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General equation of wettability: A tool to calculate the contact angle for a rough surface
Chemical Physics Letters 574 (2013) 106–111
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General equation of wettability: A tool to calculate the contact angle for a rough surface Anjishnu Sarkar, Anne-Marie Kietzig ⇑ Department of Chemical Engineering, McGill University, 3160 University Street, Montreal, QC, Canada H3A 2B2
a r t i c l e
i n f o
Article history: Received 24 March 2013 In final form 23 April 2013 Available online 2 May 2013
a b s t r a c t This work introduces a mathematical tool, namely general equation of wettability, which simplifies the determination of the contact angle for rough surfaces. A free energy minimization method is applied to an intermediate wetting state with partial liquid penetration into roughness valleys in addition to the well-known states of Wenzel and Cassie wetting. A wetting parameter is coined, which stands out as a fingerprint of a particular wetting state of a surface topology. Substitution of the appropriate wetting parameter returns the Cassie and Wenzel equations, while an implicit solution is presented for the intermediate state, which is consistent with energy minimization. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction As controversial its grammatical roots might be, the word ‘wettability’ continues to enjoy the unswerving attention of surface sciences. A mark of the surface-liquid compatibility, wettability forms the key topic in applications ranging from froth flotation to detergency industries [1,2]. Wettability is denoted as the angle subtended by the tangent lines at the solid–liquid and liquid–air interfaces, also known as the contact angle (h). Contact angle is a direct measure of the interfacial energy of a solid–liquid–air system. The contact angle of an atomically flat surface with infinite rigidity and uniform chemical composition solely depends on the surface chemistry, and is termed as Young’s contact angle. More physical surfaces with finite rigidity result in contact angle hysteresis, but still correspond to their Young contact angle (hY) [3–6]. For such a surface, the necessary energy required in building a solid– liquid interface (cSL), and hence a drop of liquid is governed by the Young’s contact angle, and in turn, the surface chemistry.
‘apparent’ horizontal base-line. Since it can be correlated with the surface chemistry (ACCA) and topology (roughness), the APCA is widely used as the signature of the interfacial energy [7–13]. The APCA is also governed by the chemical homogeneity of a rough surface. Distinct wetting modes exist for chemically homogeneous and heterogeneous surfaces, respectively, known as homogeneous and heterogeneous wetting regimes. Additionally, it is possible for chemically homogeneous surfaces to exhibit a heterogeneous wetting regime. This is possible through the existence of air pockets trapped between the surface and the liquid, which then serve as chemical heterogeneities. For a rough and chemically homogeneous surface, the homogeneous and heterogeneous wetting regimes are respectively expressed by Wenzel and Cassie– Baxter models [14,15]. In both the aforementioned models, the APCA is explicitly expressed in terms of the surface topology and the chemistry. The Wenzel’s contact angle (hW) is given as the product of the Wenzel’s roughness ratio rw and the cosine of the YCA.
cSL ¼ cSA cLA cos hY
cos hW ¼ r w cos hY
ð1Þ
The contact angle at every microscopic point on a rough surface perfectly obeys Young’s equation, and is termed as the actual contact angle (ACCA) for the concerned microscopic point [7]. To determine the ACCA, the exact position of the drop must be pinpointed with respect to a surface topology of a micrometer/nanometer length-scale, which makes it difficult to coin the ACCA and the interfacial energy. The aforementioned issue has been resolved by the use of an apparent contact angle (APCA), which is characterized by a hypothetical solid–liquid interface that runs parallel to an ⇑ Corresponding author. Fax: +1 514 398 6678. E-mail addresses: [email protected] (A. Sarkar), anne.kietzig@ mcgill.ca (A.-M. Kietzig). 0009-2614/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2013.04.055
ð2Þ
Here, rw signifies the ratio of the actual solid–liquid interfacial area on a rough surface to the solid–liquid interfacial area on a corresponding smooth surface. According to the Wenzel’s model, surface roughness preserves its chemical wettability, i.e. the hydrophilic surface becomes more hydrophilic with roughness and vice versa. For the Cassie–Baxter model, the APCA is expressed as a function of the YCA and the solid fraction f, i.e. the fraction of solid surface encompassed by the liquid.
cos hCB ¼ f cos hY þ f 1
ð3Þ
Apart from the Wenzel and Cassie states, a third wetting state is possible, where the liquid partially penetrates into the roughness structure [10,16–18]. The partial penetration or the penetration
A. Sarkar, A.-M. Kietzig / Chemical Physics Letters 574 (2013) 106–111
107
Figure 1. Penetration depth h of different wetting states indicating wetted (dark) and dry (bright) pattern surfaces. (a) Heterogeneous wetting: Cassie: h = 0; (b) heterogeneous wetting: metastable Cassie: 0 < h < c; (c) homogeneous wetting Wenzel h = c.
depth can assume multiple values, each of which corresponds to a distinct geometrical configuration of the interfaces and consequently, discrete values of the apparent contact angle. In the process of plotting contact angle data for surfaces with different chemistry and topologies, Erbil [18] has recorded cases of heterogeneous wetting state with APCA both greater and less than the Cassie–Baxter contact angle for the same surface [18]. The wetting state associated with a partial penetration inside the roughness valleys of a surface topology has been termed as an intermediate state by Erbil. Literature lacks a way to derive the APCA for such a state using classical thermodynamics. In this Letter, a novel method is designed to express the APCA for a heterogeneous regime with a finite penetration. Since they are characterized by existence of trapped air under the liquid, both the intermediate state and the Cassie state belong to heterogeneous regime. Cases of zero and partial penetration inside the roughness valleys are respectively termed as Cassie and metastable Cassie states (Figure 1) in the entire work. The term metastable is attributed to the difficulty in deducing the thermodynamic feasibility of such a wetting state. In the past few years, two parameters have been isolated as potential candidates in determining the value of contact angle, namely a one-dimensional triple line and a two dimensional contact area [19–23]. The latter allows the derivation of the Wenzel and Cassie equations via minimization of the free energy available to the area which spans the solid–liquid system. The method of free energy minimization, and hence the aforementioned equations of wetting are perfectly applicable for topologies that are uniform [21]. Using free energy minimization, a set of equations is derived in this discourse which simplifies the calculation of contact angles on rough surfaces. 2. Results and discussions A liquid drop of a given volume can orient itself in different manners on a solid surface with a well-defined surface topology. Unlike its homogeneous counterpart, where the surface under the liquid is subject to the solid–liquid (S–L) interface, the heterogeneous regime is characterized by a liquid–air (L–A) interface under the drop. The location of the L–A interface with respect to the surface topology determines the magnitudes of the mutual area available to each of the interfaces (S–L, L–A or S–A). The interfacial areas determine the equilibrium free energy available to the solid– liquid system. At thermodynamic equilibrium, the free energy available to a solid–liquid system reaches its minimum value. Depending on the location of the triple line with respect to the surface topology, the equilibrium free energy can assume different values, each corresponding to a distinct wetting state (Figure 1). Each wetting state is characterized by an apparent contact angle. Thus, the derivation of a general expression for the apparent contact angle (APCA) on a surface must be traced back to the possible locations of the L–A interface with respect to the surface topology.
Figure 2. Square pillar topology. (a) Three dimensional view showing pillar width a, pillar spacing b and pillar height c (b) top view of four pillars, which outline a unit (c) simplest building block, i.e. unit, inscribed in the dotted rectangle.
Here, the expression for the apparent contact angle is derived for a surface topology, and is shown to be valid for other topologies. A regular distribution of square pillars is chosen as the surface topology, which is characterized by cuboids with individual pillar width a (lm), inter-pillar spacing b (lm) and pillar height c (lm) (Figure 2a). To facilitate minimization of the available free energy, a simplest building block in the following is referred to as a unit. A section of four square pillars (Figure 2b) is chosen such that on repetition, it generates the entire distribution of pillars under the drop. In this first example, a unit is characterized by a distribution comprising four square pillars, each with an individual pillar width a/2 (lm), inter-pillar spacing b (lm) and height c (lm) (Figure 2c). The homogeneous and heterogeneous regimes are denoted by the extent of liquid penetrating the apex of the roughness valley, i.e. the penetration depth (Figure 1). The liquid assumes partial (0 6 h < c) and complete penetration (h = c), respectively, in the heterogeneous and homogeneous regimes. In the following, it is assumed that the heterogeneous regime is characterized by a L–A interface under the liquid that runs parallel to the pillar base. Furthermore, it is assumed that the volume of a drop is invariant, i.e. evaporation or condensation effects are ignored. The calculations are limited to a case of spherical drop shape, i.e. when the radius of the drop does not exceed the capillary length of the liquid. To establish such a condition, the drop volume (V) shares a unique relationship with the drop radius ð^r Þ, and APCA (h).
V¼
p^r3 3
ð2 þ cos hÞð1 cos hÞ2 ¼ const
ð4Þ
Furthermore, it is assumed that the volume of the liquid inside the roughness valleys plays a trivial role, as compared to the volume of the drop. It is found that there exists a threshold pillar
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A. Sarkar, A.-M. Kietzig / Chemical Physics Letters 574 (2013) 106–111
Table 1 Interfacial area terms of a unit. Heterogeneous wetting regime
Homogeneous wetting regime
unit 2 ðDAunit SL Þhet ¼ ðDASA Þhet ¼ 4ah þ a
unit ðDAunit SL Þhom ¼ ðDASA Þhom ¼ 4ac
þða þ bÞ2 2
2
2ðaþbÞ ðDAunit LA Þhet ¼ bð2a þ bÞ þ 1þcos h
2ðaþbÞ ðDAunit LA Þhom ¼ 1þcos h
The expressions for the interfacial area terms, as given in Table 1, are substituted into Eq. (7). Using Young’s equation, the free energy of a unit is expressed in terms of the L–A interfacial tension (cLA), surface chemistry (hY) and surface topology (a, b and c). Literature lacks a common expression for free energy that is invariant with a wetting state. With the introduction of a wetting parameter (jh) (Table 2), the free energy expressions Eq. (7) for the homogeneous and heterogeneous regimes (Table 1) can be expressed as a single consolidated expression Eq. (10).
Gunit ¼ cLA ða þ bÞ2 h height for correct estimation of the homogeneous wetting regime [24] (Supplementary material 1). This is why the results from this discussion are applicable for pillar heights less than 300 lm. Finally, for the ease of calculation, it is assumed that the total number of units under a drop assumes an integer value (Eq. (5)).
g¼
2 Area under the drop p^r2 sin h ¼ Area of a unit ða þ bÞ2
ð5Þ
The free energy (DG) is understood as the interfacial energy available to the system when a drop of liquid is formed, i.e. when the three interfaces are built. Given as the difference between the free energies with and without a drop (Gdrop and GNodrop, respectively), the free energy depends on the interfacial tensions and interfacial area terms Eq. (6).
DG ¼ GDrop GNo drop ¼ cSL DASL þ cLA DALA þ cSA DASA
ð6Þ
(DGunit h ),
The free energy available to a unit similarly depends on the interfacial area terms for a unit, and is expressed by using the superscript ‘unit’ equation (7). The concerned wetting state is denoted by subscript h, which further denotes the penetration depth as introduced in Figure 1.
DGunit h
unit SL DASL
¼c
unit LA DALA
þc
unit SA DASA
þc
ð7Þ
For a unit, the S–L and S–A interfacial areas are additive inverses of each other Eq. (8). This is because the S–A interfacial area lost owing to drop formation is identical to the S–L interfacial area built. unit DAunit SL ¼ DASA
ð8Þ
The L–A interfacial area for the solid–liquid system is calculated by including the contribution of the truncated sphere, and the interface under the liquid found only for the heterogeneous wetting regime. The corresponding area for a unit is expressed as the ratio of the total L–A interfacial area to the number of units under the drop Eq. (9).
DAunit LA ¼
DALA
ð9Þ
g
The area fractions for both the heterogeneous and homogeneous wetting regimes are listed in Table 1. Since the S–L (cSL) and S–A (cSA) interfacial tensions exhibit mutual dependence, they cannot be individually quantified. Assuming that Young’s equation is valid for a micrometer length scale, the aforementioned interfacial tension terms are eliminated by rearranging Eq. (1) [3].
cSA ¼ cSL cLA cos hY
ð1Þ
2 þ 1 þ jh ; 8h 2 fhom; hetg 1 þ cos h
Eq. (10) clearly indicates that the free energy available to a unit (fixed values of a and b) solely depends on the APCA and the wetting parameter. Since it takes the contribution from surface topology into consideration, the wetting parameter (jh) marks the fingerprint of a wetting state (Table 2). To minimize the free energy, the energy available to a unit is summed over all the pillars under the drop. The free energy of the solid–liquid system (DGtotal ) is evaluated by multiplying the h free energy of a unit with the total number of pillars under a drop.
DGtotal ¼ gDGunit ¼ h h
p^r2 sin2 h ða þ bÞ2
DGunit h
ð11Þ
Using Eq. (4), the radius ð^r Þ is expressed in terms of the volume (V) and APCA. The free energy (DGtotal ) is reduced to a dimensionless h quantity, namely normalized total free energy (DGtotal h; ) by division with the expression cLAp1/3(3 V)2/3. DGtotal h; ¼
DGtotal h 1 3
cLA p ð3VÞ
2
2 3
1
¼ð2þcoshÞ3 ð1coshÞ3 ð1þcoshÞ
Heterogeneous Homogeneous
2 þ1þjh 1þcosh
ð12Þ
On respectively substituting the values of 0 and c for the penetration depth h, the total free energy (DGtotal ) faithfully represents h the free energy expressions for the pure Cassie and Wenzel states available in literature [9,10]. In each of the above mentioned measures, the free energies of the homogeneous and heterogeneous regimes are separately calculated. The wetting parameter enables the integration of the free energy of all possible wetting states under the canopy of a single expression Eq. (11). For energy minimization, the derivative of the available free energy with respect to the wetting parameter (jh) is set to zero Eq. (12).
dDGtotal h; ¼0 dðcos hÞ
ð13Þ
The normalized free energy (DGtotal h; ) is incorporated into Eq. (12), which leads to an explicit correlation of jh and APCA, which is termed as the general equation of wettability Eq. (13). A detailed derivation that leads to Eq. (13) is listed in Supplementary material 2. General equation of wettability:
2ðcos h þ jh þ 1Þ ¼ ðcos h þ 2Þðcos2 h 1Þ
djh dðcos hÞ
ð14Þ
Since it lacks a term corresponding to the surface topology, this general equation of wettability provides an accurate description to the wetting state of any surface topology. The general equation can
Table 2 Expressions for wetting parameter (jh) for a square pillar topology. Wetting regime
ð10Þ
Free energy for a unit h i 2 2 2 cLA 2ðaþbÞ 1þcos h 4ah cos hY þ ða þ bÞ a ð1 þ cos hY Þ h i 2 2 cLA 2ðaþbÞ 1þcos h 4ac cos hY ða þ bÞ coshY
jh a 2 Þ ð1 þ coshY Þ 4ah cos2hY ðaþb ðaþbÞ
1 cos hY ð1 þ
4ac Þ ðaþbÞ2
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A. Sarkar, A.-M. Kietzig / Chemical Physics Letters 574 (2013) 106–111 Table 3 Derivative of jh to generate equations for the Wenzel, pure Cassie and the metastable state for square pillar topology. Wetting state
Expression
djh dðcos hh Þ
Wenzel
0
Metastable Cassie
dh 4a cos h2Y dðcos hh Þ
Pure Cassie
0
cos hW
Equation
¼ cos hY 1 þ
4ac ðaþbÞ2
Wenzel Characteristic set of equations
hM ¼ functionðh; hY ; a; b; cÞ 2 a cos hCB ¼ aþb ð1 þ cos hY Þ 1
ðaþbÞ
Cassie–Baxter
Figure 3. Deviation of compatibility parameter (u) from zero for different YCAs (beyond b/a > 4 the characteristic set of equations (Eq. (18)) is valid for all surface chemistries).
be specifically expressed for unique wetting states with the aid of appropriate subscripts (CB for h = 0, M for 0 < h < c and W for h = c, as mentioned in Table 3). For the cases of both complete penetration (h = c) and zero penetration (h = 0) of the liquid in the roughness valley, the wetting parameters (jh) for such cases assume constant values. As a result, the derivatives of jh, and consequently, the right hand side of Eq. (13) vanish (Table 3), leading to equation (14).
2ðcos h þ jh þ 1Þ ¼ 0
ð15Þ
As expected, substitution of values j0 and jc (Table 2) to Eq. (14), respectively, generate the Cassie and Wenzel equations. Thus, the Wenzel and Cassie equations are expressed under a common format, and the expression is independent of the concerned surface topology. For the metastable Cassie state, the derivative of jh is a non-zero quantity, and is instead a function of the derivative of the penetration depth with respect to APCA, dh (Table 3) (Supplementary matedh rial 3). To determine the general expression for the APCA of the metastable state, Eq. (13) is integrated with the boundary condition of the pure Cassie state (detailed derivation in Supplementary material 4). Indefinite integration of Eq. (13) generates a constant of integration (const), as given by Eq. (15).
jh ð1 þ cos hM Þ 1 3
ð1 cos hM Þ ð2 þ cos hM Þ
2 3
¼
cos hM þ 3 2
1
ð2 þ cos hM Þ3 ð1 cos hM Þ3
þ const ð16Þ
Since both the metastable Cassie and the pure Cassie states belong to the heterogeneous wetting regime, it is safe to assume that Eq. (15) is valid for zero penetration depth (hMhCB for h0). The constant of integration is evaluated with the replacement of hM with hCB Eq. (16). 1
2
const ¼ ð2 þ cos hCB Þ3 ð1 cos hCB Þ3
ð17Þ
Substituting Eq. (16) into Eq. (15), a characteristic set of equations is found which correlates the APCA of the metastable state with the penetration depth h Eq. (17). Characteristic set of equations:
4ah cos hY ð1 þ cos hM Þ ða þ bÞ2 1
þ cos hCB ð1 þ cos hM Þ 2 2
1
2
þ ð2 þ cos hCB Þ3 ð1 cos hCB Þ3 ð1 cos hM Þ3 ð2 þ cos hM Þ3 ¼ 0 ð18Þ As opposed to that of the pure Cassie state, the APCA of the metastable Cassie state can only be implicitly correlated with the penetration depth h. It is clear that as the APCA lacks an explicit solution, the expression for the total free energy is not necessarily identical to that of either of the Wenzel and the pure Cassie states. Since both the metastable Cassie and the pure Cassie states belong to the heterogeneous wetting regime, it should be possible to express the APCA of the former using an appropriate expression for the solid fraction. Since a pure Cassie state is characterized by no penetration inside the roughness valleys, the solid fraction is simply given as the ratio of total area of a square pillar top surface (a2) to the area of a unit ((a + b)2). For a metastable Cassie state, the side-walls of a square pillar are wetted by the liquid (area of 4ah), which must be taken into consideration in computing the solid fraction. Considering the extra solid–liquid contact for the metastable state, a solid fraction term is coined for the heterogeneous wetting regime, namely heterogeneous solid fraction (fCB,h). The heterogeneous solid fraction perfectly applies to the pure Cassie state (for a penetration depth of zero).
fCB;h ¼
a2 þ 4ah ða þ bÞ2 þ 4ah
ð19Þ
Accordingly, a modified Cassie–Baxter equation Eq. (19) can be formulated by substituting the modified solid fraction.
cos hCB;h ¼ fCB;h ð1 þ cos hY Þ 1 ! a2 þ 4ah ð1 þ cos hY Þ 1 ¼ ða þ bÞ2 þ 4ah
ð20Þ
In order to check the efficiency of the characteristic set of equations, the parameter hM in the left hand side of Eq. (17) is replaced with hCB,h. The resulting expression is a function of hCB,h, and is termed as compatibility parameter (u) Eq. (20).
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Figure 4. Further surface topologies, from left of right: truncated cone, cylinder, vase and flower.
Table 4 A list of wetting parameters (jh) for five surface topologies. Surface topology
Wetting state
jh
Square pillar
Heterogeneous
4ah cos2hY ðaþbÞ
Homogeneous Cone
Heterogeneous
2
a aþb
ð1 þ cos hY Þ
1 cos hY ð1 þ 4ac 2 Þ ðaþbÞ 2 02 pa 2 1 þ hc aa0 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 h ðp4a þ p2ch ða0 þ 2c ða a0 ÞÞ ða a0 Þ2 þ 4c2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 02 1 cos hY2 ½ða þ bÞ2 p4a þ p4a þ p4 ða þ a0 Þ ða a0 Þ2 þ 4c2 Þ ðaþbÞ 2 2 1 2 p4a þ p4a þ pah cos hY 4ðaþbÞ
Homogeneous Cylinder
Heterogeneous
cos hY ðaþbÞ2
ðaþbÞ
Homogeneous Vase
Heterogeneous
1 cos hY ð1 þ pac 2 Þ ðaþbÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 0 2 0 0 0 0Þ Þ Þ 1 p a2 p a2 aa c2 0 aa c2 sin1 2cðaa sin1 2ðaa 0Þðc2hÞ þ 2cðaa 1 2cðaa 4 þ 4 þ ðphÞ 2 þ 2ðaa0 Þ þ a p 4 þ 4ðaa0 Þ ðaþbÞ2 ðaa0 Þ2 þc2 ðaa Þ2 þc2 ðaa0 Þ2 þc2 ðaa0 Þ2 þc2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0
Homogeneous Flower
Heterogeneous Homogeneous
2
0
1 2ðaa 0Þðc2hÞ 2ðaa 0Þðc2hÞ cos hY : ðaa Þ2 þc2 ðaa Þ2 þc2 n 0 o n 0 o2 h2 c2 aa c2 sin1 ða þ bÞ2 þ p2 cða þ a0 Þ þ pc aa 1 cos 4 þ 4ðaa0 Þ 2p 4 þ 4ðaa0 Þ aþb
2cðaa0 Þ ðaa0 Þ2 þc2
2
½pah cos hY þ a ðp8þ2Þ ð1 þ cos hY Þ 1 cos hY 1 þ pac 2
1 ðaþbÞ2
ðaþbÞ
u¼
4ah cos hY ð1 þ cos hCB;h Þ ða þ bÞ2 1
þ cos hCB ð1 þ cos hCB;h Þ 2 2
1
2
þ ð2 þ cos hCB Þ3 ð1 cos hCB Þ3 ð1 cos hCB;h Þ3 ð2 þ cos hCB;h Þ3 ð21Þ Since u is a dimensionless and bounded parameter, it is expected to approach zero so as to have a valid characteristic equation Eq. (17). For a multitude of YCA, u approaches vanishingly low values (<0.01) at pillar spacing to width ratio higher than 4 (Figure 3). The compatibility of the energy minimized APCA for the metastable state (hM) and the modified Cassie–Baxter’s contact angle (hCB,h), shows that Eq. (17) successfully approximates the energy minimization only for a unique range of geometric factors. For the first time, an approximate is made to the APCA for a metastable Cassie state with a rigorous procedure of energy minimization. To investigate any loss of generality with respect to the surface topology, the entire procedure of energy minimization
Eqs. (5)–(18) is repeated for four distinct surface topologies. The aforementioned topologies are termed as truncated cone, cylinder, vase and flower (Figure 4). Each of the four topologies are characterized by a base diameter (a lm), spacing (b lm) and height (c lm), which are identical to that of the square pillar topology. As stated earlier, each wetting state for a given surface topology can be mapped to a unique wetting parameter (jh). The general equation of wetting Eq. (12) applies to each surface. Although an explicit solution cannot be found for the APCA of the metastable state, the APCAs for pure Cassie (hCB) and Wenzel (hW) states can be derived by appropriate substitutions (h = 0 for the heterogeneous regime and h = c for the homogeneous regime, respectively), to Eq. (12) for the surface topology in question (Table 4). Thus, determination of hW and hCB can be simplified to a two step process, in which the values of j0 and jc are substituted into Eq. (14).
A. Sarkar, A.-M. Kietzig / Chemical Physics Letters 574 (2013) 106–111
With regard to Table 4, it should be noted that the cylinder and flower geometries share the same wetting parameter, and hence the APCA for the homogeneous wetting regime. The wetting parameter for a surface topology with uniform cross section parallel to the base (square pillar, cylinder, flower) is directly proportional to the product of the circumference of the top view and the height of the pillars. For such topologies with same dimensions of a unit, the wetting parameters lead to identical values of the Wenzel contact angle.
111
etration depth of the liquid inside the roughness valleys. Since the liquid inside roughness valleys becomes significant for pillars above a threshold height, there is scope for the future derivation of a rigorous solution, which includes the contribution of the liquid inside the roughness valleys toward the total drop volume. Acknowledgements We thank Soseh Zadoorian and Jorge Lehr for their efforts proof reading the manuscript.
3. Conclusion Appendix A. Supplementary data The current discussion details the thermodynamics of all wetting states possible for a surface topology. In addition to the well-known pure Cassie and Wenzel states, the metastable Cassie state is thoroughly studied, which is characterized by a partial penetration of the liquid inside the roughness valleys. The extent of penetration, i.e. the penetration depth of the liquid forms the key parameter in characterizing the wetting state. A penetration depth dependent parameter is coined, namely wetting parameter (jh), which can be mapped to a unique wetting state corresponding to a distinct surface topology. The wetting parameter enables integration of the free energy of all possible wetting states under the canopy of a single expression. A general equation of wettability is designed (Eq. (13)), which universally applies to a possible wetting state of any surface topology. The substitutions of h = 0 and h = c are carried out to the general equation for exemplarily five topologies, which respectively generate the Cassie and Wenzel equations. Incorporation of the wetting parameter will immensely simplify the calculation of Wenzel and Cassie APCAs for a set of surface topologies. For one of the topologies, namely square pillar, the general equation of wettability is simplified using appropriate boundary conditions to generate an implicit correlation between the APCA for the metastable state and the penetration depth. The correlation, also known as the characteristic set of equations (Eq. (17)) is in good agreement with the traditional approach. While experimental verification of the phenomenon of a metastable Cassie state can be a difficult task, a possible solution might be the imaging of sub-surface characteristics in order to measure the pen-
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General equation of wettability: A tool to calculate the contact angle for a rough surface Anjishnu Sarkar, Anne-Marie Kietzig ⇑ Department of Chemical Engineering, McGill University, 3160 University Street, Montreal, QC, Canada H3A 2B2
a r t i c l e
i n f o
Article history: Received 24 March 2013 In final form 23 April 2013 Available online 2 May 2013
a b s t r a c t This work introduces a mathematical tool, namely general equation of wettability, which simplifies the determination of the contact angle for rough surfaces. A free energy minimization method is applied to an intermediate wetting state with partial liquid penetration into roughness valleys in addition to the well-known states of Wenzel and Cassie wetting. A wetting parameter is coined, which stands out as a fingerprint of a particular wetting state of a surface topology. Substitution of the appropriate wetting parameter returns the Cassie and Wenzel equations, while an implicit solution is presented for the intermediate state, which is consistent with energy minimization. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction As controversial its grammatical roots might be, the word ‘wettability’ continues to enjoy the unswerving attention of surface sciences. A mark of the surface-liquid compatibility, wettability forms the key topic in applications ranging from froth flotation to detergency industries [1,2]. Wettability is denoted as the angle subtended by the tangent lines at the solid–liquid and liquid–air interfaces, also known as the contact angle (h). Contact angle is a direct measure of the interfacial energy of a solid–liquid–air system. The contact angle of an atomically flat surface with infinite rigidity and uniform chemical composition solely depends on the surface chemistry, and is termed as Young’s contact angle. More physical surfaces with finite rigidity result in contact angle hysteresis, but still correspond to their Young contact angle (hY) [3–6]. For such a surface, the necessary energy required in building a solid– liquid interface (cSL), and hence a drop of liquid is governed by the Young’s contact angle, and in turn, the surface chemistry.
‘apparent’ horizontal base-line. Since it can be correlated with the surface chemistry (ACCA) and topology (roughness), the APCA is widely used as the signature of the interfacial energy [7–13]. The APCA is also governed by the chemical homogeneity of a rough surface. Distinct wetting modes exist for chemically homogeneous and heterogeneous surfaces, respectively, known as homogeneous and heterogeneous wetting regimes. Additionally, it is possible for chemically homogeneous surfaces to exhibit a heterogeneous wetting regime. This is possible through the existence of air pockets trapped between the surface and the liquid, which then serve as chemical heterogeneities. For a rough and chemically homogeneous surface, the homogeneous and heterogeneous wetting regimes are respectively expressed by Wenzel and Cassie– Baxter models [14,15]. In both the aforementioned models, the APCA is explicitly expressed in terms of the surface topology and the chemistry. The Wenzel’s contact angle (hW) is given as the product of the Wenzel’s roughness ratio rw and the cosine of the YCA.
cSL ¼ cSA cLA cos hY
cos hW ¼ r w cos hY
ð1Þ
The contact angle at every microscopic point on a rough surface perfectly obeys Young’s equation, and is termed as the actual contact angle (ACCA) for the concerned microscopic point [7]. To determine the ACCA, the exact position of the drop must be pinpointed with respect to a surface topology of a micrometer/nanometer length-scale, which makes it difficult to coin the ACCA and the interfacial energy. The aforementioned issue has been resolved by the use of an apparent contact angle (APCA), which is characterized by a hypothetical solid–liquid interface that runs parallel to an ⇑ Corresponding author. Fax: +1 514 398 6678. E-mail addresses: [email protected] (A. Sarkar), anne.kietzig@ mcgill.ca (A.-M. Kietzig). 0009-2614/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2013.04.055
ð2Þ
Here, rw signifies the ratio of the actual solid–liquid interfacial area on a rough surface to the solid–liquid interfacial area on a corresponding smooth surface. According to the Wenzel’s model, surface roughness preserves its chemical wettability, i.e. the hydrophilic surface becomes more hydrophilic with roughness and vice versa. For the Cassie–Baxter model, the APCA is expressed as a function of the YCA and the solid fraction f, i.e. the fraction of solid surface encompassed by the liquid.
cos hCB ¼ f cos hY þ f 1
ð3Þ
Apart from the Wenzel and Cassie states, a third wetting state is possible, where the liquid partially penetrates into the roughness structure [10,16–18]. The partial penetration or the penetration
A. Sarkar, A.-M. Kietzig / Chemical Physics Letters 574 (2013) 106–111
107
Figure 1. Penetration depth h of different wetting states indicating wetted (dark) and dry (bright) pattern surfaces. (a) Heterogeneous wetting: Cassie: h = 0; (b) heterogeneous wetting: metastable Cassie: 0 < h < c; (c) homogeneous wetting Wenzel h = c.
depth can assume multiple values, each of which corresponds to a distinct geometrical configuration of the interfaces and consequently, discrete values of the apparent contact angle. In the process of plotting contact angle data for surfaces with different chemistry and topologies, Erbil [18] has recorded cases of heterogeneous wetting state with APCA both greater and less than the Cassie–Baxter contact angle for the same surface [18]. The wetting state associated with a partial penetration inside the roughness valleys of a surface topology has been termed as an intermediate state by Erbil. Literature lacks a way to derive the APCA for such a state using classical thermodynamics. In this Letter, a novel method is designed to express the APCA for a heterogeneous regime with a finite penetration. Since they are characterized by existence of trapped air under the liquid, both the intermediate state and the Cassie state belong to heterogeneous regime. Cases of zero and partial penetration inside the roughness valleys are respectively termed as Cassie and metastable Cassie states (Figure 1) in the entire work. The term metastable is attributed to the difficulty in deducing the thermodynamic feasibility of such a wetting state. In the past few years, two parameters have been isolated as potential candidates in determining the value of contact angle, namely a one-dimensional triple line and a two dimensional contact area [19–23]. The latter allows the derivation of the Wenzel and Cassie equations via minimization of the free energy available to the area which spans the solid–liquid system. The method of free energy minimization, and hence the aforementioned equations of wetting are perfectly applicable for topologies that are uniform [21]. Using free energy minimization, a set of equations is derived in this discourse which simplifies the calculation of contact angles on rough surfaces. 2. Results and discussions A liquid drop of a given volume can orient itself in different manners on a solid surface with a well-defined surface topology. Unlike its homogeneous counterpart, where the surface under the liquid is subject to the solid–liquid (S–L) interface, the heterogeneous regime is characterized by a liquid–air (L–A) interface under the drop. The location of the L–A interface with respect to the surface topology determines the magnitudes of the mutual area available to each of the interfaces (S–L, L–A or S–A). The interfacial areas determine the equilibrium free energy available to the solid– liquid system. At thermodynamic equilibrium, the free energy available to a solid–liquid system reaches its minimum value. Depending on the location of the triple line with respect to the surface topology, the equilibrium free energy can assume different values, each corresponding to a distinct wetting state (Figure 1). Each wetting state is characterized by an apparent contact angle. Thus, the derivation of a general expression for the apparent contact angle (APCA) on a surface must be traced back to the possible locations of the L–A interface with respect to the surface topology.
Figure 2. Square pillar topology. (a) Three dimensional view showing pillar width a, pillar spacing b and pillar height c (b) top view of four pillars, which outline a unit (c) simplest building block, i.e. unit, inscribed in the dotted rectangle.
Here, the expression for the apparent contact angle is derived for a surface topology, and is shown to be valid for other topologies. A regular distribution of square pillars is chosen as the surface topology, which is characterized by cuboids with individual pillar width a (lm), inter-pillar spacing b (lm) and pillar height c (lm) (Figure 2a). To facilitate minimization of the available free energy, a simplest building block in the following is referred to as a unit. A section of four square pillars (Figure 2b) is chosen such that on repetition, it generates the entire distribution of pillars under the drop. In this first example, a unit is characterized by a distribution comprising four square pillars, each with an individual pillar width a/2 (lm), inter-pillar spacing b (lm) and height c (lm) (Figure 2c). The homogeneous and heterogeneous regimes are denoted by the extent of liquid penetrating the apex of the roughness valley, i.e. the penetration depth (Figure 1). The liquid assumes partial (0 6 h < c) and complete penetration (h = c), respectively, in the heterogeneous and homogeneous regimes. In the following, it is assumed that the heterogeneous regime is characterized by a L–A interface under the liquid that runs parallel to the pillar base. Furthermore, it is assumed that the volume of a drop is invariant, i.e. evaporation or condensation effects are ignored. The calculations are limited to a case of spherical drop shape, i.e. when the radius of the drop does not exceed the capillary length of the liquid. To establish such a condition, the drop volume (V) shares a unique relationship with the drop radius ð^r Þ, and APCA (h).
V¼
p^r3 3
ð2 þ cos hÞð1 cos hÞ2 ¼ const
ð4Þ
Furthermore, it is assumed that the volume of the liquid inside the roughness valleys plays a trivial role, as compared to the volume of the drop. It is found that there exists a threshold pillar
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Table 1 Interfacial area terms of a unit. Heterogeneous wetting regime
Homogeneous wetting regime
unit 2 ðDAunit SL Þhet ¼ ðDASA Þhet ¼ 4ah þ a
unit ðDAunit SL Þhom ¼ ðDASA Þhom ¼ 4ac
þða þ bÞ2 2
2
2ðaþbÞ ðDAunit LA Þhet ¼ bð2a þ bÞ þ 1þcos h
2ðaþbÞ ðDAunit LA Þhom ¼ 1þcos h
The expressions for the interfacial area terms, as given in Table 1, are substituted into Eq. (7). Using Young’s equation, the free energy of a unit is expressed in terms of the L–A interfacial tension (cLA), surface chemistry (hY) and surface topology (a, b and c). Literature lacks a common expression for free energy that is invariant with a wetting state. With the introduction of a wetting parameter (jh) (Table 2), the free energy expressions Eq. (7) for the homogeneous and heterogeneous regimes (Table 1) can be expressed as a single consolidated expression Eq. (10).
Gunit ¼ cLA ða þ bÞ2 h height for correct estimation of the homogeneous wetting regime [24] (Supplementary material 1). This is why the results from this discussion are applicable for pillar heights less than 300 lm. Finally, for the ease of calculation, it is assumed that the total number of units under a drop assumes an integer value (Eq. (5)).
g¼
2 Area under the drop p^r2 sin h ¼ Area of a unit ða þ bÞ2
ð5Þ
The free energy (DG) is understood as the interfacial energy available to the system when a drop of liquid is formed, i.e. when the three interfaces are built. Given as the difference between the free energies with and without a drop (Gdrop and GNodrop, respectively), the free energy depends on the interfacial tensions and interfacial area terms Eq. (6).
DG ¼ GDrop GNo drop ¼ cSL DASL þ cLA DALA þ cSA DASA
ð6Þ
(DGunit h ),
The free energy available to a unit similarly depends on the interfacial area terms for a unit, and is expressed by using the superscript ‘unit’ equation (7). The concerned wetting state is denoted by subscript h, which further denotes the penetration depth as introduced in Figure 1.
DGunit h
unit SL DASL
¼c
unit LA DALA
þc
unit SA DASA
þc
ð7Þ
For a unit, the S–L and S–A interfacial areas are additive inverses of each other Eq. (8). This is because the S–A interfacial area lost owing to drop formation is identical to the S–L interfacial area built. unit DAunit SL ¼ DASA
ð8Þ
The L–A interfacial area for the solid–liquid system is calculated by including the contribution of the truncated sphere, and the interface under the liquid found only for the heterogeneous wetting regime. The corresponding area for a unit is expressed as the ratio of the total L–A interfacial area to the number of units under the drop Eq. (9).
DAunit LA ¼
DALA
ð9Þ
g
The area fractions for both the heterogeneous and homogeneous wetting regimes are listed in Table 1. Since the S–L (cSL) and S–A (cSA) interfacial tensions exhibit mutual dependence, they cannot be individually quantified. Assuming that Young’s equation is valid for a micrometer length scale, the aforementioned interfacial tension terms are eliminated by rearranging Eq. (1) [3].
cSA ¼ cSL cLA cos hY
ð1Þ
2 þ 1 þ jh ; 8h 2 fhom; hetg 1 þ cos h
Eq. (10) clearly indicates that the free energy available to a unit (fixed values of a and b) solely depends on the APCA and the wetting parameter. Since it takes the contribution from surface topology into consideration, the wetting parameter (jh) marks the fingerprint of a wetting state (Table 2). To minimize the free energy, the energy available to a unit is summed over all the pillars under the drop. The free energy of the solid–liquid system (DGtotal ) is evaluated by multiplying the h free energy of a unit with the total number of pillars under a drop.
DGtotal ¼ gDGunit ¼ h h
p^r2 sin2 h ða þ bÞ2
DGunit h
ð11Þ
Using Eq. (4), the radius ð^r Þ is expressed in terms of the volume (V) and APCA. The free energy (DGtotal ) is reduced to a dimensionless h quantity, namely normalized total free energy (DGtotal h; ) by division with the expression cLAp1/3(3 V)2/3. DGtotal h; ¼
DGtotal h 1 3
cLA p ð3VÞ
2
2 3
1
¼ð2þcoshÞ3 ð1coshÞ3 ð1þcoshÞ
Heterogeneous Homogeneous
2 þ1þjh 1þcosh
ð12Þ
On respectively substituting the values of 0 and c for the penetration depth h, the total free energy (DGtotal ) faithfully represents h the free energy expressions for the pure Cassie and Wenzel states available in literature [9,10]. In each of the above mentioned measures, the free energies of the homogeneous and heterogeneous regimes are separately calculated. The wetting parameter enables the integration of the free energy of all possible wetting states under the canopy of a single expression Eq. (11). For energy minimization, the derivative of the available free energy with respect to the wetting parameter (jh) is set to zero Eq. (12).
dDGtotal h; ¼0 dðcos hÞ
ð13Þ
The normalized free energy (DGtotal h; ) is incorporated into Eq. (12), which leads to an explicit correlation of jh and APCA, which is termed as the general equation of wettability Eq. (13). A detailed derivation that leads to Eq. (13) is listed in Supplementary material 2. General equation of wettability:
2ðcos h þ jh þ 1Þ ¼ ðcos h þ 2Þðcos2 h 1Þ
djh dðcos hÞ
ð14Þ
Since it lacks a term corresponding to the surface topology, this general equation of wettability provides an accurate description to the wetting state of any surface topology. The general equation can
Table 2 Expressions for wetting parameter (jh) for a square pillar topology. Wetting regime
ð10Þ
Free energy for a unit h i 2 2 2 cLA 2ðaþbÞ 1þcos h 4ah cos hY þ ða þ bÞ a ð1 þ cos hY Þ h i 2 2 cLA 2ðaþbÞ 1þcos h 4ac cos hY ða þ bÞ coshY
jh a 2 Þ ð1 þ coshY Þ 4ah cos2hY ðaþb ðaþbÞ
1 cos hY ð1 þ
4ac Þ ðaþbÞ2
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A. Sarkar, A.-M. Kietzig / Chemical Physics Letters 574 (2013) 106–111 Table 3 Derivative of jh to generate equations for the Wenzel, pure Cassie and the metastable state for square pillar topology. Wetting state
Expression
djh dðcos hh Þ
Wenzel
0
Metastable Cassie
dh 4a cos h2Y dðcos hh Þ
Pure Cassie
0
cos hW
Equation
¼ cos hY 1 þ
4ac ðaþbÞ2
Wenzel Characteristic set of equations
hM ¼ functionðh; hY ; a; b; cÞ 2 a cos hCB ¼ aþb ð1 þ cos hY Þ 1
ðaþbÞ
Cassie–Baxter
Figure 3. Deviation of compatibility parameter (u) from zero for different YCAs (beyond b/a > 4 the characteristic set of equations (Eq. (18)) is valid for all surface chemistries).
be specifically expressed for unique wetting states with the aid of appropriate subscripts (CB for h = 0, M for 0 < h < c and W for h = c, as mentioned in Table 3). For the cases of both complete penetration (h = c) and zero penetration (h = 0) of the liquid in the roughness valley, the wetting parameters (jh) for such cases assume constant values. As a result, the derivatives of jh, and consequently, the right hand side of Eq. (13) vanish (Table 3), leading to equation (14).
2ðcos h þ jh þ 1Þ ¼ 0
ð15Þ
As expected, substitution of values j0 and jc (Table 2) to Eq. (14), respectively, generate the Cassie and Wenzel equations. Thus, the Wenzel and Cassie equations are expressed under a common format, and the expression is independent of the concerned surface topology. For the metastable Cassie state, the derivative of jh is a non-zero quantity, and is instead a function of the derivative of the penetration depth with respect to APCA, dh (Table 3) (Supplementary matedh rial 3). To determine the general expression for the APCA of the metastable state, Eq. (13) is integrated with the boundary condition of the pure Cassie state (detailed derivation in Supplementary material 4). Indefinite integration of Eq. (13) generates a constant of integration (const), as given by Eq. (15).
jh ð1 þ cos hM Þ 1 3
ð1 cos hM Þ ð2 þ cos hM Þ
2 3
¼
cos hM þ 3 2
1
ð2 þ cos hM Þ3 ð1 cos hM Þ3
þ const ð16Þ
Since both the metastable Cassie and the pure Cassie states belong to the heterogeneous wetting regime, it is safe to assume that Eq. (15) is valid for zero penetration depth (hMhCB for h0). The constant of integration is evaluated with the replacement of hM with hCB Eq. (16). 1
2
const ¼ ð2 þ cos hCB Þ3 ð1 cos hCB Þ3
ð17Þ
Substituting Eq. (16) into Eq. (15), a characteristic set of equations is found which correlates the APCA of the metastable state with the penetration depth h Eq. (17). Characteristic set of equations:
4ah cos hY ð1 þ cos hM Þ ða þ bÞ2 1
þ cos hCB ð1 þ cos hM Þ 2 2
1
2
þ ð2 þ cos hCB Þ3 ð1 cos hCB Þ3 ð1 cos hM Þ3 ð2 þ cos hM Þ3 ¼ 0 ð18Þ As opposed to that of the pure Cassie state, the APCA of the metastable Cassie state can only be implicitly correlated with the penetration depth h. It is clear that as the APCA lacks an explicit solution, the expression for the total free energy is not necessarily identical to that of either of the Wenzel and the pure Cassie states. Since both the metastable Cassie and the pure Cassie states belong to the heterogeneous wetting regime, it should be possible to express the APCA of the former using an appropriate expression for the solid fraction. Since a pure Cassie state is characterized by no penetration inside the roughness valleys, the solid fraction is simply given as the ratio of total area of a square pillar top surface (a2) to the area of a unit ((a + b)2). For a metastable Cassie state, the side-walls of a square pillar are wetted by the liquid (area of 4ah), which must be taken into consideration in computing the solid fraction. Considering the extra solid–liquid contact for the metastable state, a solid fraction term is coined for the heterogeneous wetting regime, namely heterogeneous solid fraction (fCB,h). The heterogeneous solid fraction perfectly applies to the pure Cassie state (for a penetration depth of zero).
fCB;h ¼
a2 þ 4ah ða þ bÞ2 þ 4ah
ð19Þ
Accordingly, a modified Cassie–Baxter equation Eq. (19) can be formulated by substituting the modified solid fraction.
cos hCB;h ¼ fCB;h ð1 þ cos hY Þ 1 ! a2 þ 4ah ð1 þ cos hY Þ 1 ¼ ða þ bÞ2 þ 4ah
ð20Þ
In order to check the efficiency of the characteristic set of equations, the parameter hM in the left hand side of Eq. (17) is replaced with hCB,h. The resulting expression is a function of hCB,h, and is termed as compatibility parameter (u) Eq. (20).
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Figure 4. Further surface topologies, from left of right: truncated cone, cylinder, vase and flower.
Table 4 A list of wetting parameters (jh) for five surface topologies. Surface topology
Wetting state
jh
Square pillar
Heterogeneous
4ah cos2hY ðaþbÞ
Homogeneous Cone
Heterogeneous
2
a aþb
ð1 þ cos hY Þ
1 cos hY ð1 þ 4ac 2 Þ ðaþbÞ 2 02 pa 2 1 þ hc aa0 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 h ðp4a þ p2ch ða0 þ 2c ða a0 ÞÞ ða a0 Þ2 þ 4c2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 02 1 cos hY2 ½ða þ bÞ2 p4a þ p4a þ p4 ða þ a0 Þ ða a0 Þ2 þ 4c2 Þ ðaþbÞ 2 2 1 2 p4a þ p4a þ pah cos hY 4ðaþbÞ
Homogeneous Cylinder
Heterogeneous
cos hY ðaþbÞ2
ðaþbÞ
Homogeneous Vase
Heterogeneous
1 cos hY ð1 þ pac 2 Þ ðaþbÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 0 2 0 0 0 0Þ Þ Þ 1 p a2 p a2 aa c2 0 aa c2 sin1 2cðaa sin1 2ðaa 0Þðc2hÞ þ 2cðaa 1 2cðaa 4 þ 4 þ ðphÞ 2 þ 2ðaa0 Þ þ a p 4 þ 4ðaa0 Þ ðaþbÞ2 ðaa0 Þ2 þc2 ðaa Þ2 þc2 ðaa0 Þ2 þc2 ðaa0 Þ2 þc2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0
Homogeneous Flower
Heterogeneous Homogeneous
2
0
1 2ðaa 0Þðc2hÞ 2ðaa 0Þðc2hÞ cos hY : ðaa Þ2 þc2 ðaa Þ2 þc2 n 0 o n 0 o2 h2 c2 aa c2 sin1 ða þ bÞ2 þ p2 cða þ a0 Þ þ pc aa 1 cos 4 þ 4ðaa0 Þ 2p 4 þ 4ðaa0 Þ aþb
2cðaa0 Þ ðaa0 Þ2 þc2
2
½pah cos hY þ a ðp8þ2Þ ð1 þ cos hY Þ 1 cos hY 1 þ pac 2
1 ðaþbÞ2
ðaþbÞ
u¼
4ah cos hY ð1 þ cos hCB;h Þ ða þ bÞ2 1
þ cos hCB ð1 þ cos hCB;h Þ 2 2
1
2
þ ð2 þ cos hCB Þ3 ð1 cos hCB Þ3 ð1 cos hCB;h Þ3 ð2 þ cos hCB;h Þ3 ð21Þ Since u is a dimensionless and bounded parameter, it is expected to approach zero so as to have a valid characteristic equation Eq. (17). For a multitude of YCA, u approaches vanishingly low values (<0.01) at pillar spacing to width ratio higher than 4 (Figure 3). The compatibility of the energy minimized APCA for the metastable state (hM) and the modified Cassie–Baxter’s contact angle (hCB,h), shows that Eq. (17) successfully approximates the energy minimization only for a unique range of geometric factors. For the first time, an approximate is made to the APCA for a metastable Cassie state with a rigorous procedure of energy minimization. To investigate any loss of generality with respect to the surface topology, the entire procedure of energy minimization
Eqs. (5)–(18) is repeated for four distinct surface topologies. The aforementioned topologies are termed as truncated cone, cylinder, vase and flower (Figure 4). Each of the four topologies are characterized by a base diameter (a lm), spacing (b lm) and height (c lm), which are identical to that of the square pillar topology. As stated earlier, each wetting state for a given surface topology can be mapped to a unique wetting parameter (jh). The general equation of wetting Eq. (12) applies to each surface. Although an explicit solution cannot be found for the APCA of the metastable state, the APCAs for pure Cassie (hCB) and Wenzel (hW) states can be derived by appropriate substitutions (h = 0 for the heterogeneous regime and h = c for the homogeneous regime, respectively), to Eq. (12) for the surface topology in question (Table 4). Thus, determination of hW and hCB can be simplified to a two step process, in which the values of j0 and jc are substituted into Eq. (14).
A. Sarkar, A.-M. Kietzig / Chemical Physics Letters 574 (2013) 106–111
With regard to Table 4, it should be noted that the cylinder and flower geometries share the same wetting parameter, and hence the APCA for the homogeneous wetting regime. The wetting parameter for a surface topology with uniform cross section parallel to the base (square pillar, cylinder, flower) is directly proportional to the product of the circumference of the top view and the height of the pillars. For such topologies with same dimensions of a unit, the wetting parameters lead to identical values of the Wenzel contact angle.
111
etration depth of the liquid inside the roughness valleys. Since the liquid inside roughness valleys becomes significant for pillars above a threshold height, there is scope for the future derivation of a rigorous solution, which includes the contribution of the liquid inside the roughness valleys toward the total drop volume. Acknowledgements We thank Soseh Zadoorian and Jorge Lehr for their efforts proof reading the manuscript.
3. Conclusion Appendix A. Supplementary data The current discussion details the thermodynamics of all wetting states possible for a surface topology. In addition to the well-known pure Cassie and Wenzel states, the metastable Cassie state is thoroughly studied, which is characterized by a partial penetration of the liquid inside the roughness valleys. The extent of penetration, i.e. the penetration depth of the liquid forms the key parameter in characterizing the wetting state. A penetration depth dependent parameter is coined, namely wetting parameter (jh), which can be mapped to a unique wetting state corresponding to a distinct surface topology. The wetting parameter enables integration of the free energy of all possible wetting states under the canopy of a single expression. A general equation of wettability is designed (Eq. (13)), which universally applies to a possible wetting state of any surface topology. The substitutions of h = 0 and h = c are carried out to the general equation for exemplarily five topologies, which respectively generate the Cassie and Wenzel equations. Incorporation of the wetting parameter will immensely simplify the calculation of Wenzel and Cassie APCAs for a set of surface topologies. For one of the topologies, namely square pillar, the general equation of wettability is simplified using appropriate boundary conditions to generate an implicit correlation between the APCA for the metastable state and the penetration depth. The correlation, also known as the characteristic set of equations (Eq. (17)) is in good agreement with the traditional approach. While experimental verification of the phenomenon of a metastable Cassie state can be a difficult task, a possible solution might be the imaging of sub-surface characteristics in order to measure the pen-
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