Hypothesis Test 14 — Entanglement Packing Length ("The Plateau Is a Packing Length")
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
report.md
# Hypothesis Test 14 — Entanglement Packing Length ("The Plateau Is a Packing Length")
**Test directory:** `tests/14-entanglement-packing/`
**Date:** 2026-10-30
---
## Source (verbatim)
From `theories/entanglement-packing-length.html`, Incoprea Theory Vol. XII, Theory 05 (Polymer Physics):
> **The claim:** "The entanglement plateau modulus of a polymer melt is set by the packing length rather than by molecular weight, so two polymers of very different molecular weight can have the same modulus."
> **Predictions:**
> - **P1** — "Modulus should collapse onto packing length across polymers."
> - **P2** — "Molecular weight should shift the plateau width, not its height."
> **Refutation:** "Modulus tracking molecular weight would remove the packing control."
This is a restatement of the Fetters packing-length correlation: G_N ≈ kT/p³ (slope −3 in log-log), independent of molecular weight.
---
## Why probable
The claim is essentially the standard, well-tested result of melt polymer physics. Witten, Milner and Wang (1989) introduced the packing length p ≡ M/(ρ·N_A·⟨R²⟩₀) = chain volume per mean-square end-to-end distance. Fetters, Lohse, Richter, Witten and Zirkel (Macromolecules 1994, 27, 4639) showed that G_N ∝ p⁻³ across polyolefins and other melts (their own fit: slope −2.89, r² = 0.9924 on 26 polymers at 413 K), and Fetters, Lohse and Graessley (J. Polym. Sci. B 1999, 37, 1023) and Fetters, Lohse, Milner and Graessley (Macromolecules 1999, 32, 6847) extended the correlation, with G_N = kT/(n_t²·p³) and n_t ≈ 20.8 across ~40 polymers spanning 2 decades of G_N. Within the tube model this is also expected: G_N = ρRT/M_e, and M_e ∝ p³. Molecular weight enters only through the width of the plateau (M_w ≫ M_c required); the plateau height is a local, chemistry-dependent quantity. Prior to testing, this hypothesis is therefore *probable* — but it is falsifiable, e.g., if bulky side groups produced systematic deviations, or if G_N tracked the M_w of the samples used.
## Method
### Data
A table of 16 polymer-melt rows (11 distinct polymer chemistries; two measurement temperatures; several rows repeat a polymer at different documented M_w to probe the M_w-independence at fixed p) was assembled with **only values traceable to published sources**:
- **G_N and p** are taken verbatim from the published tables of the Fetters compilation: Fetters, Lohse, Richter, Witten, Zirkel, *Macromolecules* **1994**, 27, 4639–4647, Table 1 (413 K) and Table 2 (298 K), doi:10.1021/ma00095a029. Full text consulted at <https://datapdf.com/connection-between-polymer-molecular-weight-density-chain.html> (which reproduces the ACS article).
- **M_w** values are taken from the studies in which each polymer's plateau modulus was measured (or, where an exact sample value is documented in a later study, from that study), each cited per row:
- Pearson, Fetters, Graessley, Ver Strate, von Meerwall, *Macromolecules* **1994**, 27, 711 (PE: "model polyethylenes ranging in molecular weight from 10³ to 10⁶") — doi:10.1021/ma00081a014.
- Colby, Fetters, Graessley, *Macromolecules* **1987**, 20, 2226 (1,4-polybutadiene: M_w 1×10³ – 1.65×10⁷) — doi:10.1021/ma00175a030; abstract via <https://pure.psu.edu/en/publications/melt-viscosity-molecular-weight-relationship-for-linear-polymers/>.
- Fetters, Graessley, Kiss, *Macromolecules* **1991**, 24, 3136 (polyisobutylene: M_w 2.7×10⁴ – 1.6×10⁶; largest tabulated sample M_w = 533 000) — doi:10.1021/ma00011a016; full text at <https://datapdf.com/viscoelastic-properties-of-polyisobutylene-melts-american-ch.html>.
- Abdel-Goad, Pyckhout-Hintzen, Kahle, Allgaier, Richter, Fetters, *Macromolecules* **2004**, 37, 8135 (1,4-polyisoprene: M_w 1.5×10³ – 3.2×10⁶) — doi:10.1021/ma030557+.
- Fuchs, Friedrich, Weese, *Macromolecules* **1996**, 29, 5893 (PMMA: samples M_w 25 000–390 000; G_N = 6.35×10⁵ Pa at 190 °C, explicitly "molecular weight independent") — doi:10.1021/ma951385m; full text at <https://datapdf.com/viscoelastic-properties-of-narrow-distribution-polymethyl.html>.
- Fujimoto, Ozaki, Nagasawa, *J. Polym. Sci. A-2* **1968**, 6, 129 (poly-α-methylstyrene: M_w 4×10⁴ – 5×10⁵) — doi:10.1002/pol.1968.160060106.
- Unidad, Abdel Goad, Bras, Zamponi, Faust, Allgaier, Pyckhout-Hintzen, Wischnewski, Richter, Fetters, *Macromolecules* **2015**, 48, 5576, Table 2 (exact sample M_w: PEP 67.3, PI 128, PS 417, PVCH 1230 kg/mol; p = 1.69/2.12/3.10/3.17/3.95/5.59 Å) — doi:10.1021/acs.macromol.5b00341; full text at <https://datapdf.com/consequences-of-increasing-packing-length-on-acs-publication.html>.
The full table with a per-row source string is in `data/polymer_table.csv`. No constant was invented; where a source documents a range, the upper end of the range is used and the range is recorded in the source column. Temperature is encoded in the polymer name; PaMS G_N was measured at 473 K per the table footnote, and PMMA G_N (0.31 MPa) is R. H. Colby's value as tabulated in Fetters 1994, with Fuchs 1996 providing the M_w documentation.
### Statistics
`code/packing_test.py` (Python; numpy/pandas/scipy/matplotlib):
1. **Slope test:** OLS regression of log₁₀(G_N [Pa]) on log₁₀(p [Å]); the claim predicts slope −3 (G_N ∝ p⁻³). Slope ± SE, R², N; two-sided t-test against −3.
2. **Molecular-weight test:** partial correlation of log₁₀(G_N) with log₁₀(M_w) controlling log₁₀(p); plus a multiple regression log₁₀(G_N) ~ log₁₀(p) + log₁₀(M_w) with an F-test of whether M_w adds explanatory power at fixed p. (A raw G_N–M_w correlation is also reported as a contrast, since M_w and p covary in this literature — high-p polymers were studied with higher-M_w samples — which is exactly the confounding the partial correlation removes.)
3. **Robustness:** same regression restricted to one row per (polymer, temperature) (N = 13).
4. **Reference check:** fitted intercept compared with the theoretical G_N = kT/(n_t²p³), n_t = 20.8, T = 413 K.
## Results
- **Fitted slope: −3.034 ± 0.135** (95 % CI [−3.30, −2.77]); **R² = 0.973**, **N = 16** polymer-melt rows. The slope differs from −3 by **1.1 %** (t = −0.25, p = 0.81).
- Robustness (one row per polymer/temperature, N = 13): slope **−3.025 ± 0.137**, R² = 0.978 — unchanged.
- Reference: Fetters et al. 1994 themselves reported slope **−2.89** (r² = 0.9924) on their full 26-polymer 413 K set.
- **Partial correlation:** r(log₁₀ G_N, log₁₀ M_w | log₁₀ p) = **−0.078** (p = 0.78, df = 13). In the multiple regression the M_w coefficient is b = −0.0069 ± 0.0242 per decade of M_w (p = 0.78); adding M_w raises R² by only 0.0002 (F = 0.08, p = 0.78). **M_w adds nothing at fixed p.** Illustratively, the table spans M_w from 67.3 kg/mol (PEP) to 16 500 kg/mol (polybutadiene), a factor of ~245, while G_N at a given p stays fixed.
- Intercept: fitted 7.101 vs theoretical 7.120 for G_N = kT/(n_t²p³) at 413 K (n_t = 20.8) — the amplitude of the p⁻³ law is also consistent.
- **Residual plot summary:** residuals from the bivariate fit are centered (mean ≈ 0) with s.d. = 0.065 decades; residual correlation with log₁₀ p is −0.0000 (by construction) and with log₁₀ M_w is −0.078 — no structure. See `figures/GN_vs_p.png` (log-log data, fitted line, and the slope −3 reference line) and `figures/residuals.png`.
## Verdict
**SUPPORTED**
The slope is −3.03 ± 0.13, consistent with −3 to within 1 % (well inside the 15 % criterion), R² = 0.97, and at fixed packing length the plateau modulus shows no correlation with molecular weight (partial r = −0.08, p = 0.78): molecular weight shifts only the plateau *width*, not its height. Both predictions (P1: collapse onto p; P2: M_w affects width, not height) hold, and the refutation condition (modulus tracking M_w) is not observed.
## Limitations
- **N = 16 rows from 11 chemistries**, assembled from the Fetters 1994 compilation plus documented M_w values; a larger table (the full Fetters 1999/handbook set, ~40 polymers) was not fully accessible free-of-charge. Rows requiring paywalled sample tables (PDMS, PEO, a-PP, s-PP, hhPP, PVE) were **excluded rather than estimated** — no constants were fabricated.
- **M_w column semantics:** for some rows M_w is the exact documented sample value; for others it is the documented upper end of the molecular-weight range used in that polymer's G_N determination (recorded in the `source` column). This is adequate for the fixed-p M_w test but limits the M_w-variance within each (p, G_N) cell.
- **Temperature mixing:** G_N and p are paired at the temperature stated in the Fetters table; PaMS G_N is at 473 K and PMMA G_N (0.31 MPa) is a tabulated Colby value, while Fuchs et al. 1996 report 0.635 MPa at 190 °C for conventional PMMA — a reminder of real literature scatter (0.3–0.7 MPa) for that polymer.
- **G_N determination methods** vary across the compilation (terminal-loss-peak integration, tan δ minimum, BSW fits), adding ~±10–20 % scatter — visible in the residual s.d. of 0.065 decades.
- The correlation is empirical; p⁻³ scaling is also implied by theory (G_N = ρRT/M_e with M_e ∝ p³), so the test does not discriminate among micro-mechanisms.
- Some source access routes (ACS, ScienceDirect, Springer) were bot-blocked; values were verified through DOI-resolved reproductions (datapdf mirrors, PSU Pure, publisher-indexed abstracts) and are flagged as such in the CSV.
## References
1. Fetters, L. J.; Lohse, D. J.; Richter, D.; Witten, T. A.; Zirkel, A. *Macromolecules* **1994**, 27, 4639–4647. doi:10.1021/ma00095a029 (Tables 1–2 used for G_N, p).
2. Fetters, L. J.; Lohse, D. J.; Graessley, W. W. *J. Polym. Sci. Part B: Polym. Phys.* **1999**, 37, 1023–1033. doi:10.1002/(SICI)1099-0488(19990515)37:10<1023::AID-POLB7>3.0.CO;2-T
3. Fetters, L. J.; Lohse, D. J.; Milner, S. T.; Graessley, W. W. *Macromolecules* **1999**, 32, 6847–6851. doi:10.1021/ma990620o
4. Unidad, H. J.; Abdel Goad, M.; Bras, A. R.; Zamponi, M.; Faust, R.; Allgaier, J.; Pyckhout-Hintzen, W.; Wischnewski, A.; Richter, D.; Fetters, L. J. *Macromolecules* **2015**, 48, 5576–5581. doi:10.1021/acs.macromol.5b00341
5. Fetters, L. J.; Lohse, D. J.; García-Franco, C. A.; Brant, P.; Richter, D. *Macromolecules* **2002**, 35, 10096–10101. doi:10.1021/ma025659z
6. Pearson, D. S.; Fetters, L. J.; Graessley, W. W.; Ver Strate, G.; von Meerwall, E. *Macromolecules* **1994**, 27, 711–719. doi:10.1021/ma00081a014
7. Colby, R. H.; Fetters, L. J.; Graessley, W. W. *Macromolecules* **1987**, 20, 2226–2237. doi:10.1021/ma00175a030
8. Fetters, L. J.; Graessley, W. W.; Kiss, A. D. *Macromolecules* **1991**, 24, 3136–3141. doi:10.1021/ma00011a016
9. Abdel-Goad, M.; Pyckhout-Hintzen, W.; Kahle, S.; Allgaier, J.; Richter, D.; Fetters, L. J. *Macromolecules* **2004**, 37, 8135–8144. doi:10.1021/ma030557+
10. Fuchs, K.; Friedrich, Chr.; Weese, J. *Macromolecules* **1996**, 29, 5893–5901. doi:10.1021/ma951385m
11. Fujimoto, T.; Ozaki, N.; Nagasawa, M. *J. Polym. Sci. Part A-2* **1968**, 6, 129–140. doi:10.1002/pol.1968.160060106
12. Roovers, J.; Toporowski, P. M. *Rubber Chem. Technol.* **1990**, 63, 734–746. doi:10.5254/1.3538286
13. Eckstein, A.; Suhm, J.; Friedrich, C.; Maier, R.-D.; Saßmannshausen, J.; Bochmann, M.; Mülhaupt, R. *Macromolecules* **1998**, 31, 1335–1340. doi:10.1021/ma971270d
## Files
- `report.md` — this report
- `code/packing_test.py` — analysis script (regression, partial correlation, plots)
- `code/results_summary.txt` — numeric summary dump
- `data/polymer_table.csv` — 16 rows: polymer, G_N_Pa, p_angstrom, Mw_kgmol, source (per-row citations)
- `figures/GN_vs_p.png` — log-log plot with slope −3 reference line
- `figures/residuals.png` — residual scatter (vs log p, vs log M_w)
- `raw/` — downloaded evidence PDFs/texts (Fetters 1994/1991/Fuchs 1996/Unidad 2015 full texts as fetched; Akron & Naples theses inspected but not used)
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Working report — not peer reviewed. Educational information, not medical advice. AI-assisted test: DeepSeek.