#
Multi-Scale Modelling of Aggregation of TiO_{2} Nanoparticle Suspensions in Water

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## Abstract

**:**

_{2}nano-additive in food products. Following the intent of relating nanomaterials atomic structure with their toxicity without having to conduct large-scale experiments on living organisms, we investigate the aggregation of titanium dioxide nanoparticles using a multi-scale technique: starting from ab initio Density Functional Theory to get an accurate determination of the energetics and electronic structure, we switch to classical Molecular Dynamics simulations to calculate the Potential of Mean Force for the connection of two identical nanoparticles in water; the fitting of the latter by a set of mathematical equations is the key for the upscale. Lastly, we perform Brownian Dynamics simulations where each nanoparticle is a spherical bead. This coarsening strategy allows studying the aggregation of a few thousand nanoparticles. Applying this novel procedure, we find three new molecular descriptors, namely, the aggregation free energy and two numerical parameters used to correct the observed deviation from the aggregation kinetics described by the Smoluchowski theory. Ultimately, molecular descriptors can be fed into QSAR models to predict the toxicity of a material knowing its physicochemical properties, enabling safe design strategies.

## 1. Introduction

_{2}used in sunscreen is made of even smaller particles with a diameter in the range of 1 to 150 nm; to avoid cellular damage from the reactive oxygen species produced during titanium dioxide photocatalytic activity, nanoparticles are commonly coated with alumina or silica, which also improve dispersion [20]. The small size of nanomaterials increases enormously the surface-to-bulk ratio, changing dramatically the macroscopic properties of the material. Due to their size, nanomaterials are hard to characterize experimentally, whereas simulations can be of great help to shed light on the physicochemical properties. However, such nanoparticles are too large to be simulated at a quantum level, hence it is necessary to make some approximations, the most common one is to consider a planar slab representative of the material [21,22]. The main problem of this approximation is that nanospheres, nano ellipsoid, and nanorods have a broader range of uncoordinated Ti sites than nanocrystals and slabs [23], furthermore, nanoparticles present a wide range of low index facets because of their curvature, so considering a single, low index slab can lead to different results, especially in an aqueous environment, where water molecule can absorb on the surface in a molecular, dissociative or even a mixture of the two ways [24,25].

## 2. Materials and Methods

#### 2.1. Molecular Models and Strategy

_{111}O

_{222}of radius 0.78 nm, Ti

_{417}O

_{834}of radius 1.50 nm and Ti

_{985}O

_{1970}of radius 2.00 nm. They were chosen to be representative of small nanoparticles found in several commercial applications, and to be calculable by classical force fields MD. Figure 1 displays the models used.

#### 2.2. Ab Initio DFT Calculations

#### 2.3. Classical Molecular Dynamics Simulations

#### 2.4. Brownian Dynamics Simulations

^{®}post-processing algorithm. The proposed implementation does not require the a priori knowledge of the number of clusters present in the simulation box, which are in fact determined by the pairwise distance between each particle. Two particles were considered as neighboring and belonging to the same aggregate if their distance is less than a threshold distance, here considered to be $3R$. Thus, two neighboring NPs must be in the attractive region of their interaction potential, while a third particle cannot be interposed between them.

## 3. Results

#### 3.1. Ab Initio DFT Characterization

#### 3.2. Potential of Mean Force for the Approach of Two Identical Titanium Dioxide Anatase Nanoparticles in Water

#### 3.3. Aggregation of Titanium Dioxide Anatase Nanoparticles Using Brownian Dynamics

_{2}NPs in water were found to form branched aggregates [55].

## 4. Discussion

_{2}NPs were built according to their stoichiometric composition. Therefore, we expect that most of the contribution to the stability of the suspension is given by the van der Waals attraction between particles, whereas the repulsion given by the electric double layer is negligible: the DLVO theory, in this case, predicts an unstable suspension which flocculates, just as we observed in our BD simulations. However, the well-established theories used to predict the van der Waals attraction fail to effectively describe NPs potential curves. In fact, the attraction potential between two identical spheres with radius R can be evaluated as [59]:

_{2}-TiO

_{2}in water [59] and d is the distance between the centres of the spheres. The interaction potentials obtained from Equation (7) and those obtained by the MD simulations are compared in Figure 6. The DLVO potential given by Equation (7) predicts an attraction well, which asymptotically tents to $-\infty $ as the surface-to-surface distance between the NPs tends to zero, i.e., $d\to 2R$ (see blue dashed lines in Figure 6). The results of the MD simulations show that the free energy minimum appears at a larger distance with respect to $d=2R$: as commented above, this is the effect of the van der Waals radii of the surface atoms, which exert a strong repulsive force opposing to further approach of the NPs, and can not be modeled by Equation (7). Steric repulsion avoids the mutual penetration of NPs as they enter the region of the potential well: a direct implementation of the obtained DLVO potential would not allow the BD simulations to converge to an equilibrium configuration. These results are in line with those previously reported in the literature [50] and reaffirm the inadequacy of DLVO theory in describing the interaction potential between NPs.

## 5. Conclusions

_{2}NP aggregation in water starting from their atomistic description. The coordinates of the atoms forming the NPs were retrieved from quantum calculations. MD simulations of pairs of NPs were performed to calculate their interaction potential (PMF) using the Umbrella Sampling technique. The results obtained were compared with the theoretical PMF predicted by the DLVO theory, finding significant discrepancies and justifying the proposed approach. The PMF was used to perform BD simulations of tens of thousands of NPs at a relatively low computational cost. Such large systems could not have been studied with regular MD simulations, so the protocol adopted was crucial for understanding the aggregation kinetic of TiO

_{2}NPs. Thus, the proposed protocol bridges the gap between the quantum description of a single particle and the observation of the microscopic clustering process.

_{2}NPs, its generality makes it suitable for a wide variety of other materials, including charged or coated particles. In fact, the NPs considered in this work are neutral, while they might present a net non-zero surface charge arising from the positive under-coordinated Ti${}^{4+}$ and Ti${}^{5+}$ sites as well as the negative under-coordinated oxygen sites. Therefore, the case reported could be considered as a limiting case of TiO

_{2}NPs at their isoelectric point, namely the value of the solution pH at which an NP has zero net electric charge. The presence of a net surface charge results in a repulsive barrier, that increases the stability of the colloidal suspension, as evidenced by experimental [66] and theoretical works [50]. From a multi-scale perspective, future work will focus on modeling under-coordinated reactive sites and their role in determining an energy barrier in PMF. This may involve moving from diffusion-limited to reaction-limited kinetics, thus allowing to test the Smoluchowski theory on a complete DLVO system. The computational bottleneck is represented by the calculation of the PMF curve and the aggregation free energy using MD and US, which is very demanding in terms of computing resources: this is unfortunately mandatory because of the inability of the DLVO theory to correctly describe the interaction between two NPs.

## Supplementary Materials

_{2}nanoparticles in water; Figure S2: Formation of clusters during the BD simulation of Ti417O834 nanoparticles with a = 0.8% reported as an example; Figure S3: Evaluation of the root mean square error (RMSE) between the theoretical aggregation kinetic, evaluated through Equations (6)–(9), and the BD simulations; Table S1: Classical Molecular Dynamics simulation parameters; Table S2: Fitting coefficients used to describe the PMFs as split-curves; Table S3: Geometry and energy results for different computational settings used in ab initio DFT calculations; Table S4: Brownian Dynamics simulation parameters; Table S5: Brownian Dynamics: number of beads.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AFE | Aggregation Free Energy |

BD | Brownian Dynamics |

DFT | Density Functional Theory |

EFSA | European Food Safety Authority |

MD | Molcular Dynamics |

NP | Nanoparticle |

PMF | Potential of Mean Force |

QSAR | Quantitative Structure Activity relationship |

US | Umbrella Sampling |

WHAM | Weighted Histogram Analysis |

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**Figure 1.**TiO

_{2}nanoparticles used in this work: (

**a**) Ti

_{111}O

_{222}, radius 0.78 nm; (

**b**) Ti

_{417}O

_{834}, radius 1.5 nm; (

**c**) Ti

_{985}O

_{1970}, radius 2 nm. Image obtained with VMD software [28], version 1.9.3.

**Figure 2.**Free Energy profiles for the aggregation of two TiO

_{2}NPs in water with chemical formula (

**A**) Ti

_{111}O

_{222}, (

**B**) Ti

_{417}O

_{834}and (

**C**) Ti

_{985}O

_{1970}. The results of the MD simulations (solid black line) were fitted with Equation (1) (dotted red line) to produce the pair-wise tabled potentials. The PMFs and their respective fit curves were represented as down-shifted considering a null interaction at $d\to \infty $.

**Figure 3.**Formation of clusters during a BD production run of 600 NPs of Ti

_{417}O

_{834}with $\varphi $ = 0.8%, reported as an example; (

**A**) Initial configuration with randomly distributed and isolated NPs; (

**B**) After 3 $\mathsf{\mu}$s, the particles formed 8 major aggregates. For a detailed view on $xy$, $xz$ and $yz$ planes, see Figure S2.

**Figure 4.**Aggregation kinetic of (

**A**) Ti

_{111}O

_{222}, and (

**B**) Ti

_{985}O

_{1970}with $\varphi $ = 7%, reporting the total number of clusters present in the simulation box (solid black line) and their average size (blue circles). Note that in the latter the NPs formed a single cluster after approximately 7 $\mathsf{\mu}$s of simulations.

**Figure 5.**Aggregation kinetic of Ti

_{111}O

_{222}with $\varphi $ = 3.5% at (

**A**) 0.2 $\mathsf{\mu}$s, (

**B**) 10 $\mathsf{\mu}$s and (

**C**) 30 $\mathsf{\mu}$s. As the simulation advances, the clusters size distribution (histograms, left-hand side) tends towards the equilibrium configuration, namely fewer and larger clusters. Note that the aggregates were colored for simplicity of representation only.

**Figure 6.**Comparison between the classic DLVO theory (dashed blue line), evaluated through Equation (7), and the PMF evaluated obtained by WHAM procedure (solid black line) for a pair of (

**A**) Ti

_{111}O

_{222}NPs, (

**B**) a pair of Ti

_{417}O

_{834}NPs and (

**C**) a pair of Ti

_{985}O

_{1970}NPs.

**Figure 7.**Comparison between the Smoluchowski theoretical model (solid red lines), the simulation data (black circles) and the modified aggregation models (shaded blue and green areas). The comparison was performed considering the configurations with $R=0.78$ nm and (

**A**) $\varphi =0.8\%$ or (

**B**) $\varphi =3.5\%$; with $R=1.50$ nm and (

**C**) $\varphi =0.8\%$ or (

**D**) $\varphi =3.5\%$; with $R=2.00$ nm and (

**E**) $\varphi =0.8\%$ or (

**F**) $\varphi =3.5\%$. To avoid numerical artifacts in the fitting procedure, only the time frames presenting more than 5 clusters were considered by the fitting algorithm.

**Table 1.**Selected physico-chemical properties for the three molecular models. SASA is the solvent-accessible surface area, E${}_{tot}$ is the total energy in eV, $\Delta {H}_{f}^{\circ}$/TiO${}_{2}$ is the standard formation enthalpy (experimental value for bulk anatase is −9.78 eV [54]), E${}_{sol}$ is the solvation energy [32] in eV (not available for the largest particle).

Property | 0.78 nm | 1.5 nm | 2.0 nm | Bulk |
---|---|---|---|---|

TiO_{2} unit formula | 111 | 417 | 985 | 4 |

SASA × ${10}^{7}$ (m${}^{2}$) | 2.15 | 4.44 | 7.14 | - |

E${}_{tot}$ (eV) | −2808.98 | −10,623.23 | −25,498.99 | −105.93 |

$\Delta {H}_{f}^{\circ}$/TiO${}_{2}$ (eV) | −8.28 | −8.45 | −8.86 | −9.46 |

E${}_{sol}$ (eV) | −0.42 | −0.34 | - | - |

**Table 2.**Aggregation free energy (AFE) and its distance from the surface ${d}_{AFE}-2R$ of NPs pairs in water, calculated using Classical MD and US technique; the aggregation free energy was evaluated as the depth of the potential well (see Figure 2).

NP Radius, R (nm) | AFE (kJ mol${}^{-1}$) | ${\mathit{d}}_{\mathbf{AFE}}-2\mathit{R}$ (nm) |
---|---|---|

0.78 | 44.18 | 0.23 |

1.50 | 54.35 | 0.35 |

2.00 | 75.56 | 0.46 |

**Table 3.**Values of the numerical coefficients ${n}_{1}$ and ${n}_{2}$ used in Equations (13) and (14), obtained by minimizing the RMSE between the modelling predictions and the simulation data. The modified theoretical model allows to obtain an analytical description of the aggregation kinetic consistent with the predictions of the multi-scale model proposed.

R (nm) | $\mathit{\varphi}$ | ${\mathit{n}}_{2}$ | ${\mathit{n}}_{3}$ |
---|---|---|---|

$0.78$ | 0.8% | 0.101 | 0.221 |

1.8% | 0.101 | 0.362 | |

3.5% | 0.134 | 0.221 | |

7.0% | 0.148 | 0.101 | |

$1.50$ | 0.8% | 0.141 | 0.181 |

1.8% | 0.134 | 0.161 | |

3.5% | 0.268 | 0.027 | |

7.0% | 0.027 | 0.094 | |

$2.00$ | 0.8% | 0.0 | 0.698 |

1.8% | 0.162 | 0.114 | |

3.5% | 0.169 | 0.067 | |

7.0% | 0.040 | 0.034 |

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**MDPI and ACS Style**

Mancardi, G.; Alberghini, M.; Aguilera-Porta, N.; Calatayud, M.; Asinari, P.; Chiavazzo, E. Multi-Scale Modelling of Aggregation of TiO_{2} Nanoparticle Suspensions in Water. *Nanomaterials* **2022**, *12*, 217.
https://doi.org/10.3390/nano12020217

**AMA Style**

Mancardi G, Alberghini M, Aguilera-Porta N, Calatayud M, Asinari P, Chiavazzo E. Multi-Scale Modelling of Aggregation of TiO_{2} Nanoparticle Suspensions in Water. *Nanomaterials*. 2022; 12(2):217.
https://doi.org/10.3390/nano12020217

**Chicago/Turabian Style**

Mancardi, Giulia, Matteo Alberghini, Neus Aguilera-Porta, Monica Calatayud, Pietro Asinari, and Eliodoro Chiavazzo. 2022. "Multi-Scale Modelling of Aggregation of TiO_{2} Nanoparticle Suspensions in Water" *Nanomaterials* 12, no. 2: 217.
https://doi.org/10.3390/nano12020217