Modeling inter-particle magnetic correlations in magnetite nanoparticle assemblies using x-ray magnetic scattering data

Magnetic nanoparticles are increasingly used in nanotechnologies and biomedical applications, such as drug targeting, MRI, bio-separation. Magnetite (Fe3O4) nanoparticles stand to be effective in these roles due to the non-toxic nature of magnetite and its ease of manufacture. To be more effective in these applications, a greater understanding of the magnetic behavior of a collection of magnetite nanoparticles is needed. This research seeks to discover the local magnetic ordering of ensembles of magnetite nanoparticles occurring under various external fields. To complete this study, we use x-ray resonant magnetic scattering (XRMS). Here we discuss the modeling of the magnetic scattering data using a one-dimensional chain of nanoparticles with a mix of ferromagnetic, anti-ferromagnetic, and random orders. By fitting the model to the experimental data, we extracted information about the magnetic correlations in the nanoparticle assembly. © 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5080155


Contents
Introduction and Background

Motivation
Magnetic nanoparticles (NPs) are quickly becoming an integral part of many industries such as biomedicine, computer hardware, and even the automotive industry [1]. The biomedical industry is exploring the use of magnetic NPs in various applications such as MRI contrasting agents, but also for additional use in drug targeting, hyperthermia, and bio-separation [2][3][4]. Also, computer systems rely on data storage and magnetic NPs could provide a better way to manufacture long term high density magnetic storage media [5]. Additionally, the engineers of suspension systems in high performance vehicles are also exploring the use of magnetic NPs to dynamically alter the viscosity of shock absorber dampening fluids [6].
Magnetite (Fe 3 O 4 ) is a commonly occurring ferromagnetic iron oxide found in many minerals, which makes it an ideal candidate for the applications mentioned above because of its non-toxic nature and how easy it is to manufacture at scale. Successfully optimizing the use of magnetite NPs for use these applications, and many others, could lead to breakthroughs in cancer diagnosis and treatment, faster computers, and safer vehicles. However, in order to accomplish this we need 1

Prior Work
a greater understanding of how assemblies of magnetite NPs behave while exposed to external magnetic fields.
The properties of bulk materials are largely determined by the arrangement of the atoms that make up their composition. Because of the greater ratio of surface area to volume, NPs potentially exhibit properties that differ from the material in bulk form. We are primarily concerned with the properties that express on the nanoscale. In particular, the particle size, how easily NPs self-assemble into regular arrangements, and how the nanospin moments behave.
The behavior of nanospin moments gives rise to a unique magnetic phase called superparamagnetism which is the tendency of sufficiently small particles to randomly flip nanospin orientation due to thermal activation. However, if the temperature is small there is not enough thermal energy for the random flip to occur within typically measured time frames and the particle is considered magnetically blocked. The temperature at which this occurs is the blocking temperature (T B ). The existing magnetic correlations and the dynamics of magnetic fluctuation are some of the characteristics traits of NP assemblies that needs to be understood. To this end, we are employing several methods of probing the characteristics of NPs to determine the necessary information for improving the application of magnetite NPs.

Prior Work
Magnetite's bulk properties are well established [7]. Because of the rising interest in the material in its nanoparticle form, recent research has focused on probing the magnetic properties of collections of NP, which may be in the form of a powder or thin monolayered assembly [8]. The particular samples we examine in this work were fabricated at Brigham Young University (BYU) using an organic solution method. The characterization of these samples begins with x-ray diffraction (XRD).
The XRD measurements reveal that the spectrum of individual particles is consistent with the crystal structure of bulk magnetite [8], as shown in Figure 2.1. Additionally, transmission electron microscopy (TEM) images were used to determine the distribution of particle sizes present in the samples and example images are also shown in Figure 2.1. The collective magnetic behavior of the samples was determined using vibrating sample magnetometry (VSM). The magnetization loops measured at various temperatures showed a smooth Langevin-type shape, characteristic of superparamagnetic behavior at high temperatures [9]. Zero-Field-Cooling (ZFC) and Field Cooling (FC) curves also indicated superparamagnetic behavior with blocking temperatures that are directly correlated to the size of NPs [10]. The prior work done on this topic suggests that for larger particles there may be inter-particle coupling at low external field values that could cause interesting magnetic ordering within the ensembles of NPs. To determine the orbital and spin contributions to the magnetic moment of our Fe 3 O 4 NPs we used x-ray magnetic circular dichroism (XMCD). Furthermore, to probe the nanoscale magnetic correlations we used x-ray resonant magnetic scattering (XRMS)

Statement Problem and Thesis
A more detailed accounting of these characteristics, specifically for the 5 and 11 nm Fe 3 O 4 NPs examined by this work, is reported in this thesis. In particular, what we thoroughly explored is the behavior of magnetite NP assemblies when exposed to the external magnetic fields that the applications mentioned above potentially require. This work seeks to discover and characterize the local magnetic ordering of magnetite NP assemblies using XRMS data collected at synchrotron facilities and a modeling process in both real and reciprocal space.

Characterization of Samples
This work examines magnetite NP assemblies of two particle diameters, one of 5 nm and another of 11 nm. These samples were prepared in collaboration with Dr. Roger Harrison of the Brigham Young University Chemistry Department. The exact procedure used to manufacture the NPs can be found here [8,10,11]. As mentioned above the NP characteristics were previously determined in our lab on campus via XRD, TEM, and VSM. The TEM imaging and ZFC/FC results are of particular interest here and later informed the parameters used in the modeling processes including the temperature and field values the data was collected at. The TEM images, shown in Fig. 2.1, show that the 5 nm diameter NP assemblies are more homogeneous in size than the 11 nm NP assemblies.
This homogeneity leads to more closely packed arrangements (as seen on the TEM images). The  NPs. When below the blocking temperature, the particles remain magnetically frozen and do not have enough thermal energy for the nanospin moment to flip in a typical observation timescale [10].
The VSM magnetization loops define the saturation and coercive points for each sample, and the hysteresis exhibited at low applied field values gives an indication of some manner of inter particle coupling. This coupling could result in interesting magnetic ordering in the samples. The following

XMCD and XRMS Diffraction Images
In addition to the magnetic characterization detailed above, the bulk of the data used in this work is comprised of x-ray resonant magnetic scattering (XMRS) diffraction images. These XRMS images were collected at opposite helicities of circular polarized light whose energy was tuned to the L 3 resonant edge of Fe for magneto-optical contrast. These data were collected at the Stanford Synchrotron Radiation Lightsource with an optical setup detailed here [12]. When the x-ray energy is tuned to a resonant edge it provides necessary magneto-optical contrast to provide magnetic scattering information. Additionally, the use of partially coherent x-ray light leads to coherent about the local magnetic ordering as a function of temperature and applied field for a given particle size. Figure 2.3 shows example diffraction images and the resulting speckle pattern. However, in this thesis work, we did not focus on the speckle patterns themselves, but instead we utilized signals constructed from the images in a manner detailed below.

Preparation of Data
We use the XRMS speckle patterns to construct 1D signals by first integrating them azimuthally and then examining the changes between data collected at opposite helicities of the incident x-ray light.
To reduce the 2D diffraction images we angularly integrate around the center of the diffraction ring to project the signal into 1D. To separate the charge scattering signal from the magnetic scattering   On the other hand, empirical models make no attempt to use physical laws to explain phenomena, but instead only fit observed data and utilize extrapolation or interpolation to predict outcomes. This work examines the behavior of magnetite NP assemblies through both a mechanistic simulation of particle assemblies in real space and through an empirical fitting process of the XRMS data collected in reciprocal space.
As mentioned in section 2.2 the number of particles illuminated by the x-ray beam is sufficiently large (in the order of millions) to make direct simulation computationally impossible. In order to model the magnetic behavior of the NP assemblies we make two main simplifications so the problem becomes tractable. The first assumption is that the physics governing the NP interactions can largely be captured by using a 1D representation of the NP assembly as a chain of spherical particles.
Given the high number of NPs illuminated, which averages the multiple lattice orientations, the 2D magnetic correlations function is nearly isotropic and its radial part can therefore be represented by a 1D model. Second, we assume that the lengths associated to the various magnetic correlations die off rapidly enough to ensure that these chains can be finite in length. We account for the variance in particle spacing and diameter present in the samples by averaging multiple chain models together before Fourier transforming the resulting charge density function and comparing the result to the data that was shown in section 2.2.1.

1D Nanoparticle Chains
We begin the modeling process by constructing 1D chains of spherical NPs using a particle diameter and spacing estimated from the distributions measured from the TEM images of the samples shown in we project spheres of the given diameters and spacing into a 1D CDF as illustrated in Figure 3.2a.
We then construct and average ten additional chains, each with values uniformly sampled from the observed distribution of particle diameters and spacing. The weighted summation of this collection of 1D chains serves as the representation of the NP assembly in real space. The resulting average CDF is then a function of position r and fitting parameters, θ θ θ , and takes the form where φ i is the 1D chain for a given diameter and spacing. P(i) is the probability of that chain occurring based on the observed distribution of particle size and spacing measured from the TEM images discussed in Section 2.1. θ θ θ is a vector of parameters used in the model and fitting methods described in chapter 4.
From the chains used to construct the CDF we create "magnetic density functions" (MDF) representing various magnetic orders, including ferromagnetic (FM) and anti-ferromagnetic ordering   The resulting MDF for the FM and AF ordering are respectively:   Table 3.1. The Fourier transforms of C and M,C(θ θ θ ; q) andM(θ θ θ ; q) respectively, are then used to construct the charge intensity, I c and R m signals ready for fitting to the data as follows (derivation in Appendix A): where Re and Im are the real and imaginary parts of complex quantities, and α and β are the real and imaginary components of the complex index of refraction for magnetite.

Empirical Gaussian Fit
In addition to the real space model discussed above, we also make use of an empirical model composed of a sequence of Gaussian packets to fit the signals I c and R m directly to the reciprocal space scattering data and extract values for the correlations and relative amounts of FM and AF ordering present in the samples. This method relies on an examination of the features present in the q-space charge scattering signal I c (q) and magnetic ratio signal R m (q). For each sample, both the I c (q) and R m (q) signal exhibit a primary peak located near q * . In the case of I c (q) this peak corresponds to the average inter-particle distance, and for R m (q), this peak corresponds to a FM ordering as every particle is aligned with its neighbors. In the case of I c (q), the position of the peak does not noteably shift at all with the applied field. Because of this, we first use a simple fit of the charge scattering signal to find the value of q * which then determines the initial value of the FM peak position parameter. Another region of interest in the R m signal is located around q * /2. This is Parameter Description Constraints of note because half the distance of q * in reciprocal space translates into to a period in real space that is twice as long as the average inter particle spacing and which therefore corresponds to AF ordering. To model the R m (q) data we then sum the Gaussian functions for FM, AF and random order with a baseline shift as follows: where the variables are described in detail in Table 3.2

Levenberg-Marquardt
The Levenberg-Marquardt (LM) algorithm [14,15] is a robust and efficient method for solving non-linear least squares problems and is generally considered to be the best starting point for non-linear parameter optimization. Broadly speaking the LM algorithm takes best elements of the Gauss-Newton and gradient descent methods by interpolating between the two through the use of a damping coefficient. This work uses the implementation of the LM algorithm found in the LMfit python library [16] 3.

Basin Hopping
One of the significant problems we encountered during the search for best fit parameters was with the LM algorithm falling into local minima rather than the ideal global minimum that we were looking for. The reciprocal space model was particularly vulnerable to local minima because the summation of the variables used often meant there were different combinations of parameter values that yielded the same result. The constraints we imposed on the parameters helped to alleviate this.
But we needed to employ a basin hopping algorithm, on top of the LM fitting process, to jostle fit results out of flat or otherwise shallow anomalies in the profile likelihoods for our parameters of interest.
The basin hopping algorithm [17] works by taking the last best fit results and applying a small random perturbation to the parameter values. These new perturbed values are then used as the initial values for a subsequent run the LM fitting process. We used the implementation of basin hopping found in the LMfit python library [16] with the number of iterations set at 200 hops. We found that this value was sufficient to arrive at the global minimum within the constraints.

Results and Analysis
The great British statistician, George Box, is often attributed with the saying "all models are wrong, but some are useful" [18]. This aphorism points to one of the core concerns at the heart of any modeling process: how reliable are a model's results? Here we examine the results of applying the two models described above and the methods we used to analyze their reliability and quantify their error. We first examine the Fisher information matrix (FIM) associated with the model for a given data point and, when applicable, construct profile likelihoods for our parameters of interest.

Fisher Information Matrix
The Fisher Information Matrix (FIM) formalism is a method intended to determine how well the model parameters are constrained by the data. A more in-depth discussion of the method and Fisher information in general can be found here [19,20]. The essence of the technique is

Profile Likelihood
The FIM method of determining the error and confidence in the results was generally sufficient for the nano-chain model, but due to the distinct non-linearity of the empirical fitting model we utilized the profile likelihood method to quantify the error ranges for the ratio of AF to FM ordering present in the samples. The profile likelihood method uses the f-test to compare an alternate model to the best fit result and determine to a given confidence value how much of the difference between the two models is the result of a significant change in parameters and not just due to the loss of a degree of freedom.
where χ 2 0 is the residual of the best fit, and χ 2 f the residual of a test model. Additionally, P f ix is the number of fixed parameters in the alternate model, N the number of data points, and P the number of parameters in the best fit model. In our case, for the construction of the profile likelihood we fixed one parameter in the test models, and there are N = 152 data points and P = 9 parameters in the best fit model, leading to F(1, 143) = 3.907. Using this value to solve the above equation for χ 2 f gives us all the models whose parameter result falls within the 95% confidence interval.

Model Results
This work applies the nano chain model to the NP11 sample and uses the FIM to calculate the standard error in the parameters of interest. We also examine how reducible the nanochain model is and what constraints the data imposes upon the transformation from parameter space to data space using the methods described above. We apply the Gaussian empirical fit to both NP11 (at 20 K, 280 K) and NP5 (at 15 K, 300 K) samples and construct profile likelihood to estimate the confidence intervals.

Nanochain Model
The nanochain model is a mechanistic model and is constructed such that each of the parameters has interpreted meaning. Our evaluation of the FIM eigensystem revealed that four of the original 12 parameters of the nanochain model are poorly constrained by the data.

Empirical Gaussian Fit
We applied the Gaussian fit model to data from both the NP11 and NP5 samples, and found that the

Chapter 5 Conclusion
In this work we have established the validity of computational modeling techniques to probe interparticle magnetic correlations in assemblies of magnetic NPs using XRMS data. Our modeling of the 11 nm and 5 nm NP assemblies at temperatures above and below their respective blocking temperatures (T B ), shows that when the applied field (H) approaches remanence, the nanospin carried by the individual NPs tends to be randomly oriented though a non-negliable AF contribution is present in the samples at temperatures below the blocking temperature. This behavior is consistent across both the real space nanochain and empirical q-space fitting models and confirmed the superparamagnetic behavior of the material at T > T B . Additionally, it appears that the AF correlations are stronger for the 11 nm NPs compared to the 5 nm with the former's AF/FM ratio 3 times larger.
Further work expanding the nanochain model from a 1D representation, to a 2D model to fit the original 2D speckle patterns should provide greater resolution and certainty in the magnetic ordering and correlations parameters. These methods could also be applied to other magnetic NP assemblies to study the dependence of magnetic correlation on the particle size and concentration during the self-assembly process.
A c and A m . The scattering intensity I observed at the detector is related to the scattering amplitude through The scattering intensity is comprised of a pure charge term, |A c | 2 , a pure magnetic term, |A m | 2 , and a cross term whose sign is dependent on the polarization of incident light. We can exploit the difference polarization has on the scattering to extract the magnetic contribution using a quantity we call "magnetic ratio R m " (modified from the standard dichroic ratio) R m = I + − I − √ I + + I − Where I + and I − are the intensities associated with their respective helicities. The magnetic component A m is typically small with respect to A c which reduces the dichroic ratio to: The numerator can be further reduced but breaking the complex quantities A c and A m into their real and imaginary components. The results of the x-ray absorption spectroscopy (XAS) collected on the magnetite nanoparticles shows that at the energy chosen for our XRMS measurements (the third peak of our observed L 3 -Fe