Controlling the Size of Nanoparticles Using a Magnetic Field: A Sphere Packing Approach

Abstract

We present an analytical framework that predicts and controls nanoparticle size through external magnetic fields, uniting first-principles thermodynamics with a sphere packing approach. Calibrated to diamagnetic silver nanoparticles (20 nm at zero field and 5 nm at 250 mT), the model yields a closed-form relation between radius and field that reproduces the observed shift in most-probable size. Within the limits of classical capillarity and spherical demagnetization, the field lowers the nucleation barrier and drives the distribution toward smaller particles. Our results are robust for radii above 3 nm (5740 atoms). Below this scale non-extensive effects likely dominate, as discussed in detail in Supplementary Information. The approach generalizes to both diamagnetic and paramagnetic systems and the limitations expected for very small or ferromagnetically ordered nanoparticles are discussed.

Figures (7)

• Figure 1: (a) Sketch of the Free Energy landscape (b) The formation of NPs (c) Typical NPs radii distribution based on the experiments done by Kthiri et al kthiri2021novel.
• Figure 2: (a) Sphere packing for $r/a=2$ (left) and $r/a = 35$. The $r/a = 2$ diagram is exact, while the diagram $r/a = 35$ is an illustration, obtained by randomly distributing the atoms inside the NP. (b) A comparison between the optimal packing fraction, an analytical approximation of packomania's datapack and Eq. \ref{['eq:n']}. (c) A residuals plot comparing the performance of all fits.
• Figure 3: (a) A heatmap demonstrating the impact of the orientation angle distribution on how the radius transforms under a magnetic field with $\sigma = 0$ rad (deterministic orientation). (b) The field-radius relation from Eq. \ref{['eq:3']} with the respective work of formation barrier height relative to the highest experimental point at $r = 20$ nm, $\mathcal{B}=0$ mT. The normalized Free Energy Barrier Height is $\frac{\Delta\mathcal{F}}{\Delta\mathcal{F}_{max}}$ where $\Delta\mathcal{F}_{max}$ is the Work of Formation from Eq. \ref{['eq:1']} corresponding to the $r=20$ nm, $\mathcal{B}=0$ mT case (c) Comparison between theory and experiments for AgNPs. The histograms are made with experimental data from Kthiri et al kthiri2021novel. The red solid curve contains the two anchor data points from which parameters are extracted, namely $(r_1, \mathcal{B}_1=0$ mT) and $(r_2, \mathcal{B}_2=250$ mT). The solid curves are calculated with $\sigma = 0$ rad (deterministic orientation) and the dashed with $\sigma=1.2$ rad (Gaussian-averaged orientation). The green curves are for starting points beyond the $\mathcal{B} = 0$ mT histogram (d) The uncertainty bands for the central and extreme experimental points for $E_s$ (main) and $\Delta\mu$ (inset). The bands are estimated based on a $\pm20\%$ deviation from the fit values obtained via Eqs. \ref{['Es']} and \ref{['delta_mu']}.
• Figure S1: (a) The change in the average of the angle factor with respect to $\sigma$ (b) The effect of the orientation factor on the field-radius relationship.
• Figure S2: (a) A demonstration diagram of a NP with a perfectly packed core and an imperfectly packed surface of thickness $\delta$ (b) The effect of the surface thickness $\delta$ on the packing fraction $\phi_d$