Universal scaling laws for dynamical-thermal hysteresis

Abstract

Dynamic hysteresis, the rate-dependent lagged response of materials to external fields, underpins applications from energy-efficient transformers to gas storage systems. A fundamental yet unresolved question is how the hysteresis loop area $A$ scales with the field sweep rate $R$. Here, we reveal that a competition between the field sweep and thermal fluctuations governs a universal crossover between two scaling regimes: $A - A_0 \propto R^{1/3}$ for $R < R^*$ and $A - A_0 \propto R^{2/3}$ for $R > R^*$, where $A_0$ is the quasi-static area and the crossover rate $R^* \propto T/T_c$ depends on the temperature $T$ and the material's critical temperature $T_c$. We demonstrate these scaling laws universally across experiments of magnetic materials, simulations of Ising and metal-organic framework models, and analytical solutions of a stochastic Langevin equation. This framework not only resolves the long-standing non-universality of reported scaling exponents but also provides a direct design principle for the application of dynamic hysteresis.

Figures (20)

• Figure 1: Systems and their hysteresis curves. (a) Illustration of the experimental setup. (b) Magnetic induction - magnetic field ($B$-$H$) curves measured in experiments. (c) Illustration of the generalized Ising model with $l=3$ (dashed square). (d) Magnetization - magnetic field ($m$-$H$) curves measured in simulations of the generalized Ising model ($l=13$). (e) Illustration of the MOF model. (f) Fraction of guest particles adsorbed in the host MOF, $N_{\rm ads}/N$, as a function of their chemical potential $\mu$. In (b,d,f), $R$ values are indicated in the figure legend.
• Figure 2: Universal scaling of hysteresis dispersion in experiments and simulations. (a) $(A-A_0)/(A^*-A_0)$ as a function of $R/R^*$, where $A^* \equiv A(R^*)$. We include results from the Langevin equation, MOF models, Ising models and two experimental magnetic materials (nanocrystalline and FeCoV alloy). (b) $R^*/R_0$ as a function of $T/T_c$ (see SI Table \ref{['SItab:Temperature']} for the list of $T$, $T_c$ and other model parameters).
• Figure 3: Langevin dynamics. (a) Dynamical hysteresis by solving Eq. (\ref{['eq:LE2']}) with $T=0$. The solid (dashed) black curve is the quasi-static equation $H=a_2\phi+a_4\phi^3$, and the two solid circles denote the spinodal points. (b) Hysteresis area $A-A_0$ versus $R$ with $T = 0$. (c) Dynamical-thermal hysteresis for Eq. (\ref{['eq:LE2']}) with $T=2$ and its corresponding scaling (d). We set $a_2=-400$, $a_4=948$ and $\lambda=0.0005$.
• Figure 4: Scaling collapse according to Eq. (\ref{['eq:CUS21']}). Data are obtained by numerically solving the Langevin equation Eq. (\ref{['eq:LE2']}), with $a_2=-400$, $a_4=948$ and $\lambda=0.0005$. The fitting parameter $c_0=4$. Solid lines, dashed lines and dotted lines correspond to $R=0.01$, $R=1$ and $R=100$, respectively.
• Figure 5: Histogram of the exponent $\alpha$ reported in the literature (see Table \ref{['tab:coefficient']} for the list).