SEC.Models.LkmLinear

SEC.Models.LkmLinear#

SEC.Models.LkmLinear.py

PDF of the Lumped Kinetic Model (LKM) with a linear isotherm, computed via characteristic function (CF) inversion using FFT (PDE-based).

Assumptions#

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

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

  • Mass transfer between mobile and stationary phases follows a linear driving force with rate constant k_MT (standard LKM convention: k_MT = k_STLC / (1-ε))

Model parameters#

Pe : Peclet number Pe = u*L/D_ax t0 : dead time t0 = L/u (mobile-phase transit time) k_MT : mass-transfer rate constant [1/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*(s + k_MT*R) / (Pe*(s + k_MT)))) )

References

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

  • Felinger et al. (2004), J. Chromatogr. A 1043:149–157 (SDM≡LKM equivalence)

  • Validated against STLC PDE solver in molass-researcher/experiments/19_sdm_upgrade/19g, 19h

lkm_linear_cf(w, Pe, t0_s, k_s, R)#

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

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

lkm_pdf(x, Pe, t0, k_MT, R, timescale=None)#

PDF of the LKM (Lumped Kinetic 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).

  • k_MT (float) –

    Mass-transfer rate constant [1/time], standard LKM convention: dq/dt = k_MT × (a·c − q) (ε = mobile-phase porosity).

    Some LKM solvers normalise the rate internally by (1−ε). For example, the STLC package implements dq/dt = k/(1−ε) × (a·c − q), so its user-facing k differs from k_MT. Convert with k_MT = k_solver / (1 − ε) before passing to this function.

  • 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]. Override if the default produces edge artefacts (e.g. very wide peaks).

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

Linearity assumption

This model assumes a linear isotherm (constant Henry coefficient a). At high sample loads the isotherm becomes non-linear (Langmuir or anti-Langmuir), invalidating R as a fixed parameter. Use this model only in the linear (dilute) regime typical of analytical SEC-SAXS.

Relationship to the D=0 (Thomas) model

In the limit Pe → ∞ this CF reduces to the Thomas / kinetic model, which has a closed-form Bessel I₁ solution. At typical SEC Peclet numbers (Pe ~ 100–500) the D=0 approximation introduces a systematic k_MT underestimation of ~3–4% (validated in experiment 19h). This PDE-CF implementation avoids that bias.

Equivalence to the stochastic–dispersive model (Felinger 2004)

Felinger et al. (J. Chromatogr. A 1043:149–157, 2004) proved that the LKM is mathematically identical to the stochastic–dispersive model (SDM) in linear chromatography, provided exponential sojourn times and Poisson-distributed adsorption events are assumed. The parameter mapping is:

  • k_MT (molass) ≡ kd (Felinger desorption rate constant)

  • Nm (Felinger mass transfer units) = k_MT × k' × t0 where k' = R - 1 (retention factor).

This equivalence holds only for linear isotherms (b=0 in EDM). For nonlinear chromatography (fronting or tailing peaks), the LKM and SDM diverge.

Examples

>>> import numpy as np
>>> from molass.SEC.Models.LkmLinear import lkm_pdf
>>> t = np.linspace(0.1, 20.0, 500)
>>> y = lkm_pdf(t, Pe=500, t0=2.0, k_MT=1.667, R=4.0)
>>> np.trapz(y, t)   # ≈ 1.0