Molecular-Dynamics Simulation of Forces between Colloidal

Chapter 15

Molecular-Dynamics Simulation of Forces between Colloidal Nanoparticles 1,2

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Downloaded by UNIV OF CALIFORNIA SANTA CRUZ on October 14, 2014 | http://pubs.acs.org Publication Date: December 14, 2004 | doi: 10.1021/bk-2005-0890.ch015

Kristen A. Fichthorn and Yong Qin 1

Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802 [email protected] [email protected] 2

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Introduction/Background Nanoparticles hold great promise for a diverse array of materials applications, ranging from electronic circuits to bulk materials with novel mechanical properties to biological materials. Many applications involve colloidal nanoparticles, whose effective use in nanotechnology hinges on their selective assembly or their stabilization against aggregation. Various methods have been used to stabilize colloidal nanoparticles; however all involve dispersant molecules such as surfactants or polyelectrolytes. Not only do these dispersants alter the chemistry and physics of nanoparticle systems, but since they occupy a significant massfractionof a nanoparticle system, they produce a tremendous waste stream during processing. An improved understanding of the forces between "bare" colloidal nanoparticles could lead to new and environmentally beneficial strategies for engineering colloidal nanoparticle suspensions. Historically, the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory has been used to describe electrostatic and van der Waals interactions in colloidal systems [1]. However, the assumptions of DLVO theory do not apply to nanoparticles. Further, recent studies suggest that forces that are not taken into account by DLVO theory, such as solvation and depletion, could be important in colloidal nanoparticle systems. From a theoretical point of view, it is now possible to simulate colloidal nanoparticles using large-scale, parallel molecular dynamics (MD). These studies can yield atomic-scale detail that is not currently accessible with experimental methods and they can be used to resolve the origins and magnitudes of forces between colloidal nanoparticles. The goal of this study is to apply MD simulations to investigate the relative magnitudes of solvation and van der Waals forces in a model colloidal nanoparticle system. 128

© 2005 American Chemical Society

In Nanotechnology and the Environment; Karn, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2004.

129


Materials/Methods We use parallel MD to simulate two solid nanoparticles immersed in a liquid. The bulk solvent is simulated as 108,000 Lennard-Jones atoms in a cubic cell with periodic boundary conditions. To study the influence of particle size on interparticle forces, we simulate spherical nanoparticles of two different sizes. The small spherical nanoparticles are composed of 64 atoms and the average diameter is 1.67 nm. The large spherical nanoparticles are composed of 2048 atoms and the average diameter is 6.01 run. The surfaces of the spherical nanoparticles are rough, as shown in Fig. 1. To investigate the influence of surface roughness, the nanoparticles are rotated so that they contact from a different angle and have different contacting surfaces. Finally, since the shape of a nanoparticle can influence solvation forces, we also study cubic nanoparticles, which have 2744 atoms arranged in a face-centered-cubic (fee) structure, such that the contacting surfaces have the smooth and flat fcc(l 11) structure. Particle-solvent interactions can further affect the solvation force between nanoparticles. In the simulations, nanoparticles can be either solvophilic or solvophobic. Solvophilic means that the solvent molecules have a stronger attraction to atoms in the nanoparticles than they do to themselves. To model this case, we set the solid-fluid attraction to be five times stronger than fluid-fluid attraction. For solvophobic nanoparticles, the interaction between the solvent molecules and the nanoparticle atoms is five times weaker than the solventsolvent attraction.

Results/Discussion Several existing theories can be used to calculate van der Waals forces. These theories rely on the assumption that the van der Waals attraction between two colloidal particles is the sum of pair-wise additive dispersion forces between the atoms comprising the objects. Using the assumption of a continuum solid, Hamaker [2] and Bradley [3] derived expressions for the van der Waals interaction between two macroscopic, spherical particles. In both of these formulae, the scaling of the interaction with particle separation is different than the 1/r scaling between two atoms. Since nanoparticles are an intermediate case between single atoms and macroscopic colloidal spheres, it is unclear whether these macroscopic theories can describe van der Waals interactions between them. We calculated and compared van der Waals forces between the spherical nanoparticles using direct evaluation of the pair-wise dispersion forces, as well as with Hamaker's and Bradley's equations. Due to the small size and irregular surfaces of the nanoparticles, results with different theories vary considerably. 6



130 Both Hamaker's and Bradley's equations fail for short nanoparticle separations. The discrepancy between Hamaker's formula and direct evaluation arises from the irregular surface structure of the nanoparticles and Hamaker's equation predicts van der Waals forces well at separations greater than 3 fluid layers (about 1 nm). In both the small and large nanoparticle systems, Bradley's equation always underestimates the van der Waals force. The free energy change of the nanoparticle system with particle separation is calculated from a variant of the thermodynamic integration (TI) method [4]. In the TI method, the derivative of the free energy with respect to a nanoparticle separation δ is calculated directly and then integrated to yield ΑΑ(δ),

where δ, and Sj are states of the system with different nanoparticle separations. Accounting only for solvation forces in the derivative of the potential £/, we have

M^^dS(r {F (S)-F (S)) AB

where F^

s

and Fg

AtS

s

BiS

s

,

(2)

are the forces acting on particles A and Β due to the

solvent and Ρ^β is the unit vector pointing from particle A to particle 5. The ensemble-average quantity in the integral of Eq. (2) is the solvation force. Solvation forces for solvophilic and solvophobic nanoparticles have been calculated for the different nanoparticle systems. In all the solvophilic nanoparticle systems, the solvation forces oscillate between attraction and repulsion. We find that the phase of the oscillations for the spherical nanoparticles is influenced by surface roughness and depends on their relative orientation. The oscillatory behavior is caused by the solvent's ordering near the surface. This effect is particularly evident for the cubic nanoparticles, which exhibit the strongest solvation forces. In Fig. 2, we show the solvation-force profile for the cubic nanoparticle along with the solvent-density profile from a slice of the simulation box surrounding the nanoparticles for a nanoparticle separation of 3.5σ. The dark red contours in the solvent-density plot indicate a higher solvent density than the bulk and solvent layering in the region between the two nanoparticles. We find that solvation forces for solvophobic nanoparticles are always attractive. In this case, solvent molecules are repelled from the interparticle region and the density there is lower than the bulk density. Because the solventparticle interaction is different from the solvophilic case, the solvophobic solvation forces are orders of magnitude smaller than the solvophilic solvation



Figure 1: Snapshot of spherical 64- and 2048-atom nanoparticles.

600.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Separation (σ)

Figure 2: van der Waals and solvation forces for the solvophilic, cubic nanoparticle (left) and the solvent density profile for the cubic nanoparticle at a separation of about 3.5σ(right).


132 forces and van der Waals forces. Nevertheless, the attractive solvation forces enhance the attractive van der Waals forces between nanoparticles so they can aggregate more easily. A comparison of solvation forces and van der Waals forces (assuming that the solid is A1 0 and the solvent is Ar) for the cubic, solvophilic nanoparticles is shown in Fig. 2, where we see that the solvation force is comparable to the van der Waals forces. This indicates that solvation forces may be beneficial in preventing nanoparticles from aggregating and that stable nanoparticle dispersions may be achieved in suitable nanoparticle-solvent systems.


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References nd

1. Israelachvili, J. N . (1992) "Intermolecular and Surface Forces", 2 Ed., New York: Academic Press. 2. Hamaker, H. C. (1937) Physica IV 10, 1058. 3. Bradley, R. S. (1932) Phil. Mag. 13, 853. 4. Mezei, M . and Beveridge, D. L. (1986) Ann. Ν. Y. Acad. Sci. 482, 1.


Molecular-Dynamics Simulation of Forces between Colloidal

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