SEC.Models.EdmLinear

SEC.Models.EdmLinear#

SEC.Models.EdmLinear.py

PDF of the Equilibrium Dispersive Model (EDM) with a linear isotherm, computed via characteristic function (CF) inversion using FFT.

This is the k_MT → ∞ limit of the LKM: when mass transfer is instantaneous the stationary and mobile phases are always in local equilibrium, and the only band-broadening mechanism is axial dispersion.

Assumptions#

  • Linear isotherm: q = a * c (constant Henry coefficient; no overloading)

  • Axial dispersion parametrised by the Peclet number Pe = u*L/D_ax

  • Instantaneous mass transfer (equilibrium dispersive model)

Model parameters#

Pe : Peclet number Pe = u*L/D_ax t0 : dead time t0 = L/u (mobile-phase transit time) R : retention factor R = t_R / t0 = 1 + F*a (F = (1-ε)/ε, a = Henry coeff.)

Transfer function (Laplace domain)#

H(s) = exp( Pe/2 * (1 - sqrt(1 + 4*s*t0*R / Pe)) )

This is the k_MT → ∞ limit of the LKM transfer function:

LKM: Phi = (4*s*t0/Pe) * (s + k_MT*R) / (s + k_MT) EDM: Phi = (4*s*t0*R / Pe) [k_MT → ∞]

The elution profile is a perfect Gaussian with:

μ₁ = t0 * R σ² = 2 * t0² * R² / Pe

This agrees with the classical moment formula (Qamar 2014, Table 1, Dirichlet BCs, linear isotherm) and is validated in:

molass-researcher/experiments/27_qamar_2014_paper/27b_edm_lkm_grm_consistency.ipynb

Note on edm_func (molass-legacy)#

molass_legacy.SecTheory.Edm.edm_func implements the full EDM with a quadratic isotherm (Langmuir linearisation, b ≠ 0) via the Hopf-Cole analytic solution (Rehman 2022). Its b=0 branch is mathematically equivalent to edm_pdf but uses a finite-column outlet-concentration formula that introduces a small systematic mean shift (~0.5%) absent from this CF approach.

Use edm_pdf for:

  • Linear-regime SEC-SAXS analysis (analytical and comparative work)

  • Model-hierarchy checks: EDM < GRM < LKM

Use edm_func for:

  • Overloading / nonlinear isotherm studies (b ≠ 0)

References

  • Lapidus & Amundson (1952), J. Phys. Chem. 56:984

  • Qamar et al. (2014), Comput. Chem. Eng. 71:383, Table 1

  • Rehman et al. (2022), Thermal Science 26:4253 (nonlinear EDM)

  • Validated in molass-researcher/experiments/27_qamar_2014_paper/27b

edm_linear_cf(w, Pe, t0_s, R)#

Characteristic function of the EDM elution profile (scaled coordinates).

Parameters are in scaled time units (see edm_pdf for the scaling convention). Do not call this directly; use edm_pdf instead.

edm_pdf(x, Pe, t0, R, timescale=None)#

PDF of the EDM (Equilibrium Dispersive Model) elution profile with linear isotherm.

Uses FFT-based characteristic function inversion via FftInvPdf.

Parameters:

  • x (array_like) – Time array (physical time units, e.g. seconds or minutes).

  • Pe (float) – Peclet number Pe = u*L/D_ax. Controls axial dispersion width. Typical SEC range: 100–1000.

  • t0 (float) – Dead time — mobile-phase transit time (same units as x).

  • R (float) – Retention factor R = t_R / t0 = 1 + F*a (F = phase ratio = (1-ε)/ε, a = Henry coefficient).

  • timescale (float or None, optional) – Time rescaling factor for the internal FFT grid. If None (default), chosen automatically as 80 / (t0 * R), mapping the peak position t_R = t0*R to position 80 on the FFT grid [0, 1024].

Returns:

Normalised PDF evaluated at each point in x (integral ≈ 1). Multiply by the peak area (c_inj × t_inj) to obtain absolute concentration units.

Return type:

ndarray

Notes

The profile is analytically a Gaussian with:

μ₁ = t0 * R σ² = 2 * t0² * R² / Pe

This is the narrowest of the three linear-isotherm SEC models:

EDM (this function) — axial dispersion only GRM (GrmLinear.edm_pdf) — axial + film/pore diffusion LKM (LkmLinear.lkm_pdf) — axial + lumped mass transfer

The hierarchy EDM < GRM < LKM holds at all finite k_MT.

Examples

>>> import numpy as np
>>> from molass.SEC.Models.EdmLinear import edm_pdf
>>> t = np.linspace(0.1, 30.0, 500)
>>> y = edm_pdf(t, Pe=400, t0=5.0, R=3.0)
>>> np.trapz(y, t)   # ≈ 1.0