Investigating Magnetic Nanoparticle–Induced Field Inhomogeneity via Monte Carlo Simulation and NMR Spectroscopy

Abstract

Magnetic nanoparticles (MNPs) perturb magnetic field homogeneity, influencing transverse relaxation and the full width at half maximum (FWHM) of nuclear magnetic resonance (NMR) spectra. In Nuclear Magnetic Resonance (NMR), this appears as decay of the free induction decay (FID) signal, whose relaxation rate determines spectral FWHM. In D₂O containing MNPs, both nanoparticles and solvent molecules undergo Brownian motion and diffusion. Under a vertical main field ($B_0$), MNPs respond to their magnetization behavior, evolving toward a dynamic steady state in which the time-averaged distribution of local field fluctuations remains stable. The resulting spatial magnetic field can thus characterize field homogeneity. Within this framework, Monte Carlo simulations of spatial field distributions approximate the dynamic environment experienced by nuclear spins. NMR experiments confirm that increasing MNP concentration and particle size significantly broadens FWHM, while stronger $B_0$ enhances sensitivity to MNP-induced inhomogeneities.

1. Introduction

MNPs (magnetic nanoparticles) are rapidly developing materials with significant application potential, particularly due to their exceptional physical properties, such as nanoscale size and superparamagnetic characteristics. These properties have led to their extensive use in biomedicine [1], MRI (magnetic resonance imaging) [2], magnetic fluids [3], and data storage [4]. In NMR (Nuclear Magnetic Resonance), MNPs have emerged as effective contrast agents [5], enhancing imaging quality by altering tissue relaxation times. Specifically, iron oxide magnetic nanoparticles create localized magnetic field inhomogeneities, accelerating spin dephasing in tissues and reducing $T_2$ relaxation time. This reduction in $T_2$ time results in darker appearances of the target regions in MRI images, thereby improving contrast [5,6]. Additionally, in medicine and pathology, FWHM (full width at half maximum) and chemical shift parameters in NMR spectroscopy provide insights into tissue properties and lesions [7,8].

In practice, MNPs are coated with surfactants and uniformly dispersed in a base solution, forming a stable colloidal suspension. When the main magnetic field is applied, the magnetized MNPs in the solution tend to form clusters, thereby altering the surrounding magnetic field distribution and creating static spatial field distributions that induce local magnetic field inhomogeneities. This inhomogeneity is reflected in MRS (magnetic resonance spectroscopy) as a broadening of the FWHM. The FWHM of the spectrum is positively correlated with the transverse relaxation rate $r_2$ of deuteron. In the outersphere motional averaging regime (MAR) theory [9,10], a single MNP shortens the transverse relaxation time $T_2$ of deuteron. Changes in the physical properties of MNPs (size [11], shape [12,13,14,15,16], composition [17,18,19], crystal structure [20,21], surface properties [22,23,24,25]) influence the proton relaxation process, and MNPs can thus be utilized in controlling this process.

Previous studies have focused on the effect of MNPs on the spatial magnetic field under different conditions. For instance, Yuri Matsumoto found that relaxation in compact clusters is similar to that in individual MNPs with comparable particle size and cluster diameter [26]. Gang Liu and other scholars pointed out that the polymerization of MNPs will reduce the symmetry of the magnetic field, thus leading to an inhomogeneous magnetic field in space, based on which changing the size and shape of the MNPs will further exacerbate the inhomogeneity of the spatial magnetic field [27].

Recent studies have employed Monte Carlo and kinetic Monte Carlo simulations to capture the collective behavior of MNP ensembles. For example, Martin et al. modeled iron oxide nanoparticle assemblies while accounting for size dispersion, anisotropy, and dipolar interactions, demonstrating how these factors critically modulate local field inhomogeneities [28]. Likewise, Papadopoulos et al. applied kinetic Monte Carlo methods to optimize nanoparticle design for biomedical hyperthermia, highlighting the importance of dynamic clustering and time-dependent field perturbations [29]. These works illustrate that advanced simulation frameworks can provide microscopic insights into field distributions that directly impact NMR observables. In parallel, recent NMR investigations have quantitatively linked MNP-induced clustering and temperature effects to relaxation enhancements. For instance, Bok et al. experimentally observed that transverse relaxation strongly depends on nanoparticle spacing and aggregation state, providing direct evidence of the role of mesoscopic structure in FWHM broadening [30]. Such results underscore that both simulations and experiments converge on the idea that MNP-induced field inhomogeneities are highly sensitive to particle size, concentration, and magnetic field strength.

Current studies have described the effects of MNPs under different main magnetic fields, but a systematic framework that directly links simulation-based field inhomogeneity to experimental spectral broadening remains limited. Addressing this gap forms the central motivation of our present study. In pulsed NMR, such inhomogeneity influences the decay of the FID (free induction decay) signal, and the corresponding decay rate determines the FWHM in the frequency domain [31]. From the perspective of Bloch theory, this FID reflects transverse relaxation, and the broadening of the FWHM caused by MNPs is analogous to that produced by an inhomogeneous main magnetic field [32].

In this work, we extend our earlier microstructure-based studies of magnetic fluids by establishing a quantitative link between magnetic field inhomogeneity and spectral broadening in NMR. Specifically, we demonstrate a linear relationship between FWHM and the degree of field inhomogeneity, thereby connecting the time-domain relaxation process to its frequency-domain manifestation. The system evolves toward a statistical dynamic equilibrium, where the probability distribution of local field perturbations remains stable. Under these conditions, the spatial magnetic field distributions obtained from Monte Carlo simulations can approximate the dynamic field sampling experienced by nuclear spins. To characterize the variability in magnetic field strength, we combine point-dipole-based Monte Carlo simulations with magnetic resonance experiments, with the field inhomogeneity represented by Lorentzian-like histograms. This integrated simulation–experiment framework allows us to systematically investigate how MNP concentration, particle size, and main magnetic field strength influence spatial field uniformity and the observed NMR spectra. Importantly, the present study also explores two main field strengths (1.41 T and 1.88 T), extending beyond earlier works that typically considered only single-field conditions.

2. Materials and Methods

In pulsed NMR, the recorded signal is the FID, whose decay rate is determined by the effective transverse relaxation time $T_2^{\ast }$. The corresponding linewidth in the frequency domain is given by the FWHM, which for a single-exponential decay takes a Lorentzian. According to Bloch theory, magnetic field inhomogeneity accelerates transverse relaxation and thereby broadens the FWHM. In dispersions containing MNPs, local field perturbations vary with particle size, concentration, and magnetization, resulting in resonance frequency dispersion and spectral broadening. Although both nanoparticles and solvent molecules undergo Brownian motion and diffusion, the system reaches a statistical dynamic equilibrium in which the time-averaged distribution of field fluctuations remains stable over the FID timescale. Under this assumption, Monte Carlo simulations based on the point-dipole approximation can approximate the dynamic field sampling experienced by nuclear spins. The inhomogeneity is quantified by histograms of local field distributions, which follow a Lorentzian line shape consistent with experimental FWHM measurements. This framework allows us to systematically link concentration, particle size, and magnetic field strength to spectral broadening observed in NMR.

2.1. Monte Carlo Simulation Based on Magnetic Charge Theory

This paper uses a two-dimensional Monte Carlo simulation based on magnetic charge theory to model the spatial distribution of magnetic fields. A single MNP is treated as a magnetic dipole. Due to its nanoscale size, it is assumed to have a single-domain structure, meaning that in the absence of a main magnetic field, the magnetic moment of the entire particle remains constant in both magnitude and direction. In applications, MNPs are encapsulated by surfactants and dispersed in a base liquid. The repulsive potential energy between particles, $u_{ij}^v$, arises from the overlap of surfactant molecular layers. When a magnetic field $H_0$ is applied, the MNPs become magnetized, resulting in a magnetic field interaction potential energy $u_i^H$. Additionally, there is a magnetic dipole interaction potential energy between the MNPs, denoted as $u_{ij}^m$. Assuming the MNPs in the base fluid are primarily influenced by these three energies under a uniform magnetic field $H_0$, the energy of each particle $i$ and $j$ can be expressed as the sum of the aforementioned three potential energies. To describe the motion of MNPs using state transition probabilities, the Metropolis sampling algorithm is employed to simplify computations. If the current state of the system is $x\left(n\right)$, and after a change, the new state becomes $x(n+1)$, with the corresponding energy transitioning from $E\left(n\right)$ to $E(n+1)$, the change in energy is given by $\mathsf{\Delta }E=E(n+1)-E\left(n\right)$. In such a system, changes typically occur in the direction of decreasing entropy, or equivalently, decreasing energy. When the system undergoes a state change, if the energy decreases (i.e., $\mathsf{\Delta }E<0$)), the state change is deemed acceptable. If the energy remains the same or increases (i.e., ($\mathsf{\Delta }E\ge 0$)), the acceptance of the state change is determined probabilistically. Reasonable state changes will be accepted. Otherwise, the state change will be rejected and the system will continue to evaluate the next potential state change. This process will repeat accordingly. It should be emphasized that the present simulations are restricted to two-dimensional modeling. The rationale for adopting a two-dimensional framework, as well as its implications and limitations, is explicitly discussed in the Conclusions.

2.2. Instruments and Samples Used in the Experiment

Water-dispersible Fe₃O₄ nanoparticles (SHP series, Ocean Nanotech, San Diego, CA, USA) with diameters ranging from 5–30 nm (SHP-5 to SHP-30) were used. Samples were dispersed in deuterium oxide (D₂O) to minimize proton background. Magnetization data were acquired using a SQUID-based MPMS (Quantum Design) to obtain M(H) and ZFC/FC curves. NMR spectra were measured on benchtop Spinsolve 60 and Spinsolve 80 spectrometers (Magritek), corresponding to $B_0$ = 1.41 T and 1.88 T, respectively. The use of identical instruments ensured consistency across experiments. Since the gyromagnetic ratio of deuterons is fixed, the $B_1$ field was constant across measurements.

2.3. Experimental Design

The experimental design included three components: (i) magnetization measurements, (ii) size- and concentration-dependent NMR studies, and (iii) field-strength-dependent NMR studies.

(i) Magnetization was characterized by M(H) hysteresis loops (−2 T to 2 T), zero-field-cooled (ZFC) and field-cooled (FC) curves (280–320 K), and temperature-dependent magnetization M(T) measurements, all performed using the MPMS.

(ii) NMR spectra were recorded for nanoparticles of different diameters (5–30 nm, concentration 0.025 mg/mL) and for 5 nm particles at concentrations from 0.01–0.25 mg/mL.

(iii) To probe field-strength effects, spectra were acquired on Spinsolve 60 and 80 instruments, and FWHM and chemical shift values were converted to Hz for comparability.

This systematic approach allowed us to investigate the role of size, concentration, and external field strength on spectral broadening and chemical shifts.

3. Results and Discussion

To ensure reproducibility and transparency of the experimental results, detailed datasets are provided in the Appendices. Appendix A presents data for various samples and concentrations of MNPs obtained using the Spinsolve 60 spectrometer, while Appendix B contains corresponding data acquired on the Spinsolve 80 system.

This section systematically investigates the influence of MNP properties on magnetic field homogeneity by directly comparing Monte Carlo (MC) simulation results with experimental NMR data. Our central hypothesis is that the NMR spectral linewidth broadening (FWHM), a measure of transverse relaxation, is quantitatively linked to the width of the MNP-induced local magnetic field distribution as calculated by our simulations.

The MRS of pure deuterated water without MNPs typically exhibits a narrow and symmetrical Lorentzian line shape, reflecting a relatively uniform magnetic field and a slow decay of the FID signal. The introduction of MNPs disrupts the uniformity of the local magnetic field, leading to a faster loss of transverse coherence in the FID and, consequently, a pronounced broadening of the FWHM in the frequency domain. In Bloch’s relaxation framework, this broadening results from inhomogeneous local fields that cause spins at different spatial locations to precess at slightly different Larmor frequencies, producing frequency dispersion.

MNPs generate position-dependent induced magnetic fields, the characteristics of which are determined by particle size, concentration, spatial distribution, external magnetic field strength, and ambient temperature. In solution, both MNPs and solvent deuteron undergo rapid Brownian motion and molecular diffusion, but the overall system reaches a statistical dynamic equilibrium. Under this equilibrium, the time-averaged effect of the fluctuating local fields on the spins can be described by the same field distribution that would be obtained from a static spatial snapshot, as in Monte Carlo simulations.

Comparing the magnetic resonance spectra of pure deuterated water and deuterated water solutions containing MNPs reveals a significant increase in FWHM. The measured FWHM can be explained as a combination of the intrinsic FWHM of the deuterons and the broadening effect caused by field inhomogeneities induced by the MNPs. As shown in Figure 1, the total FWHM reflects the sum of these two contributions.

Figure 1 presents the MRS of pure deuterated water and deuterated water containing MNPs. The spectrum of pure D₂O shows a narrow, symmetrical line shape, indicating a relatively uniform magnetic field and slow transverse relaxation. Upon the addition of MNPs, the induced local magnetic fields cause dispersion in the precession frequencies of nuclear spins, accelerating the dephasing of transverse magnetization in the time domain and leading to significant FWHM broadening in the frequency domain. This linewidth enhancement reflects the dynamic influence of local field inhomogeneity on spin relaxation processes.

Since the magnetization strength of MNPs determines the magnitude of the local magnetic field they induce, its variation with temperature and external magnetic field directly influences the degree and dynamics of spatial field inhomogeneity. In the NMR context, changes in magnetization strength alter the time-dependent local field $\mathsf{\Delta }B$ experienced by nuclear spins, affecting the dephasing rate of transverse magnetization in the FID signal. According to Bloch and FID theory, faster dephasing in the time domain manifests as broader linewidths in the frequency domain. Therefore, when evaluating the impact of MNP physical properties on magnetic field inhomogeneity, it is essential to consider not only the static relationship between magnetization and external parameters, but also how these variations drive dynamic relaxation processes. Figure 2 illustrates the dependence of MNP magnetization strength on external field and temperature, providing the basis for linking particle properties to both the magnitude and the temporal behavior of field inhomogeneity.

Figure 2 illustrates the relationship between the magnetization intensity of magnetic nanoparticles and both the external magnetic field and ambient temperature. To better illustrate the relationship between magnetization and the external magnetic field, the inset of Figure 2a provides an enlarged view of the variation in magnetization within the field range of 0.75 T to 2.1 T. DC moment fixed center (Ctr) indicates that the MPMS measures the DC magnetization by using the fixed position method to obtain the magnetic moment at the center position. Due to their superparamagnetic nature, the magnetization behavior of MNPs can be described by Langevin Theory.

The temperature-dependent magnetization measured using the MPMS reveals a reduction in magnetization at elevated temperatures due to enhanced thermal fluctuations disrupting the alignment of nanoparticle magnetic moments. This behavior is consistent with the corresponding decrease in FWHM observed in NMR spectra since weaker magnetization leads to smaller local field inhomogeneities. Furthermore, the slight non-linear deviations observed in the fitted curves may arise from factors such as particle size distribution, inter-particle interactions, and the transition between single- and multi-domain regimes. These effects, although beyond the scope of the simplified Langevin model, reflect the intrinsic complexity of MNP ensembles and account for the minor discrepancies between experimental data and theoretical predictions.

As shown in Figure 2a, in weak magnetic fields, magnetization intensity exhibits an approximately linear dependence on the magnetic field, i.e., $M=\chi H$, where $\chi$ is the magnetic susceptibility. In strong magnetic fields, magnetization gradually approaches saturation, displaying nonlinear growth that follows the Langevin function.

Figure 2b presents the relationship between temperature and magnetization intensity. For superparamagnetic MNPs, increasing temperature enhances thermal fluctuations, disrupting the alignment of magnetic moments and reducing magnetization intensity. These experimental observations will be further validated by subsequent NMR results.

Building on this, Figure 3a quantifies how the local magnetic field inhomogeneity varies with MNP mass concentration for 25 nm particles, over the range 0.01–0.25 mg/mL. The increase in concentration enhances the $\mathsf{\Delta }B$ experienced by nuclear spins, leading to faster transverse magnetization dephasing in the FID signal. In the frequency domain, this manifests as pronounced broadening of the FWHM, consistent with the paramagnetic broadening described by Bloch theory. Notably, the concentration-dependent FWHM measured experimentally agrees well with the width of the $\mathsf{\Delta }B$ distribution obtained from Monte Carlo simulations, establishing a direct quantitative link between simulated field inhomogeneity and spectral broadening.

Figure 3b further correlates these inhomogeneity values with the transverse relaxation rate ($R_2$). The strong positive correlation reflects the dynamic origin of relaxation enhancement: larger $\mathsf{\Delta }B$ amplitudes accelerate FID, which translates into higher R₂ and broader spectral lines. This relationship holds systematically across different concentrations and is further confirmed under both 1.41 T and 1.88 T main magnetic fields, where stronger fields enhance sensitivity to MNP-induced perturbations.

3.1. Effect of MNP Concentration on Field Homogeneity

We first examined the impact of MNP concentration on the magnetic field distribution and the corresponding NMR linewidth.

3.1.1. Simulation of Concentration Effects

This study uses the Monte Carlo method to simulate a two-dimensional system of individual particles. The simulation parameters are initial particle size $d_0$ = 20 nm, number of particles $N$ = 625, magnetic flux density intensity generated by the main magnetic field $B_0$ = 14,100 Gs (1.41 T), and temperature T = 300 K. The particle size follows a lognormal distribution with a mean of $d_0$ and a standard deviation of $\sigma =0.2$. The system boundary length is $L\approx 60 d_0$, the MNP saturation magnetization $M_s=4.168\times {10}^5\mathrm{A}/\mathrm{m}$, and the number density of active agent molecules per unit surface area $n_s$ = ${10}^{18}/{\mathrm{m}}^2$. The thickness of the active agent is $\delta =0.15 d_0$. A periodic boundary condition is applied, allowing particles that move beyond the system boundary to re-enter from the opposite side. The cutoff distance for particle interaction is set to $r_{coff}=10 d_0$.

In the Monte Carlo simulations, we considered N = 625 nanoparticles, which provided a statistically sufficient ensemble size for averaging local field perturbations while keeping the computational load tractable. The simulation grid size of 5500 nm × 5500 nm was chosen to be much larger than the particle diameter, thereby minimizing boundary effects and ensuring that the simulated region represented a bulk-like environment. For the calculation of dipolar interactions, a cutoff distance of 10 d₀ (where d₀ is the particle diameter) was applied, following common practice in nanoparticle simulations to balance computational efficiency and accuracy.

To ensure the reliability of the simulation results, we performed convergence tests by monitoring the stability of the model. The results were found to converge after approximately 16,000 steps. Accordingly, all simulations were carried out with a total of 20,000 steps to guarantee consistent and stable outcomes.

Using the above parameters, we will simulate the effects of different $d_0$ and $H_0$ on spatial magnetic field generation. Particle concentration is controlled by varying the number of particles $N$. Monte Carlo simulations are conducted by varying one variable while keeping all other conditions constant. Once the energy of system stabilizes, several points in the magnetic field space are selected, and the magnetic field strength at each point, resulting from the superposition of fields from MNPs in the cluster, is calculated. The grid size for the simulation results is set to 5500 nm × 5500 nm, with a spacing of 200 nm between grid points. A histogram of the frequency distribution is created by counting the occurrences of each magnetic field strength at the grid points.

To study the effect of different MNP particle sizes on magnetic field generation, the number of MNPs is varied while keeping all other parameters constant. The results are shown in Figure 4.

Figure 4a shows the frequency histograms of the spatial magnetic field distribution for simulations with different numbers of MNPs (N = 225, 400, 625, 900). Figure 4b presents the FWHM of these frequency histograms as a function of the number of MNPs, along with its fitting curve.

3.1.2. Experimental Validation of Concentration Effects

Following the experimental design, magnetic resonance measurements were performed for MNP suspensions with concentrations ranging from 0.01 mg/mL to 0.25 mg/mL, using particles of 25 nm diameter. The resulting spectra are presented in Figure 5, where panel (a) shows representative MRS at different concentrations, panel (b) plots the relationship between concentration and FWHM, and panel (c) illustrates the corresponding variation in chemical shift.

The resulting spectra are presented in Figure 5, where panel (a) shows representative MRS at different concentrations, panel (b) plots the relationship between concentration and FWHM, and panel (c) illustrates the corresponding variation in Chemical shift.

3.1.3. Discussion

Simulation of Concentration Effects shows that as the number (and thus concentration) of MNPs increases, the spatial magnetic field intensity distribution becomes more dispersed. A linear relationship exists between the number of MNPs and the FWHM of the frequency histogram of the spatial magnetic field distribution. This indicates that a higher concentration of MNPs makes the spatial magnetic field more inhomogeneous, leading to a broadening of the FWHM in the NMR magnetic resonance spectrum.

Experimental Validation of Concentration Effects shows that from the time-domain perspective, increasing MNP concentration enhances local magnetic field variations, leading to faster FID signal decay due to a greater spread of spin precession frequencies. In the frequency domain, this effect manifests as a progressive broadening of the FWHM, as shown in Figure 5b. Additionally, the increased field perturbations cause a measurable shift in the resonance frequency, reflected in the concentration-dependent chemical shifts in Figure 5c. These trends are consistently observed across experiments with different particle sizes, confirming that MNP-induced field inhomogeneity is a key driver of both linewidth broadening and resonance frequency shifts.

We believe that the quantitative agreement between the linear trend predicted by the MC simulations and the linear trend observed in the NMR experiments supports our model. This suggests that the dominant mechanism for line broadening in this regime is the increased density of magnetic dipoles, which additively contributes to the overall field inhomogeneity. This corroboration helps establish a link between microscopic particle distribution and macroscopic NMR observables.

3.2. Effect of MNP Particle Size on Field Homogeneity

To investigate the effect of MNPs with different radii on the magnetic field, NMR experiments were conducted on samples with the same mass concentration but varying particle sizes. Samples SHP-05 to SHP-30 correspond to MNPs with diameters ranging from 5 to 30 nm. Because particle count varies at a fixed mass concentration, this study seeks to eliminate its influence. To standardize the number of particles,

Table 1. Translational relationship of magnetic nanoparticle concentration.

Sample	Concentration (mg/mL)	Molar Concentration (nmol/mL)	$\mathbf{Particles}\mathbf{per}\mathbf{Unit}\mathbf{Volume}({\mathbf{N}/\mathbf{m}}^3$)
SHP-05	5	34.5	$207.69\times {10}^{20}$
SHP-10	5	4.3	$25.886\times {10}^{20}$
SHP-15	5	1.35	$8.127\times {10}^{20}$
SHP-20	5	0.55	$3.311\times {10}^{20}$
SHP-25	5	0.29	$1.7458\times {10}^{20}$
SHP-30	5	0.17	$1.0234\times {10}^{20}$

lists the molar concentration and particle count per unit volume for the selected SHP samples; to eliminate the interference caused by variations in particle count among samples, this study aimed to standardize the number of particles. First, for samples SHP-05 to SHP-30, a linear relationship between mass concentration and FWHM was established under identical mass concentration conditions (R² ≥ 0.978). Given the known molar concentration for each sample, this relationship enabled us to derive the mass concentration corresponding to a standardized particle number of 5 × 10¹⁸/$m^3$ for different particle sizes, as shown in Table 1. Since mass concentration and FWHM have a one-to-one correspondence, the FWHM for samples with varying particle sizes under uniform particle count conditions can be determined accordingly. As shown in Table 1.

For an individual MNP, the magnetic moment ($\mu$) is expressed as the product of the saturation magnetization ($M_S$) and the particle volume (V), i.e.,

$$\mu =M_{S}V=M_S{\displaystyle \frac{4}{3}}\pi R^3.$$

(1)

In practice, however, MNPs tend to aggregate into clusters, which enhances the effective magnetic moment through dipolar interactions. This clustering affects the spatial magnetic field distribution and contributes to NMR linewidth broadening. Figure 6 shows the relationship between particle size and FWHM, along with the saturation magnetization for different particle sizes, indicating that both individual MNP magnetization and clustering effects influence the spatial magnetic field distribution.

Figure 6a presents the particle size and corresponding saturation magnetization ($M_s$) of a single MNP, demonstrating a positive correlation between particle size and $M_s$. Figure 6b illustrates the relationship between particle radius and FWHM for MNPs with the same number concentration (${Conc}_N$ = 5 × 10¹⁷/${\mathrm{m}}^3$, 1 × 10¹⁸/${\mathrm{m}}^3$, 5 × 10¹⁸/${\mathrm{m}}^3$). The insets in Figure 6b are included to clearly show the trends at the lower concentrations (5 × 10¹⁷/m³ and 1 × 10¹⁸/m³), where the dependence of FWHM on particle size may not be apparent in the main plot.

Discussion on Concentration

The relationship between FWHM and particle size reveals two distinct behavioral regimes depending on the MNP concentration.

At low concentrations (${Conc}_N$ = 5× 10¹⁷/${\mathrm{m}}^3$), the FWHM increases with particle size in a manner that agrees well ($R^2$ ≥ 0.98) with a model based on the magnetic moment of a single, isolated MNP. This suggests that in this dilute regime, the spatial magnetic field inhomogeneity is primarily determined by well-dispersed, individual particles or small, loosely bound clusters. The corresponding NMR line broadening can be adequately interpreted through standard paramagnetic broadening mechanisms.

Conversely, at higher concentrations (${Conc}_N$ = 5 × 10¹⁸/${\mathrm{m}}^3$), the FWHM grows much more steeply with increasing particle size, causing the fit to the single-particle model to deteriorate significantly. This pronounced deviation strongly suggests that inter-particle interactions and collective magnetic behaviors are becoming dominant. While direct quantitative analysis of clustering was not performed, this behavior is highly indicative of enhanced particle aggregation. The formation of such clusters would create larger magnetic entities with combined dipolar fields, producing a broader distribution of local magnetic fields than would be expected from individual particles alone. This would, in turn, lead to faster transverse relaxation and the observed sharp increase in spectral linewidth.

Furthermore, the magnetic behavior of MNPs is a complex interplay of multiple factors. Smaller, single-domain particles are known to experience strong surface effects that can reduce their saturation magnetization. In contrast, larger particles may transition to a multi-domain state, which also alters their magnetic response. Therefore, the observed NMR line broadening is a convolution of single-particle properties, collective magnetic effects like clustering, and intrinsic size-dependent phenomena.

3.3. Effect of Main Magnetic Field Strength

Finally, we explored the systematic dependence of line broadening on the main magnetic field strength, $B_0$.

3.3.1. Simulation of $B_0$

To study the effects of different main magnetic fields, the strength of $B_0$ was varied while keeping all other parameters constant. The simulation results are presented in Figure 5.

Figure 7a shows the frequency histograms of the spatial magnetic field distribution when the magnetic flux density strength is set $B_0$ = 15,000 to 16,000 Gs. Figure 5b displays the relationship between the FWHM of the frequency distribution histogram and the magnetic flux density strength, along with its fitting curve.

3.3.2. Experimental Validation of $B_0$

To further investigate the effects of different main magnetic fields on the FWHM and chemical shifts in the MRS, experiments were performed using the Spinsolve 80 according to the experimental design, with MNP concentrations ranging from 0.01 to 0.25 mg/mL for 25 nm particles. The results are shown in Figure 8, where panel (a) presents the dependence of FWHM on MNP concentration under different main magnetic field strengths and panel (b) shows the corresponding chemical shifts.

3.3.3. Discussion About the Strength of $B_0$

Figure 7 shows that as the strength of the $B_0$ increases, the frequency histogram of the spatial magnetic field intensity broadens, and the peak of the fitted curve also rises. This implies that different main magnetic field strengths cause different chemical shifts, i.e., changes in resonance frequency in NMR. Additionally, there is a linear relationship between the $B_0$ and the FWHM of the frequency histogram of the spatial magnetic field strength. This indicates that a stronger main magnetic field makes the magnetic field distribution more inhomogeneous, which is reflected in a broader FWHM in the MRS.

From the time-domain perspective, increasing the main magnetic field enhances the magnetization of MNPs in the unsaturated regime, thereby strengthening local magnetic field gradients experienced by the spins. This increases the spread of Larmor precession frequencies and accelerates FID, which manifests in the frequency domain as greater FWHM broadening. As shown in Figure 8a, a stronger main magnetic field leads to a larger increase in FWHM at the same MNP concentration.

Similarly, Figure 8b shows that the resonance frequency shift in deuterium increases with MNP concentration, with a greater shift observed under stronger main magnetic field. This behavior is consistent with the magnetization–field relationship in Figure 2a, where a higher $B_0$ enhances the effective magnetic moment of MNPs, amplifying spatial magnetic field perturbations. Both the FWHM broadening and chemical shift variations can be attributed to the same mechanism: stronger main magnetic fields intensify MNP-induced spatial field inhomogeneity. This effect is observed consistently in both simulations and experiments, highlighting the role of MNP magnetization in modulating local magnetic environments and NMR spectral features.

3.4. Practical Implications and Potential Applications

The present study demonstrates that both the concentration (0.01–0.25 mg/mL) and particle size (5–30 nm) of magnetic nanoparticles critically influence NMR spectral properties, providing insights for practical applications. Specifically, concentrations of 0.05–0.1 mg/mL with particle sizes of 20–30 nm produced the most pronounced spectral broadening, indicating their potential utility as negative contrast agents in MRI or as sensitive probes in magnetic resonance spectroscopy where enhanced responsiveness to local field inhomogeneity is desirable. In contrast, lower concentrations (<0.05 mg/mL) and smaller particles (5–10 nm) are suitable for applications requiring minimal magnetic perturbation, such as long-term in vivo circulation or studies prioritizing narrow linewidths. Furthermore, achieving high T₂ contrast is not simply a matter of increasing concentration, as higher concentrations of larger particles can induce clustering and unpredictable signal loss. Instead, moderate concentrations of mid-sized particles (15–20 nm) offer an optimal balance between relaxation enhancement and colloidal stability. Overall, these findings highlight that careful tailoring of nanoparticle size and concentration provides a practical strategy to modulate relaxation properties and spectral resolution.

4. Conclusions

This study systematically investigates the effect of magnetic nanoparticles (MNPs) on spatial magnetic field uniformity through Monte Carlo simulations and NMR experiments. The spatial distribution of MNPs under external magnetic fields was simulated, and the resulting field inhomogeneity was analyzed via frequency histograms, while experimental measurements of M(H) and M(T) were combined with NMR to assess FWHM and chemical shifts across MNP concentrations (0.01–0.25 mg/mL), particle sizes (5–30 nm), and magnetic flux densities (1.41–1.88 T).

In this study, a two-dimensional (2D) Monte Carlo simulation framework was adopted to model the spatial magnetic field distribution induced by MNPs. Each nanoparticle was treated as a point dipole with its magnetic moment determined by particle size, while the dipolar interactions and external field effects were projected onto a 2D plane. This choice was made primarily to reduce computational complexity and to allow clear statistical characterization of local field fluctuations. Although this approach captures the main statistical features of field inhomogeneity, long-range dipolar interactions and clustering effects may be underestimated, motivating future 3D simulations for more realistic ensembles. In our simulations, the magnetic field inhomogeneity is quantified by the standard deviation ($\mathsf{\Delta }B$) of the field distribution. According to Bloch theory, this $\mathsf{\Delta }B$ directly contributes to the FID rate and thus scales linearly with the FWHM in the frequency domain. While we did not perform an absolute calibration between $\mathsf{\Delta }B$ and FWHM, the consistent trends observed across concentration, particle size, and magnetic field strength demonstrate that the simulated distributions capture the same physical mechanism underlying the experimental spectra.

The results show that MNP-induced field inhomogeneity is closely related to particle properties. Magnetization follows Langevin theory, with FWHM trends reflecting the combined effects of concentration, particle size, temperature, and field strength. Increasing MNP concentration amplifies field inhomogeneity, leading to broader FWHM and larger chemical shifts, with near-linear dependence. At low concentrations, FWHM increases with particle size, consistent with single-particle magnetic moments, whereas at higher concentrations, clustering accelerates FWHM growth and weakens this correlation. Temperature elevation enhances thermal fluctuations, reducing magnetization and narrowing FWHM.

This work confirms that both nanoparticle concentration (0.01–0.25 mg/mL) and size (5–30 nm) are decisive factors governing NMR spectral characteristics, with direct implications for their practical deployment. Our findings indicate that intermediate concentrations (approximately 0.05–0.1 mg/mL) combined with larger particles (20–30 nm) lead to marked spectral broadening, making them attractive candidates for applications where strong T₂-weighted contrast or heightened sensitivity to local field variations is advantageous. Conversely, smaller particles (5–10 nm) at lower concentrations (<0.05 mg/mL) exert minimal influence on the magnetic environment, which may be preferable in cases where maintaining narrow linewidths or reducing magnetic disturbance is critical, such as prolonged in vivo tracking. Importantly, excessively high concentrations of large particles risk inducing aggregation and unstable relaxation behavior, whereas mid-sized particles (15–20 nm) at moderate concentrations provide a more stable compromise between relaxation enhancement and colloidal stability. Taken together, these results underscore the need for tailored selection of nanoparticle size and concentration to balance spectral resolution with functional performance in biomedical and spectroscopic applications.