# Simulation of Colloidal Stability and Aggregation Tendency of Magnetic Nanoflowers in Biofluids

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Model

_{1}to x

_{K}of arbitrary size (Figure 1), and (1) thereby becomes:

_{j}and x

_{k}form a particle of volume υ = ${x}_{j}+{x}_{k}$ between preset grid volumes x

_{i-1}and x

_{i}(Figure 1), then this is arithmetically split between these neighboring grid volumes. For the preservation of numbers and mass within the whole spectrum, this split is defined by $\omega $ in (10) as follows:

#### 2.2. Experimental Procedure

_{0}) was within 3 min after mixing the IONfs with PBS or SBFand the second (t

_{24}) after 24-h of incubation at 37 °C.

## 3. Results and Discussion

#### 3.1. Model Validation

#### 3.1.1. Constant Kernel

#### 3.1.2. Linear Kernel

_{1}is the modified Bessel function of first class and order one. Using (13), we obtain the benchmark solution for the number concentration, and the comparison with the numerical solution is shown in Figure 3b.

#### 3.1.3. Quadratic Kernel

_{1}is the modified Bessel function of first class and order one. It is important to point out that for this kernel, the mass is not conserved for $\tau >1$ due a loss of mass from finite size clusters (sol particles) to the infinite cluster (gel or super particle) [25]. Using (13), we obtain the benchmark solution for the number concentration, and the comparison with the numerical solution is shown in Figure 3c.

#### 3.1.4. Brownian Kernel

#### 3.2. Aggregation Experiments

_{0}and t

_{24}. The average size value by number of 15 consecutive measurements of IONfs in PBS at t

_{0}and t

_{24}are 16.22 ± 8.034 nm and 79.60 ± 19.25 nm, respectively. A significant increase in size was reported after 24-h incubation in PBS, dictated by the presence of salts in the colloidal solution (Figure 4a). For the sample of IONfs dispersed in SBF at t

_{0}, the average size value by number is 25.80 ± 9.01 nm, whereas at t

_{24}it is at 35.62 ± 11.71 nm. It is worth mentioning that the presence of FBS adds a second peak in these measurements, which was determined at 5.22 ± 1.43 nm, by measuring the plain SBF without IONfs. Again, there is an apparent increase in size that corresponds to the formation of protein corona (PC), but not that substantial to indicate aggregation. This can be confirmed by the ζ-potential decrease before and after treatment with 10% FBS in PBS from −27 mV to −19 mV. The latter can be attributed to the attractive interactions induced between the carboxylic acid of IONfs and the protein’s functional groups present in FBS (e.g., NH

_{2}). PC forms when nanoparticles are in close contact with proteins, such as albumins, which adsorb on their surface, further stabilizing them in the medium [27]. According to our data, IONfs seem to aggregate more in plain PBS than in SBF; this behavior can be attributed to the formation of PC, which averts the adsorption of the salts due to the higher affinity for the proteins rather than the salt counter ions [28]. In low nanoparticle concentrations, like the one used in our research, aggregation phenomena stemming from the PC formation are minimal (Figure 4b). FBS is a biological fluid that contains more than 3700 different proteins, with 12 orders of magnitude differences in their relative concentrations. Among all of the proteins in the serum, albumin represents the most abundant fraction and accounts for ~60% mass of the total proteins in the solution (1 mM), with globulins being the second most abundant (~40%) and all the rest representing less than 1% [23]. The observation of aggregation blocking due to PC is consistent with previous studies [29].

#### 3.3. Simulations

#### 3.3.1. Effects of Ionic Strength

^{−20}J. It can be seen that, as expected, the introduction of dispersants of a higher molarity weakens the effects of EDL, as the Debye length increases and consequently imposes exponential decreasing of the EDL potential. Therefore, attractive forces (Van der Waals) remain dominant, as in the case of PBS, where the total potential energy remains negative for all distance values (Figure 6b). Consequently, the aggregation rate is higher in the cases of a higher molarity, and the PBS case is almost identical to having no EDL potential, as also observed in the past [32]. In addition, molarities as low as 1 mM induce lower aggregation rates than in the experiment (Figure 6a).

#### 3.3.2. Effects of van der Waals Forces

^{−20}J for IONps [33], and reach values of up to 4.5 × 10

^{−20}J especially for maghemite NPs [34]. In the case of PBS, it seems that the changes within the range of Hamaker constants studied are marginal, i.e., there is no apparent change in the aggregation rate and the results after 24 h are almost identical (Figure 7a). In this case, the total potential energy is again dominated by van der Waals forces and shows no repulsive effects for any surface distance (Figure 7b).

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**TEM images of IONfs at two different magnifications Scale bars are at 50 nm (

**left**) nm and 10 nm (

**right**).

**Figure 3.**Validation of the model by comparison with the analytical expressions for (

**a**) constant kernel, (

**b**) linear kernel, (

**c**) quadratic kernel, and (

**d**) Brownian kernel. Comp—computational results; an—analytical expressions.

**Figure 5.**Simulations for a 24 h period (

**a**) comparison with experimental results and (

**b**) potential energies. Tot—total; vdw—van der Walls; edl—electric double layer; mag—magnetic; comp—computational; exp—experiment.

**Figure 6.**Effects of ionic strength on (

**a**) particle size distribution and (

**b**) total potential energy.

**Figure 7.**Effects of Hamaker constant for PBS on (

**a**) particle size distribution; (

**b**) total potential energy (values of A are in 10

^{−20}J).

**Figure 8.**Effects of Hamaker constant for NaCl (10 mM) on (

**a**) particle size distribution; (

**b**) total potential energy (values of A are in 10

^{−20}J).

Parameter | Value |
---|---|

Μ ^{1} | 0.6864 cP |

T ^{2} | 310 K |

z_{c} ^{3} | 1 |

ε_{Γ} ^{4} | 78.5 |

Φ ^{5} | −0.027 V |

M ^{6} | 0.7346 A/m |

A ^{7} | 3.1 × 10^{−20} J |

PBS ^{8} | KCl 2.68 mM KH _{2}PO_{4} 1.47 mMNaCl 136.89 mM K _{2}HPO_{4} 8.10 mM |

^{1}viscosity of solvent;

^{2}temperature;

^{3}valence of counter ion;

^{4}dielectric constant of the solvent;

^{5}surface potential (approx. zeta potential);

^{6}intensity of magnetization;

^{7}Hamaker constant;

^{8}phosphate buffered saline solution.

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**MDPI and ACS Style**

Neofytou, P.; Theodosiou, M.; Krokidis, M.G.; Efthimiadou, E.K.
Simulation of Colloidal Stability and Aggregation Tendency of Magnetic Nanoflowers in Biofluids. *Modelling* **2022**, *3*, 14-26.
https://doi.org/10.3390/modelling3010002

**AMA Style**

Neofytou P, Theodosiou M, Krokidis MG, Efthimiadou EK.
Simulation of Colloidal Stability and Aggregation Tendency of Magnetic Nanoflowers in Biofluids. *Modelling*. 2022; 3(1):14-26.
https://doi.org/10.3390/modelling3010002

**Chicago/Turabian Style**

Neofytou, Panagiotis, Maria Theodosiou, Marios G. Krokidis, and Eleni K. Efthimiadou.
2022. "Simulation of Colloidal Stability and Aggregation Tendency of Magnetic Nanoflowers in Biofluids" *Modelling* 3, no. 1: 14-26.
https://doi.org/10.3390/modelling3010002