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Langmuir 2006, 22, 3662-3671
Characterization of Watermarks Formed in Nano-Carpet Effect J.-G. Fan and Y.-P. Zhao* Nanoscale Science and Engineering Center, Faculty of Engineering, Department of Physics and Astronomy, The UniVersity of Georgia, Athens, Georgia 30602 ReceiVed NoVember 30, 2005. In Final Form: January 25, 2006 In the spreading of a water droplet on an aligned silicon nanorod array surface, a precursor rim was detected moving ahead of the contact line. In this process, nanorods were bundled by the capillary force to form clusters, and a watermark developed on the surface after water evaporated. The size of the watermark, Rmax p , corresponding to the maximum ∝ radius of the precursor rim, followed a simple power law relationship with the volume of water droplet Ω, Rmax p Ωβ. The scaling exponent β increased when the nanorod height decreased, but all in the vicinity of 1/3. This behavior was attributed to the competition of evaporation and spreading of a water droplet during the spreading process. The size of the bundled nanorod cluster formed by the capillary force not only depended on the nanorod height but also on the location in the watermark. The cluster size almost remained as a constant near the center, and then it decreased with the distance from the center. This phenomenon can be qualitatively interpreted through the change of the total free energy during the precursor invading the nanorod array, by considering the contribution from the mechanical energy change due to the bending and clustering of nanorods.
I. Introduction Spreading of liquids on solid surfaces has been studied extensively on smooth,1-4 rough,4-7 and porous surfaces.8-9 Most investigations have addressed the evolution of a liquid drop over time, and it is believed that the drop size versus time follows a simple power law, R ∝ tR, where R is usually 1/10 or 1/8 depending on the dominant driving force.1-4,6-10 However, these studies do not involve any obvious physical change of the substrate morphologies during the spreading of liquid drops, since the interaction between liquid and solid usually is not strong enough to overcome the mechanical strength of the bulk solid and is not able to change the substrate structure. Recently, with the increasing potential applications of nanotechnology in chemical, biological, environmental and material sciences, one-dimensional nanostructures, such as nanowires, nanotubes, and nanorods, have been widely investigated. These structures provide exceptionally high surface areas and high aspect ratios compared to conventional flat or microstructured surfaces. At the same time, a higher aspect ratio means that these structures are more apt to be deformed by external forces. Both superhydrophobic and superhydrophilic surfaces have been reported on nanostructures of high aspect ratio.11-13 Many different studies on nanostructures with hydrophilic properties show that the structures of nanotubes or nanorods change after wetting by a liquid.11,14-19 We call this structure change the “nano-carpet effect”. Most of the studies * To whom correspondence should be addressed. (1) Tanner, L. H. J. Phys. D: Appl. Phys. 1979, 12, 1473. (2) Dahuber, A. A.; Troian, S. M.; Reisner, W. W. Phys. ReV. E 2001, 64, 031603. (3) Elyousfi, A. B. A.; Chesters, A. K.; Cazabat, A. M.; Villette, S. J. Colloid Interface Sci. 1998, 207, 30. (4) Cazabat, A. M.; Stuart, M. A. C. Prog. Colloid Polym. Sci. 1987, 74, 69. (5) Yost, F. G.; Rye, R. R.; Mann, J. A. J. R. Acta Mater. 1997, 45, 5337. (6) Gerdes, S.; Cazabat, A. M.; Stro¨m, G.; Tiberg, F. Langmuir 1998, 14, 7052. (7) Apel-Paz, M.; Marmur, A. Colloids Surf. A 1999, 146, 273. (8) Bacri, L.; Brochard-Wyart, F. Eur. Phys. J. E 2000, 3, 87. (9) Starov, V. M.; Zhdanov, S. A.; Kosvintsev, S. R.; Sobolev, V. D.; Velarde, M. G. AdV. Colloid Interface 2003, 104, 123. (10) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. (11) Lau, K. K. S.; Bico, J.; Teo, K. B. K.; Chhowalla, M.; Amaratunga, G. A. J.; Milne, W. I.; McKinley, G. H.; Gleason K. K. Nano Lett. 2003, 3, 1701. (12) Fan, J.-G.; Tang, X.-J.; Zhao, Y.-P. Nanotechnology 2004, 15, 501. (13) Tsoi, S.; Fok, E.; Sit, J. C.; Veinot, J. G. C. Langmuir 2004, 20, 10771.
have been concentrated on the aligned carbon nanotubes (CNTs). Two different processes have been shown for the structure change. Some studies have shown that after immersing into a liquid, and drying, the aligned nanowires (i.e., CNTs) bundled together to form cellular network structures,11,14-16 whereas other studies showed that the bundling of nanorods occurred during the spreading of a liquid droplet.17-19 Lau et al.11 found that CNTs with a length of 2 µm formed pyramid-like bundles after the spreading of water on the surface. The authors attributed this bundling to be a result of the attractive capillary forces that arose during the evaporative drying.11 A similar pyramid-like CNT cluster was observed by Nguyen et al. during the CNT purification process.14 In their study, the length of the CNTs was ∼1 µm and the diameter of the bundled CNTs was less than 1 µm. Chakrapani et al.15 observed the formation of a cellular network after immersing and drying a 50 µm long aligned CNT array. The average size of the cell domain was about 100-200 µm. Both freeze-dried experiment and in-situ optical microscope observation suggested that the cellular network formed during the evaporation process. Recently Journet et al. also observed the formation of pyramid-like CNT clusters due to the evaporative drying of ethanol during the alkanethiol fictionalization process.16 The length of the CNTs was in the order of a few micrometers (typically 2 µm). On the other hand, Liu et al. also observed the cellular network formation on an aligned CNT array (h ) 19 µm).17 The size of the cellular domain ranged from 30 to 60 µm. Their observation of the color changing of the CNT array during spreading suggested that the pattern formed during the spreading process. Correa-Duarte et al. attempted to culture cells on the cellular patterns formed by CNTs.18 They found that for small CNT length (∼3 µm), a pyramid-like CNT cluster formed. (14) Nguyen, C. V.; Delzeit, L.; Cassell, A. M.; Li, J.; Han, J.; Meyyappan, M. Nano Lett. 2002, 2, 1079. (15) Chakrapani, N.; Wei, B.; Carrillo, A.; Ajayan, P. M.; Kane, R. S. Proc. Natl. Acad. Sci. 2004, 101, 4009. (16) Journet, C.; Moulinet, S.; Ybert, C.; Purcell, S. T.; Bocquet, L. Europhys. Lett. 2005, 71, 104. (17) Liu, H.; Li, S.; Zhai, J.; Li, H.; Zheng, Q.; Jiang, L.; Zhu, D. Angew. Chem., Int. Ed. 2004, 43, 1146. (18) Correa-Duarte, M. A.; Wagner, N.; Rojas-Chapana, J.; Morsczeck, C.; Thie, M.; Giersig, M. Nano Lett. 2004, 4, 2233. (19) Fan, J.-G.; Dyer, D.; Zhang, G.-G.; Zhao, Y.-P., Nano Lett. 2004, 4, 2133.
10.1021/la053237n CCC: $33.50 © 2006 American Chemical Society Published on Web 03/09/2006
Watermarks Formed in Nano-Carpet Effect
However, for the CNT length of 35 and 50 µm, cellular networks with average domain sizes of 5-15 and 5-60 µm formed. All of these observations have revealed two important features for a liquid interacting with a CNT array: (1) For a short CNT array (h < 3 µm), pyramid-like clusters have formed, but for a much longer CNT, cellular network structures have generally formed; (2) the cluster size, or the cellular domain size, depends on the length (or the aspect ratio) of the CNT. The longer the CNT array, the larger the cluster size or the domain size. A similar bundling effect was also observed during water spreading on an aligned silicon nanorod array (h ) 890 nm).19 The bundling happened during the spreading of the water droplet, and a circular disk appeared instantly after the water drop touched the silicon nanorods. Also, the surface appearance change indicated a change of the underlying structure, which was the bundling of the nanorods. After the water evaporated, a circular watermark remained on the surface with a shinning disk in the center. The scanning electron microscope (SEM) images across the watermark showed three distinct regions: In the center, nanorods bundled together to form micrometer-sized channels. From the center to the edge of the shining disk, nanorods bundled and tilted toward the center. Apart from the edge of the shining disk, nanorods bundled together in a manner similar to that of the center, but the average size of the bundled clusters became smaller when the distance was further away from the center. The bundling of nanorods could be attributed to the capillary force acting between silicon nanorods and water, which had a magnitude estimated to be on the order of nano-Newtons.19 Although all of the above studies have pointed out that the capillary force of the liquid acting between nanorods/nanowires is the driving force for the pattern formation, a detailed characteristic of the watermark formed either due to evaporative drying or spreading, such as the size and other morphological characterization, is still missing. In this paper, we will further demonstrate that the nano-carpet effect for Si nanorod arrays is caused by the liquid spreading process. We have investigated in detail how the height of the nanorod array along with the volume of the water droplet affects the watermark size and have also studied the bundling of nanorods as a function of rod height. A simple model, based on the free energy change during water invading the nanorods, has been proposed and is qualitatively consistent with the experimental results. The organization of the paper is as follows: In section II, we give a brief introduction to the experimental method used to prepare the Si nanorod arrays and the characterization techniques. In section III, we will discuss the watermark size as a function of the nanorod height as well as water droplet volume. A simple model based on droplet spreading and evaporation is proposed to qualitatively interpret the observed results. Section IV gives a detailed characterization of the nanorod clusters for different rod height. A model based on interfacial free energy and the mechanical bending energy of the cluster is developed. In the end, we conclude our observations and explanations. II. Experiments Silicon nanorod samples were fabricated by glancing angle deposition (GLAD). A detailed description of this method can be found elsewhere.12,20-22 In brief, in an electron beam evaporation system (Torr International, Inc.), a RCA-1 cleaned p-type Si (100) substrate was installed in such a way that its surface normal had an angle φ ) 86° with the incoming vapor. Then the evaporation system (20) Robbie, K.; Brett, M. J. J. Vac. Sci. Technol. A 1997, 15, 1460. (21) Sit, J. C.; Vick, D.; Robbie, K.; Brett, M. J. J. Mater. Res. 1999, 14, 1197. (22) Zhao, Y.-P.; Ye, D.-X., Wang, G.-C.; Lu, T.-M. Proceedings of the 48th SPIE Annual Meeting, 2003, Vol. 5219-11.
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Figure 1. (a) Top view and (b) cross section of silicon nanorods grown by glancing angle deposition. The deposition conditions were the following: incident angle 86°, deposition rate 0.2 nm/s, and the total normal film thickness 2000 nm. The nanorod height in (b) is about 890 nm. was pumped down to a base pressure of around 10-6-10-7 Torr. During the deposition, a stepper motor rotated the substrate azimuthally about its surface normal at 0.05 rev/s, and the film thickness and deposition rate were monitored by a quartz crystal microbalance (QCM). We prepared nanorod samples with normal film thicknesses of 200, 500, 700, 1300, and 2000 nm, and later, the real nanorod height was determined by a field emitting scanning electron microscope (SEM: LEO 982). To form watermarks, water droplets with controlled volumes were dispensed onto a silicon nanorod surface at room temperature (25 °C, humidity 23%), using a contact angle measurement system (DataPhysics, OCA 20). The structures of the as-deposited samples, as well as the samples treated with water were examined by SEM, while the size of the watermark was measured by a measurescope (Unitron TMS 6618). Figure 1 shows the top-view and cross-section SEM images of the Si nanorod sample with 2000 nm normal film thickness. The nanorods had an average diameter of 103 ( 14 nm at the top and 22 ( 4 nm near the substrate. The height of the nanorods was 890 ( 43 nm, and the rod-rod separation was 131 ( 20 nm. Thus, the nanorods covered about 47% of the top surface. Table 1 summarizes the morphological parameters of the as-deposited nanorod samples of different rod heights, as measured from the cross-section and top-view SEM images or their power spectra (not shown here).
III. Size Characterization of the Watermarks When a 2 µL water droplet made contact with the 890 nm silicon nanorod surface, it spread out quickly and the base of the water droplet reached a maximum within 3 s. The contact angle was measured to be about 3°.12 Figure 2 shows three snapshots of the water spreading fronts, taken from the top of the contact angle setup with a CCD camera of 210 fps.23 Upon immediate impact, a circular puddle formed on the nanostructured surface (23) Fan, J.-G.; Zhao, Y.-P. unpublished.
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Table 1. Nanorod Height and Coverage Measured from the Cross-Sectional and Top View SEM Images normal film thickness, d (nm)
nanorod height, h (nm)
nanorod diameter, D (nm)
nanorod separation, d0 (nm)
rod coverage,a f (%)
200 500 700 1300 2000
50 ( 4 115 ( 10 421 ( 20 649 ( 27 890 ( 43
n/a 77 ( 13 66 ( 11 61 ( 10 103 ( 14
n/a 177 (PS)b 184 (PS)b 70 (PS)b 131 ( 20
n/a 41 46 46 47
a The rod coverage on the surface was obtained by the method introduced in ref 12. b “PS” means value obtained though power spectra.
The watermark is the result of the water droplet spreading on nanorod array surface. Since the precursor rim has a larger radius than that of the contact line. The maximum radius of the watermark (Figure 2d) is determined by the precursor rim rather than the contact line. This phenomenon is very similar to the rim formation during liquid droplet spreading on a rough surface,4,7 or to the stain left after liquid droplet spreading on a porous substrate.24-29 For both rough surface spreading and porous structure spreading, the capillary flow through a groove or a channel plays a very important role. Conceptually, the nanorod array surface can be treated as a rough surface with high aspect ratio pillars, or as a three-dimensional porous network with connected open capillaries. According to the classical theory derived by Washburn,30 the driving force for the spreading is due to the capillary pressure
Pc )
Figure 2. Snapshots of the water spreading fronts, taken from the top of the spreading water droplet with a CCD camera of 210 fps at (a) t ) 0 s (arbitrarily determined); (b) t ) 0.72 s, and (c) t ) 1.68 s, respectively. Rc and Rp represent the radii of the contact line and precursor rim, respectively. (d) An orange watermark formed on 890 nm Si nanorods by a 3 µL water droplet.
(Figure 2(a), t ) 0 s). The circular boundary of the puddle is defined as the contact line, or the three-phase line of the water droplet, and the radius of the contact line is denoted as Rc. After less than 0.2 s, in addition to the spreading of the contact line, a small dim ring appeared in front of the contact line and the width of the ring, and the radius of the contact line, started to grow with time. This ring is due to water imbibed into the nanorod structure, and we name it as the precursor rim, with its radius denoted as Rp (Figure 2(b), t ) 0.72s). When the radius of the contact line spread further, the width of the precursor rim also kept growing (Figure 2(c), t ) 1.68 s). In about 2 s, the radius of the contact line Rc reached a maximum, whereas the width of the precursor rim kept increasing for another 2 s. Both the precursor front and the contact line remained at the maximum positions for about 2 min, and then both Rc and Rp receded slowly. Occasionally, the contact line would stop for a few seconds before withdrawing further. In about 3 min, the water droplet totally evaporated, with an orange disklike watermark left on the surface, as can be seen in Figure 2d. If one examines the watermark carefully, one can see the existence of several rings. Those rings are due to the contact line pinning during the receding process (drying).
2γ cos(θe + φ) re
(1)
where γ is the surface tension of the liquid, θe is the contact angle between the liquid and the surface of the nanorod, φ is the nanorod inclination angle, and re is the effective radius of the capillaries. Thus, the effective capillary radius re plays a crucial role for the dynamics of the precursor rim. For the nanorod array, the opening between two nanorods is equivalent to the cross-section of a capillary tube, therefore the effective capillary radius re is determined by the average height of the nanorods and the average gap between the nanorods. To understand how the height of the nanorod affects the formation of the precursor rim, water drops with volumes of 0.1 to 4 µL were used to form watermarks on nanorod substrates with rod heights of 50, 115, 421, 649, and 890 nm, respectively. Figure 3 shows some representative optical images of the watermarks formed on silicon nanorod surfaces having heights of 890, 649, 421, and 115 nm, respectively. There are several distinct features: (1) The color of the as-deposited nanorod samples differs from each other, as nanorod heights of 890, 649, 421, and 115 nm, appear yellowgreen, green, dark-green, and gray, respectively. (2) The colors of the watermark also differ from those of the as-deposited surfaces and watermarks with different nanorod height. They are orange, yellow, light blue, and dark gray at decreasing nanorod height, respectively. These colors indicate there are characteristic light wavelengths which are reflected for different samples, due to the film thickness and specific effective dielectric constant. (3) For the samples with rod heights of 890, 649, and 421 nm, there exists a shining circular inner-disk in the center, which suggests that nanorod structures formed in the center are different from those outside the inner-disk, as pointed out by Fan et al. in ref 19. However, on the silicon nanorods with 50 nm rod height, the watermark could not be detected by the naked eyes. After coating with a thin layer of Chromium film (∼10 nm), dim dark disks appeared on positions where the water drops placed, meaning that watermark also formed on very short silicon nanorod substrates. The chromium coating simply enhanced the contrast of optical imaging. (4) The larger the volume of the water droplet is, the larger the watermark size will be. This is true for all the samples with different rod heights. The diameter of the watermark, 2Rmax p , measured by the measurescope, is plotted in Figure 4 as a function of the nanorod (24) Kissa, E. J. Colloid Interface Sci. 1981, 83, 265. (25) Marmur, A. J. Colloid Interface Sci. 1988, 124, 301. (26) Borhan, A.; Rungta, K. K. J. Colloid Interface Sci. 1993, 158, 403. (27) Danino, D.; Marmur, A. J. Colloid Interface Sci. 1994, 166, 245. (28) Modaressi, H.; Garnier, G. Langmuir 2002, 18, 642. (29) Seveno, D.; Ledauphin, V.; Martic, G.; Voue´, M.; De Coninck, J. Langmuir 2002, 18, 7496. (30) Washburn, E. W. Phys. ReV. 1921, 17, 273.
Watermarks Formed in Nano-Carpet Effect
Langmuir, Vol. 22, No. 8, 2006 3665
Figure 3. Optical micrographs of the watermarks formed on silicon nanorod substrates with heights of (a) 890, (b) 649, (c) 241, and (d) 115 nm, respectively, and with water droplet volumes selected from 1.5 to 4 µL. The volume decreases from left to right for each sample.
similar to the apparent contact angle change as a function of the nanorod height observed previously.12 In ref 12, the apparent contact angle decreased almost exponentially for increasing nanorod height at h < 150 nm, whereas for taller nanorods, the contact angle reached a constant value. When the diameter of the watermark is plotted as a function of water volumes in loglog scale, as shown in Figure 5a, a power law relationship is observed
Rmax ∝ Ωβ p
Figure 4. Plot of the watermark size as a function of the nanorod height for different water volumes. For different volumes, the 2Rmax p ∼ h relationship follows almost the same trend.
height for different water droplet volume Ω. The diameter of the watermark increases almost monotonically with the nanorod height. More specifically, for nanorod height increasing from 115 to 421 nm, the watermark size almost doubles for all water droplet volumes. But, the changes in watermark sizes for the heights of 421, 649, and 890 nm, are very small. This trend is
(2)
where β is the scaling exponent. It is found that β ) 0.283 ( 0.002, 0.306 ( 0.010, 0.310 ( 0.008, and 0.322 ( 0.018 for samples with rod height of 890, 649, 421, and 115 nm, respectively. All the values of the exponent β are in the vicinity of 1/3. Moreover, with the increment of nanorod height, β decreases slightly as shown in Figure 5b. For reference, a 500 nm thin amorphous Si (a-Si) film on Si substrate was fabricated under the same deposition conditions without substrate tilting. Water droplets with different volumes were dispensed onto the reference sample. The side views of the droplets were taken when the drops stabilized, and the diameter of the drop base is also proportional to the maximum lateral dimension of the drop in the image. The plot of the drop base diameter versus volume
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we assume that there are two connected processes, the evaporation during the spreading on a smooth surface, and the water invasion into nanorod array. The volume of water on the nanorod array contains two parts, the water above the nanorods (Ω) and the water imbibed into the nanorod channels (Ωp). The assumption of a smooth surface (or a heterogeneous smooth surface) is based on the experimental observation of the precursor film spreading ahead of the contact line. This precursor layer effectively “lubricates” the surface by filling into the nanorod channels. In fact, the spreading law on such a surface agreed with that on a smooth surface.23 Usually, Ωp , Ω, and the volume Ωp inside the channel can be neglected. For example, a 3 µL water droplet on the 890 nm silicon nanorods, which has the largest watermark size, will have a maximum watermark diameter of about 10 mm. Thus 2 Ωp ) πRmax h*(1 - f) p
(7)
where Rmax is the maximum radius of the watermark, h is the p height of the nanorods, and f is the nanorods coverage. Using the data in Table 1, Ωp is found to be about 3.7 × 10-2 µL. This is much less than 3 µL. This volume could be important for later stage of evaporation. Since the water contact angle on nanorods is very small, eq 5 can be simplified
π Ω ) Rc3θ 4 Figure 5. (a) log-log plot of the watermark diameter versus water droplet volume on a 500 nm thick Si film and Si nanorod array substrates with height of 890, 649, 241, and 115 nm, respectively. ∝ Ωβ0 has been observed; (b) the plot of power A power law 2Rmax p law exponent β as a function of the nanorod height h. The exponent β decreases monotonically with increasing nanorod height.
in Figure 5a gives β ) 0.332 ( 0.005, which is larger than all of the β values obtained for nanorod samples. Since the volume of the water droplet is relatively small, the shape of the water droplet on the a-Si film can be treated as a spherical cap, and the volume of the water drop can be expressed as
1 Ω ) πh(3Rc2 + H2) 6
(3)
where H is the height of the spherical cap. For a spherical cap
θ H ) Rc tan 2
()
(4)
where θ is the contact angle. Substituting eq 4 into eq 3, one obtains
Ω)
πRc3 θ θ 3 + tan2 tan 6 2 2
(
)
(5)
Therefore
Rc ∝ Ω1/3
(6)
This result agrees very well with the scaling exponent for the droplet on a flat surface. However, the derivation of eq 6 is based on the assumption that there is no liquid evaporated during the spreading. However, from the dynamic observation, as shown in Figure 2, the watermark formation process can be viewed as the competition of spreading with evaporation, as well as spreading inside the nanorod arrays. To simplify the problem,
(8)
The time derivative of Ω gives
dRc π 3 dθ dΩ 3 ) πRc2θ + Rc dt 4 dt 4 dt
(9)
The change of the water volume should equal to the volume evaporated. For simplicity, we assume that the water evaporates at a constant rate on the surface
dΩ ) -JπRc2 dt
(10)
where J ) dH/dt is the evaporation rate and πRc2 is the approximate total surface area of the water droplet with a spherical cap shape. Substituting eq 10 into eq 9
dRc dθ ) -4J Rc + 3θ dt dt
(11)
In addition, for a water droplet with small contact angle, with its size decreasing due to evaporation, the change of the base radius Rc is simply related to the change of the height H of the droplet by
θ dH ) dRc 2
(12)
Equation 12 can be directly derived from eq 4. Thus
dRc 2J )dt θ
(13)
For a nonvolatile liquid, the droplet spreading on a smooth surface is governed by the capillary force31 (31) de Gennes, P. G.; Brochard-Wyart, F.; Que´re´, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, WaVes; Springer: New Yrok, 2003.
Watermarks Formed in Nano-Carpet Effect
dRc V* ) (cos θe - cos θ)θ dt 3ln
Langmuir, Vol. 22, No. 8, 2006 3667
(14)
is Rmax c
xπ4Ω 3
where V* ≡ γ/µ, µ is the viscosity of the liquid, and ln is a dimensionless coefficient. For water, V* ) 70 m/s, ln ) 20, and θe is the equilibrium contact angle, which is very small in our case, and eq 14 can be simplified as
dRc V* 3 ) θ dt 6ln
(15)
Combining eq 13 with eq 15, the change of Rc due to the competition of spreading and evaporation can be written as
dRc V* 3 2J ) θ dt 6ln θ
(16)
Equations 11 and 16 are the two basic equations we use to solve the dynamics of Rc. When Rc reaches the maximum, dRc/dt ) 0, corresponding to a contact angle θ/c at Rmax c , eq 16 gives 4
θ/c )
x
12lnJ . V*
Rmax ) c
(
)
2 4 3 - θ/4 c /θ0
1/6
θ/-1/3 ≈ c
() 2 3
xπ4Ω
1/6
3
1/3 / -1/3 θc
0
(19)
where Ω0 ) π/4 R03θ0 is the initial volume of the water droplet. This approximation is carried out under the condition that θ/c , θ0. Equation 19 shows that the maximum radius of the contact line scales with both the drop volume Ω0 and the apparent contact angle θ/c . The power law relationship between the maximum drop size and drop volume partly explains why all the scaling exponents β in Figure 5 are in the vicinity of 1/3, and the deviations from 1/3, observed for nanorods with different heights, are due to the capillary spreading inside the nanochannel. From the dynamic spreading measurement, we have confirmed that the precursor front obeyed the general Washburn relationship (Figure 2)23-30
(17) δ ) Rp - R c )
The angle θ/c is the apparent contact angle that was measured in ref 12. Combining eq 11 with eq 16 yeilds
V* 3 2J dRc 6lnθ - θ dθ ) Rc V*θ4 2J 2ln
1/3 0
(
)
reγ cos θe 2µ
1/2
t1/2
(20)
that is
(18)
Equation 18 can be solved analytically, and the expression for
Rp ) Rc + δ
(21)
When the radius of the precursor rim reaches a maximum, dRp/dt )0
Figure 6. SEM top-view images of the bundled nanorod clusters formed from the center to the edge of the watermark for different substrates. Panels a-c are from the watermark on h ) 115 nm nanorod arrays; Panels d-f are from the watermark on h ) 649 nm nanorod arrays; and panels g-i are from the watermark on h ) 890 nm nanorod arrays.
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Fan and Zhao
(
)
dRc 1 reγ cos θ |t)tp ) dt 2 2µ
1/2
tp-1/2
(22)
where tp is the time when Rp reaches the maximum. The property dRc/dt|t)tp < 0 indicates that the contact line is in the receding process; that is, when Rc is receding, Rp can reach the maximum. Thus, Rp takes longer to reach the maximum compared to that of Rc. This is consistent with our experimental observation described above. Inserting eq 22 into eq 16 and assuming θ/p , 1, one has
θ/p ≈ 4J and
Rpc ) Rmax c )
(
2θ/4 p
3θ/4 c
() x 2 3
1/3 3
-
(
2µ reγ cos θe
)
1/2
tp1/2
(23)
) () x
1/6 / θc /4 θp θ/p
≈
2 3
1/3 3
()(
4 1/3 1 Ω π 0 4J
1/3
4 1/3 /-1/3 Ω θ π 0 p
)
reγ cos θe 2µ
1/6
tp1/6
(24)
where Rpc and θ/p are the radius of the contact line and the contact angle at time tp. Thus
≈ Rmax p
() x 2 3
1/3 3
()(
4 1/3 1 Ω π 0 4J
1/3
)
reγ cos θe 1/6 1/6 tp + 2µ reγ cos θe 1/2 1/2 tp (25) 2µ
(
)
Equation 25 shows that the capillary flow inside the nanorod array modifies the maximum radius for the precursor rim, leading to two major qualitative conclusions. First, the effective capillary radius re ∝ h1/2; therefore, eq 25 shows that Rmax increases p monotonically with the nanorod height, which is qualitatively consistent with the results in Figure 4. However, as will be discussed later, the relationship re ∝ h1/2 will not be held for very long nanorods due to the bending and clustering of the nanorods, and the behavior of the watermark on the longer nanorod will be affected. Second, since the dynamic behavior of the contact line is expected to be the same for different nanorod heights, according to eqs 11 and 16, to satisfy eq 22, taller nanorods require a longer time tp to reach the maximum. Thus, the taller will be. For the droplet the nanorods are, the larger the Rmax p volume dependence, roughly the maximum radius of the watermark (precursor rim) obeys the Ω01/3 law, except that a constant term appears in eq 25. This extra term is closely dependent on the nanorod height as discussed above. With the increment of the rod height, the contribution of this constant term becomes more and more important, since it will increase with a much faster rate with respect to the rod height, as compared to the volume dependent term. Thus, with the increase of the rod height, the Rmax ∼ Ω0 relationship will deviate further from the p Ω01/3 law. Eventually, eq 25 predicts that the exponent β will decrease with the nanorod height, which qualitatively agrees with the results in Figure 5. It should be noted that, although the pinning of the contact line was observed during the liquid drop receding on the silicon nanorod surface, the above model does not include any pinning mechanism. The pinning mechanism is still unclear to us.
IV. Morphological Characterization of the Watermarks Previously, we have reported the formation of nanorod clusters in the watermark areas.19 Since the driving force to form the
Figure 7. Determination of the fractal dimension of the watermark morphology using the perimeter-area plot. (a) A representative plot of perimeter L versus area A of the bundled nanorod clusters in the log-log scale. The plot was derived from an SEM image at a distance r ) 2.35 mm from the watermark center, with the rod height of 890 nm, and a water droplet volume of 2.5 µL. The linear fit gives a slope of 0.89. (b) The plot of the fractal dimension Df at distance r from the center, of the same watermark.
clusters is due to the capillary force, the detailed characterization on the morphology of the clusters will bring out more information regarding the water invading process. Therefore, detailed SEM characterization of watermarks formed at nanorod surfaces with different rod heights has been carried out. Figure 6 shows the representative top-view SEM images taken at three different regions of the watermark on the nanorod sample with nanorod heights of 115, 649, and 890 nm, respectively. The volume of the water droplet used to form those watermarks is fixed, Ω ) 2.5 µL. Three general characteristics are observed: (1) For the same watermark, the size of the bundled nanorod clusters decreases from the center of the watermark toward the edge of the watermark. (2) The bundled clusters are irregularly shaped. In fact, those clusters are all isolated and surrounded by water channels made by water pathways during the invasion process. These water channels are percolated through the entire observation area. (3) Both the cluster size and channel size are closely dependent on the nanorod heights. In fact, they increase as the nanorod height increases. One could speculate that the morphology of the nanorod clusters will be very similar to those formed in two-dimensional percolation theory, due to the nature of the invading process. Or more specifically, the morphology of the bundled nanorod clusters can be described by the invasion percolation (IP) model.32,33 The (32) Meakin, P. Fractals, Scaling, and Growth Far from Equilibrium; Cambridge University Press: Cambridge, England, 1998. (33) Sahimi, M. Applications of Percolation Theory; Taylor & Francis Ltd.: London, 1994.
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Figure 8. Use of power spectrum to determine the fractal dimension and the cluster size of the nanorod bundles: (a) the original SEM image at the center (r ) 0 mm) of the watermark formed on the 890 nm Si nanorod array with a water droplet volume of 2.5 µl; (b) the level-setting image of panel a. The white islands represent the bundled nanorod clusters, and the black channels are the water pathway. (c) The circular averaged power spectra of panels a and b plotted in log-log scale.
Figure 7a plots the perimeters versus their areas of the bundled clusters of the watermark on the 890 nm nanorods at r ) 1.97 mm in log-log scale. From the curve fitting in Figure 7a, the value of the fractal dimension Df at r ) 1.97 mm is obtained to be Df ) 1.8 ( 0.2. This result is consistent with the fractal
dimension of IP model.35,36 In addition, the fractal dimension across the entire watermark has been calculated from the L-A plots, and in Figure 7b, the fractal dimension Df is plotted as a function of the distance r from the center of the watermark on the 890 nm nanorod surface. From the near center to the far edge, the fractal dimension almost remains as a constant, Df ∼ 1.8. This result shows that the pattern or cluster morphology across the watermark is the same, which implies that the invasion percolation could be the main mechanism for the pattern formation. The invariance of Df across the watermark is due to the universal property of the invasion percolation process. Another independent way to characterize the morphology of a clustered surface is to calculate the two-dimensional power spectrum of the surface. However, the SEM top-view images not only show the lateral characteristics of the clusters, such as size, shape, distance, but also contain their vertical characteristics, such as bending and height. Thus, the power spectrum obtained from a direct Fourier transformation of the original SEM image will mix the height information with the cluster information, and will make the interpretation more complicated. To obtain only the cluster information, we have to level-set the image as a black and white image. An example is shown in Figure 8. Figure 8a is the original SEM image taken from the center of a watermark on 890 nm high rod surface, and Figure 8b shows the blackand-white image of Figure 8a after the level-setting. This image only gives the lateral distribution and shape of the clusters. According to Amar et al., at large k, the power spectrum for a
(34) Voss, R. F.; Laibowitz, R. B.; Allessandrini, E. I. Phys. ReV. Lett. 1982, 49, 1441.
(35) Lo´pez, R. H.; Vidales, A. M.; Zgrablich, G. Physica A 2003, 327, 76. (36) Ferer, M.; Bromhal, G. S.; Smith, D. H. Physica A 2004, 334, 22.
IP model has been proposed to model the displacement of a wetting fluid in a porous medium, by the injection of another nonwetting fluid. In the absence of viscous or gravitational forces, the displacement is controlled solely by the capillary pressure, and the invasion front separating the two fluids advances by penetrating the pore throat at the front with the largest size.32,33 Physically, the driving force in the IP model is the capillary pressure difference between the two fluids, which is similar to that of the water penetrating the nanorod array. The differences between the conventional invasion percolation and the invasion in nanorods are (1) the two fluids in the nanorod array are water and air and (2) the porous network becomes the openings in the randomly distributed nanorod array. Since in the present study one can only measure the static SEM images of those invaded surfaces, one can only compare the static characteristics of the cluster morphology with that of the IP model. One most important parameter of those characteristics is the fractal dimension, Df, of the clusters. The fractal dimension, Df, can be determined by measuring the perimeter L and area A of the fractal clusters34
L ∝ ADf/2
(26)
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Fan and Zhao
clustered surface obeys a power law37
P(k) ∼ k-Df
(27)
Usually a circular average of the two-dimensional power spectrum is used to obtain the fractal dimension, so the circular averaged power spectra of Figure 8, panels a and b, are both calculated and plotted in log-log scale in Figure 8c. Both power spectra have a maximum at kc ≈ 0.546 µm-1. However, at k > kc, both spectra have a power law tail, and the exponents are very different. For the power spectrum obtained from the original SEM image (Figure 8a), the exponent is -2.39, whereas the exponent obtained from the level-setting image (Figure 8b) is -1.81. The large exponent obtained from the original SEM image is the result of the mixing of the height information (grey scale) with the lateral information in the power spectrum, or in other words, one has to treat the surface as a true topographic surface. However, the exponent (the fractal dimension) obtained from the level-setting image (two-level surface) is in agreement with those obtained by the perimeter-area method described above. Besides the fractal dimension Df of the clusters, the diameter or the area of the cluster is another important parameter to characterize the cluster morphology. This parameter can be derived from the power spectrum as well. For all of the levelsetting SEM images, either from different locations of the same watermark, or from watermarks on different nanorod surface, all of the two-dimensional power spectra have the same ring-like structure, indicating that there is a characteristic length scale on the surface. The center of the power spectrum ring kc is directly related to the average cluster-cluster distance ξ on the surface by kc ) 1/ξ.19 Since the coverage of the nanorod f on the surface remains unchanged both before and after water invasion, and the clusters are closely packed nanorods, the average diameter of the clusters is proportional to the average separation ξ of the clusters, i.e., ξ ) 1/kc represents the average diameter of a cluster, and ξ2 is proportional to the average area of a cluster. Figure 9 plots the ξ2 obtained from kc versus the distance r from the center of the watermark for three different nanorod surfaces with heights of 115, 649, and 890 nm, respectively. For all three different rod heights, the ξ2 almost remains as a constant from the center of the watermark to a relative large distance r from the center and then starts to decrease. This is consistent with the observation in Figure 6. The ξ2 values near the center of the watermark are 0.024, 1.17, and 2.56 µm2, for rod heights of 115, 649, and 890 nm, respectively; that is, the cluster size increases monotonically with the rod height. At the edge of the watermark, ξ2 decreases almost linearly with r, and the decreasing rates (slope) obtained from the linear fitting are -0.037, -0.18, and -1.24 µm2/mm for rod heights of 115, 649, and 890 nm, respectively. That is, the absolute value of the slope increases with the rod height. To understand why both the cluster size and the cluster size decreasing rate at the edge of a watermark depend strongly on the nanorod height, we propose a model for the clustering, from the free energy point of view. Before the water invaded into the nanostructure, as shown in Figure 10a, the total free energy for a cluster with N nanorods can be written as
Fi ) NπDhγSV +
x3 2 Nd0 γSV 2
(28)
where D and d0 are the diameter and average separation of the nanorods, respectively, and γSV is the interfacial energy between the solid and the vapor (air). The x3/2 in the second term of eq (37) Amar, J. G.; Family, F. F.; Lam, P.-M. Phys. ReV. B 1994, 50, 8781.
Figure 9. Plot of the cluster area ξ2 versus the distance r from the center of the watermark for nanorod heights of 115, 649, and 890 nm, respectively. Near the center of the watermark, the area ξ2 almost remains constant, while at the edge of the watermarks, the area ξ2 decreases almost linearly with r.
Figure 10. Sketches showing (a) the arrangement of the nanorods before water invasion and (b) the bundling of the nanorods after water invasion with a general water height level l from the substrate.
28 is obtained by assuming a unit cell of a close-packed lattice. When liquid invades the structure with a height of l as shown in Figure 10b and the nanorods start to bundle together, the total free energy changes to
D2 γ + 4 SV x3 2 D2 Nd0 γSL - Nπ γSL + Em (29) 2 4
Ff ) NπDlγSL + NπD(h - l)γSV + Nπ
where γSL is the interfacial energy between the solid and the liquid and Em is the mechanical energy gained through nanorod clustering. Assuming that all nanorods in a cluster are closely packed, and using a model described in ref 38, the mechanical energy for an N-cluster bundle can be written as
π2EYD4(d0 - D)2 (N - 1)2 Em ) 32h3
(30)
where EY is the Young’s modulus of the nanorod material. Thus, the total energy change during the spreading and bundling processes is
Watermarks Formed in Nano-Carpet Effect
(
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)
x3 2 D2 d0 - π (γSL - γSV) + 2 4 π2EYD4(d0 - D)2 (N - 1)2 (31) 3 32h
∆F ) NπDl(γSL - γSV) + N
where γ cos θe ) γSV - γSL according to Young’s theory of a flat surface. In order for the process to be energy favorable, d∆F/dN ) 0
Dlγ cos θe +
(
)
x3 2 D2 d γ cos θe ) 2π 0 4 πEYD4(d0 - D)2 (N - 1) (32) 16h3
The total cluster area can be written as
(
)
x3 2 D2 3 h γ cos θe d 2π 0 4 , πEYD2(d0 - D)2 (33)
16 Dl + Acluster ) (N - 1)D2 )
center of the watermark, the liquid level in the nanorods starts to decrease. At distance r from the center of the watermark, the height l of the liquid level in the nanorod becomes
l ) h - (r - r0) tan R Then eq 34 becomes
ξ2 ) 16[Dh - D(r - r0) tan R + x3d02/(2π) - D2/4]h3γ cos θe π2EYD2(d0 - D)2 ) ξ02 -
x
(
ξ)
)
x3 2 D 3 d h γ cos θe 2π 0 4 . π2EYD2(d0 - D)2
16 Dl +
(34)
For the clusters in the center of the watermark, l ) h, and eq 34 shows that the average size ξ of the cluster in the center of the watermark monotonically increases with the nanorod height, which is consistent with our observation with different nanorod heights in Figure 9. The decreasing of the cluster size on the edge of the watermark can also be explained by eq 34. According to the studies on the liquid capillary flow in a groove, the capillary front in the groove can be approximated by linearly decreasing the liquid level with an angle R, which is different from the equilibrium contact angle θe.39 If we make a similar assumption to the liquid front inside the nanorod array, as shown in Figure 11, the behavior of the clustering at the edge of the watermark can be estimated. As shown in Figure 11, at distance r0 from the
Figure 11. Sketch showing the water invasion front inside the nanorod array. The water height level l at the edge of the invasion front deceases linearly with a constant angle R with respect to the substrate surface. For simplicity, the bundling of nanorods is not illustrated, but it should be like the cluster depicted in Figure 10.
ξ02
(r - r0) tan R
h + x3d02/(2πD) - D/4
(36)
where ξ0 is the cluster size in the center of a watermark. Equation 36 predicts that if one plots the ξ2 versus r, at the edge of the watermark, the average area of the clusters should decrease linearly with the distance r. The slope of the linear decline
-
i.e., the cluster-cluster distance ξ can be expressed as 2
(35)
ξ02 tan R h + x3d02/(2πD) - D/4)
is determined by the shape and separation of the nanorods. The absolute value of the slope increases monotonically with the nanorod height according to eq 34. This prediction is consistent with the experimental results shown in Figure 9.
V. Conclusions In summary, we have given a very detailed description of the watermark formation on silicon nanorod arrays. It has been demonstrated that the watermark formation on silicon nanorods is caused by a spreading precursor, and the precursor rim determines the final size of the watermark. For nanorods with different heights, the watermark size follows a simple power law relationship with the water droplet volume. A simple model based on the spreading and evaporation on a smooth surface, and the Washburn’s capillary flow inside nanorod array has been proposed to qualitatively interpret the experimental observations. The detailed morphology of the watermark exhibits the bundling of the nanorods. The bundled cluster has a fractal dimension of ∼1.8, which is consistent with the invasion percolation model. The cluster size is also closely dependent on nanorod height. Near the center of the watermark, the cluster size almost remains constant, while at the edge it decreases with the distance from the center. This phenomenon can be qualitatively interpreted through the change of the total free energy during the precursor invading the nanorod array. Due to the small dimensions of the nanorods, the mechanical energy change due to the bending and clustering of nanorods must be taken into account. Acknowledgment. This work is supported by NSF ECS0304340 and The University of Georgia Research Foundation Junior Faculty Research Fund. The authors thank Will Verdecchia for proofreading the manuscript. LA053237N (38) Zhao, Y.-P.; Fan, J.-G. Appl. Phys. Lett. (L05-07336R), in press. (39) Rye, R. R.; Mann, J. A.; Yost, F. G.; Langmuir 1996, 12, 555.
