Log-Normal Spectrum in Rheology

• Log-normal spectrum in rheology is a model that represents relaxation time distributions and particle size polydispersity using a log-transformation to normalize the data.
• The approach is grounded in the maximum entropy principle and information geometry, ensuring finite moments and a smooth transition from monodisperse to polydisperse regimes.
• It accurately captures experimental viscoelastic responses in polymers and suspensions, enabling universal scaling of rheological data with minimal, physically interpretable parameters.

A log-normal spectrum in rheology characterizes the distribution of relaxation times or particle sizes relevant for viscoelastic response or suspension flow phenomena. The log-normal form emerges both as an information-theoretic extremum for relaxation spectra in linear viscoelasticity and as a natural descriptor for particle size polydispersity in noncolloidal suspensions. Its theoretical foundation, empirical fit to polymeric and particulate systems, and connections to alternative spectral models (e.g., fractional Maxwell or discrete mixture spectra) make it a central concept for the parameterization and analysis of complex rheological materials.

1. Information Geometry and the Logarithmic Relaxation Time Variable

In the context of linear viscoelasticity, the generalized Maxwell model expresses the relaxation modulus $G(t)$ as a superposition of exponential decays: $G(t) = \int_0^\infty d\tau\; e^{-t/\tau} h(\tau),$ where $h(\tau)$ is the relaxation spectrum. The information-geometric approach shows that the family of exponentials $P_M(t|\tau) = \frac{1}{\tau} e^{-t/\tau}$, regarded as probability densities in $t$, have a Fisher information metric $I(\tau) = 1/\tau^2$, implying that logarithmic time $\xi = \ln(\tau/\tau_0)$ is the unique natural coordinate. In this variable, the Fisher metric is flat ($I'(\xi) = 1$), making statistical distance Euclidean in $\xi$. Thus, the corresponding spectrum $H(\xi) = \tau_0 e^\xi h(\tau_0 e^\xi)$ defines the relaxation modulus as

$$G(t) = \int_{-\infty}^{\infty} d\xi\; e^{-t/(\tau_0 e^\xi)} H(\xi).$$

This transformation underpins the interpretation of $\xi = \ln\tau$ as the canonical variable for representing continuous relaxation spectra (Uneyama, 2 Sep 2025).

2. Maximum Entropy Principle and the Log-Normal Form

By maximizing the Shannon entropy for the normalized spectrum $\tilde H(\xi) = H(\xi)/G_0$ under constraints of normalization and fixed variance, the unique maximizer is the Gaussian in $\xi$: $$\tilde H(\xi) = \frac{1}{\sqrt{2\pi}\sigma} \exp\Bigl[-\frac{(\xi-\mu)^2}{2\sigma^2}\Bigr].$$ Translating back, $h(\tau)$ becomes log-normal: $$h(\tau) = \frac{G_0}{\tau\sqrt{2\pi}\sigma} \exp\Bigl[-\frac{(\ln(\tau/\tau_0) - \mu)^2}{2\sigma^2}\Bigr].$$ Here, $\mu = \langle \xi \rangle$ is a typical log-relaxation time and $\sigma^2 = \langle (\xi - \mu)^2 \rangle$ the log-space variance. The variance $\sigma^2$ parameterizes the breadth of the spectrum, interpolating continuously from monodisperse (Maxwellian) to broad polydisperse regimes (Uneyama, 2 Sep 2025).

3. Linear Viscoelastic Response Functions and Frequency-Domain Behavior

From the log-normal spectrum $H(\xi)$, the relaxation modulus $G(t)$ and the storage and loss moduli $G'(\omega)$, $G''(\omega)$ are constructed as: $$\begin{aligned} G(t) &= \int_{-\infty}^{\infty} d\xi\; e^{-t/(\tau_0 e^\xi)} H(\xi), \ G'(\omega) &= \int_{-\infty}^\infty d\xi\; H(\xi) \frac{(\tau_0 e^\xi \omega)^2}{1 + (\tau_0 e^\xi \omega)^2}, \ G''(\omega) &= \int_{-\infty}^\infty d\xi\; H(\xi) \frac{\tau_0 e^\xi \omega}{1 + (\tau_0 e^\xi \omega)^2}. \end{aligned}$$ In the low-frequency limit, $G'(\omega) \sim G_0\langle \tau^2 \rangle \omega^2$ and $G''(\omega) \sim G_0\langle \tau \rangle \omega$, confirming true terminal behavior. The absence of pathologies (e.g., divergent moments) is guaranteed, and moments follow $$\langle \tau^p \rangle = \tau_0^p e^{p^2 \sigma^2/2}$$. This log-normal spectrum suffices to capture broad viscoelastic spectra showing a single, symmetric peak in $\ln\omega$ (Uneyama, 2 Sep 2025).

4. Comparison with Fractional Maxwell Model and Other Distributions

The fractional Maxwell model exhibits power-law scaling and a logarithmic spectrum of the form: $$H_{\rm FM}(\xi) = \frac{G_0}{2\pi} \frac{\sin(\alpha\pi)}{\cosh(\alpha\xi) + \cos(\alpha\pi)},$$ which, for small $\alpha$, broadens and approaches a Gaussian, suggesting the log-normal spectrum as a generic approximation for broad, symmetric relaxation spectra. The effective width $\sigma_{\rm eff} = \sqrt{1+\cos(\alpha\pi)}/\alpha$ diverges for $\alpha \to 0$, bridging the gap between a finite-width log-normal and the strictly scale-free power law of the fractional Maxwell case. However, the log-normal cannot strictly reproduce power-law tails or the critical divergence of mean relaxation time (Uneyama, 2 Sep 2025).

5. Application to Experimental Data: HDPE Melt and Solid

Experimental viscoelastic data for high-density polyethylene (HDPE), both in melt (time–temperature superposed at $T_\mathrm{ref}=140^\circ\mathrm{C}$) and partially crystalline solid ($T_\mathrm{ref}=100^\circ\mathrm{C}$) states, were shown to be accurately described by the log-normal spectrum. Parameters were tuned as follows:

State	$G_0$ (Pa)	$\tau_0$ (s)	$\sigma$
HDPE Melt	$1.1 \times 10^6$	$6.5 \times 10^{-5}$	$3.4$
HDPE Solid	$1.2 \times 10^8$	$1.0 \times 10^{-3}$	$9.0$

In both cases, the log-normal model captured the shapes of $G'(\omega)$ and $G''(\omega)$ over multiple decades, providing evidence for its practical utility in real systems where relaxation spectra are broad but unimodal (Uneyama, 2 Sep 2025).

6. Log-Normal Size Distributions in Suspension Rheology

In dense non-Brownian suspensions, particle radii drawn from a (truncated) log-normal distribution serve as a model for polydispersity. For $a \in [a_-,a_+]$,

$$p(a) = \frac{p_0(a)}{\Phi\!\Bigl(\frac{\ln a_+ - \mu}{\sigma} \Bigr) - \Phi\!\Bigl(\frac{\ln a_- - \mu}{\sigma}\Bigr)},$$

where $p_0(a)$ is the unnormalized log-normal density and $\Phi$ is the standard normal cumulative. The polydispersity index $$\alpha = \sqrt{\operatorname{Var}(a)}/\langle a \rangle$$ up to $\alpha \approx 0.3$ controls the increase in the maximum flowable (jamming) volume fraction $\phi_m$. For all polydispersities, reduced flow curves $\eta_r = (1 - \phi/\phi_m)^{-2}$ and scaled normal stresses collapse onto the monodisperse master curves when $\phi$ is replaced by $\phi/\phi_m$. Even a carefully chosen bidisperse mixture with the first three moments matched reproduces the log-normal system’s rheology, emphasizing the dominance of low-order moments and $\phi_m$ (Pednekar et al., 2017).

7. Advantages, Limitations, and Domains of Applicability

Advantages

• The log-normal spectrum is uniquely justified by information-theoretic (maximum entropy) arguments with minimal constraints, requiring only three parameters ($G_0$, $\tau_0$, $\sigma$) that possess direct physical interpretations.
• It always yields finite first and second moments, ensuring true terminal flow and exclusion of unphysical power-law divergences.
• It flexibly interpolates between monodisperse (Maxwell) and highly polydisperse (broad) relaxation times or particle sizes, with parameters easily estimated from data (Uneyama, 2 Sep 2025).

Limitations

• The spectrum is necessarily symmetric in $\ln\tau$, so genuine asymmetry in the experimental distribution (e.g., multiple peaks or skewness) cannot be captured without superposing multiple log-normals or adding further constraints.
• Strict power-law tails or gel-like criticality in $G(t) \sim t^{-\alpha}$ are not reproducible except as broad approximations.
• Critical-gel viscoelastic behavior at low frequency, characteristic of some materials, is not embodied except as an asymptotic case where $\sigma \gg 1$ (Uneyama, 2 Sep 2025).

Applicability

• Suitable when only the mean and variance of the relaxation time or particle-size distribution are constrained and a single, symmetric peak is observed in $\ln\omega$ or $\ln a$.
• Appropriate as a baseline (maximally unbiased) model; more elaborate spectral features (asymmetry, multimodality) should be incorporated only if empirically required.
• Enables universal scaling of rheological data in polydisperse suspensions once the appropriate $\phi_m$ is accounted for, permitting direct comparison across distributions (Uneyama, 2 Sep 2025, Pednekar et al., 2017).

The log-normal spectrum thus provides a theoretically and practically robust baseline for the parametrization and interpretation of both viscoelastic relaxation and particulate suspension rheologies, grounded in information theory, statistical geometry, and empirical agreement with broad classes of experimental systems.