Dispersions of Zirconia Nanoparticles Close to the Phase Boundary of Surfactant-Free Ternary Mixtures

The achievement of a homogeneous dispersion of nanoparticles is of paramount importance in supporting their technological application. In wet processing, stable dispersions were largely obtained via surfactant or surface functionalization: although effective, the use of dispersant can alter, or even impair, the functional properties of the resulting nanostructured systems. Herein, we report a novel integrated modeling and experimental approach to obtain stable ZrO2 nanoparticle (NP) dispersions at native dimensions (about 5 nm) in homogeneous ternary mixtures of solvents (i.e., water, ethanol, and 1,2-dichlorobenzene) without any further surface functionalization. A miscibility ternary diagram was computed exploiting the universal quasi-chemical functional-group activity coefficient (UNIFAC) model, which was then experimentally validated. Dynamic light scattering (DLS) on these mixtures highlights that nanometric structures, resembling nanoemulsion droplets, form close to the mixture two-phase boundary, with a size that depends on the ternary mixture composition. ZrO2–NPs were then synthesized following a classic sol–gel approach and characterized by XRD and Raman spectroscopy. ZrO2–NPs were dispersed in HCl and mixed with different mixtures of ethanol and 1,2-dichlorobenzene (DCB), obtaining homogeneous and stable dispersions. These dispersions were then studied by means of DLS as a function of DCB concentration, observing that the nanoparticles can be dispersed at their native dimensions when the mass fraction of DCB was lower than 60%, whereas the increase of the hydrophobic solvent leads to the NPs’ agglomeration and sedimentation. The proposed approach not only offers specific guidelines for the design of ZrO2–NPs dispersions in a ternary solvent mixture but can also be extended to other complex solvent mixtures in order to achieve stable dispersions of nanoparticles with no functionalization.


S1. UNIFAC model equations and UNIFAC-LLE parameters for the ternary mixture of water, 1,2dichlorobenzene and ethanol
The mixtures leading to phase separations can give rise to two liquid phases, and , characterised by a and composition, respectively, expressed via the molar fraction of the -th component. These compositions represent the unknowns of the problem, thus leading to a total of unknowns, where is the number of species in the system. In this 2 case 3 species are present, namely DCB, water, and ethanol, thus the total number of unknowns is 6. Once equilibrium is reached at a given temperature, , pressure, , and composition (expressed in vector terms), the same species in the two phases, and , must have the same fugacity, namely: The fugacity of the -th component in a liquid mixture (phase ) can be expressed as: where is the activity coefficient of the species in the mixture, and the fugacity of the pure species in liquid phase. Therefore, at equilibrium, it must be: This will enforce a number of equations equal to 3. As a consequence of mass conservation, the stoichiometric relationships must hold true: In the following the explicit dependency of the variables on temperature, pressure, and composition will be dropped for clarity purposes. The combinatorial part is expressed as: where the parameters and are characteristics of the pure species considered and are indications of area and volume fractions, respectively. is an ensemble of pure-species variables is a characteristic constant equal to 10 for the UNIFAC model.
The aforementioned parameters can be expressed as: where and are representative of the molecule van der Waals volumes and surface areas, expressed by means of group contributions as: where is the number of different groups in molecule , the number of groups of type in the -th molecule, and and represent the group contributions to molecular volume and area, respectively. It should be noted that only pure species parameters characterise the combinatorial part of the activity coefficient.
The residual part of the activity coefficient takes into account binary interactions between the molecules in the mixture. No further (ternary) parameter is necessary to take into account the interaction of three molecules together, even for systems with three or more components. 2 The residual part is expressed as: is the total number of different group types in the mixture, is the molar fraction of the -th group in the mixture, and is the group-interaction parameter. The quantity ln Γ ( ) (which can be seen as the interactions of the groups within the same molecule) is computed as from Equation (s8) Table S1 for the groups present in the ternary mixture of 1,2-dichlorobenzene, water, and ethanol. The groups constituting each molecule are reported in Table S2. For the sake of example, DCB is formed by 4 aromatic carbons with and hydrogen atom (CArH) and 4 aromatic carbons with a chlorine atom (CArCl). and green) correspond to the composition of the three solvent mixtures discussed in Table   1, respectively TM1, TM2 and TM3.  In the case of polydisperse samples, the field autocorrelation function is a sum of 1 ( ) the exponential decays corresponding to each of the species in the population:

S9
where the weights are proportional to the relative scattering from each species and the characteristic decay times are proportional to the species size .
Exploiting the properties of the exponential, it is easy to demonstrate that the integral of the correlation function: is equal to the intensity-averaged decay time . deviates from the exponential decay expected for monodisperse particles.

S29
To estimate the typical cluster size and concentration we make the simplifying assumption that in our samples are present only two populations, the particles and the clusters.
In this case Eq.S18 will become: Where the subscripts p and c refer to the particles and clusters respectively.
The can be easily obtained by fitting the first part of the field correlation function for the nanoparticles in the original aqueous solvent (H 2 O + 0.1 M NaCl). Our fit gives a value of , which corresponds to a typical particle radius . = 27.8 = 4.66 The weights and can be estimated by looking when the correlation function departs significatively from the single exponential behaviour expected for monodisperse particles (see figure S15). Once and are known, the value of can be easily calculated , , by inverting Eq.2. To estimate the fraction of particles associated in clusters is useful to recall that: S30 ∝ ( ) 3 #( 20) The fraction of particles associated in cluster is then: The results are reported in the Table S6.