Effect of albumin mediated clustering on the magnetic behavior of MnFe₂O₄ nanoparticles: experimental and theoretical modeling study

First-principles calculations based on spin-polarized density functional theory were performed for an ultra-small MnFe₂O₄ particle bonded to albumin protein using the Vienna Ab Initio Simulation Package (VASP) [34, 35]. The electronic charge density and the local potential were expressed in plane wave basis sets. Electronic relaxation was performed with accuracy of 10⁻⁴ eV, while the Ionic relaxation with 10⁻³ eV/atom. The Perdew–Burke–Ernzerhof approximation was used to calculate the exchange correlation functional.The interactions between the electrons and ions were described using the projector-augmented-wave method.

The overall three-dimensional structure of bovine serum albumin (BSA), shown by x-ray crystallography, has a heart shape [36]. Structurally, albumin consists of three homologous domains I, II, and III. Each domain contains two sub-domains (A and B), which contains 4 and 6 α-helices, respectively. In our model, the particle is covered partially by sections of the albumin protein (section IIB) at the top and bottom face.

The modeled particle consists of 4 atomic layers of Mn ferrite (inverse spinel), covered by sections of the BSA protein. The relaxed structures are shown in figure 1 [37]. After the ionic relaxation, the size of the Mn ferrite particle is 1.6 nm × 0.7 nm × 0.6 nm. A Mn ion is found to create bond only with O atoms, whereas Fe ions create bonds with O, C and N atoms. A Mn atom shows an increased magnetic moment by ∼0.6 μ_B. Fe atom bonded with C shows a decreased moment by ∼0.05 μ_B. Fe atom bonded with Ν shows a decreased magnetic moment by ∼0.27 μ_B. No systematic behavior has been found for the Fe atoms bonded with O, as some of them show an increased and some of them a decreased magnetic moment with respect to the uncoated case. This can be attributed to the different environment constructed around the magnetic ion.

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As we can see in figure 1(b) the albumin is partially coating the Mn ferrite nanoparticle giving room for the touching between the nanoparticles in the formed clusters as described in section 2.1. The mean value of the magnetic moment of the bonded atoms is increased by 1.8% with respect to the uncoated case. Due to the general expansion and distortion of the cell, neighboring magnetic atoms are found at larger distances and displaced positions with respect to the uncoated case. In this case, the mean magnetic moment of the distorted layer of atoms is decreased by 2.8% and that of the inner atoms is decreased by 1.75%. Overall the mean value the surface spins has an increased magnetic moment of 0.07 μ_Β with respect to the uncoated case. The general distortion affects also the environment of inner atoms leading to a reduction of the mean moment close to 0.17 μ_Β with respect to the uncoated case. In order to estimate the variation of the magnetic anisotropy energy (MAE), the energy of the system is calculated by rotating all spins coherently along different spin axis including LS orbit coupling.

In table 1 the results of the electronic structure calculations of the mean magnetic moment per atom type, the total magnetic moment and MAE energy for the uncoated and the coated with albumin particle are reported.

From table 1 we can see that (a) the coated particle has smaller magnetic moment per atom for each type of atoms resulting to smaller total magnetic moment by 3%. This reduction is further enhanced with increasing number of bonded atoms at the nanoparticle surface according to our calculations and (b) the coated particle has lower MAΕ compared to the uncoated one. The relative difference is approximately 7.5%.

In the uncoated nanoparticles the coordination symmetry is greatly reduced for the metal cations at the surface due to missing of some coordination oxygen atoms. In the coated with albumin nanoparticles, the adsorbed ligands can be considered that they take the positions of the missing oxygen atoms. Even though the surface is spatially distorted in terms of number of neighbors, the surrounding environment tends to recover the bulk phase which makes the crystal field of the surface metal ion more closely resembling that of the bulk material. Importantly, these changes account for the reduction in the anisotropy of the coated sample.

The macroscopic magnetic behavior of the coated assembly of MnFe₂O₄ nanoparticles has been studied and compared with that of the uncoated assembly of MnFe₂O₄. In both cases we take input from the electronic structure calculations for the nanoparticles anisotropy and magnetization.

We model the assembly of MnFe₂O₄ nanoparticles coated with albumin as a group of well separated small clusters of nanoparticles as shown in figure 2.

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The clusters are surrounded by the long albumin molecules so they are not touching each other while the nanoparticles in the clusters can be in physical contact. This model is considered based on the experimental evidence (i.e. high temperature irreversibility of the ZFC-FC magnetization curve) of the formation of clusters when the uncoated MnFe₂O₄ nanoparticles were dispersed in the albumin solution. We consider N ultra-small spherical ferrimagnetic nanoparticles with core/surface morphology that takes into account also surface effects, located randomly on the nodes of a cubic lattice inside a box of 10α × 10α × 10α where α is the smallest inter-particle distance. The clusters of nanoparticles are created by dividing the box into eight regions with size 5α × 5α × 5α each and variable particle concentration (p₁ = 30%, p₂ = 40%, p₃ = 50%, p₄ = 60%, p₅ = 70%, p₆ = 60%, p₇ = 40%, p₈ = 50%) (figure 2) for each cluster, but under the constraint that the total concentration is p = 50% the same as in the case of the simulated uncoated assembly as described in [9].

In our simulations each nanoparticle is described by a set of three classical spin vectors one for the core $\vec{s}$_1i and two for the surface $\vec{s}$_2i and $\vec{s}$_3i with magnetic moment m_n = M_nV_n/M_sV, i = 1, ..., N (total number of particles), n = 1 stands for the core and n = 2, 3 for the 'up' and 'down' surface sublattices of the nanoparticle, respectively. V is the particle volume and M_S its saturation magnetization. V_n and M_n are the volume and the saturation magnetization of the core, the 'up' and the 'down' surface sublattices spins.

The total energy of the system [9] for the N nanoparticles is:

$$\begin{eqnarray}&&\begin{array}{l}E=-\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{N}\left[{J}_{c1}\left({\mathop{s}\limits^{\longrightarrow}}_{1i}\cdot {\mathop{s}\limits^{\longrightarrow}}_{2i}\right)+{J}_{c2}\left({\mathop{s}\limits^{\longrightarrow}}_{1i}\cdot {\mathop{s}\limits^{\longrightarrow}}_{3i}\right)\right.\\ \,\,+\,\left.{J}_{srf}\left({\mathop{s}\limits^{\longrightarrow}}_{2i}\cdot {\mathop{s}\limits^{\longrightarrow}}_{3i}\right)\right]\\ \,\,-\,\displaystyle \sum _{i=1}^{N}{K}_{c}{V}_{1}{\left({\mathop{s}\limits^{\longrightarrow}}_{1i}\cdot {\hat{e}}_{1i}\right)}^{2}-\displaystyle \sum _{i=1}^{N}{K}_{srf}\left[{V}_{2}{\left({\mathop{s}\limits^{\longrightarrow}}_{2i}\cdot {\hat{e}}_{2i}\right)}^{2}\right.\\ \left.\,\,+\,{V}_{3}{\left({\mathop{s}\limits^{\longrightarrow}}_{3i}\cdot {\hat{e}}_{3i}\right)}^{2}\right]\\ \,\,-\,\displaystyle \frac{1}{2}g\displaystyle \sum _{i,j=1,i\ne j}^{N}\left(\displaystyle \sum _{n=1}^{3}{m}_{ni}\cdot {\mathop{s}\limits^{\longrightarrow}}_{ni}\right){D}_{ij}\left(\displaystyle \sum _{n=1}^{3}{m}_{nj}\cdot {\mathop{s}\limits^{\longrightarrow}}_{nj}\right)\\ \,\,-\,\displaystyle \frac{1}{2}{J}_{{\rm{inter}}}\displaystyle \sum _{\langle i,j\rangle }^{\,}\left[\left({\mathop{s}\limits^{\longrightarrow}}_{2i}\cdot {\mathop{s}\limits^{\longrightarrow}}_{3j}\right)+\left({\mathop{s}\limits^{\longrightarrow}}_{3i}\cdot {\mathop{s}\limits^{\longrightarrow}}_{2j}\right)\right]\\ \,\,-\,\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{n=1}^{3}{\mu }_{0}H{m}_{ni}\left({\mathop{s}\limits^{\longrightarrow}}_{ni}\cdot {\hat{e}}_{h}\right).\end{array}\end{eqnarray} \tag{ 1 }$$

The first, second and third energy term in equation (1) in the square brackets describe the nearest-neighbor Heisenberg exchange interaction between the core spin and the two surface spins, and the exchange interaction between the surface spins, respectively. The fourth and the fifth terms give the anisotropy energy for the core and the surface spins (in the square brackets) assumed uniaxial with ê_1i, ê_2i, ê_3i being the anisotropy random easy-axes direction. The sixth term gives the dipolar interactions among all spins in the nanoparticles where the magnetic moments of the three macrospins of each particle are defined as m₁ = M₁V₁/M_sV, m₂ = M₂V₂/M_sV, and m₃ = M₃V₃/M_sV and D_ij is the dipolar interaction tensor. The next term describes the inter-particle exchange interactions between the nanoparticles in contact inside each cluster, where 〈i, j〉 denotes summation over nearest neighbors. The last term is the Zeeman energy (ê_h being the direction of the magnetic field).

The energy parameters of the equation (1), as they are entered in the simulations, have been normalized by the factor 20 × K_C V_1, where V₁ is the core volume of the nanoparticle, so they are dimensionless.

The parameters of the intra-particle exchange coupling constants between the core spin and the surface spins are j_c1 = 0.5, j_c2 = 0.45, and the exchange coupling constant between the surface spins j_srf = −1.0, as were calculated in [9] for uncoated nanoparticles system. Here we set j_srf = −0.8 for the albumin coated clusters of nanoparticles, because our DFT calculations have shown that the bonding with the albumin causes expansion of the neighboring magnetic atoms at the surface of the Mn ferrite nanoparticles and consequently reduction of the effective exchange coupling strength.

The effective anisotropy constants for the core is taken k_C = 0.05 and for the surface is k_srf = 1.0 which is larger than in the core taking into account the reduced surface symmetry [9] for the simulated uncoated sample. For the albumin coated sample, our DFT calculations indicate that the surface anisotropy is reduced due to the existence of the albumin (see section 2.3.1), consequently the effective surface anisotropy constant is taken here k_srf = 0.8, that is 20% reduction from the surface anisotropy of the uncoated nanoparticles. We consider larger reduction of the surface anisotropy comparing with the DFT calculations to include different percentage of the surface coating and higher surface volume since the modeled particle here is of 2 nm size.

There is not exact microscopic model for the calculation of the exchange coupling constants between the nanoparticles. So we treat them as free parameters. Thus we consider the inter-particle exchange coupling constant between the nanoparticles in contact in the cluster j_inter = −0.80 larger than that of the uncoated case (j_inter = −0.50), since the presence of albumin brings closer the trapped nanoparticles in the clusters and enhances the fraction of the shell that comes into contact with the neighboring shells of these nanoparticles. The reduced dipolar strength is taken [9] g = μ₀(M_sV)²/(4πd³20K_cV₁) ~ 3 (where d is the smallest distance between two nanoparticles equal to the particle's diameter d = 2 nm) for both systems.

The saturation magnetization ratios have been extracted from atomic scale calculations for the spinel structure of a 2 nm diameter MnFe₂O₄ nanoparticle, assuming 0.5 nm surface thickness, and they are m₁ = 0.1, m₂ = 0.5, m₃ = 0.4 in case of the uncoated nanoparticles assembly [9]. According to previous experimental results of [27] and our DFT calculations (see section 2.3.1) the surface magnetic moments are reduced due to the existence of albumin so we consider m₂ = 0.45 and m₃ = 0.4 that is 10% reduction from the initial value, since we considered average larger coverage than the one at the DFT calculations. We also expect that the clustering inside the albumin can reduce the total magnetization. The external magnetic field is H. The thermal energy is k_BT (where T is the temperature).

The magnetic configuration was obtained by the Monte Carlo simulation technique, with the implementation of the Metropolis algorithm [38]. The Monte Carlo simulations results for a given temperature and applied field were averaged over 60 samples with various spin configurations, realizations of the easy-axes distribution and different spatial configurations for the nanoparticles. The calculation of the (ZFC)/(FC) magnetization curves is a three steps process: (1) the system is cooled at a constant temperature rate dΤ = 0.05 from temperature T = 5 to T = 0.01 at zero applied field H = 0 (2) the sample is heated from temperature T = 0.01 to T = 5 at the same constant temperature rate under the application of a magnetic field H_app and the ZFC magnetization curve is monitored and (3) by cooling down the system in the presence of the applied field up to T = 0.01 we calculate the FC magnetization curve.

The demagnetization curve DCD and isothermal remanence IRM curve are calculated using the same procedure as the experimental DCD and IRM measurements described in section 2.2. The ΔM(H) plots were calculated using the expression ΔM(H) = M_DCD(H)—(1–2M_IRM (H)) [39].