J. Phys. Chem. B 2006, 110, 4283-4290
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Orientation of a Nanocylinder at a Fluid Interface Eric P. Lewandowski,† Peter C. Searson,†,‡ and Kathleen J. Stebe*,†,‡ Department of Chemical & Biomolecular Engineering and Department of Materials Science & Engineering, Johns Hopkins UniVersity, Baltimore, Maryland 21218 ReceiVed: NoVember 2, 2005; In Final Form: December 28, 2005
A nanocylinder placed on a fluid interface can assume an end-on or side-on orientation, or it can immerse itself in the surrounding bulk phases. Any of these orientations can satisfy a mechanical force balance when the particle is small enough that gravitational effects are negligible. The orientation is determined by the surface energies of the fluid-solid, fluid-vapor, and vapor-solid surfaces. A comparison of the energy of each state allows phase diagrams to be defined in terms of the scaled aspect ratio x ) 2L/πr and the contact angle θo, where L and r denote the nanocylinder length and radius, respectively. Line tension can also influence the orientations by changing the equilibrium contact angle θ and by increasing the energetic cost of the contact line. Phase diagrams accounting for positive line tensions Σ are also constructed. These phase diagrams can be divided into two classes. In the first, over some range of x and Σ, nanocylinders can be driven from side-on to end-on orientations with increasing Σ. This transition terminates at a triple point where the sideon, end-on, and immersed energies are the same. In the second class, there is no triple point and, for a range of Σ values, nanocylinders of all aspect ratios x prefer an end-on orientation. In all cases, for high enough Σ, line tension drives a wetting transition similar to that already noted in the literature for spherical particles. The zero line tension predictions are compared favorably to experiment, in which functionalized gold nanowires made by template synthesis are spread at aqueous-gas interfaces, immobilized using a gel-fixation technique, and observed by scanning electron microscopy. The small aspect ratio particles (disks) were in an end-on configuration, while the longer nanowires were in a side-on orientation, in agreement with the theory.
Introduction Particles at fluid interfaces assume orientations dictated by their wetting boundary conditions and the forces applied to them. For example, when a spherical particle is placed on a planar fluid interface with a depth larger than the particle diameter, the liquid-vapor interface intersects the particle at the equilibrium contact angle. The weight of the particle causes the interface to deflect, creating an upward surface tension force on the particle that balances its weight. For sufficiently small particles, the interface remains flat, with the particle immersing itself to a depth at which it satisfies its contact angle. This is the zero Bond number limit, where the Bond number Bo is defined as the ratio of gravitational forces to surface tension forces acting on the particle. If line tensions are positive, they can drive wetting transitions at critical values for the line tension at which the energetic cost of the contact line becomes large enough to force the particles under the fluid.1,2 In the small Bo limit, particles create weak deflections of the liquid-vapor interface. If another particle is present on the interface within the capillary length over which the deflections persist, asymmetric menisci form which create net attractive forces between the particles.3-7 For spherical particles, these effects have been thoroughly studied and are exploited in a variety of applications ranging from Pickering emulsions,8-11 in which particles are used to stabilize emulsions, to colloidisomes,12,13 or liquid marbles,14 in which particle monolayers at interfaces impart strong * To whom correspondence should be addressed. E-mail:
[email protected]. † Department of Chemical & Biomolecular Engineering. ‡ Department of Materials Science & Engineering.
resistance to interface rupture. Classically, the retention of particles at vapor-fluid interfaces has also been exploited in froth flotation in the mining industry to collect partially wet particles of metal ore.15,16 Renewed interest in these systems has been generated by the advent of nanotechnology. For example, ordered nanoparticle aggregates (multilayers or monolayers) can be formed in evaporating thin liquid films17,18 either on solid supports or on liquid subphases, with subsequent deposition of the ordered structures onto solid substrates. Similarly, particles at interfaces are forced into ordered arrays by application of surface pressure on a Langmuir trough.19 Other particle assembly techniques that exploit particle association with interfaces rely on particles being trapped within a liquid lens placed on solid surfaces20-26 or in liquid drops in emulsions.27,28 The particles are drawn into ordered aggregates by contraction of capillary bridges upon removal of the suspending solvent by evaporation or other methods. In this paper, we study small cylindrical particles at planar fluid interfaces both experimentally and theoretically. Chan et al.16 analyzed cylinders in their side-on orientations in terms of the deflection that they create at fluid interfaces and the interactions between the particles caused by the interface deflections. Neumann and collaborators29,30 analyzed the sideon orientation in terms of mechanical and thermodynamic equilibrium for finite Bo, identifying critical radii beyond which particles become too heavy to remain at the surface. Prior theoretical work on axisymmetric bodies at fluid interfaces31 also considers the shape of the meniscus made by either a heavy sphere or cylinder side-on at the interface. Neumann and collaborators29,30 also studied theoretically cylinders with com-
10.1021/jp0563282 CCC: $33.50 © 2006 American Chemical Society Published on Web 02/10/2006
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plex wetting patterns and identified a series of metastable states resulting from the pinning of contact lines at wet-nonwet boundaries. We are interested in the behavior of a partially wet cylindrical particle at fluid-fluid interfaces, in particular, a cylinder that is finite in length in the limit of negligible Bo. The particle can assume either an end-on orientation, in which the cylinder axis is perpendicular to the undisturbed fluid interface, or a side-on orientation, in which the cylinder axis is parallel to the plane of the undisturbed fluid interface. Transitions between these states as a function of particle aspect ratio, contact angle, and line tension are considered theoretically. Experiments are performed as a function of particle aspect ratio to ascertain the orientation at the surface. Gold nanowires fabricated by templated electrodeposition and functionalized with octadecane thiol are spread at liquid-air interfaces, immobilized using a gelfixation technique, and imaged by scanning electron microscopy (SEM). Theory Consider a cylindrical particle of radius r and length L placed at a liquid-vapor interface that is planar in the absence of the particle. Let θ be the equilibrium contact angle formed at the three-phase contact line. The particle can orient itself either endon or side-on at the interface. An equilibrium force balance can be satisfied in either orientation; the orientation with the lower free energy is the one that is realized. Below, the mechanical force balances are discussed, with particular attention on the small Bo limit. This is followed by a free energy analysis of the particle to describe the phase transitions of the particle orientations at the interface in terms of surface energies of the particles and line tension at the three-phase contact line. Force Balance on a Particle in the End-on Orientation. The vertical force balance requires the sum of the weight of the particle, the buoyant force of the surrounding fluids, and the surface tension force exerted at the three-phase contact line to be zero. For the end-on orientation, let y be the length of the cylinder that lies beneath the liquid-vapor interface. The force balance then requires
πr2g{(FV - FS)L + y(FL - FV)} - 2πrγLV cos θ ) 0 (1a) where Fi (i ) S, L, V) denotes densities, with subscripts S, L, and V referring to the solid particle, the liquid, and the surrounding vapor phases, respectively, and γLV denotes the liquid-vapor surface tension. This equation can be recast into nondimensional form
{( ) (
)}
FV y FL FV Bo FS L 12 FL r FS r FS FS
) cos θ
(1b)
where Bo is defined as
FLgr2 Bo ) γLV
(1c)
This parameter is typically negligible for small particles, so the force balance requires
cos θ ) 0
(1d)
This dictates that the interface must be flat, since gravitational forces that would bend the interface are negligible. Furthermore, it requires that only particles with contact angles θ ) 90° can exist at equilibrium in a planar fluid interface intersecting along
Figure 1. Nanocylinder orientations that satisfy the mechanical force balance. (a) The end-on orientation for θ > 90°. (b) The end-on orientation for θ < 90°. (c) The side-on orientation for θ < 90°. Along the cylindrical faces, the nanoparticle immerses to form a wetting angle φ. (d) The side-on orientation for θ < 90°. A saddle-shaped meniscus with zero net curvature forms at the ends so that FγLV,ends/2LγLV is zero.
the length of the cylinder. The depth of immersion is not determined in this case, since the surface energy of the liquidsolid and vapor-solid surfaces are identical for this circumstance. For all other contact angles, the particle can achieve equilibrium in a planar interface only by immersing itself either up to the top edge of the particle defined by the intersection of the cylindrical sides and the circular top of the particle for θ < 90°, so that y ) L, as shown in Figure 1a, or to the bottom edge of the particle θ > 90°, so that y ) 0, as shown in Figure 1b. At these edges, the contact angle is not defined in terms of the balance of the surface energies but rather remains pinned and pivots to satisfy the macroscopic force balance which requires θ ) 90°. Force Balance on a Particle in the Side-on Orientation. In a side-on orientation, a cylinder can immerse itself so that the contact line along the axis intersects the cylinder with a wetting angle φ defined from the submerged pole to the interface, as shown in parts c and d of Figure 1. The vertical force balance requires
{r2Lg[(FV - FS)(π - φ) + (FL - FS)φ] + Fgrav,ends} 2γLVL sin(φ + θ) + FγLV,ends ) 0 (2a) where Fgrav,ends and FγLV,ends denote gravitational and surface tension forces associated with the menisci present at the ends of the cylinder. This expression can be recast into dimensionless form
{( ) ( )
}
FV FL FV φ Fgrav,ends FS 1 -1 + Bo π + 2 FL FS FS FS π F gπr2L S FγLV,ends sin(φ + θ) + ) 0 (2b) 2LγLV When Bo is negligible
FγLV,ends 2LγLV
- sin(φ + θ) ) 0
(2c)
Equation 2c can be solved if the fluid interface is planar along
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the cylindrical faces intersecting the cylinder with contact angle θ, so that
φ + θ ) 180°
(2d)
and if
FγLV,ends 2LγLV
)0
(2e)
Equations 2d and 2e can both be satisfied for θ ) 90° since the interface can remain planar along the cylindrical sides as well as at the ends of the cylinder. However, when θ < 90°, the interface cannot remain planar at the ends. Rather, the liquid will climb the ends, locally deforming the interface. Since gravity is too weak to balance this local capillary rise, the fluid will climb until the end faces of the cylinder are completely wet. Even in this circumstance, FγLV,ends/2LγLV can be zero if a saddle-shaped meniscus is formed at the ends with zero net curvature, as shown in Figure 1c. A similar analysis can be made for θ > 90°. The saddle-shaped menisci at both ends of the cylinder match to the otherwise planar interface along the length of the cylinder via a zero-curvature transition region. This transition region allows the liquid-vapor surface to bend smoothly upward to meet the menisci on the ends. In the free energy analysis of cylindrical rods developed below, we assume the menisci adopt such saddle shapes and that the cylindrical regions lie in a planar interface. The transition region is not analyzed further. Modified Young Equation for Side-on Cylindrical Particle. In the absence of line tension, the contact angle θo is determined by the balance of energies at the three-phase contact line
cos θo )
γVS - γLS γLV
(3)
The effects of line tension σ in modifying this balance for cylinders in a side-on orientation for contact lines constrained to move in the φ direction can be derived as follows for partially wet particles. The free energy of the particle at the surface can be expressed for θ < 90°
aSO ) γVS2(π - φ)rL + γLS(2φrL + 2πr2) + γLV(As - 2rL sin φ) + σ(4(π - φ)r + 2L) (4) Differentiating this expression with respect to φ and setting it equal to zero yields a balance of forces at the three-phase contact line
cos θ ) cos θ0 +
4Σ πx
(5a)
where
in eq 2d are implicit in that the prevailing equilibrium contact angle θ is by determined θo, Σ, and x according to eq 5a. Free Energy Analysis of Particle Orientations. For θ ) 90°, the cylinder can situate itself either end-on or side-on or immerse itself in the liquid phase. The cylinder can also expel itself from the liquid to rest in the vapor phase. The Helmholtz free energy for each orientation is denoted aE, aSO, aU and aO, where the subscripts E, SO, U, and O denote end-on, side-on, under (or the submerged state), and over (corresponding to the particle in upper phase), respectively. The free energies can be written in terms of the respective surface energies for each interface and the line tension can be given by σ.
aE ) γVS2πr(L - y) + γLS2πry + γLV(As - πr2) + πr2(γVS + γLS) + σ2πr (6a) aSO ) γVS2(π - φ)rL + γLS2φrL + γLV(As - 2rL sin φ) + πr2(γVS + γLS) + σ(4r + 2L) (6b) aU ) γLS(2πrL + 2πr2) + γLVAs
(6c)
aO ) γVS(2πrL + 2πr2) + γLVAs
(6d)
These equations can be recast into dimensionless form by dividing through by the reference energy associated with the hole made by the cylinder in the end-on state in the liquidvapor interface, πr2γLV
aSO′ )
( ) (
) (
(7a)
2γVS L As + 1 + 2 - 1 + 2Σ γLV r πr
(7b)
a U′ )
γLS 2L As +2 + 2 γLV r πr
(7c)
a O′ )
γVS 2L As +2 + 2 γLV r πr
(7d)
a E′ )
(
)
( (
) )
In these expressions, the primes indicate a dimensionless energy, and As denotes the area of the liquid-vapor interface in the absence of the particle. For θ ) 90°, the surface energies of the vapor-liquid and vapor-solid surfaces are equal, so that the U and O states are energetically equivalent. To establish the conditions under which particles remain at the fluid interface, we compare the free energies at the interface to the free energy of the submerged particles, i.e., ∆aE-U′ and ∆aSO-U′
∆aE-U′ ) aE′ - aU′ ) -1 + 2Σ σ Σ) rγLV
(5b)
and the aspect ratio of the nanocylinder is
2L x) πr
(5c)
For high aspect ratio cylinders, θ approaches θo. A similar analysis can be made for θ > 90°, which again yields eq 5a. For θ ) 90°, such an argument yields a contact angle independent of Σ. The effects of line tension on the force balance
)
2γVS As 4 2L L 2L 1+ + +Σ + γLV r π πr πr2 πr
∆aSO-U′ ) aSO′ - aU′ ) -x + Σ
(π4 + x)
(8a) (8b)
When these energy differences are negative, the interface is the lower energy state. This is always the case for Σ ) 0. By remaining at the interface, the particle creates a hole in the liquid-vapor interface that reduces the energy of the system. For positive line tension, there is a critical value in for line tension that drives a wetting transition for the particle from the end-on state to under the interface (Σc,E-U) or from the side-on state to under the interface (Σc,SO-U). These critical line tensions
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are defined when the energy cost of creating the contact line exactly balances the energy decrease created by having the particle span an area of liquid-vapor interface
Σc,E-U ) Σc,E-O ) Σc,SO-U ) Σc,SO-O )
1 2
(9a)
x 4 +x π
4 +x-2 π
(
)
(10)
When ∆aSO-E′ ) 0, particles can orient in either the end-on or side-on state. For Σ ) 0, this coexistence criterion is simply given by x ) 1. For x > 1, the hole created in the liquidvapor interface is larger in the side-on state, which is then preferred. For x < 1, the hole created in the liquid-vapor interface in the end-on state is larger, so end-on is preferred. Line tension changes this criterion by imposing an energetic cost to the contact line that resists this transition for large x, favoring the end-on state. Below, we develop the free energy arguments for arbitrary θ and use them to develop a phase diagram for the preferred particle orientation in the absence of line tension. Thereafter, the response of the system for finite Σ is described. First, we consider partially wet particles (θ < 90°), for which the U state is always preferred to the O state. Therefore, the free energy in these orientations normalized by πr2γLV can be written
( ) (
)
γVS γLS 2L As + - 1 + 2Σ (11a) +1 + a E′ ) γLV γLV r πr2 aSO′ )
γVS γLS (πx - φx) + (xφ + 2) + γLV γLV As 4(π - φ) + x (11b) - x sin φ + Σ 2 π πr
(
(
) (
)
To determine whether the particles will remain at the interface, the free energies in each orientation must be compared to the submerged state U
∆aEU′ ) aE′ - aU′ ) cos θ0 - 1 + 2Σ ∆aSO-U′ ) aSO′ - aU′ ) x(θ cos θ0 - sin θ) + Σ
(12a)
(4θπ + x) (12b)
Equations 12a and 12b indicate that in the absence of line tension, only perfectly wet particles (i.e., with θ ) 0) prefer to submerge themselves rather than remain at the interface. The free energy difference between the two orientations determines which orientation is preferred at the interface
)
When Σ ) 0, eq 13 indicates that the coexistence of an end-on and a side-on orientation occurs at
(9b)
Furthermore, since the U and O states are equivalent for θ ) 90°, so Σc,E-U ) Σc,E-O, where the subscript E-O denotes transition from the end-on state to over the interface, and Σc,SO-U ) Σc,SO-O, where the subscript SO-O denotes a transition from the side-on state to over the interface. To ascertain whether the side-on or end-on state is energetically favored, the energy difference between the two orientations must be considered
∆aSO-E′ ) aSO′ - aE′ ) 1 - x + Σ
∆aSO-E′ ) aSO′ - aE′ ) (θx - 1) cos θ0 + (1 - x sin θ) + 4θ + x - 2 (13) Σ π
x)
1 - cos θ0 sin θ0 - θ0 cos θ0
(14a)
where θ ) θo for Σ ) 0. For any x greater than this coexistence value, the particle orients side-on. For any x less than this value, the particle orients end-on. For general Σ, the coexistence occurs for
4θ π x) θ cos θ0 - sin θ + Σ
(
cos θ0 - 1 + Σ 2 -
)
(14b)
where θ is related to θo by eq 5a. For partially nonwet particles (θ > 90°), the U state is never preferred over the O state and so is not discussed further here. The dimensionless free energy in the E and SO orientations can be written as
a E′ )
(
)
γVS γLS As (πx + 1) + + - 1 + 2Σ γLV γLV πr2
aSO′ ) -φx cos θ0 +
(15a)
γVS As (xπ + 2) + 2 - x sin φ + γLV πr 4φ Σ + x (15b) π
(
)
The relevant free energy differences are
∆aE-O′ ) aE′ - aO′ ) -cos θ0 - 1 + 2Σ
(16a)
∆aSO-O′ ) aSO′ - aO′ ) -φx cos θ0 - x sin φ + 4φ + x (16b) Σ π
(
)
Equations 16a and 16b indicate that for Σ ) 0, the particles will not submerge for any x unless θ ) 180°. The energy difference shows that the preferred orientation at the interface is
∆aSO-E′ ) aSO′ - aE′ ) Σ
(4φπ + x - 2) + 1 - x sin φ -
φx cos θ0 + cos θ0 (17)
When Σ ) 0, eq 17 indicates that coexistence between an end-on and a side-on orientation occurs for
x)
1 + cos θ0 sin θ0 + (π - θ0) cos θ0
(18a)
where θ ) θo for Σ ) 0. For any x greater than this coexistence value, the particle orients side-on. For any x less than this value, the particle orients end-on. For arbitrary line tension, the coexistence is defined by
x)
1 + cos θ0 + Σ
(4φπ - 2)
sin θ + (π - θ) cos θ0 - Σ
(18b)
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Figure 2. Phase diagrams of nanocylinder orientations for dimensionless line tension Σ ) 0. x is the aspect ratio of the cylinder (2L/πr) and θo is the contact angle.
where θ is the contact angle defined in the modified Young equation, eq 5a. In the analysis given above, the contact angle θo is defined in terms of Young’s equation. Real surfaces exhibit contact angle hysteresis, with significant differences between advancing and receding contact angles. If a particle were uniformly wet before being placed on the interface, the receding contact angles would prevail. If it were uniformly dry, the advancing contact angle would prevail. If, however, heterogeneous surface energies were established with partial prewetting, a more complex analysis would be needed to account for this heterogeneity. Phase Diagrams of Particle Orientations. The coexistence criteria derived above can be used to develop phase diagrams for the nanocylinder in terms of its preferred orientation (sideon or end-on) or location (under or over the interface) as a function of contact angle θ and line tension Σ. When Σ ) 0, particles remain at the interface for any 0° < θo < 180°. Their preferred orientation is summarized in Figure 2 in terms of the values of θ° and x that define the coexistence between side-on and end-on orientations as given in eqs 14a and 18a. x ) 1 delineates the aspect ratio at which the area of the rectangular hole (2rL) created in the liquid-gas interface by a nanocylinder immersed to θο ) 90° is equal to the area of the circular hole (πr2) made in the interface by a cylinder in an end-on orientation. Thus, for θο ) 90°, the particle remains end-on for all x < 1 and side-on for all x > 1. For partially wet or nonwet particles, longer cylinders are needed for these areas to balance, and the curve is symmetric about θο ) 90°. When Σ > 0, the phase diagram becomes more complex, since the coexistence aspect ratio x depends explicitly on Σ, as the energetic cost of the contact line increases with particle length, and implicitly on Σ through the modified Young’s equation which defines θ(θo, Σ). The behavior of the system is summarized in Figure 3, where the various orientations are given for x as a function of Σ for fixed θo. In Figure 3a, θo ) 45°. The vertical intercept of this figure corresponds to x ) 1.93, which is the side-on to end-on coexistence value defined in Figure 2 for θo ) 45°. Increasing Σ from zero at large x drives a side-on to under transition. Increasing Σ from zero at small x drives an end-on to under transition. However, for a finite range of x (1.93 < x < 27.4), increasing Σ from zero results in a side-on to end-on transition. The upper limit for this transition corresponds to the point where the energies for three states are the same. We define this point as a triple point xT,ΣT. The value for xT as a function of θo is shown in Figure 3b. This curve diverges to infinite x as θo approaches 43.4°; i.e., for any θo < 43.4°, there is no triple point. There is an interesting implication to the absence of a triple point, explored in Figure 3c, where θo is fixed at 30°. In this case, a regime exists in which the end-on state is preferred for
Figure 3. (a) The phase diagram for θo ) 45°. For x < 27.4 there are transitions from side-on to end-on to under as preferred orientations as Σ increases. There is a triple point where side-on, end-on, and under coexist. (b) The value for the triple point xT vs θo. For any θo < 43.4 ° there is no triple point. (c) The phase diagram in the absence of a triple point. θo ) 30°. A range of Σ appears in which the end on is preferred for all x.
all cylinder aspects ratios x for line tensions bounded below by the coexistence curve for side-on to end-on transition, and above by the coexistence curve for the end-on to under transition. Experimental Section Protocol. Alkane thiol functionalized gold nanowires prepared by template synthesis are spread at the interface of an aqueous gel. The particle location is fixed by curing the gel and creating a PDMS stamp of the interface. When the PDMS is peeled away from the gel phase, nanowires are retained in it. The stamp is subsequently imaged under the SEM to infer the contact angle and orientation of the nanowires at the gel/air interface. Below, details are given for each step of the experiment. Nanowire Synthesis. Nanowires are formed by templatedirected electrochemical synthesis.32-34 A Cu film of thickness 600 nm is evaporated onto the branched side of an alumina template (Whatman Anodisc) with nominal pore diameter of 200 nm. The Cu film acts as a working electrode in a threeelectrode cell. A 10 µm sacrificial layer of nickel is subsequently plated into the pores of the membrane at -800 mV (Ag/AgCl), from solution containing 1.6 M nickel sulfamate (Univertical Chemical Co.), 20 g/L NiCl (Alfa Aesar), and 20 g/L H3BO3 (Alfa Aesar). Gold nanowires are then plated into the template
4288 J. Phys. Chem. B, Vol. 110, No. 9, 2006 to the desired length at -880 mV (Ag/AgCl) (Technic 434 HS). The working electrode and nickel sacrificial layer are dissolved in 30% nitric acid solution (EMD Chemicals). The alumina membrane is subsequently dissolved in 2 M NaOH (Sigma Aldrich) solution, thereby freeing the nanowires, which are rinsed with water, centrifuged, decanted, and resuspended five times. The nanowires are then functionalized in 2 mM octadecane thiol (ODT) in ethanol solution under agitation for 24 h. This suspension is centrifuged and rinsed with ethanol five times to remove excess ODT, and resuspended in 2-propanol (Sigma Aldrich). The effective radius used in our discussion below was determined by averaging the radius for 48 nanowires, yielding a value of r ) 148 ( 24 nm. We study nanowires of length L ) 64 ( 16 nm (averaged over 50 nanowires) and L ) 692 ( 50 nm (averaged over 50 nanowires) with approximate values of x ) 0.28 ( 0.42 and x ) 2.98 ( 0.23, respectively. Gel-Trapping Technique. The protocol used for the gel trapping technique is based on previous work by Paunov.35 An aqueous solution containing 2 wt % of Kelcogel, a gellan gum (supplied by CPKelco), is prepared in deionized water (Millipore) heated to 95 °C. The gellan gum is centrifuged to release any air bubbles and cooled to 50 °C. This solution is poured into a Petri dish and cooled to 4 °C. Nanowires are spread onto this gellan solution-air interface from suspension in 2-propanol using a 10 µL Hamilton gastight syringe in one injection. The solution is cooled for 30 min to allow it to gel. Subsequently, a 10:1 mixture of PDMS Sylgard 184 elastomer/curing agent (Dow Corning) is prepared and centrifuged to release air bubbles. This mixture is then poured over the gellan gum and allowed to cure for 48 h, forming a PDMS negative stamp of the air-gellan solution interface. This stamp is placed in a 95 °C water bath for 5 min to rinse off excess gellan gum. A 2-3 nm layer of platinum is sputtered on the stamp for SEM preparation. The interface stamp is then imaged under a JEOL 6700F SEM.
Lewandowski et al.
Figure 4. (a) Plan view SEM image of nanowires in the alumina template. The cross section of the nanowires is not circular but is an irregular prism-like shape with nominal radius r ) 148 ( 24 nm. (b) SEM side view of a short (approximately 50 nm) Au nanowire segment above the nickel which is the sacrificial layer. The interface between the Au and Ni is sharp.
Comparison of Experiments to Theory The alumina templates have irregular prism-shaped holes, forming nanowires that are not perfect cylinders, as shown in Figure 4a. The nanowires formed by template synthesis are shown in Figure 4b. The nanowires have relatively sharp ends, as is apparent in the figure, in which the boundary between the Au nanowires and the sacrificial nickel segments is shown. Images of the PDMS inverse stamp created from nanowires spread at the aqueous-gas interface are presented in Figure 5. The disk-shaped nanowires with aspect ratio x ) 0.28 are always end-on, while the longer nanowires x ) 3.0 are always sideon. While ODT self-assembled monolayers (SAMs) adsorbed on macroscopic gold surfaces yielded contact angles of 118° for water/air/SAM interfaces (as measured using a Rame-Hart contact angle goniometer), the contact angle on the nanowire itself is apparently less than this value, as the nanowires in a side-on orientation are retained in the PDMS stamp up to what appears to be roughly their equatorial cross section, suggesting that θ is roughly 90°. The expressions for free energy developed above for perfect cylinders can be used to guide our interpretation of the results, despite the more complex cross section and somewhat irregular ends of the nanowires. The results in Figure 5 are consistent with the transition from end-on to side-on orientation predicted in Figure 3 for Σ ) 0 as the length of a nanowire is increased at fixed cross section. While prior experimental studies of cylindrical nanoparticles at fluid interfaces are limited, micrometerscale cylinders have been used to coat surfaces of 10 µm scale
Figure 5. (a) SEM image of disklike nanowires (Lav ) 64 nm) embedded in a PDMS replica of the aqueous-air interface illustrating an end-on orientation. (b) SEM image of nanowires (Lav ) 692 nm) in a PDMS replica of the aqueous-air interface illustrating a side-on orientation.
drops to create dispersions.36 Similarly, there have been a number of studies in which nanorods have been spread at
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Figure 6. Phase diagram for partially wet nanowires (θo ) 85°). For all Σ > 0.45, the nanowires would be immersed in the subphase.
TABLE 1: Nanocylinders at Interfaces in a Side-on Orientation reference
interface
length (nm)
Acharya et al.38 Kim et al.40 Martin et al.39 Whang et al.37 Noble et al.36
aqueous/gas aqueous/gas aqueous/gas aqueous/gas aqueous/oil
5 20 3000 >1000 10000-70000
diameter (nm) x ) 2L/πr 1.5 5 350 40 400-200
4 5 11 >32 6.4-220
aqueous-gas interfaces in order to study their interactions at interfaces and to create ordered assemblies upon deposition on solid substrates.37-40 In each of these studies, x . 1, and the nanowires remain in a side-on orientation. Details are given in Table 1. These results may provide some insight into the probable order of magnitude of line tension. In Figure 6, a phase diagram for a partially wet nanowire for θo ) 85° corresponding to those inferred for our nanowire is presented. For Σ > 0.45, all nanowires would be immersed in the subphase or expelled into the upper phase. Line tension values σ of order 10-6 N have been reported for a number of three-phase systems inferred either from experiments in which the depth of immersion of solid spherical particles has been studied as a function of particle diameter or from the apparent contact angle measured as a function of liquid-phase drop radius on solid surfaces.41-46 For nanowires of the dimension studied here, assuming a surface tension for the liquid-vapor interface of 70 mN/m, Σ would be roughly 102, so no particles would be retained at the interface if line tension were of this magnitude. In contrast, theoretical predictions for line tension range from 10-10 to10-12 N,47-49 and measurements of line tension in this range have been reported by some experimentalists.50,51 For line tensions of this magnitude, Σ would range from 10-2 to 10-4, and so nanowires would remain at the interface, as has been observed in this study and those reported in Table 1. We note that the application of the expressions for free energy developed for perfect cylinders must be made with care. The modified Young equation derived in eq 5 required that the wetted contour at the ends of the nanocylinder in the side-on orientation change with angle φ. An analogous argument for a noncircular end can be made accounting for changes in contact line location with arclength tracing along the edges of the ends. However, if the ends are sufficiently rough or irregular to provide pinning sites for the contact line, these arguments would not be valid. Conclusions Phase diagrams for particle orientations are developed for cylindrical objects at fluid interfaces based on the energies of the solid-fluid surfaces and the fluid-fluid interface. Predicted transitions from end-on to side-on orientations as a function of cylinder aspect ratio for zero line tension compare favorably to
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