arXiv:1801.07468v1 [cond-mat.soft] 23 Jan 2018 Adsorption isotherms for charged nanoparticles Alexandre P. dos Santos ∗ Instituto de F´ ısica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil Amin Bakhshandeh † Instituto de F´ ısica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil Alexandre Diehl ‡ Departamento de F´ ısica, Instituto de F´ ısica e Matem´ atica, Universidade Federal de Pelotas, Caixa Postal 354, CEP 96010-900, Pelotas, RS, Brazil Yan Levin § Instituto de F´ ısica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil We present theory and simulations which allow us to quantitatively calculate the amount of sur- face adsorption excess of charged nanoparticles to a charged surface. The theory is very accurate for weakly charged nanoparticles and can be used at physiological concentrations of salt. We have also developed an efficient simulation algorithm which can be used for dilute suspensions of nanopar- ticles of any charge, even at very large salt concentrations. With the help of the new simulation method, we are able to efficiently calculate the adsorption isotherms of highly charged nanoparti- cles in suspensions containing multivalent ions, for which there are no accurate theoretical methods available. I. INTRODUCTION The interaction between lipid membranes, DNA, elec- trodes, and other charged surfaces with nanoparticles is of fundamental importance in biochemistry, biophysics, and diagnostic medicine. It is well known that salt can modify significantly the interaction between biomolecules in aqueous suspensions, affecting their stability [1–11]. The Derjaguin-Landau-Verwey-Overbeek (DLVO) the- ory [12] attributes the stability of suspensions to the competition between electrostatic and dispersive, van der Waals (vdW), forces. Electrostatic repulsion between colloidal particles prevents them from coming into a close contact at which strong dispersion forces can make the particles stick together, resulting in flocculation and pre- cipitation. The vdW attraction is very short-ranged and is only weakly affected by the presence of electrolyte. On the other hand, the Coulomb repulsion between like charged particles is strongly susceptible to the presence of electrolyte, which screens the electrostatic interactions. The DLVO theory provides a qualitative understanding of stability of colloidal systems in suspensions contain- ing 1:1 electrolyte. The theory, however, is not able to account for either the ionic specificity (Hofmeister ef- fect) [13–17], like-charge attraction [18–21], or the re- versal of the electrophoretic mobility often observed in * alexandre.pereira@ufrgs.br † bakhshandeh.amin@gmail.com ‡ diehl@ufpel.edu.br § levin@if.ufrgs.br suspensions containing multivalent ions [22–24]. In this paper, we will explore the interaction between nanopar- ticles and charged surfaces. Our goal is to quantitatively calculate the adsorption isotherms for dilute suspensions of nanoparticles in solutions containing large – physio- logical concentrations – of electrolyte. When studying Coulomb systems the starting point is often the Poisson-Boltzmann (PB) equation. Indeed, it has been observed that PB equation can very accu- rately describe the density profiles of monovalent ions near a charged wall. However, since the PB equation does not take into account either electrostatic correlations or steric repulsion between ions it is bound to fail if used to calculate the adsorption isotherms of charged nanoparti- cles near a charged wall [25]. Nevertheless, we will show that a very simple modification of PB equation can ex- tend its validity to study an important class of weakly charged nanoparticles, allowing us to quantitatively cal- culate their adsorption isotherms. For a more strongly charged nanoparticles, or if solution contains multivalent ions, we will present a simulation method which allows us to obtain adsorption isotherms at infinite dilution of nanoparticles, which are often of great practical interest. The paper is organized as follows: in Section II, we introduce a modified PB (mPB) equation which allows us to accurately calculate the density profiles of weakly charged nanoparticles near a charged surface. In Sec- tion III we present an efficient Monte Carlo (MC) simu- lation method which can be used to obtain the adsorption isotherms for very dilute suspensions of nanoparticles at large salt concentrations. In section IV, we compare the theory with the simulations and discuss suspensions con-