Molecular simulation of nanoparticle diffusion at ﬂuid interfaces

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In Molecular simulation of nanoparticle diffusion at ﬂuid interfaces molecular dynamics simulations are used to exam- ine the transport properties of a nanoparticle in both bulk solvent and at a liquid–liquid interface. Speciﬁcally it aims to address the effect of interfacial adsorption on the diffusion of nanoparticles (at low concentration).

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Nội dung Text: Molecular simulation of nanoparticle diffusion at ﬂuid interfaces

Chemical Physics Letters 495 (2010) 55–59 Contents lists available at ScienceDirect Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett Molecular simulation of nanoparticle diffusion at ﬂuid interfaces D.L. Cheung Department of Chemistry and Centre for Scientiﬁc Computing, University of Warwick, Coventry CV4 7AL, UK a r t i c l e i n f o a b s t r a c t Article history: Using molecular dynamics simulations the transport properties of a model nanoparticle in solution are Received 7 April 2010 studied. In bulk solvent the translational diffusion coefﬁcients are in good agreement with previous sim- In ﬁnal form 24 June 2010 ulation and experimental work, while the rotational diffusion is more rapid than in previous simulations. Available online 30 June 2010 When the nanoparticle is adsorbed at a liquid–liquid interface it becomes strongly attached to the inter- face. This leads to highly anisotropic motion with in-plane diffusion being several orders of magnitude larger than out-of-plane diffusion. By contrast the rotational diffusion is only slightly changed when the particle is adsorbed at the interface. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction lations have addressed the stability of nanoparticles at interfaces [13–17], interactions between nanoparticles [18], orientational Motivated by the long-standing observation that solid particles behaviour of anisotropic nanoparticles [19,17], and the self-assem- may adhere to liquid interfaces [1,2] the behaviour of nanoparti- bly of nanoparticles at ﬂuid interfaces [20,21]. These simulations cles at ﬂuid interfaces has, over the past few years, become an area have largely focused on the static properties of nanoparticles at of active interest [3]. Much of this interest has been driven by the interfaces, while only recently has simulation be turned to the dy- potential use of liquid interfaces as a template for the formation of namic properties of nanoparticles at liquid interfaces [22]. dense nanoparticle structures, such as monolayers and mem- In this paper molecular dynamics simulations are used to exam- branes. The modiﬁcation of interfacial properties by the adhesion ine the transport properties of a nanoparticle in both bulk solvent of nanoparticles may also be used to stabilize the formation of and at a liquid–liquid interface. Speciﬁcally it aims to address the large-scale structures. effect of interfacial adsorption on the diffusion of nanoparticles It has long been recognized that the dynamic and transport (at low concentration). Understanding the translational diffusion properties of nanoparticles, in the bulk [4–8] and at interfaces, of nanoparticles at liquid interfaces, in particular in relation to par- may be signiﬁcantly different to those of larger particles. The trans- ticle size, is important in their use as tracer particles; due to the lational motion of particles at interfaces is important in under- highly inhomogeneous nature of the interfacial region the diffusive standing the interfacial self-assembly processes and in the use of behaviour of nanoparticles may be different to that in the bulk. nanoparticles as tracer particles in micro- and nanorheology Study of the rotational diffusion of nanoparticles, which has been [9,10]. The orientational dynamics of nanoparticles is also worthy poorly studied bulk solvents as well as at interfaces, will provide of investigation. In particular functionalization of particles ad- a guide to the applicability of interfacial techniques of particle syn- sorbed on liquid–liquid interfaces has been used as a route to the thesis [11] to nanoparticles and to nanoparticle catalysis. The formation of micron-sized Janus particles [11]; the extension of applicability of the continuum Stokes–Einstein(–Debye) relations this technique to smaller particles, however, relies on the rota- to nanoparticles at interfaces will also be investigated. The simula- tional diffusion of the particles being sufﬁciently slower than the tion methodology and model is outlined in the following section, reaction rates. The orientational motion of nanoparticles also has along with description of the analysis of the simulation trajecto- implications for their use as catalysts [12]; as edges and points ries. The results are then presented in Section 3 before some brief on nanoparticle surfaces are typically the most active sites for conclusions in Section 4. catalysis the orientation of these sites relative to the interface will determine their activity. 2. Methodology Due to the ability of molecular simulations to directly access the nanometre lengthscales appropriate to nanoparticles, it provides a 2.1. Model and simulation details natural tool for the investigation of these systems. Previous simu- The solvent was modelled as a Lennard-Jones ﬂuid, interacting E-mail address: david.cheung@warwick.ac.uk with the potential 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.06.074
56 D.L. Cheung / Chemical Physics Letters 495 (2010) 55–59 Table 1 C ‘ ¼ hP‘ ð^eðtÞ ^eð0ÞÞi; ð6Þ Summary of principal properties of the nanoparticles studied in this work. Rc is P nanoparticle radius, N cluster is number of atoms in nanoparticle, hR2g i ¼ ð1=N cluster Þ i r 2i where P‘ ðxÞ is a Legendre polynomial and e ^ðtÞ is the vector joining is the nanoparticle radius of gyration and I is the nanoparticle moment of inertia. the two atoms farthest apart in the nanoparticle. At long times Rc =r N cluster hR2g i=r2 I=mr2 C ‘ ðtÞ decays exponentially ðC ‘ ðtÞ / expðt=su Þ [7]) and for ‘ ¼ 1 the decay time su may be related to the Dr via [27] 1.5 55 1.31 48 2.0 141 2.47 232 1 1 2.5 249 3.64 604 Dr ¼ lim log C 1 ðtÞ ¼ ; ð7Þ 2 t!1 2su 3.0 459 5.46 1672 giving us two routes to the rotational diffusion coefﬁcient. It should be noted that the ﬁrst route, through C x ðtÞ is more convenient as it ( h 6 i 12 involves the integral of a rapidly decaying function [4]. 4 rr rr r 6 r cut ; VðrÞ ¼ ð1Þ 0 r > r cut ; 3. Results pﬃﬃﬃ with rcut ¼ 2:5r for like particles and r cut ¼ 6 2r for unlike particles (i.e. the interaction is cut-off at the potential minima). The nanopar- 3.1. Nanoparticles in bulk solvent ticle was modelled as a rigid cluster of Lennard-Jones sites, con- structed from a FCC lattice, containing particles within a distance We ﬁrst present results for nanoparticles in a one-component Rc of the origin (details of the simulated nanoparticles are presented bulk solvent. Fig. 1 shows the solvent density proﬁle around nano- in Table 1). The interaction between the sites in the nanoparticle particles in a bulk solvent. For the smallest particle considered the pﬃﬃﬃ and the solvent particles is given by Eq. (1) with r cut ¼ 6 2r. For surrounding ﬂuid is highly structured, with a number of peaks simulations of the nanoparticle in bulk solvent, the nanoparticle (spacing 1r). On increasing the Rc the ﬂuid structure becomes was initially placed in the centre of the simulation cell and the sol- weaker; for the largest particle Rc ¼ 3r the density proﬁle tends vent particles were randomly placed around it, while for the inter- monotonically to a constant value far from the particle surface. face simulations, the nanoparticle was placed in the interface The centre-of-mass motion of the nanoparticle, as quantiﬁed by between the two liquid components. the mean-squared-displacement ðDr 2 Þ and velocity autocorrelation The system was studied using constant-NVT molecular dynam- function ðC v ðtÞÞ are shown in Fig. 2. As may be expected the trans- ics simulations, using the LAMMPS simulation package [23]. All sim- lational motion of the nanoparticle depends on the particle radius ulations were performed at temperature T ¼ kB T= ¼ 1 (using a with smaller particles diffusing more rapidly (Fig. 2a). The velocity Nosé–Hoover thermostat [24]) and solvent density autocorrelation function (Fig. 2b) decays more rapidly for the q ¼ qr3 ¼ 0:69. For the single component system this is well smallest particles. The rotational motion of the particle (Fig. 2c within the liquid region for the truncated Lennard-Jones ﬂuid and d) also shows a strong dependence on particle size, with both [25] and this density is well above the demixing transition for the orientational correlation function ðC 1 ðtÞÞ and angular velocity the two-component mixture. A timestep of ds ¼ 0:005s was used correlation function ðC x ðtÞÞ decaying more slowly as Rc increases. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s ¼ mr2 = and the simulations consisted of 6 106 MD steps The diffusion coefﬁcients are summarized in Table 2. For large (106 MD steps equilibration and 5 106 steps for data gathering). particles the Dt may be given by the Stokes–Einstein (SE) relation kB T 2.2. Analysis Dt ¼ ; ð8Þ nT pRc g The translational motion of the nanoparticle may be analyzed where g is the viscosity and nT depends on the boundary conditions through the usual self-diffusion coefﬁcient (for slip boundaries nT ¼ 4 while for stick boundaries or nT ¼ 6). For the present system the viscosity, calculated from RNEMD simula- 1 d 1 d tions [28], is g ¼ 1:1mr1 s1 , giving Dt ¼ 0:032 0:016r2 s1 for Dt ¼ lim hjrc ðtÞ r c ð0Þj2 i ¼ lim Dr2 ; ð2Þ 6 t!1 dt 6 t!1 dt stick boundary conditions and Dt ¼ 0:048 0:024r2 s1 for slip, in where r c ðtÞ is the centre-of-mass position of the nanoparticle at reasonable agreement with simulation. This is contrary to recent time t and the angled brackets denote averaging over different time simulation work [8]. This difference may arise due to the lack of origins. Dt is extracted from the slope of the mean-squared diffusion at long times. Dt may also be found from the velocity autocorrela- 1.25 tion function C v ðtÞ ¼ hv ðtÞ v ð0Þi from [26] Z 1 1 1 Dt ¼ dtC v ðtÞ: ð3Þ 3 0 The orientational motion of the nanoparticle may be quantiﬁed 0.75 g(r) in terms of the angular velocity correlation function 0.5 C x ðtÞ ¼ hxðtÞ xð0Þi: ð4Þ Analogously to translational motion, the integral of C x ðtÞ may 0.25 be used to deﬁne the rotational diffusion coefﬁcient Z 1 0 kB T C x ðtÞ kB T 0 2 4 6 8 Dr ¼ dt ¼ sx ; ð5Þ I 0 C x ðt ¼ 0Þ I r-Rc / σ where I is the particle moment of inertia (as the particles are rigid Fig. 1. Normalized solvent density proﬁles around nanoparticle with radius and approximately spherical I ¼ Ixx ¼ Iyy ¼ Izz ) and sx is the angular Rc ¼ 1:5r (solid curve, black), Rc ¼ 2r (dotted curve, red), Rc ¼ 2:5r (dashed curve, velocity relaxation time. The orientational motion may also be char- green), Rc ¼ 3r (dot-dashed curve, blue). (For interpretation of the references to acterized through the orientational correlation function colour in this ﬁgure legend, the reader is referred to the web version of this article.)
D.L. Cheung / Chemical Physics Letters 495 (2010) 55–59 57 a b [22]. The present results are also in good agreement with experi- 120 1 0.04 mental measurements, with Dt 0:2 cm2 s1 for 1 nm dye particles 100 0.8 Cv (t)/Cv(0) Cv(t) 80 0.02 [32]. Experimentally the orientational behaviour of nanoparticles is 2 0.6 Δr / σ 60 0 less well characterized; many previous studies have been on ferro- 0.4 0 5 10 15 2 40 ∗ ﬂuids in which, for the particle sizes considered in this work, the ori- 0.2 t/τ 20 entational behaviour is dominated by magnetic interactions [33]. 0 0 For larger particles the reorientation times are typically of the order 0 100 200 300 400 500 0 10 20 30 40 c 1 d 1 of 1–100 ns for a 20 nm particle; as the orientational relaxation 0.8 0.8 0.04 Cω (t) Cω(t)/Cω(0) times scale as R3c this is consistent with the simulated reorientation 0.02 0.6 0.6 C1 (t) times of 50–150 ps, which are also in the same range as previous 0 0.4 0.4 0 5 10 15 20 simulation results [7]. ∗ 0.2 0.2 t/τ 0 0 3.2. Nanoparticles at liquid interfaces 0 50 100 150 200 250 0 20 40 60 80 ∗ ∗ t/τ t/τ When the nanoparticle is adsorbed at a liquid–liquid interface the solvent structure around the particle becomes highly aniso- Fig. 2. (a) Mean-squared displacement of uniform nanoparticle with radius Rc ¼ 1:5r (solid curve, black), Rc ¼ 2r (dotted curve, red), Rc ¼ 2:5r (dashed curve, tropic. Two-dimensional maps of the solvent around the nanopar- green), Rc ¼ 3r (dot-dashed curve, blue). (b) Normalized velocity autocorrelation ticle (Fig. 3a) show a sizeable depletion region between the two function for uniform nanoparticle (symbols as in (a)). Inset shows unnormalized solvent components [15], with the interface width being compara- C v ðtÞ. (c) Orientation correlation function C 1 ðtÞ. Symbols as in (a). (d) Normalized ble to the nanoparticle size for Rc ¼ 1:5r. The width of the interface angular velocity correlation function C x ðtÞ. Symbols as in (a). Inset shows unnormalized C x ðtÞ. (For interpretation of the references to colour in this ﬁgure may be estimated from the density proﬁles of the individual com- legend, the reader is referred to the web version of this article.) ponents, speciﬁcally from the distance between the points at which the density of the A component is 90% and 10% of its bulk value [34], yielding an interfacial width w ¼ 1:97r. In experimen- Table 2 tal units this corresponds to a w 6:7 Å, which is typical for simple Diffusion coefﬁcients (translational and rotational) and rotational correlation times ﬂuid interface. The perturbation in the ﬂuid caused by the nano- for nanoparticles in bulk solvent. Figures in parentheses give estimated error in last particle may also be seen; in particular the presence of a high den- quoted digit. sity layer near the surface of the particle. The layering of the Rc =r DMSD t =r2 s1 DVCF t =r2 s1 Dur =s1 Dx r =s 1 su =s sx =s solvent may be seen more clearly in the normalized density pro- 1.5 0.038(5) 0.037(5) 0.075(1) 0.07(2) 6.7(3) 3.5(4) ﬁles (Fig. 3b). It is noticeable that at the interface there is a peak 2.0 0.026(4) 0.024(4) 0.031(3) 0.03(1) 16.1(7) 7.2(4) in the density proﬁles near the particle surface for all Rc , whereas 2.5 0.017(2) 0.018(3) 0.014(3) 0.02(1) 26. (2) 10.5(4) for in bulk solvent (in Fig. 1 the peak is absent for larger particle 3.0 0.013(1) 0.013(2) 0.010(2) 0.010(2) 50. (5) 17.5(6) radii). a attractive interactions between the nanoparticle and solvent parti- cles in this work or differences in viscosity. Despite the lack of attractive interactions the SE relation with stick boundary condi- tions gives a better agreement with simulation, due to the solvent structure around the nanoparticle. The values of Dt are in good agreement with previous simulations [4]. The Stokes–Einstein–De- bye relation kB T Dr ¼ ð9Þ nR pR3c g predicts that the rotational diffusion coefﬁcient should scale as R3 c , which is well reproduced by particles studied in this work. Use of nR ¼ 8, appropriate for stick boundaries, gives values of Dr substan- tially smaller than calculated from simulation (Dr 0:010s1 for b 1 Rc ¼ 1:5r and 0:001s1 for Rc ¼ 3r), while the simulation results are recovered for nr 1:19. This rescaling has been previously ob- 0.8 served in molecular simulations [29] and for tracer diffusion in col- loidal suspensions [30]. The values of Dr calculated here are 0.6 g(r) somewhat larger than those from previous simulations [4]. The ori- entational ðsu Þ and angular velocity ðsx Þ relaxation times are also 0.4 listed in Table 2. su is in all cases signiﬁcantly larger than sx and 0.2 is in good agreement with previously reported values for similar sized nanoparticles [4]. sx is signiﬁcantly larger; this may be due 0 to the larger density of cluster or the neglect of the cluster’s internal 0 2 4 6 8 degrees of freedom in this work. r-Rc / σ The present results may be compared to those of atomistic sim- ulations and experimental measurements. Taking parameters for li- Fig. 3. (a) Solvent density maps around Rc ¼ 1:5r (left) and Rc ¼ 3r (right) quid argon ðr ¼ 3:395 Å; =kB ¼ 116:8 K; m ¼ 39:948 amu, and nanoparticles. (b) Normalized density proﬁles (for A-component) around nanopar- ticle of radius Rc ¼ 1:5r (solid line, black), Rc ¼ 2r (dotted line, red), Rc ¼ 2:5r s 2:2 ps [31]). The simulated translational diffusion coefﬁcients (dashed line, green), and Rc ¼ 3r (dot-dashed line, blue). (For interpretation of the are then Dt 2 105 cm2 s1 ðRc ¼ 1:5rÞ to Dt ¼ 0:5 105 references to colour in this ﬁgure legend, the reader is referred to the web version of cm2 s1 ðRc ¼ 3rÞ, in good agreement with previous simulations this article.)
58 D.L. Cheung / Chemical Physics Letters 495 (2010) 55–59 a 60 b magnitude larger than in Ref. [16]), giving bDF ¼ 8:5 33:9, 0.06 although it has been shown that Eq. 12 is likely to underestimate 50 0.05 40 0.04 the detachment energy, due to the neglect of line tension and cap- 30 0.03 illary waves [16,17]. Similar conclusions may be drawn from the 20 0.02 velocity autocorrelation function (Fig. 4), with C kv ðtÞ differing mark- 10 0.01 edly from C ? ? v ðtÞ. For all Rc C v ðtÞ has a negative region, indicating Δr 2 / σ 2 0 Cv (t) 0 that the particle is oscillating in the vicinity of the interface. 0 50 100 150 200 0 10 20 30 40 50 30 The translational diffusion coefﬁcients for the nanoparticle ad- 25 0.006 sorbed at the interface are presented in Table 3. These are in good 20 agreement with previous simulations [22] and ﬁnd that the in- 0.004 15 plane diffusion coefﬁcients are larger that the nanoparticle diffu- 10 0.002 sion coefﬁcient in bulk solvent. This increased diffusion may arise 5 0 0 due to the lower viscosity of the interfacial region; from RNEMD 0 50 100 150 200 0 20 40 60 80 100 simulations [28,36] the viscosity in the interfacial region is found t/τ t/τ to be approximately four times smaller that in the bulk ðgI 0:25mr1 s1 Þ; it should be noted the shear rate in the inter- Fig. 4. (a) Mean-squared-displacements for nanoparticle of radius Rc ¼ 1:5r (top) and Rc ¼ 3r (bottom) at liquid–liquid interface. Solid line (black) denotes total facial region is rapidly varying, so the effective viscosity acting on MSD, dotted line (red) MSD in plane of interface, dashed line (green) MSD the nanoparticle is larger than this, though still smaller than in the perpendicular to interface and dot-dashed line (blue) MSD for nanoparticle in bulk bulk solvent. Alternatively as the overall diffusion coefﬁcient is lar- solvent. (b) Velocity ACF for nanoparticle of radius Rc ¼ 1:5r (top) and Rc ¼ 3r gely unchanged when the particle is adsorbed at the interface, the (bottom) at liquid–liquid interface. Solid line (black) denotes C v ðtÞ, dotted line (red) C xy z in-plane diffusion coefﬁcient increases due to the lack of diffusion v ðtÞ, dashed line (green) C v ðtÞ and dot-dashed line (blue) C v ðtÞ for nanoparticle in bulk solvent. (For interpretation of the references to colour in this ﬁgure legend, the normal to the interface. reader is referred to the web version of this article.) Experimental measurements ﬁnd much smaller (one to two or- ders of magnitude) diffusion coefﬁcients for CdTe nanoparticles on water–toluene interface [37]. These were performed on nanoparti- Due to the presence of the interface, the motion of the nanopar- cle monolayers rather than for a single nanoparticle in the present ticle becomes anisotropic. It is useful to consider the motion of the case and are performed in solvents of different viscosities which nanoparticle parallel and perpendicular to the interface lead to a lower diffusivity [38]. 1 D 2 E 2 Adsorption at the liquid–liquid interface may also affect the Dkt ¼ ðxðtÞ xð0ÞÞ þ ðyðtÞ yð0ÞÞ ; ð10aÞ rotational motion of the nanoparticle. Shown in Fig. 5a is the orien- 4 1 D 2 E tational correlation function C 1 ðtÞ, for nanoparticles at the inter- D?t ¼ ðzðtÞ zð0ÞÞ : ð10bÞ face. For the smallest nanoparticles studied there is little 2 difference between the rotational diffusion in the bulk and at the Similarly the velocity autocorrelation function may be decom- interface with the decay being exponential in both cases. For posed into Rc ¼ 3r there is a difference between C 1 ðtÞ at the interface and Z 1 Z 1 k 1 bulk, which is most noticeable at long times where there appears Dkt ¼ dtC v ðtÞ ¼ dthv xy ðtÞ v xy ð0Þi; ð11aÞ to be a deviation from an exponential decay for the particle at 2 0 2 Z 1 Z the interface. The variation in C x ðtÞ is consistent with this ? D?t ¼ dtC v ðtÞ ¼ dthv z ðtÞ v z ð0Þi: ð11bÞ (Fig. 5b). For Rc ¼ 1:5r C x ðtÞ decays more slowly at the interface, 0 indicating that the rotational diffusion is slightly more rapid. This Shown in Fig. 4a is the mean-squared displacement for nano- situation reverses as Rc increases. Considering separately the corre- particles adsorbed at the interface. The effect of the interface on lation functions of the x; y, and z components of x shows only a the nanoparticle diffusion may be clearly seen, with diffusion in slight difference between C ax ðtÞ ða ¼ x; y; zÞ, with C xx ðtÞ C yx ðtÞ the plane of the interface being signiﬁcantly larger than perpendic- and C zx ðtÞ showing a slightly slower decay. ular to the interface, consistent with the particle being strongly The calculated rotational diffusion coefﬁcients and orientational bound to the interface. This implies that the detachment energy relaxation times are presented in Table 3. Despite the differences in of the particle from the interface is signiﬁcantly larger the kB T even the orientational correlation functions between the bulk and inter- for the smallest particle studied, larger than in previous work face, Dr is very similar for both cases. For the Rc ¼ 1:5r particle Dr [16,17] due to the larger interfacial tension in this study. The and sx are larger and su is smaller at the interface indicating more detachment energy may be estimated from [35] rapid orientational motion. This more rapid orientational motion is due to the lower effective viscosity of the interfacial region. As the DF ¼ pR2 ð1 þ cos hÞc; ð12Þ particle size increases. As Rc increases the difference between the where c is the interfacial tension and h is the contact angle (which, interface and bulk results decreases, with Dinterfacer Dbulk r for the due to the symmetry between the two solvent components, is largest particles. The trend towards the bulk values for diffusion h ¼ p=2). In reduced units br2 c ¼ 1:2 (which is up to an order of coefﬁcients may be understood as the particle becomes larger than Table 3 Diffusion coefﬁcients and orientational relaxation times for nanoparticle at liquid–liquid interface. Figures in parenthesis give estimated error in last quoted digit. Rc =r MSD VACF Du Dx su sx Dt k Dt Dt k Dt 1.5 0.035(3) 0.053(4) 0.036(3) 0.053(4) 0.09(1) 0.09(1) 5.7(5) 4.3(1) 2.0 0.031(1) 0.047(2) 0.028(1) 0.042(2) 0.032(4) 0.027(1) 16. (1) 7.6(3) 2.5 0.024(2) 0.036(3) 0.025(2) 0.038(3) 0.018(4) 0.020(1) 28. (3) 12.7(4) 3.0 0.012(2) 0.018(2) 0.013(1) 0.019(2) 0.008(2) 0.010(1) 66. (6) 15.5(9)
D.L. Cheung / Chemical Physics Letters 495 (2010) 55–59 59 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a b of su and sx with srot ¼ I=kB T (the time taken for a free rotator 1 0.06 to rotate by one radian). ‘Small-step’ rotational diffusion is indi- 0.8 cated by srot sx and srot su [27]. While sx is smaller than 0.6 0.04 srot for all Rc , only for the largest cluster is su signiﬁcantly larger 0.4 than srot , implying that this model may not be appropriate for 0.2 0.02 the rotational diffusion of smaller clusters. The precise determina- Cω (t) 0 C1 (t) 0 tion of the reorientation mechanism may form the basis for future 0 10 20 30 40 50 0 10 20 30 40 50 1 work. 0.8 0.0015 The translational diffusion coefﬁcients found in this work are 0.6 comparable to experimental values. Rotational motion of nanopar- 0.001 0.4 ticles, both in bulk solvent and at interfaces, has not been so well 0.2 0.0005 characterized. In particular the orientational relaxation time, 0 which for the particles studied in this work is of the order of 0 0 50 100 150 200 250 0 10 20 30 40 50 100 ps, provides a guide to the rates of reaction for the patterning t/τ t/τ of nanoparticle surfaces or nanoparticle catalyzed reactions at interfaces. While the present study was restricted to (approxi- Fig. 5. (a) C 1 ðtÞ for nanoparticle of radius Rc ¼ 1:5r (top) and Rc ¼ 3r (bottom). Solid line (black) denotes nanoparticle at liquid–liquid interface and dotted line mately) spherical, more complex nanoparticle geometries are of (red) denotes nanoparticle in bulk solvent. (b) C x ðtÞ for Rc ¼ 1:5r (top) and Rc ¼ 3r interest for many applications and the method used may be easily (bottom) nanoparticles at liquid–liquid interface. Solid line denotes C x ðtÞ, dotted applied to these systems. line (red) C xx ðtÞ, dashed line (green) C yx ðtÞ, and dot-dashed line C zx ðtÞ. C x ðtÞ for bulk system shown by double-dot-dashed (magenta) line. (For interpretation of the Acknowledgements references to colour in this ﬁgure legend, the reader is referred to the web version of this article.) This work was funded by UK EPSRC and ERC and computational facilities were provided by the Centre for Scientiﬁc Computing, the interfacial width and more of its surface is in contact with University of Warwick. The author is grateful to Dr. Stefan Bon ‘bulk’ solvent. for useful discussions during this work. 4. Conclusions References [1] W. Ramsden, Proc. Roy. Soc. Lond. A 72 (1903) 156. In this paper the transport properties of spherical nanoparticles [2] S.U. Pickering, J. Chem. Soc. Trans. 91 (1907) 2001. have been studied using molecular dynamics simulations. In bulk [3] F. Bresme, M. Oettel, J. Phys. Cond. Matter 19 (41) (2007) 413101/1. solvent the translational diffusion coefﬁcients are comparable with [4] D.M. Heyes, M.J. Nuevo, J.J. Morales, J. Chem. Soc., Faraday Trans. 94 (1998) 1625. previous simulation and experimental studies. The rotational diffu- [5] D.M. Heyes, M.J. Nuevo, J.J. Morales, A.C. Branka, J. Phys. Cond. Matter 10 (45) sion however is somewhat more rapid than in previous work, (1998) 10159. which may arise due to the more dense nanoparticles studied in [6] A. Tuteja, M.E. Mackay, S. Narayanan, S. Asokan, M.S. Wong, Nano Lett. 7 (5) (2007) 1276. this work. [7] C.D. Daub, D. Bratrk, T. Ali, A. Luzar, Phys. Rev. Lett. 103 (20) (2009) 207801/1. When the particle is adsorbed at a liquid–liquid interface the [8] Z. Li, Phys. Rev. E 80 (6) (2009) 061204/1. translational diffusion becomes highly anisotropic, with Dkt D? t . [9] J. Wu, L.L. Dai, Appl. Phys. Lett. 89 (9) (2006) 094107/1. Indeed due to the strong attachment of the nanoparticle to the [10] J. Wu, L.L. Dai, Langmuir 23 (8) (2007) 4324. [11] L. Hong, S. Jiang, S. Granick, Langmuir 22 (23) (2006) 9495. interface, arising due to the decrease in the interfacial free energy [12] S. Crossley, J. Faria, M. Shen, D.E. Resasco, Science 327 (5961) (2010) 68. between the two solvent components, there is virtually no diffu- [13] F. Bresme, N. Quirke, Phys. Rev. Lett. 80 (1998) 3791. sion perpendicular to the interface. This effective two-dimensional [14] F. Bresme, N. Quirke, J. Chem. Phys. 110 (1999) 3536. [15] F. Bresme, N. Quirke, Phys. Chem. Chem. Phys. 1 (1999) 2149. diffusion leads to an in-plane diffusion coefﬁcient that is signiﬁ- [16] D.L. Cheung, S.A.F. Bon, Phys. Rev. Lett. 102 (6) (2009) 066103/1. cantly larger than the diffusion coefﬁcient in the bulk (as the diffu- [17] D.L. Cheung, S.A.F. Bon, Soft Matter 5 (20) (2009) 3969. sion normal to the interface is negligible the overall diffusion [18] F. Bresme, H. Lehle, M. Oettel, J. Chem. Phys. 130 (21) (2009) 214711/1. [19] F. Bresme, J. Faraudo, J. Phys. Cond. Matter 19 (37) (2007) 375110/1. coefﬁcient for the particle at the interface is essentially the same [20] M. Luo, O.A. Mazyar, Q. Zhu, M.W. Vaughn, W.L. Hase, L.L. Dai, Langmuir 22 as in the bulk). The in-plane diffusion coefﬁcient is affected by (14) (2006) 6385. the difference between the viscosities of the two solvent compo- [21] M. Luo, L.L. Dai, J. Phys. Cond. Matter 19 (37) (2007) 375109/1. [22] Y. Song, M. Luo, L.L. Dai, Langmuir 26 (1) (2010) 5. nents; the identical viscosities in the present case leads to more ra- [23] S. Plimpton, J. Comput. Phys. 117 (1) (1995) 1. http://lammps.sandia.gov. pid in-plane diffusion than if the solvent viscosities were different. [24] W.G. Hoover, Phys. Rev. A 31 (1985) 1695. This, along with the present simulations being performed on a sin- [25] B. Smit, J. Chem. Phys. 96 (1992) 8639. [26] J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids, second edn., Academic gle nanoparticle (effectively inﬁnite dilution), may explain the dif- Press, London, 1986. ference between the present simulation results and experimental [27] D.J. Tildesley, P.A. Madden, Mol. Phys. 48 (1983) 129. measurements on nanoparticle monolayers (in which the nanopar- [28] F. Muller-Plathe, Phys. Rev. E 59 (1999) 4894. ticle diffusion was slower on the interface). [29] M. Vergeles, P. Keblinski, J. Koplik, R. Banavar, Phys. Rev. Lett. 75 (2) (1995) 232. By contrast the rotational diffusion of the particle is almost iso- [30] G.H. Koenderick, Z. Zhang, D.G.A.L. Aarts, M.P. Lettinga, A.P. Philipse, G. Nägele, tropic. For the smallest particles studied the diffusion (both trans- Faraday Discuss. 123 (2003) 335. lational and rotational) is more rapid at the interface than in bulk [31] G.A. Fernandez, J. Vrabec, H. Hasse, Fluid Phase Equilibr. 221 (1–2) (2004) 157. [32] H. Ow, D.R. Larson, M. Srivastava, B.A. Baird, W.W. Webb, U. Wiesner, Nano solvent as the sit in the low-density region between the two ﬂuid Lett. 5 (1) (2005) 113. components. On increasing particle size the diffusion coefﬁcients [33] J. Connolly, T.G. St Pierre, J. Magn. Magn. Mater. 225 (1–2) (2001) 156. tend towards their bulk values, indicating that the rotational mo- [34] L.L. Dai, Mol. Simul. 35 (10–11) (2009) 773. [35] P. Pieranski, Phys. Rev. Lett. 45 (7) (1980) 569. tion of nanoparticles is the same at the interface and in the bulk. [36] T.J. Müller, F. Müller-Plathe, ChemPhysChem 10 (13) (2009) 2305. This tendency towards bulk-like behaviour may be understood as [37] Y. Lin, A. Böker, H. Skaff, D. Cookson, A.D. Dinsmore, T. Emrick, T.P. Russell, more of the particle surface is in contact with the bulk solvent. Langmuir 21 (1) (2005) 191. [38] M. Negishi, T. Sakaue, K. Yoshikawa, Phys. Rev. E 81 (2) (2010) 020901/1. The rotational diffusion may be characterized by taking the ratios