Transient Network Theory (TNT) — Mathematical Foundations & Verification¶

This document records the published governing equations for each TNT model variant, cross-referenced against the RheoJAX implementation in rheojax/models/tnt/. Sources are cited inline; web-accessible links are listed at the end.

1. Green-Tobolsky / Tanaka-Edwards (Basic Transient Network)¶

Primary references:

Green, M.S. & Tobolsky, A.V. (1946). J. Chem. Phys. 14, 80-92.
Tanaka, F. & Edwards, S.F. (1992). Macromolecules 25, 1516-1523.
Lodge, A.S. (1956). Trans. Faraday Soc. 52, 120-130.

1.1 Chain Distribution / Conformation Tensor ¶

Green & Tobolsky (1946) conceived a network of chains with temporary junctions that are steadily created and destroyed. Tanaka & Edwards (1992) formalised this using the conformation tensor

\[\mathbf{S} = \frac{\langle \mathbf{R} \otimes \mathbf{R} \rangle}{R_0^2}\]

where \(\mathbf{R}\) is the end-to-end vector of a network strand and \(R_0 = \sqrt{\langle R^2 \rangle_0}\) is the equilibrium root-mean-square distance. At equilibrium \(\mathbf{S} = \mathbf{I}\).

1.2 Creation and Destruction Rates ¶

Constant-rate (basic model):

Destruction (breakage) rate: \(\beta = 1/\tau_b\)
Creation (reformation) rate: \(g_0 = 1/\tau_b\)

At equilibrium the two rates balance, maintaining \(\mathbf{S} = \mathbf{I}\).

Force-dependent detachment (Tanaka-Edwards, Bell):

If a force \(f\) acts on a chain bound at time \(t_0\), the breakage rate at time \(t\) is (Tanaka & Edwards 1992, Eq. 2; see also Bell 1978):

\[\beta(f) = \omega_0 \exp\!\left(-\frac{W_b - f \cdot a}{k_B T}\right) = \beta_0 \exp\!\left(\frac{f \cdot a}{k_B T}\right)\]

where \(\omega_0\) is the thermal vibration frequency, \(W_b\) is the bond dissociation energy, and \(a\) is the activation length. In the conformation-tensor formulation this becomes:

\[\beta(\mathbf{S}) = \frac{1}{\tau_b}\exp\!\bigl[\nu\,(\text{stretch} - 1)\bigr], \qquad \text{stretch} = \sqrt{\operatorname{tr}(\mathbf{S})/3}\]

with \(\nu\) a dimensionless force-sensitivity parameter (\(\nu = 0\) recovers constant breakage).

Implementation: _kernels.py, functions breakage_constant, breakage_bell, breakage_power_law.

1.3 Constitutive Equation (Conformation Tensor Evolution)¶

\[\frac{d\mathbf{S}}{dt} = \boldsymbol{\kappa}\cdot\mathbf{S} + \mathbf{S}\cdot\boldsymbol{\kappa}^T + g_0\,\mathbf{I} - \beta(\mathbf{S})\,\mathbf{S}\]

where \(\boldsymbol{\kappa} = (\nabla\mathbf{v})^T\) is the velocity gradient tensor. The first two terms are the upper-convected derivative (affine deformation); the third is chain creation; the fourth is chain destruction.

Equivalently, writing as relaxation toward equilibrium:

\[\frac{d\mathbf{S}}{dt} = \boldsymbol{\kappa}\cdot\mathbf{S} + \mathbf{S}\cdot\boldsymbol{\kappa}^T - \frac{\mathbf{S} - \mathbf{I}}{\tau_b}\]

This is mathematically identical to the upper-convected Maxwell (UCM) model.

Implementation: _kernels.py, tnt_single_mode_ode_rhs (lines 588-638) and build_tnt_ode_rhs factory (lines 823-881).

1.4 Stress Tensor ¶

\[\boldsymbol{\sigma} = G\,(\mathbf{S} - \mathbf{I}) + 2\eta_s\,\mathbf{D}\]

where \(G \approx n_{\text{chains}} k_B T\) is the network modulus and \(\mathbf{D} = (\boldsymbol{\kappa}+\boldsymbol{\kappa}^T)/2\).

In simple shear (\(\boldsymbol{\kappa} = \dot\gamma\,\mathbf{e}_x\otimes\mathbf{e}_y\)):

\[\begin{split}\sigma_{xy} &= G\,S_{xy} + \eta_s\dot\gamma \\ N_1 &= G(S_{xx} - S_{yy}) \\ N_2 &= G(S_{yy} - S_{zz}) = 0 \quad\text{(upper-convected)}\end{split}\]

Implementation: _kernels.py, stress_linear_xy, stress_fene_xy.

1.5 Simple Shear Component Equations ¶

\[\begin{split}\frac{dS_{xx}}{dt} &= 2\dot\gamma\,S_{xy} - \frac{S_{xx}-1}{\tau_b} \\[4pt] \frac{dS_{yy}}{dt} &= -\frac{S_{yy}-1}{\tau_b} \\[4pt] \frac{dS_{zz}}{dt} &= -\frac{S_{zz}-1}{\tau_b} \\[4pt] \frac{dS_{xy}}{dt} &= \dot\gamma\,S_{yy} - \frac{S_{xy}}{\tau_b}\end{split}\]

1.6 Steady-State Solutions (Simple Shear)¶

\[\begin{split}S_{xy} &= \tau_b\dot\gamma \\ S_{xx} &= 1 + 2(\tau_b\dot\gamma)^2 \\ S_{yy} &= S_{zz} = 1\end{split}\]

Flow curve (Newtonian):

\[\sigma_{xy} = (G\tau_b + \eta_s)\,\dot\gamma = \eta_0\,\dot\gamma\]

First normal stress difference:

\[N_1 = 2G(\tau_b\dot\gamma)^2, \qquad \Psi_1 = 2G\tau_b^2\]

Implementation: _kernels.py, tnt_base_steady_conformation, tnt_base_steady_stress, tnt_base_steady_n1.

1.7 Relaxation Modulus ¶

\[G(t) = G\,\exp(-t/\tau_b)\]

Single Maxwell exponential.

Implementation: _kernels.py, tnt_base_relaxation.

1.8 SAOS Moduli ¶

\[\begin{split}G'(\omega) &= G\,\frac{(\omega\tau_b)^2}{1+(\omega\tau_b)^2} \\[4pt] G''(\omega) &= G\,\frac{\omega\tau_b}{1+(\omega\tau_b)^2} + \eta_s\omega\end{split}\]

Implementation: _kernels.py, tnt_saos_moduli.

1.9 Parameters ¶

Symbol	Units	Description
\(G\)	Pa	Network modulus (\(\approx n_\text{chains} k_B T\))
\(\tau_b\)	s	Bond lifetime (reciprocal breakage rate)
\(\eta_s\)	Pa·s	Solvent viscosity

Derived: \(\eta_0 = G\tau_b + \eta_s\) (zero-shear viscosity).

2. Tanaka-Edwards Model with Force-Dependent Breakage ¶

Primary references:

Tanaka, F. & Edwards, S.F. (1992). Macromolecules 25, 1516-1523.
Bell, G.I. (1978). Science 200, 618-627.

2.1 Breakage Rate (Kramers/Bell)¶

From Tanaka & Edwards, the thermally-activated breakage rate in the presence of a force \(f\) is:

\[\beta(f) = \omega_0\,\exp\!\Bigl[-\frac{W_b}{k_BT}\Bigr] \cdot\exp\!\Bigl[\frac{f\cdot a}{k_BT}\Bigr] = \frac{1}{\tau_b}\,\exp\!\Bigl[\frac{f\cdot a}{k_BT}\Bigr]\]

In the conformation tensor description, the end-to-end distance proxy is the stretch ratio \(\lambda = \sqrt{\operatorname{tr}(\mathbf{S})/3}\), giving the coarse-grained Bell form:

\[\beta(\mathbf{S}) = \frac{1}{\tau_b}\,\exp\!\bigl[\nu(\lambda - 1)\bigr]\]

At equilibrium (\(\lambda=1\)): \(\beta = 1/\tau_b\).

Power-law variant:

\[\beta(\mathbf{S}) = \frac{1}{\tau_b}\,\lambda^m\]

Implementation: _kernels.py, breakage_bell (line 71), breakage_power_law (line 103).

2.2 ODE System (Bell breakage, simple shear)¶

\[\begin{split}\frac{dS_{xx}}{dt} &= 2\dot\gamma S_{xy} + g_0 - \beta(\mathbf{S})\,S_{xx}\\ \frac{dS_{yy}}{dt} &= g_0 - \beta(\mathbf{S})\,S_{yy}\\ \frac{dS_{zz}}{dt} &= g_0 - \beta(\mathbf{S})\,S_{zz}\\ \frac{dS_{xy}}{dt} &= \dot\gamma\,S_{yy} - \beta(\mathbf{S})\,S_{xy}\end{split}\]

with \(g_0 = 1/\tau_b\) (creation rate unaffected by force).

Key consequence: Force-dependent breakage produces shear thinning (faster breakage at high strain reduces effective relaxation time) and stress overshoot with non-exponential relaxation.

Implementation: _kernels.py, build_tnt_ode_rhs(breakage_type="bell").

3. Cates Reptation-Reaction Model (Living Polymers)¶

Primary references:

Cates, M.E. (1987). Macromolecules 20, 2289-2296.
Cates, M.E. (1990). J. Phys. Chem. 94, 371-375.
Turner, M.S. & Cates, M.E. (1991). Langmuir 7, 1590.
Chen, V., Drucker, C.T., Love, C., Peterson, J. & Peterson, J.D. (2024). arXiv:2407.07213 (analytic series solution).

3.1 Physical Mechanism ¶

Wormlike micelles are entangled living polymers that relax stress via two mechanisms operating simultaneously:

Reptation (curvilinear diffusion along the tube): time scale \(\tau_\text{rep} \sim L^3/(\pi^2 D)\).
Reversible scission (random breaking/recombination): time scale \(\tau_\text{break}\).

Breaking reorganises tube segments: interior segments become end segments (which relax quickly) and vice versa.

3.2 Reptation Spectrum (Unbreakable Chains)¶

For permanent entangled polymers (Doi-Edwards):

\[G(t) = G_0 \sum_{p\;\text{odd}} \frac{8}{\pi^2 p^2} \exp\!\left(-\frac{p^2 t}{\tau_\text{rep}}\right)\]

3.3 Breaking Parameter ¶

\[\zeta = \frac{\tau_\text{break}}{\tau_\text{rep}}\]

3.4 Effective Relaxation Time (Fast-Breaking Limit)¶

When \(\zeta \ll 1\) (fast-breaking), Cates (1987) showed that the stress relaxation becomes near-single-exponential with the geometric mean relaxation time:

\[\boxed{\tau_d = \sqrt{\tau_\text{rep}\cdot\tau_\text{break}}}\]

Physical derivation: Reptation requires diffusion over contour length \(L\). Breaking cuts the micelle every \(\tau_\text{break}\) into pieces of size \(\sim\sqrt{D\tau_\text{break}}\). Setting this equal to the tube escape distance gives \(\tau_d \sim \sqrt{\tau_\text{rep}\,\tau_\text{break}}\).

Implementation: _kernels.py, tnt_cates_effective_tau (line 560).

3.5 Stress Relaxation ¶

Fast-breaking regime (\(\zeta \ll 1\)):

\[G(t) \approx G_0\,\exp\!\left(-\sqrt{2t/\tau_\text{break}}\right)\]

This stretched exponential is well approximated by a single Maxwell mode with time constant \(\tau_d\).

Slow-breaking regime (\(\zeta \gg 1\)):

Standard reptation spectrum (Eq. in Sec. 3.2).

3.6 Constitutive Equation (Fast-Breaking UCM)¶

In the fast-breaking limit the system reduces to a single-mode UCM with \(\tau_b \to \tau_d\):

\[\frac{d\mathbf{S}}{dt} = \boldsymbol{\kappa}\cdot\mathbf{S} + \mathbf{S}\cdot\boldsymbol{\kappa}^T - \frac{\mathbf{S}-\mathbf{I}}{\tau_d}\]

\[\boldsymbol{\sigma} = G_0(\mathbf{S}-\mathbf{I}) + 2\eta_s\mathbf{D}\]

Implementation: cates.py delegates to the base single-mode ODE with \(\tau_b = \tau_d = \sqrt{\tau_\text{rep}\tau_\text{break}}\).

3.7 SAOS Moduli ¶

Single Maxwell mode with \(\tau_d\):

\[\begin{split}G'(\omega) &= G_0\,\frac{(\omega\tau_d)^2}{1+(\omega\tau_d)^2}\\[4pt] G''(\omega) &= G_0\,\frac{\omega\tau_d}{1+(\omega\tau_d)^2} + \eta_s\omega\end{split}\]

3.8 Cole-Cole Semicircle ¶

Plotting \(G''\) vs \(G'\) parametrically in \(\omega\) (with \(\eta_s=0\)) gives a perfect semicircle:

\[\left(G' - \frac{G_0}{2}\right)^{\!2} + (G'')^2 = \left(\frac{G_0}{2}\right)^{\!2}\]

Centre at \((G_0/2,\;0)\)
Radius \(G_0/2\)
Passes through the origin (\(\omega\to 0\)) and \((G_0,0)\) (\(\omega\to\infty\))

Deviations indicate intermediate breaking, branching (Y-junctions), or polydispersity.

More precisely, at arbitrary \(\zeta\):

\[G^*(\omega) \approx \frac{G_0}{1 + \sqrt{\tau_\text{break}/(2i\omega)}}\]

(Turner & Cates 1991).

Implementation: Verified via tnt_saos_moduli_vec called with effective \(\tau_d\).

3.9 Steady Shear (Non-Monotonic Flow Curve)¶

\[\sigma_{xy} = G_0\,\frac{\tau_d\dot\gamma}{1+(\tau_d\dot\gamma)^2} + \eta_s\dot\gamma\]

The network contribution is non-monotonic with a maximum at \(\dot\gamma = 1/\tau_d\) (constitutive instability leading to shear banding for \(\text{Wi}_d > 1\)).

3.10 Parameters ¶

Symbol	Units	Description
\(G_0\)	Pa	Plateau modulus (\(\sim k_BT/\xi^3\), \(\xi\) = mesh size)
\(\tau_\text{rep}\)	s	Reptation time
\(\tau_\text{break}\)	s	Mean scission/breakage time
\(\eta_s\)	Pa·s	Solvent viscosity

Derived: \(\tau_d = \sqrt{\tau_\text{rep}\tau_\text{break}}\), \(\eta_0 = G_0\tau_d + \eta_s\).

4. Sticky Rouse Model ¶

Primary references:

Baxandall, L.G. (1989). Macromolecules 22, 1982.
Leibler, L., Rubinstein, M. & Colby, R.H. (1991). Macromolecules 24, 4701-4707.
Rubinstein, M. & Semenov, A.N. (1998). Macromolecules 31, 1373-1385.
Rubinstein, M. & Semenov, A.N. (2001). Macromolecules 34, 1058-1068.

4.1 Physical Picture ¶

A flexible chain of \(N\) Kuhn segments carries \(N_s\) evenly spaced reversible association sites (“stickers”). Between stickers, the chain behaves as a Rouse sub-chain. The sticker lifetime \(\tau_s\) imposes a minimum effective relaxation time for all modes.

4.2 Rouse Mode Spectrum ¶

For a standard Rouse chain, mode \(k\) has:

\[\tau_{R,k} = \frac{\tau_{R,1}}{k^2}, \qquad k = 1,2,\ldots,N\]

where \(\tau_{R,1}\) is the longest Rouse time.

4.3 Effective Relaxation Times (Sticker Renormalisation)¶

Sticker exchange slows all modes faster than \(\tau_s\). Two formulations appear in the literature:

Additive (Leibler-Rubinstein-Colby, 1991; used in RheoJAX docs):

\[\tau_{\text{eff},k} = \tau_{R,k} + \tau_s\]

Max (phenomenological, used in RheoJAX implementation):

\[\tau_{\text{eff},k} = \max(\tau_{R,k},\;\tau_s)\]

Both produce the same qualitative physics: modes with \(\tau_{R,k} < \tau_s\) are pinned at \(\tau_s\), while slow modes (\(\tau_{R,k} > \tau_s\)) are essentially unaffected.

Note

Discrepancy found: The RheoJAX documentation (tnt_sticky_rouse.rst line 27) states \(\tau_{\text{eff},k} = \tau_{R,k} + \tau_s\) (additive), whereas the Python source (sticky_rouse.py lines 14-15) uses \(\max(\tau_{R,k},\tau_s)\). Both are valid physical approximations. The additive form more faithfully follows Leibler-Rubinstein-Colby; the max form is more commonly used in recent computational works. Recommend unifying docs and code.

4.4 Multi-Mode Constitutive Equation ¶

Each mode \(k\) evolves independently:

\[\frac{d\mathbf{S}_k}{dt} = \boldsymbol{\kappa}\cdot\mathbf{S}_k + \mathbf{S}_k\cdot\boldsymbol{\kappa}^T - \frac{\mathbf{S}_k - \mathbf{I}}{\tau_{\text{eff},k}}\]

Total stress:

\[\boldsymbol{\sigma} = \sum_{k=1}^{N} G_k\,(\mathbf{S}_k - \mathbf{I}) + 2\eta_s\mathbf{D}\]

Implementation: _kernels.py, tnt_multimode_ode_rhs.

4.5 Relaxation Modulus ¶

\[G(t) = \sum_{k=1}^{N} G_k\,\exp\!\left(-\frac{t}{\tau_{\text{eff},k}}\right)\]

At intermediate times (when multiple modes have \(\tau_{\text{eff},k} \approx \tau_s\)), this yields Rouse-like power-law decay \(G(t) \sim t^{-1/2}\).

Implementation: _kernels.py, tnt_multimode_relaxation.

4.6 SAOS Moduli ¶

\[\begin{split}G'(\omega) &= \sum_k G_k\,\frac{(\omega\tau_{\text{eff},k})^2} {1+(\omega\tau_{\text{eff},k})^2}\\[4pt] G''(\omega) &= \sum_k G_k\,\frac{\omega\tau_{\text{eff},k}} {1+(\omega\tau_{\text{eff},k})^2} + \eta_s\omega\end{split}\]

Characteristic signatures:

\(G' \sim \omega^{1/2}\) at intermediate frequencies (Rouse regime)
Sticky plateau at \(1/\tau_s < \omega < 1/\tau_{R,N}\)
Terminal flow (\(G' \sim \omega^2\)) at \(\omega \ll 1/\tau_{\text{eff},1}\)

Implementation: _kernels.py, tnt_multimode_saos_moduli.

4.7 Parameters ¶

Symbol	Units	Description
\(G_k\)	Pa	Modulus of mode \(k\)
\(\tau_{R,k}\)	s	Rouse relaxation time of mode \(k\)
\(\tau_s\)	s	Sticker lifetime
\(\eta_s\)	Pa·s	Solvent viscosity

Derived: \(G_N^{(0)} = \sum_k G_k\) (plateau modulus), \(\eta_0 = \sum_k G_k\tau_{\text{eff},k} + \eta_s\).

5. Loop-Bridge Model (Telechelic Networks)¶

Primary references:

Annable, T., Buscall, R., Ettelaie, R. & Whittlestone, D. (1993). J. Rheol. 37, 695-726.
Tanaka, F. & Edwards, S.F. (1992). Macromolecules 25, 1516-1523.
Leibler, L., Rubinstein, M. & Colby, R.H. (1991). Macromolecules 24, 4701-4707.
Bell, G.I. (1978). Science 200, 618-627.

5.1 Physical Picture ¶

Telechelic chains have associating end-groups that can exist in two populations:

Bridges (fraction \(f_B\)): Both ends attached to different junctions — load-bearing.
Loops (fraction \(f_L = 1 - f_B\)): Both ends on the same junction — non-load-bearing.

The dynamic equilibrium between these populations determines the modulus and produces the distinctive shear-thickening-then-thinning behaviour observed experimentally (Annable et al. 1993).

5.2 Bridge Fraction Kinetics ¶

\[\frac{df_B}{dt} = \underbrace{\frac{1-f_B}{\tau_a}}_{\text{loop}\to\text{bridge}} - \underbrace{f_B\,\beta(\mathbf{S})}_{\text{bridge}\to\text{loop}}\]

where:

\(\tau_a\) = loop-to-bridge association time
\(\beta(\mathbf{S})\) = force-dependent bridge detachment rate

Equilibrium bridge fraction (at \(\mathbf{S}=\mathbf{I}\), \(\beta = 1/\tau_b\)):

\[f_{B,\text{eq}} = \frac{\tau_b}{\tau_a + \tau_b}\]

Implementation: loop_bridge.py, _loop_bridge_ode_rhs (line 146).

5.3 Force-Dependent Detachment (Bell Model)¶

\[\beta(\mathbf{S}) = \frac{1}{\tau_b}\,\exp\!\Bigl[ \nu\bigl(\sqrt{\operatorname{tr}(\mathbf{S})/3} - 1\bigr)\Bigr]\]

At equilibrium: \(\beta = 1/\tau_b\). Under stretch: detachment accelerates exponentially.

5.4 Conformation Tensor Evolution (Bridges Only)¶

Bridges evolve under flow with creation and force-activated destruction:

\[\frac{d\mathbf{S}}{dt} = \boldsymbol{\kappa}\cdot\mathbf{S} + \mathbf{S}\cdot\boldsymbol{\kappa}^T + g_0\,\mathbf{I} - \beta(\mathbf{S})\,\mathbf{S}\]

with \(g_0 = 1/\tau_b\).

Implementation: loop_bridge.py, _loop_bridge_ode_rhs (lines 150-161).

5.5 Stress Tensor ¶

Only bridges contribute to elastic stress:

\[\boldsymbol{\sigma} = f_B\,G\,(\mathbf{S} - \mathbf{I}) + 2\eta_s\,\mathbf{D}\]

Implementation: Loop-bridge predict methods multiply the base stress by \(f_B\).

5.6 Shear Thickening Mechanism ¶

The non-monotonic viscosity arises from competing effects:

Low shear rates (\(\dot\gamma \ll 1/\tau_b\)): Bridges are at equilibrium; viscosity \(\eta \approx f_{B,\text{eq}} G\tau_b + \eta_s\).
Moderate shear rates: Flow-induced stretching increases the detachment rate \(\beta\), but the bridge fraction can increase if the reattachment rate (\(1/\tau_a\)) exceeds the enhanced detachment. More bridges → higher modulus → shear thickening.
High shear rates (\(\dot\gamma \gg 1/\tau_b\)): Force-activated detachment dominates; bridges are destroyed faster than they form. Bridge fraction drops → shear thinning.

This produces the characteristic non-monotonic viscosity observed by Annable et al. (1993) in HEUR thickeners: weak thickening followed by strong thinning.

5.7 SAOS Moduli (Linearised)¶

In the linear regime (\(\mathbf{S} \approx \mathbf{I}\)), the loop-bridge model reduces to a single Maxwell mode with effective modulus \(f_{B,\text{eq}} G\) and relaxation time \(\tau_b\):

\[\begin{split}G'(\omega) &= f_{B,\text{eq}}\,G\, \frac{(\omega\tau_b)^2}{1+(\omega\tau_b)^2}\\[4pt] G''(\omega) &= f_{B,\text{eq}}\,G\, \frac{\omega\tau_b}{1+(\omega\tau_b)^2} + \eta_s\omega\end{split}\]

5.8 Parameters ¶

Symbol	Units	Description
\(G\)	Pa	Network modulus (all bridges active)
\(\tau_b\)	s	Bridge lifetime (detachment timescale)
\(\tau_a\)	s	Loop-to-bridge association time
\(\nu\)	dimensionless	Force sensitivity (Bell exponent)
\(f_{B,\text{eq}}\)	dimensionless	Equilibrium bridge fraction
\(\eta_s\)	Pa·s	Solvent viscosity

Derived: \(\eta_0 = f_{B,\text{eq}}\,G\,\tau_b + \eta_s\), \(f_{B,\text{eq}} = \tau_b/(\tau_a+\tau_b)\).

Verification Notes ¶

Discrepancy: Sticky Rouse Effective Time ¶

The documentation uses the additive form \(\tau_{\text{eff},k} = \tau_{R,k} + \tau_s\), while the code uses max: \(\tau_{\text{eff},k} = \max(\tau_{R,k},\tau_s)\). Both are physically motivated but numerically distinct. Recommend aligning the documentation with the code (or vice versa) and adding a note about the alternative.

All Other Equations: Verified ¶

Green-Tobolsky/Tanaka-Edwards constitutive equation, steady states, SAOS moduli, and relaxation modulus match published forms exactly.
Cates geometric-mean \(\tau_d\), Cole-Cole semicircle, and UCM constitutive equation match Cates (1987, 1990) and Turner & Cates (1991).
Loop-Bridge kinetics, Bell detachment, and stress coupling to bridge fraction match Tanaka & Edwards (1992) and Annable et al. (1993).
Bell force-dependent breakage matches Bell (1978) Kramers form.

Sources ¶

Web-accessible references: