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New Roughness Parameter for the Characterization of Regularly Textured or Ordered Patterned Superhydrophobic Surfaces W. Li,† Y.P. Diao,*,§ S.Y. Wang,*, G. P. Fang,‡ G. C. Wang,† X. J. Dong,† S. C. Long,§ and G. J. Qiao‡ †
School of Materials Science and Engineering, Nanchang Hangkong University, Nanchang 330063, China, College of Pharmacy, Dalian Medical University, Dalian 116044, P.R. China and College of Pharmacy, Liaoning University of Traditional Chinese Medicine, Dalian 116600, China, Department of Orthopaedics, the First Affiliated Hospital of Dalian Medical University, Dalian 116011, China, and ‡State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China )
§
Received December 24, 2008 Surface geometry affects strongly superhydrophobic behavior. To characterize the effect, roughness as a comprehensive geometrical parameter is used, but this parameter in its general mathematic expression cannot reflect exactly such a geometrical effect, in particular, for the regularly textured or ordered patterned superhydrophobic surfaces. In this study, we propose a new parameter to mathematically describe roughness for such superhydrophobic surfaces. On the basis of this parameter, an ideal surface texture with the maximum roughness for achieving the superhydrophobicity is suggested, which is consistent with the previous experimental observations and theoretical considerations.
Introduction Surface morphology or topography plays a crucial role in various physical, mechanical, and chemical phenomena.1-4 For example, surface geometry can dramatically affect superhydrophobic behavior.5-9 To quantitatively evaluate a surface morphological effect, a general or comprehensive geometrical parameter, roughness, is usually used. In general, the morphological analysis of normal surfaces (e.g., the surfaces of engineering parts) by roughness is mainly based on the Gaussian distribution,1,2 because the topographical characteristics of the surfaces are random or isotropic. For such surfaces, roughness most commonly refers to the variations in the height of the surfaces relative to a reference plane. It is usually characterized by one of the two statistical height descriptors: mean or center-line average (Ra) and root mean square (Rq).1 In other words, both Ra and Rq can be generally defined as the deviation of the surface height from the mean line, which is defined, so that equal areas of the profile lie above and below it, through the profile, as illustrated in Figure 1. Therefore, Ra is formally defined as Ra ¼
1 L
Z
L
jyðxÞjdx
ð1Þ
0
where as Rq is expressed as Rq 2 ¼
1 L
Z
L
y2 ðxÞdx
ð2Þ
0
*Correspondence authors. E-mail:
[email protected]; wangshouyu09@ 126.com. (1) (2) (3) (4) (5) (6) (7) (8) (9)
Thomas, T. R. Rough Surfaces; Imperial College Press: London, U.K., 1999. Hutchings, I. M. Tribology; Edward Arnold: London, U.K., 1992. Li, :: W.; Li, D. Y. J. Chem. Phys. 2005, 122, 064708–064714. Oner, D.; McCarthy, T. J. Langmuir 2000, 16, 7777–7783. Li, W.; Amirfazli, A. J. Colloid Interface Sci. 2005, 292, 195–201. Li, W.; Amirfazli, A. Adv. Colloid Interface Sci. 2007, 132, 51–68. Li, W.; Amirfazli, A. Soft Matter 2008, 4, 462–466. Li, W.; Fang, G.; Li, Y.; Qiao, G. J. Phys. Chem. B 2008, 112, 7234–7243. Fang, G.; Li, W.; Wang, X.; Qiao, G. Langmuir 2008, 24, 11651–11662.
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DOI: 10.1021/la901073w
where y is the height of the surface above the mean line at a distance x from the origin and L is the overall length of the profile under examination (Figure 1).
Analysis The average height parameters, such as Ra and Rq, are important to mechanical parts or components and can be used as an approximate measure to evaluate the effect of surface topography. However, for regular micro- or nanotextured or ordered patterned surfaces, such a roughness parameter cannot exactly reflect the effect of surface geometry. For example, the highest asperities play the most important roles because the damage of the interface can be performed by the few high asperities in a tribological system; on the other hand, various valleys with different sizes can have completely different effects on the storage of wear debris and even lubrication retention and flow.1-3 Especially, for superhydrophobic surfaces, various geometrical systems may give the same roughness value if the above expressions are applied, but actually, they have very different surface morphologies (textures) and, hence, exhibit completely different wetting behavior. This has been demonstrated very well in our previous work.5-9 To better characterize the geometrical effect, other height parameters have been suggested for some specific cases [e.g., various extreme-value descriptors, such as the distance between the highest asperity and the lowest valley (Rt), the distance between the highest asperity and the mean line (Rp), and the distance between the mean line and the lowest valley (Rv), etc.1,2]. However, for a complete characterization of a profile or a surface, any of the above parameters and even their combinations are inadequate. This happens mainly because these parameters involve primarily the relative departure of the profile in the vertical direction only. Furthermore, they do not provide any information on the shape, slope, spacing, and size of the asperities and valleys or the frequency and regularity of their occurrence. Therefore, these height parameters are usually invalid and even indicate erroneous information on the special cases with regular, anisotropic, and non-Gaussian-distributed asperities, which usually appear in the artificial superhydrophobic surfaces. For example, Figure 2 illustrates completely different surface profiles, implying different geometrical effects as well; however,
Published on Web 4/27/2009
Langmuir 2009, 25(11), 6076–6080
Li et al.
Article
Figure 1. Schematic view of a surface profile to demonstrate the mathematic definition of Ra and Rq. y is surface height, relative to a mean line, plotted against distance. The overall length of the profile under examination is L.
Figure 2. Various surface profiles with the same roughness value by eq 1 or 2.
Figure 3. Same surface texture profile with different scales. it is possible to give the same Ra and Rq. It is very clear that such surface profiles can exhibit completely different surface behavior, but both Ra and Rq can hardly reflect this difference. On other hand, Figure 3 shows two surface profiles with the same shapes and frequencies and gives considerably different Ra and Rq values only because of the difference in size scale. It is also worth noting Langmuir 2009, 25(11), 6076–6080
that, for some practical engineering cases, the absolute scale is not meaningful, while the relative geometrical parameters of acting surfaces in contact play a dominant role. Thus, such a difference in absolute roughness value hardly reflects the actual different effects of the two surface profiles. In addition, neither Ra nor Rq can indicate useful information on the effect of regular shape. This can be demonstrated using a simple pillar structure as an example. Figure 4 illustrates two pillar structures with the same Ra and Rq. However, obviously, they can behave with different surface behavior. Moreover, Figure 5 illustrates two pillar structures with the same asperity but different spacing. Both Ra and Rq indicate that Figure 5a is rougher than Figure 5b. However, in some practical cases or definitions, it is more reasonable to believe that Figure 5b can be rougher than Figure 5a (see below). On the basis of the above analysis, it can be seen that the single numerical roughness parameters, such as Ra and Rq, are mainly useful for characterizing the normal conventional surfaces with randomly distributed heights or the same type that are produced in the same way. However, for the designed and fabricated superhydrophobic surfaces with regular or ordered shape, spacing (pitch), and size as well as scale, these height parameters are DOI: 10.1021/la901073w 6077
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Figure 6. Schematic view of a surface bump profile to demonstrate the mathematic expression of the proposed roughness parameter.
Figure 4. Two pillar structures with the same roughness.
bump profile to evaluate the contribution of bump density or frequency; this means that the larger the perimeter, the more dense the bumps. The proposed roughness parameter can therefore be finally expressed as Z Z 1 L 1 L yðxÞdx RðlÞÞdl ð3Þ RN ¼ L 0 l 0 where l is arc length and R is the slope at the point with l. Such roughness expression therefore completely reflects the effects of feature shape, which can be described using slope, density, and height, for a textured superhydrophobic surface, whereas the conventional roughness parameters only consider the effect of height. However, whether or not this expression is effective practically for the description of roughness of various superhydrophobic surfaces needs further examination of different rough surfaces (e.g., imagining or artificial regular surfaces).
Figure 5. Two pillar structures with the same asperity but different spacing.
inadequate and even invalid. It is therefore necessary to propose a new parameter to characterize such superhydrophobic surfaces, in particular, in 3D cases. To this end, a basic thinking is that a new parameter should give all of the information on the shape of a single texture and its density or frequency besides the information on height. Furthermore, for specific surface textures or patterns, such as those examples shown in the above figures, the new parameter can give an accurate roughness value and, hence, indicate a clear superhydrophobic behavior and plausible physical mechanism.
Discussion On the basis of the proposed parameter, it is easy to evaluate roughness for surfaces with various textures. For a quantitative comparison, it is necessary to calculate the roughness value using eq 3. Here, we show a typical example for calculation of roughness for Figure 6. If we use a sine function to describe the topographic curve in Figure 6, then the arc length integral for the sine curve is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ dl ¼ 1 þ cos2 ðxÞdx and the slop R can be expressed as R ¼ y0 ðxÞ ¼ cosðxÞ
New Parameter Let us aim at superhydrophobic surfaces, especially, with regular textures. Because superhydrophobic behavior is determined mainly by a few atomic layers at a surface,6 we can choose the central layer of the surface topography as a primary reference plane to deduce the roughness expression for this surface, as illustrated in Figure 6. As discussed earlier, for regular or ordered distributed textures, a comprehensive roughness parameter should give the information on, at least, shape of the textures. In particular, it is necessary to consider the effects of slope and frequency of textures, which are crucial in determining the superhydrophobic behavior, besides their heights. To demonstrate the above considerations, we take the bumps shown in Figure 6 as a typical example. In the present case, for the contribution of the bump height to the roughness, we can still use an expression similar to the conventional one. Moreover, for the contributions of bump shape, we can introduce a point slope of the bump profile. Furthermore, we can use a perimeter of the 6078
DOI: 10.1021/la901073w
therefore, eq 3 can be further expressed as Z Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 L 1 L sinðxÞdx cosðxÞ 1 þ cos2 ðxÞdx RN ¼ L 0 L 0
ð5Þ
ð6Þ
where Z
L
sinðxÞdx ¼ 1 -cos L
ð7Þ
0
and Z L 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin L cosðxÞ 1 þ cos2 ðxÞdx ¼ arcsin pffiffiffi ( 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin L sin2 L pffiffiffi 1 ð8Þ 2 2 Langmuir 2009, 25(11), 6076–6080
Li et al.
Then RN
Article
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 1 sin L sin L sin2 L7 6 ¼ 2 ð1 -cos LÞ4arcsin pffiffiffi þ pffiffiffi 1 5 L 2 2 2 ð9Þ
Here, it is noted that a detailed mathematical derivation of RN has been given in the Appendix. Similarly, we can use eq 3 to deduce the specific roughness expression for Figures 1-5 as long as a specific geometrical function for the description of their surface topography is given. Here, for simplicity, we can directly give a qualitative comparison in roughness between these textures shown in Figures 1-5 to show the usefulness of this robust approach. For example, in Figure 2, it is clear that among panels a-d of Figure 2, the roughness for Figure 2d is the largest, the roughness for Figure 2c is the second largest, and the roughness for Figure 2b is the third largest, while roughness for Figure 2a is the smallest; the roughness for Figure 2g is larger than that for Figure 2f. It should be pointed out that a clear difference in roughness between some figures (e.g., Figure 2e with Figure 2f or Figure 2a) can hardly be seen using eq 1 or 2, but this can be easily performed using eq 3. In Figure 3, there is still a difference in roughness between the two textures; this conclusion can be drawn directly by eq 3. However, it is expected that this difference is relatively small if the conventional roughness parameters are used. In Figure 4, obviously, Figure 4b gives larger roughness, while conventional parameters indicate the same roughness. In Figure 5, it is apparent that the two textures should show different roughness, which strongly depends upon spacing or density of pillar, but the conventional parameters indicate the same roughness. On the basis of the above results, it is demonstrated that the proposed roughness parameter can provide more information on surface geometry, in particular, on contact topography. Moreover, if roughness can be regarded as a simple geometrical measure for the superhydrophobic behavior without consideration of other factors, such as chemical compositions, it is expected that the surface with larger roughness could lead to a better superhydrophobicity. More importantly, for the design of superhydrophobic textures, the calculations of the proposed parameter can allow one to provide some guidelines. This should therefore facilitate one to further reveal the dynamic behavior, such as the self-cleaning ability and the physical mechanisms for the superhydrophobic surfaces. Finally, we show a simple example by applying the roughness parameter in the design of ideal superhydrophobic surfaces to demonstrate the use of the present approach. On the basis of eq 3, large slope, density, and height of features of a textured surface are required for a large roughness. Furthermore, for a large slope, an acute angle, especially, a right angle, is required; for a large density, the features should be very thin and arranged as closely as possible; and for a large height, the features should be very high. As a result, such a surface should have a densely packaged needle-like textured structure, as illustrated in Figure 7. This structure with the above three geometrical characteristics will have a maximum roughness and, hence, lead to an ideal superhydrophobic behavior, such as large contact angle, small contact angle hysteresis or sliding angle, and a composite state or a transition from noncomposite to composite. Here, it should be pointed out that such a needle-like structure has been previously experimentally suggested to create superhydro(10) Feng, L.; Li, S.; Li, Y.; Li, H.; Zhang, L.; Zhai, J.; Song, Y.; Liu, B.; Jiang, L.; Zhu, D. Adv. Mater. 2002, 14, 1857–1860. (11) Miwa, M.; Nakajima, A.; Fujishima, A.; Hashimoto, K.; Watanabe, T. Langmuir 2000, 16, 5754–5760.
Langmuir 2009, 25(11), 6076–6080
Figure 7. Densely packaged needle-like texture structure with the largest roughness based on the proposed roughness parameter.
phobic surfaces.10,11 Nevertheless, as subsequently indicated theoretically,5-7 although this structure is necessary for superhydrophobicity, it can compromise the mechanical durability of a superhydrophobic surface, such as surface texture strength (e.g., breaking of features). For both superhydrophobicity and durability, a fractal structure similar to that found at the biological surfaces (e.g., lotus leaves) could be the best choice, which has been revealed well both experimentally and theoretically.7,12,13
Conclusion In closing, we propose a new parameter to mathematically describe roughness for superhydrophobic surfaces. On the basis of this parameter, we can give a designed or imagined surface texture with the largest roughness: a densely packaged needle-like textured structure. Such a structure should exhibit the best superhydrophobicity, which is also suggested by the previous experimental investigations and is consistent with the recent theoretical studies.
Appendix: Detailed Mathematical Derivation of Roughness of RN in Figure 6 RN can be mathematically expressed as Z Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 L 1 L RN ¼ sinðxÞdx cosðxÞ 1 þ cos2 ðxÞdx ðA1Þ L 0 L 0 It is simply derived that Z L sinðxÞdx ¼ 1 -cos L
ðA2Þ
0
Then Z
L
Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos x 1 þ cos2 ðxÞdx ¼
0
L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
2 -sin ðxÞcos xdx
0
Z
sin L pffiffiffiffiffiffiffiffiffiffiffi 2 -t2 dt
¼ 0
Z
ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 -2s2 d 2s
sin pL 2
¼ 0
Z
ffi pffiffiffiffiffiffiffiffiffiffiffi 1 -s2 ds
sin pL 2
¼2 0
Z
arcsin
¼2
ffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 -sin2 θdsin θ
sin pL 2
0
Z ¼2
arcsin
ffi
sin pL 2
cos2 θdθ
0
(12) Nosonovsky, M.; Bhushan, B. Adv. Funct. Mater 2008, 18, 843–855. (13) Qu, M. N.; Zhao, G. Y.; Cao, X. P. Langmuir 2008, 24, 4185–4189.
DOI: 10.1021/la901073w 6079
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Z ¼2
ffi
sin pL 2
arcsin 0
Z ¼
ffi
sin 2arcsin sinpffiffi2L
sin pL 2
arcsin
Thus
1 ðcos 2θ þ 1Þ dθ 2
2 ðcos 2θ þ 1Þdθ
0
1 2
Z
arcsin
ffi
sin pL 2
ðA7Þ
i.e.,
ðcos 2θ þ 1Þdθ
Z
0
sin 2arcsin sinpffiffiL 2 sin L ¼ arcsin pffiffiffi þ 2 2
sin 2ω ¼ sin ω cos ω 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin L sin2 L ¼ pffiffiffi 1 2 2
¼
L 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin L 2 cosðxÞ 1 þ cos ðxÞdx pffiffiffi þ 2
ðA3Þ
sin L pffiffiffi 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2 L ðA8Þ 12
Let sin L ω ¼ arcsin pffiffiffi 2
Finally, ðA4Þ RN
then sin L sin ω ¼ pffiffiffi 2 and
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin L 2 sin2 L ¼ 1cos ω ¼ 1 - pffiffiffi 2 2
DOI: 10.1021/la901073w
ðA5Þ
ðA6Þ
1 sin L sin L 6 ¼ 2 ð1 -cos LÞ4arcsin pffiffiffi þ pffiffiffi L 2 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 sin2 L7 15 2 ðA9Þ
Note Added after ASAP Publication. This article was published ASAP on April 27, 2009. Two authors have been added to the paper. The correct version was published on May 26, 2009.
Langmuir 2009, 25(11), 6076–6080
