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Modeling Assembly of Polymer-Grafted Nanoparticles at Oil-Water Interfaces
Xin Yong*
Department of Mechanical Engineering
State University of New York at Binghamton, Binghamton, New York 13902, United States
Supporting Information: Adsorption of Rigid Cores
In order to facilitate the understanding of the interfacial behavior of polymer-grafted
nanoparticles, we first examine the adsorption of the hydrophobic rigid core. A simulation box of
size Lx  Ly  Lz = 10  10  20 (in units of rc3 ) is used, where the interfaces align with the x-y
plane. Half of the box is filled with water beads and the other half is oil. The rigid core that is
initially placed in the water phase is driven to the interface by the thermal motion. Once the core
enters the interfacial region, which is dictated by the presence of water/oil density gradient, it
breaches the interface due to its favorable interaction with the oil beads, and obtains an equilibrium
position almost instantaneously (it takes less than 100 dimensionless units of time) under surface
tension. It is noteworthy that the electrostatic interactions in experiments can prevent the particle
from breaching the interface, and the relaxation to equilibrium can be slow due to nanoscale
surface heterogeneities or defects.1,2 The fast breaching and equilibration observed herein are
attributed to the fact that the rigid core used in our simulations is rather smooth and no electrostatic
interactions are considered.
It is well understood that the adsorption requires the particle to be wetted by both liquids.3,4
To quantitatively characterize the degree of hydrophobicity of the core, we define the three-phase
contact angle  with respect to the water phase as shown in Figure S1. The value of  is thus
calculated as a function of the equilibrium positions of the core center, the interface position, and
the effective core radius, which is given by the position of the first peak in the radial distribution
function between the core and water beads (see Figure S2). For the hydrophobic core used in our
simulations, the effective radius is approximately 2.8 rc . By varying the core-water interaction
parameter aco-w, we effectively adjust the hydrophobicity of the core. As shown in Figure S1b, the
contact angle increases when aco-w increases; in other words, when the core becomes more
hydrophobic. By tuning aco-w, we are able to obtain a wide range of contact angles to represent
rigid cores with different wettabilities by water.
According to Young’s equation, the surface tension of an oil-water interface should also
influence the contact angle. Thus, we model the adsorption of rigid cores for different interaction
parameters between oil and water, ao-w, which determines the interfacial tension.5 Figure S3a
shows no discernible dependence of contact angle on the oil-water interaction parameter in the
range from 50 to 200 k BT / rc . Notably, the cores do not bind to the interface irreversibly when the
interfacial tension is weak, e.g., ao-w = 40. Instead, the hydrophobic cores are perturbed by thermal
fluctuation to detach from the interface and shuttle to the bulk oil phase as demonstrated in Figure
S3b. On the other hand, if the surface tension is large, the cores irreversibly adsorb to the interface
and then are randomly distributed at the interface (Figure S3b).
References
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(1)
Kaz, D. M.; McGorty, R.; Mani, M.; Brenner, M. P.; Manoharan, V. N. Physical Ageing
of the Contact Line on Colloidal Particles at Liquid Interfaces. Nat. Mater. 2012, 11, 138–
142.
(2)
Colosqui, C. E.; Morris, J. F.; Koplik, J. Colloidal Adsorption at Fluid Interfaces: Regime
Crossover from Fast Relaxation to Physical Aging. Phys. Rev. Lett. 2013, 111.
(3)
Binks, B. P. Particles as Surfactants—similarities and Differences. Curr. Opin. Colloid
Interface Sci. 2002, 7, 21–41.
(4)
Garbin, V.; Crocker, J. C.; Stebe, K. J. Nanoparticles at Fluid Interfaces: Exploiting
Capping Ligands to Control Adsorption, Stability and Dynamics. Journal of Colloid and
Interface Science, 2012, 387, 1–11.
(5)
Groot, R. D.; Warren, P. B. Dissipative Particle Dynamics: Bridging the Gap between
Atomistic and Mesoscopic Simulation. J. Chem. Phys. 1997, 107, 4423–4435.
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Supplementary Figures
Figure S1. (a) A spherical particle adsorbed at a planar oil-water interface has an equilibrium
position determined by the contact angle  (measured with respect to the aqueous phase, such that
 < 90 corresponds to a particle preferentially wetted by water). (b) Contact angle of the
hydrophobic rigid core at the oil-water interface with ao-w = 100 as a function of the core-water
interaction parameter aco-w, while keeping the core-oil interaction parameter aco-o as 25. The two
insets are the order parameter contours in the y-z plane across the center of the rigid core. Error
bars indicate the temporal variations of the instantaneous contact angle.
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Figure S2. Radial distribution functions between the rigid core and water beads as a function of
core-water interaction parameter aco-w , which dictates the hydrophobicity of the core. The inset
is the water density contour in the y-z midplane across the center of the rigid core for aco-w = 100.
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Figure S3. (a) Contact angle of the hydrophobic rigid core (aco-w = 35) as a function of oil-water
interaction parameter ao-w, which determines the surface tension of the interface. The inset is the
order parameter contour in the y-z plane across the center of the rigid core. (b) Snapshots of the
equilibrium states of oil-water biphasic systems containing 50 rigid cores for different oil-water
interaction parameters ao-w. Pink and cyan dots represent oil and water, respectively. Error bars
indicate the temporal variations of the instantaneous contact angle.
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Figure S4. (a) Dependence of time-averaged radius of gyration of the polymer-grafted nanoparticle
on the grafted chain length Lc. (b) Mean square displacement of the polymer-grafted nanoparticle
in bulk water as a function of the polymer chain length Lc. Each line is an average for four
independent runs. (c) Diffusion coefficient of the polymer-grafted nanoparticle is inversely
proportional to the square root of its radius of gyration. The red solid line represents a least square
fitting. Error bars indicate the variations among four independent runs.
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Figure S5. (a) Time evolution of the positions of the geometric centers of polymer-grafted
nanoparticles with Lc = 10 in the z-direction. The solid lines represent the trajectories of the
nanoparticles that adsorbed to either of the two interfaces. The dotted lines represent the
trajectories of those remaining in the bulk. (b) Snapshots of the system at time t = 60000. The inset
is the top-down view of the upper interface.