A predictive model of iron oxide nanoparticles flocculation tuning Z-potential in aqueous environment for biological application

Abstract

Iron oxide nanoparticles are the most used magnetic nanoparticles in biomedical and biotechnological field because of their nontoxicity respect to the other metals. The investigation of iron oxide nanoparticles behaviour in aqueous environment is important for the biological applications in terms of polydispersity, mobility, cellular uptake and response to the external magnetic field. Iron oxide nanoparticles tend to agglomerate in aqueous solutions; thus, the stabilisation and aggregation could be modified tuning the colloids physical proprieties. Surfactants or polymers are often used to avoid agglomeration and increase nanoparticles stability. We have modelled and synthesised iron oxide nanoparticles through a co-precipitation method, in order to study the influence of surfactants and coatings on the aggregation state. Thus, we compared experimental results to simulation model data. The change of Z-potential and the clusters size were determined by Dynamic Light Scattering. We developed a suitable numerical model to predict the flocculation. The effects of Volume Mean Diameter and fractal dimension were explored in the model. We obtained the trend of these parameters tuning the Z-potential. These curves matched with the experimental results and confirmed the goodness of the model. Subsequently, we exploited the model to study the influence of nanoparticles aggregation and stability by Z-potential and external magnetic field. The highest Z-potential is reached up with a small external magnetic influence, a small aggregation and then a high suspension stability. Thus, we obtained a predictive model of Iron oxide nanoparticles flocculation that will be exploited for the nanoparticles engineering and experimental setup of bioassays.

Appendix A

Hydrodynamic resistance

$D_\infty$ is the diffusion coefficient in the absence of any interparticle forces given by

$$D_\infty =D_i+D_j=\frac{k_B T}{6\pi \mu }\frac{\left( r_i+r_j\right) }{\left( r_i r_j \right) }$$

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where D _i and D _j are the diffusion coefficients of particles i and j, respectively, $k_B$ is the Boltzmann constant, T is the temperature, $\mu$ is the viscosity of the medium. Denoting a function of particle separation $D_{i,j} = b k_B T$ as the relative diffusion coefficient between particles i and j, we adopted the formulation by Jacobson (2005) for $\frac{D_\infty }{D_{i,j}}$, since the adaption of Van der Waals/viscous collision correction factor from aerosol particles theory, thus generating a so called viscous-force correction factor to the diffusion coefficient in the continuum regime.

$$\frac{D_\infty }{D_{i_j}}=1+\frac{2.6 r_i r_j}{\left( r_i+r_j\right) ^2}\sqrt{\frac{\left( r_i r_j\right) }{\left( r_i+r_j\right) \left( r-r_i-r_j\right) }}+\frac{\left( r_i r_j\right) }{\left( r_i+r_j\right) \left( r-r_i-r_j\right) },$$

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where r is the centre-to-centre distance between two flocs or, generally, aggregations, i.e. interparticle distance. Please, note that b is the relative mobility of the particles; for two spheres moving along their centreline, it can be obtained from the exact solution of Stokes’ equation (Spielman 1970).

Van der Waals interaction

Hamaker’s formula for the London-van der Waals attractive interaction energy Vvdw between spherical particles is

$$\frac{V_{{\hbox{vdw}}}}{k_B T}=-\frac{A}{6k_B T}\left[ \frac{2r_i r_j}{r^2-\left( r_i+r_j\right) ^2}+\frac{2r_i r_j}{r^2-\left( r_i-r_j\right) ^2}+\log \left( \frac{r^2-\left( r_i+r_j\right) ^2}{r^2-\left( r_i-r_j\right) ^2}\right) \right],$$

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where A is the Hamaker constant, $k_B$ is the Boltzmann constant, T is the temperature and r is the interparticle distance.

Electrostatic interaction

In the case of thin double layers, symmetrical electrolytes and dimensionless interparticle separations $\kappa l$ ${>}4$ (where $\kappa$ is the inverse Debye-Huckel length, a measure of the double-layer thickness), the linear superposition approximation can be used

$$\begin{aligned} V_{{\hbox{el}}}= & \, \varepsilon \left( \frac{k_B T}{e}\right) ^2 Y_i Y_j\frac{r_i r_j}{r}e^{-\kappa l} \text{, } \text{ for } \kappa l\ge 4 \\ Y_i= & \, 4\tanh \left( \frac{\psi _i}{4}\right) \text{, } \text{ for } \kappa l\ge 10 \text{ and } \varPhi _i <8 \\ \varPhi _i= & \, \frac{ze\psi _{0i}}{k_B T} \end{aligned},$$

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where z represents the valence of the electrolyte dissociated in the solution, and $\psi _{0i}$ represents the surface potential of the particle i. It is assumed that the potential of one particle remains undisturbed in the presence of the others. The inverse Debye-Huckel length $\kappa$ is given by $\kappa ^{-1}\cong 2.8\times 10^{-8} I^{-0.5}$, where I is the ionic strength of the solution.

This formulation is useful when interparticle distance is greater than 36 nm. For smaller distances, the Derjaguin approximation (Chung and Hogg 1985) can be used

$$V_{{\hbox{el}}}=\varepsilon \frac{r_i r_j}{r_i+r_j}\psi _i\psi _j\log \left( 1+e^{kappa l}\right).$$

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Bell et al. (1970) defined $V_{{\hbox{el}}}$ according to the following formulation:

$$V_{{\hbox{el}}}=4\pi \varepsilon \left( \frac{\left( k_B T\right) }{ze}\right) ^2\frac{r_i r_j}{r_i+r_j}Y_i Y_j e^{-\kappa l}.$$

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Magnetic interaction

Chan et al. (1985) provided a formula for calculating the magnetic interaction

$$\frac{V_{{\hbox{mag}}}}{k_B T}=s\left( x\right) \left( -\frac{x^2}{3}\frac{1}{1+7x^2/150}\right) +\left( 1+s\left( x\right) \right) \left[ -2x+\log \left( 6x^2\right) -\frac{2}{3x}-\frac{7}{9x^2}\right]$$

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where

$$s\left( x\right) =e^{-\log 2\left( \frac{x}{2.4}\right) ^8}$$

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and

$$x=\frac{1}{k_B T}\frac{\pi d_i^3 d_j^3\chi _i\chi _j B^2}{144\,\mu _0 r^3}$$

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where $\mu _0$ is the permeability of free space, $\chi$ is the magnetic susceptibility, B is the strength of the magnetic field, d is the particle diameter and r is the separation of the particles. But, this formulation does not seem consistent, since it does not incorporate the sphericity of flocs or aggregations. In order to consider this issue, we used the following formulation (Bell et al. 1970) for the x factor:

$$x=\frac{1}{k_B T}\frac{\pi ^2}{36} d_i^3 d_j^3\frac{\chi _i}{1+\chi _i}\frac{\chi _j}{1+\chi _j}\frac{B^2}{\mu _0^2 r^3}.$$

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