 410 L/(mol·s) | 930 L/(mol·s) |
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| kt₀ | 2.4×10⁷ L/(mol·s) | 9.2×10⁶ L/(mol·s) |
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| Reactivity ratio | rA = 0.52 | rB = 0.46 |
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This matches **styrene (A) / methyl methacrylate (B)** nearly exactly:
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- Styrene MW = 104, rA ≈ 0.52; MMA MW = 100.12, rB ≈ 0.46 (literature values)
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- Initiator MI = 164 g/mol → **AIBN** (MW = 164.21), kd = 6.77×10⁻⁶ s⁻¹ (≈60°C)
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---
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## Model Components
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### 1. Kinetic Mechanism (`batch.m`)
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Uses the **terminal model** (Mayo-Lewis). The ODE system:
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- `dcI` — initiator decomposition (1st order, kd)
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- `dcA`, `dcB` — monomer consumption via homo- and cross-propagation
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- `dcA_R`, `dcB_R` — radical balance for A-centered and B-centered radicals (quasi-steady-state not assumed — radicals are integrated explicitly)
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Cross-propagation from reactivity ratio definitions:
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```
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kp_AB = kp_AA / rA
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kp_BA = kp_BB / rB
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```
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Cross-termination by geometric mean:
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```
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kt_AB = sqrt(kt_AA * kt_BB)
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```
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### 2. Gel/Glass Effect
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Rate constants are modified by polymer weight fraction (wp) via an empirical diffusion-limited formula:
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```
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kp = 1 / (1/kp0 + exp(C_eta * wp) / kp_D)
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kt = 1 / (1/kt0 + exp(C_eta * wp) / kt_D) + C_RD * kp * (1 - wp)
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```
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Where `C_eta = 25`, `C_RD = 180`. This captures the **Trommsdorff (gel) effect** and **glass effect** empirically. More rigorous alternatives use the Vrentas-Duda free volume theory (see Ref. [1] below).
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### 3. Post-processing (`cumulative.m`)
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Computes using the method of moments:
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- Instantaneous copolymer composition (F_A) — Mayo-Lewis equation
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- Cumulative composition (F_A_c) via numerical integration (trapz)
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- Number- and weight-average chain lengths (n_N, n_W) and PDI
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- Termination split between combination (ktc) and disproportionation (ktd), with ratio C_t = 1000 (i.e., predominantly disproportionation)
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---
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## Bugs and Code Quality Issues
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1. **Parameter duplication** — rate constants, densities, and molar masses are hardcoded independently in `batch.m`, `cumulative.m`, and `main.m`. Changing a parameter in one place does not update the others. Any adaptation should centralize parameters.
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2. **No Arrhenius temperature dependence** — all rate constants are fixed (implicitly ~60°C). Adding `k = A * exp(-Ea/RT)` would make the model temperature-aware.
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3. **f = 0.5 is fixed** — initiator efficiency should ideally be conversion-dependent (decreases at high conversion due to cage effect).
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4. **No chain transfer** — to monomer (Cm), to solvent, or to chain transfer agent (CTA). This limits applicability to systems without deliberate MWD control.
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5. **trapz integration in cumulative.m** is first-order accurate and sensitive to the ODE time step. Consider switching to cumulative Simpson's rule or an analytic running integral.
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6. **No semibatch/continuous mode** — the model is batch-only. Semibatch addition of monomer is a common industrial process for composition control.
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---
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## Physical Limitations
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- **Terminal model only** — penultimate unit effect is not included. For rA×rB << 1 or rA×rB >> 1 (highly alternating or blocky), the penultimate model may be needed.
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- **Bulk/solution only** — no heterogeneous phase (see the companion emulsion repo for that).
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- **No diffusion-controlled propagation at high conversion** — at very high wp (glass effect), kp can also become diffusion-limited, but this requires the `kp_D` term and glass-effect parameters to be carefully tuned.
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- **Quasi-bulk initiator decomposition** — f is assumed constant; in real systems it can drop from ~0.5 to ~0.1 at high conversion.
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---
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## Relevance and Adaptation Suggestions for This Project
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### High Relevance
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This is the most directly applicable model for radical copolymerization simulation. The Mayo-Lewis formalism with gel effect is standard for batch bulk/solution copolymerization of styrene, acrylates, methacrylates.
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### Suggested Adaptations
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1. **Python port** — convert to Python with `scipy.integrate.solve_ivp` (use `method='Radau'` or `'BDF'` as `ode15s` equivalent). Example structure:
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```python
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from scipy.integrate import solve_ivp
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import numpy as np
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def batch(t, c, params):
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cI, cA, cA_R, cB, cB_R = c
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# ... kinetics with params dict
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return [dcI, dcA, dcA_R, dcB, dcB_R]
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sol = solve_ivp(batch, [0, 7200], c0, method='Radau', args=(params,), dense_output=True)
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```
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2. **Parameterize for other monomer pairs** — the structure is general. By changing rA, rB, kp_AA, kp_BB, kt_AA, kt_BB, MA, MB you can simulate any comonomer pair (e.g., styrene/BA, MMA/BA, VAc/ethylene).
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3. **Add chain transfer** — add `dcA -= ktr_A * cA * cR` terms and a transfer constant Cm to control Mn independently of conversion.
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4. **Add temperature as a variable** — couple to an energy balance ODE for non-isothermal simulation.
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5. **Connect to emulsion model** — the batch.m kinetics can be embedded inside the particle phase of the emulsion model to simulate **emulsion copolymerization**.
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6. **Validate against mcPolymer** — the mcPolymer Monte Carlo simulator in the `mcPolymer/` directory can provide reference MWDs and composition distributions to validate this deterministic model.
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---
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## Key Literature
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- [1] [Modeling Insight into the Diffusion-Limited Cause of the Gel Effect in Free Radical Polymerization](https://consensus.app/papers/details/2b17519cf178593bbdac2f04de0236b8/?utm_source=claude_code) — G.A. O'Neil et al., *Macromolecules* 1999. Vrentas-Duda free volume theory for gel effect; more rigorous than the empirical exponential form used here.
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- [2] [Free radical polymerizations associated with the Trommsdorff effect under semibatch reactor conditions](https://consensus.app/papers/details/a415b988d9035f139b3e737769d50ae0/?utm_source=claude_code) — Ray et al., *Polym. Eng. Sci.* 1995. Gel and glass effects in batch and semibatch, MMA/PS systems.
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- [3] [Pseudo-Homopolymerization Approach To Predict the MWD in Copolymerization](https://consensus.app/papers/details/0e35d806f31b54368e906820e0b9b718/?utm_source=claude_code) — Zapata-González et al., *I&EC Res.* 2018. More efficient approach to MWD in copolymerization using the PHP+RQSSA method.
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- [4] Odian, G. *Principles of Polymerization*, 4th ed., Wiley 2004. Chapters 6–7 for copolymerization fundamentals and kinetics.
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- [5] Dotson, N.A. et al. *Polymerization Process Modeling*, VCH 1996. Standard reference for the moment method used in `cumulative.m`.
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---
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*Notes written: 2026-06-26*
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