^{1}, Pedro González-Mozuelos

^{2}and Mónica Olvera de la Cruz

^{3,a)}

### Abstract

In a previous theoretical and simulation study [G. I. Guerrero-García, E. González-Tovar, and M. Olvera de la Cruz, Soft Matter6, 2056 (2010)], it has been shown that an asymmetric charge neutralization and electrostatic screening depending on the charge polarity of a single nanoparticle occurs in the presence of a size-asymmetric monovalent electrolyte. This effect should also impact the effective potential between two macroions suspended in such a solution. Thus, in this work we study the mean force and the potential of mean force between two identical charged nanoparticles immersed in a size-asymmetric monovalent electrolyte, showing that these results go beyond the standard description provided by the well-known Derjaguin-Landau-Verwey-Overbeek theory. To include consistently the ion-size effects, molecular dynamics (MD) simulations and liquid theory calculations are performed at the McMillan-Mayer level of description in which the solvent is taken into account implicitly as a background continuum with the suitable dielectric constant. Long-range electrostatic interactions are handled properly in the simulations via the well established Ewald sums method and the pre-averaged Ewald sums approach, originally proposed for homogeneous ionic fluids. An asymmetric behavior with respect to the colloidal charge polarity is found for the effective interactions between two identical nanoparticles. In particular, short-range attractions are observed between two equally charged nanoparticles, even though our model does not include specific interactions; these attractions are greatly enhanced for anionic nanoparticles immersed in standard electrolytes where cations are smaller than anions. Practical implications of some of the presented results are also briefly discussed. A good accord between the standard Ewald method and the pre-averaged Ewald approach is attained, despite the fact that the ionic system studied here is certainly inhomogeneous. In general, good agreement between the liquid theory approach and MD simulations is also found.

G. I. G.-G. thanks Prateek Kumar Jha and Rastko Sknepnek for their kind help to use GPUs in MD simulations. The authors thank William Kung for proof-reading our manuscript. MD simulations were in part performed on Quest cluster at Northwestern University. A part of GPU simulations presented in this work were executed on GTX 480 GPUs provided by NVIDIA through their Professor Partnership Program. This material is based upon work supported by the Office of the Director of Defense Research and Engineering (DDR&E) and the Air Force Office of Scientific Research (AFOSR) under Award No. FA9550-10-1-0167, the National Science Foundation (NSF) grant (Grant No. DMR-0520513) of the Materials Research Science and Engineering Center program at Northwestern University and by the Conacyt-México grant (Grant No. 152532).

I. INTRODUCTION

II. MODEL, SIMULATIONS, AND THEORY

III. RESULTS AND DISCUSSION

IV. CONCLUDING REMARKS

### Key Topics

- Nanoparticles
- 96.0
- Electrostatics
- 24.0
- Electrolytes
- 22.0
- Computer simulation
- 10.0
- Colloidal systems
- 9.0

## Figures

Mean force (a) and potential of mean force (PMF) (b) between two neutral nanoparticles. The triangles correspond to MD results for *F*(*r*) and *W*(*r*) when the nanoparticles are immersed in a 1:1 size-asymmetric electrolyte at 1M. The data represented by filled and empty triangles are obtained, respectively, by using the standard Ewald method and the pre-averaged Ewald approach. The empty circles represent MD results for a system with the same parameters for *d* _{ M }, *d* _{+}, *d* _{−}, and ρ_{+} = ρ_{−}, but with *l* _{ b } = 0 instead of *l* _{ b } = 0.714 nm. The lines show the corresponding HNC predictions of *F*(*r*) and *W*(*r*): dashed lines represent *l* _{ b } = 0.714 nm and solid lines represent *l* _{ b } = 0. Here, and in the rest of the figures, the numerical uncertainties are smaller than the size of the symbols.

Mean force (a) and potential of mean force (PMF) (b) between two neutral nanoparticles. The triangles correspond to MD results for *F*(*r*) and *W*(*r*) when the nanoparticles are immersed in a 1:1 size-asymmetric electrolyte at 1M. The data represented by filled and empty triangles are obtained, respectively, by using the standard Ewald method and the pre-averaged Ewald approach. The empty circles represent MD results for a system with the same parameters for *d* _{ M }, *d* _{+}, *d* _{−}, and ρ_{+} = ρ_{−}, but with *l* _{ b } = 0 instead of *l* _{ b } = 0.714 nm. The lines show the corresponding HNC predictions of *F*(*r*) and *W*(*r*): dashed lines represent *l* _{ b } = 0.714 nm and solid lines represent *l* _{ b } = 0. Here, and in the rest of the figures, the numerical uncertainties are smaller than the size of the symbols.

Mean force (a) and PMF (b) between two identical charged nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M. Circles and squares correspond, respectively, to MD results for *z* _{ M } = −9 and *z* _{ M } = +9. Filled and empty symbols represent simulation data obtained, respectively, with the standard Ewald method and the pre-averaged Ewald approach. Solid and dashed lines display, respectively, the corresponding HNC predictions for *z* _{ M } = −9 and *z* _{ M } = +9.

Mean force (a) and PMF (b) between two identical charged nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M. Circles and squares correspond, respectively, to MD results for *z* _{ M } = −9 and *z* _{ M } = +9. Filled and empty symbols represent simulation data obtained, respectively, with the standard Ewald method and the pre-averaged Ewald approach. Solid and dashed lines display, respectively, the corresponding HNC predictions for *z* _{ M } = −9 and *z* _{ M } = +9.

Contributions to the PMF between two identical charged nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M. The nanoparticle valence is *z* _{ M } = −9 in (a) and *z* _{ M } = +9 in (b). Triangles, circles, and squares correspond, respectively, to *W*(*r*) and its repulsive core and electrostatic contributions obtained via MD simulations. Filled and empty symbols represent, respectively, the data obtained using the standard Ewald method and the pre-averaged Ewald approach. Lines joining symbols are intended as an eye-guide.

Contributions to the PMF between two identical charged nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M. The nanoparticle valence is *z* _{ M } = −9 in (a) and *z* _{ M } = +9 in (b). Triangles, circles, and squares correspond, respectively, to *W*(*r*) and its repulsive core and electrostatic contributions obtained via MD simulations. Filled and empty symbols represent, respectively, the data obtained using the standard Ewald method and the pre-averaged Ewald approach. Lines joining symbols are intended as an eye-guide.

Mean force (a) and PMF (b) between two identical charged nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M. Circles and squares correspond, respectively, to MD results for *z* _{ M } = −36 and *z* _{ M } = +36. Filled and empty symbols represent simulation data obtained, respectively, with the standard Ewald method and the pre-averaged Ewald approach. Solid and dashed lines display, respectively, the corresponding HNC predictions for *z* _{ M } = −36 and *z* _{ M } = +36.

Mean force (a) and PMF (b) between two identical charged nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M. Circles and squares correspond, respectively, to MD results for *z* _{ M } = −36 and *z* _{ M } = +36. Filled and empty symbols represent simulation data obtained, respectively, with the standard Ewald method and the pre-averaged Ewald approach. Solid and dashed lines display, respectively, the corresponding HNC predictions for *z* _{ M } = −36 and *z* _{ M } = +36.

Contributions to the PMF between two identical charged nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M. The nanoparticle valence is *z* _{ M } = −36 in (a) and *z* _{ M } = +36 in (b). Triangles, circles, and squares correspond, respectively, to *W*(*r*) and its repulsive core and electrostatic contributions obtained via MD simulations. Filled and empty symbols represent, respectively, the data obtained using the standard Ewald method and the pre-averaged Ewald approach. Lines joining symbols are intended as an eye-guide.

Contributions to the PMF between two identical charged nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M. The nanoparticle valence is *z* _{ M } = −36 in (a) and *z* _{ M } = +36 in (b). Triangles, circles, and squares correspond, respectively, to *W*(*r*) and its repulsive core and electrostatic contributions obtained via MD simulations. Filled and empty symbols represent, respectively, the data obtained using the standard Ewald method and the pre-averaged Ewald approach. Lines joining symbols are intended as an eye-guide.

Liquid theory predictions using the HNC closure for the mean force and the PMF between two identical nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M. Panes (a) and (c) correspond to anionic nanoparticles, whereas panes (b) and (d) correspond to cationic nanoparticles. Solid, dotted, short-dashed, long-dashed, dotted-dashed and dotted-double-dashed lines represent the data for |*z* _{ M }| = 3, 9, 18, 27, 36, 45, respectively.

Liquid theory predictions using the HNC closure for the mean force and the PMF between two identical nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M. Panes (a) and (c) correspond to anionic nanoparticles, whereas panes (b) and (d) correspond to cationic nanoparticles. Solid, dotted, short-dashed, long-dashed, dotted-dashed and dotted-double-dashed lines represent the data for |*z* _{ M }| = 3, 9, 18, 27, 36, 45, respectively.

Maximum reversed value of the mean force, *F**, and maximum reversed value of the PMF, *W**, between two identical nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M as a function of the nanoparticle valence *z* _{ M }. Symbols and solid lines correspond, respectively, to MD data and HNC results. Filled and empty symbols correspond to calculations using the standard Ewald method and the pre-averaged Ewald approach, respectively.

Maximum reversed value of the mean force, *F**, and maximum reversed value of the PMF, *W**, between two identical nanoparticles immersed in a 1:1 size-asymmetric electrolyte at 1M as a function of the nanoparticle valence *z* _{ M }. Symbols and solid lines correspond, respectively, to MD data and HNC results. Filled and empty symbols correspond to calculations using the standard Ewald method and the pre-averaged Ewald approach, respectively.

## Tables

Parameter values used in the MD simulations and the integral equation approach.

Parameter values used in the MD simulations and the integral equation approach.

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