An analytical solution for a partially wetting puddle and the location of the static contact angle

Journal of Colloid and Interface Science 348 (2010) 232–239

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An analytical solution for a partially wetting puddle and the location of the static contact angle M. Elena Diaz a, Javier Fuentes b, Ramon L. Cerro c,*, Michael D. Savage d a

Departamento de Ingenieria Quimica y Textil, Universidad de Salamanca, Plaza de los Caidos 1-5, Salamanca 37008, Spain Process Systems Enterprise, 6th Floor East, 26-28 Hammersmith Grove, London W6 7HA, UK c Chemical and Materials Engineering, University of Alabama in Huntsville, Huntsville, AL 35899, USA d School of Physics and Astronomy, Leeds University, Leeds, LS2 9JT, UK b

a r t i c l e

i n f o

Article history: Received 24 November 2009 Accepted 13 April 2010 Available online 18 April 2010 Keywords: Static contact angle Transition region Interface shape Molecular forces Wetting 2D drop

a b s t r a c t A model is formulated for a static puddle on a horizontal substrate taking account of capillarity, gravity and disjoining pressure arising from molecular interactions. There are three regions of interest – the molecular, transition and capillary regions with characteristic film thickness, hm, ht and hc. An analytical solution is presented for the shape of the vapour–liquid interface outside the molecular region where interfacial tension can be assumed constant. This solution is used to shed new light on the static contact angle and, specifically, it is shown that. (i) There is no point in the vapour–liquid interface where the angle of inclination, h, is identically equal to the static contact angle, ho, but the angle at the point of null curvature is the closest with the difference of O(e2) where e2 = ht/hc is a small parameter. (ii) The liquid film is to O(e) a wedge of angle ho extending from a few nanometers to a few micrometers of the contact line. A second analytical solution for the shape of interface within the molecular region reveals that cos h 2 2 has a logarithmic variation with film thickness, cos h ¼ cos ho  ln½1  hm =2h . The case, hm = 0, is of special significance since it refers to a unique configuration in which the effect of molecular interactions vanishes, disjoining pressure is everywhere zero and the vapour–liquid interface is now described exactly by the Young–Laplace equation and includes a wedge of angle, ho, extending down to the solid substrate. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction The idea of a transition region between a capillary meniscus and a wetting film was first introduced by Derjaguin [1]. A solution for the transition from a meniscus in a horizontal capillary tube to an adsorbed film was derived by Derjaguin et al. [2], using a simple model that ignored gravity. Their formulation was based on the work of Sheludko [3] who assumed disjoining pressure to be a function of film thickness and isotropic (independent of the angle of inclination of the vapour–liquid interface) – an assumption justified on the basis of very small angles of inclination near the wetting film. The authors also recognized that molecular forces affect interfacial tensions yet assumed these to be constant and equal to the bulk surface tensions since the transition zone is sufficiently far from the solid surface.

* Corresponding author. Address: Chemical and Materials Engineering, University of Alabama in Huntsville, EB 131 Huntsville, AL 35899, USA. Fax: +1 256 8246839. E-mail addresses: [email protected] (M. Elena Diaz), [email protected] (J. Fuentes), [email protected] (R.L. Cerro), [email protected] (M.D. Savage). 0021-9797/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2010.04.030

Renk et al. [4] introduced gravity by considering a horizontal meniscus between two flat and horizontal plates with the meniscus pinned on the lower and extending to a continuous wetting film, of uniform thickness, on the upper. The meniscus was formed by the combined effects of capillarity, suction by gravity and multilayer adsorption and its shape was represented by the augmented Young–Laplace (AYL) equation which takes account of disjoining pressure arising from molecular interactions yet surface tension remains constant. A solution to the first-order problem revealed the presence of a sharp transition layer in which the curvature falls from a constant value in the capillary region to zero in the adsorbed film. In turn this led to a more detailed solution using the method of matched asymptotic expansions to achieve a smooth transition as curvature falls rapidly to zero and the vapour–liquid interface approaches the film with zero slope. The effect of gravity on the transition of a wetting liquid was fully taken into account by Adamson and Zebib [5] who matched an infinite meniscus to an adsorbed film on a vertical plate. Here the combined effects of hydrostatic pressure and adsorption caused thinning of the film with distance up the plate. Using an inverse cube for the adsorption potential they showed that the

M. Elena Diaz et al. / Journal of Colloid and Interface Science 348 (2010) 232–239

extent of the transition from the film to the meniscus is of similar magnitude to the thickness of the film. Wayner [6] presented the first attempt to model the curvature and shape of the vapour–liquid interface close to the contact line using both the AYL equation and a spatially varying surface tension computed from a thermodynamic expression for the chemical potential of the vapour–liquid interface developed by Dzyaloshinskii et al. [7]. All the above references refer to wetting films whereas our particular interest is with non-wetting liquids as in the case of a sessile drop on a horizontal surface or the capillary rise, of finite extent up a vertical plate. Here the challenge is to determine the transition from an outer capillary/gravity region to a partially wetting film where molecular interactions give rise to disjoining pressure and possibly interfacial tensions varying with film thickness. Assuming interfacial tension to be constant and gravity negligible, Gomba and Homsy [8] derived an analytical solution for the shape of a droplet deposited on a substrate in equilibrium with a thin film, thus enabling them to predict the effect of droplet size on the apparent contact angle. All macroscopic observations reveal the existence of an apparent contact angle [9] that in static equilibrium is referred to as the macroscopic static contact angle, ho. This angle has a mathematical definition as the slope angle of the vapour–liquid interface at the point of intersection of a solution of the Young–Laplace (YL) equation with the solid surface. The origin of this definition is not clear but it was known to Scriven [10], Benner et al. [11] and indeed other authors who have used it as a boundary condition for analytical and numerical solutions of the Y–L equation. In turn, macroscopic contact angles can be measured with great accuracy by matching the digitized image of the vapour–liquid interface with a solution of the Y–L equation which is extended to intersect the solid surface [12]. One of the main aims of this paper is to determine where, on the vapour–liquid interface of a sessile drop/puddle, the angle of inclination is equal to the static contact angle ho. We consider a ‘fluid slice’, Fig. 1, that represents a vertical cross section through a puddle standing on top of a horizontal surface. A three-region model for the liquid film is shown in Fig. 2, consisting of an outer capillary region dominated by capillarity and gravity; an inner molecular region where molecular interactions give rise to disjoining pressure and spatially varying, interfacial free energies; and a transition region where surface tension is assumed constant. It is in the transition region where disjoining pressure competes with hydrostatic pressure, film curvature is negligible and the vapour–liquid interface is almost linear. Section 2 includes relevant background and mathematical modeling that leads, in Section 3, to an analytical solution for the shape of the vapour–liquid interface in the transition/capillary regions. Section 4 is devoted to a mathematical analysis of a wedge-shaped transition

Fig. 2. Definition of successive regions based on film thickness. The regions are not to scale and have been drawn to emphasize their characteristics.

zone where the angle of inclination at each point on the interface is approximately equal to the static contact angle. Finally Section 5 considers the fully augmented Young–Laplace equation for the shape of the vapour–liquid interface within the molecular region where interfacial tension varies with disjoining pressure. An analytical solution is derived and its significance is discussed. 2. Background/formulation of the model 2.1. Young’s equation In his original publication on the cohesion of fluids, Young [13] described static contact angle equilibrium as a balance of forces at the three-phase contact line due to interfacial tensions r, rSV, rSL at the vapour–liquid, solid–vapour and solid–liquid interfaces respectively

r cos ho ¼ rSV  rSL

ð1Þ

Using an intermolecular force model, Rayleigh [14], confirmed Young’s equation as a macroscopic relationship that holds away from the contact line. Benner et al. [11] referred to Young’s equation not being valid in a contact region near the solid where the liquid–vapour meniscus cannot be precisely defined. Doubts about the validity of Young’s equation arose with the derivation of alternative equations for ho based on various intermolecular force models [15–17]. The issue was subsequently resolved by Keller and Merchant [18] who used the method of matched asymptotic expansions to validate Young’s equation and deal with the alternative equations. They showed that the leading term in the outer expansion for the interface satisfies the Y–L equation whilst that in the inner (boundary layer) expansion satisfies an integral equation. Matching the solutions of these two equations confirmed that the slope angle, of the leading term in the outer expansion at the solid boundary, is that given by Young’s equation. It is this key result that leads to a precise mathematical definition for ho – as a boundary condition for solutions of the Y–L equation at the point of intersection with the solid surface where film thickness, h, is zero

h ¼ h0 when h ¼ 0

Fig. 1. Definition of variables with the apparent contact line as origin.

233

ð2Þ

The current interpretation of Young’s equation is that of a macroscopic relationship between macroscopic, experimentally observable, thermodynamic variables and the angle ho. The interfacial tensions r, rSV, rSL refer to the constant values taken by the specific, interfacial, Gibbs free energies gVL, gSV, and gSL away from the contact line.

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With the static contact angle defined by condition (2), our aim in Section 4 is to show that, somewhat surprisingly, there is no point on the vapour–liquid interface where the angle of inclination is identically equal to ho. 2.2. Three-region model The three distinct regions of the liquid film–molecular, transition and capillary – are shown in Fig. 2. Within the molecular region the equation for the shape of the vapour–liquid interface is the fully augmented Young–Laplace equation (FAYL) that takes account of disjoining pressure and spatially varying interfacial free energy;

g VL ðh; hÞ2H ¼ Pðh; hÞ  ðpL  pV Þ L

ð3Þ

V

where p , p are pressures in the liquid and vapour phases respectively and, gVL(h, h), is related to disjoining pressure, P(h, h) as proposed by Derjaguin et al. [19],

g VL ðh; hÞ ¼ r þ

Z

½6

Pðh; hÞdh

ð4Þ

h

This indicates that interfacial free energy gVL(h, h) is a function of both film thickness, h, and h, the angle of inclination of the vapour/liquid interface such that, far from the contact line, the integral term in (4) vanishes, gVL attains a constant value, r, and Eq. (3) reduces to one form of the AYL equation. The calculation of disjoining pressure by the integration of molecular forces was obtained by Sheludko [3] for a liquid film of uniform thickness in contact with a uniform solid substrate. Subsequently, Miller and Ruckenstein [16] and later Jameson and del Cerro [17] considered a liquid wedge, of angle h, with a vapour–liquid interface extending down to a three-phase contact line on a solid substrate. Taking account of only London-Van der Waals forces, disjoining pressure is expressed in terms of Hamaker con½6 stants, Aij , indicating their use in a Lennard–Jones potential where the exponent of the binary interaction is 6,

ð5Þ

The indices ij refer to interactions between molecules in phases i and j and the function G[6](h) is given by G½6 ðhÞ ¼ 12 þ 34 cosðhÞ  14 cos3 ðhÞ. Details of the integrations for a range of exponents were given by Fuentes [20]. In the case of a partially wetting puddle the shape of the vapour–liquid interface is unknown a priori, thereby precluding the derivation of an exact expression for disjoining pressure. We shall, therefore, assume that a suitable model for disjoining pressure and gVL is given by Eq. (5) with ho replacing h. The basis of this assumption is the shape of the liquid film within the transition region. This is shown, in Section 3, to be a wedge with a vapour–liquid interface that is ‘‘almost linear”, of inclination ho and extending into the outer part of the molecular region. By introducing a molecular film thickness, hm, defined by 2

½6

hm ¼

1

½6

dg VL A G½6 ðhÞ  ASL ¼ Pðh; hÞ ¼ LL 3 dh 6ph

½6

ALL G½6 ðh0 Þ  ASL 6pr

ð6Þ

where the right hand side is always positive for partially wetting fluids (0 < ho < 90°), the expression for P(h) becomes

PðhÞ ¼ rðh2m =h3 Þ

ð7Þ

It is not surprising that disjoining pressure is directly proportional to surface tension, this follows from Eq. (5) by comparing the first two terms. Table 1 gives data for the n-alkanes on PTFE with macroscopic contact angles from 21° (hexane) to 46° (hexadecane) and for dispersive liquids with contact angles from 7° (tert-butyl naphthalene on PE) to 52° (diiodomethane on PE). In each case, hm, is of the order of 1010 m and is associated with the molecular cutoff length, D, defined by Israelachvili [21]. Interfacial free energy gVL is found by integrating Eq. (5) and imposing the condition g VL ! r as h ! 1 to give

"

g VL ðhÞ ¼ r

 2 # 1 hm 1 2 h

ð8Þ

Table 1 Characteristic two-dimensional sessile drop thicknesses of n-alkanes on PTFE and dispersive liquids on PDMS and PE.

ra n-alkanes on PTFE Heptane Octane Nonane Decane Undecane Dodecane Tetradecane Hexadecane

(103 N/m)

ASLb (1020 J)

ALLb (1020 J)

hoa (°)

20.3 21.8 22.9 23.9 24.7 25.4 26.7 27.6

4.03 4.11 4.18 4.25 4.28 4.35 4.38 4.43

4.31 4.49 4.66 4.81 4.87 5.03 5.09 5.22

21.0 26.0 32.0 35.0 39.0 42.0 44.0 46.0

4.80 4.76

5.35 5.23

5.15 4.80 6.23 5.48 5.25 5.42 5.21

Dispersive liquids on PDMS Methylnaphthalene 39.8 Tert-butyl 33.7 naphthalene Liq. Paraffin 32.4 Hexadecane 27.6 Dispersive liquids on PE Diiodomethane 50.8 Bromonaphalene 44.4 Methylnaphthalene 39.8 Bromobenzene 36.3 Tert-butyl 33.7 naphthalene a b c

ht (108 m)

e (103 m)

hC (103 m)

Lcc (103 m)

hL (109 m)

hU (106 m)

8.34 9.19 9.67 9.99 9.52 9.72 9.32 9.43

3.21 3.22 3.13 3.11 2.92 2.90 2.79 2.79

7.12 6.34 5.61 5.33 4.88 4.67 4.46 4.33

0.63 0.80 0.99 1.10 1.23 1.33 1.41 1.49

1.74 1.78 1.80 1.82 1.84 1.85 1.88 1.91

1.17 1.23 1.25 1.28 1.22 1.23 1.18 1.19

12.18 10.99 9.71 9.26 8.46 8.09 7.76 7.59

52.0 49.0

2.10 3.05

1.00 1.28

2.40 2.87

1.75 1.56

1.99 1.88

0.34 0.46

4.29 4.91

6.03 5.22

40.0 36.0

10.33 7.42

3.14 2.57

4.81 4.67

1.36 1.18

1.98 1.91

1.31 1.00

8.96 8.47

7.18 5.93 5.35 5.63 5.23

52.0 36.0 27.0 13.0 7.0

5.18 5.97 2.68 5.50 1.76

1.57 2.16 1.45 2.76 1.69

3.75 4.47 3.95 8.80 8.57

1.12 1.08 0.93 0.36 0.23

1.27 1.75 1.99 1.58 1.88

0.67 0.82 0.45 0.87 0.39

4.29 7.43 7.66 13.78 16.10

hm (1011 m)

Experimental values for interfacial tension, r, and contact angles were determined by Fox and Zisman [24]. Hamaker constants were computed by Hough and White [25]. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi LC ¼ r=ðqgÞ.

M. Elena Diaz et al. / Journal of Colloid and Interface Science 348 (2010) 232–239

Clearly gVL will be close to its limiting value when

 2 1 hm 1 2 h

ð9Þ

R1 where R is the radius of the puddle as seen from above. This second curvature is assumed to be small and safe to neglect. For the fluid slice, the hydrostatic pressure term is given by

pL  pV ¼ qgðhC  hÞ

and specifically, gVL will be within 0.5% of r when 1/2(hm/h)2 = 103; h = 10hm which is between 1 and 2 nm for the liquid/solid combinations in Table 1. This gives the extent of the molecular region beyond which it is safe to assume that interfacial tension has attained its limiting value. Each of the three regions – molecular, transition and capillary has a characteristic film thickness, hm, ht and hC respectively where hC is the maximum film thickness and ht is defined in Section 4 to be the film thickness at the position of null curvature. Table 1 gives data for various liquids and substrates showing hm  Oð1010 mÞ; ht  Oð108 mÞ; hc  Oð103 mÞ. For the capillary region [23], where capillary and gravity forces dominate, a ffi second characteristic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length is the capillary length, LC ¼ r=ðqgÞ.

235

ð11Þ

where h(z), the thickness of the liquid film, becomes a constant, hC, on top of the puddle where curvature is zero. The form of the curvature equation, valid outside the molecular region where surface tension is constant, follows from Eq. (3) by making use of Eqs. (7), (10), and (11) 2



d cos h hm hC  h : ¼ 3 dh L2C h

ð12Þ

Eq. (12) is referred to as the augmented Young–Laplace equation (AYL), the solution of which, will give the shape of the interface in both the transition and capillary regions. 3. Solution of the AYL

2.3. Augmented young–laplace equation The two-dimensional liquid slice is shown in Fig. 1 with an (r, u) coordinate system centered at the contact line. Cartesian variables z and h(z), as well as the curvature of the curve h(z) are given by Scriven [10]:

zðrÞ ¼ r cos u; dh ¼ tan h; dz

d cos h dh

d cos h hC  h ¼ dh L2C

ð13Þ

Integrating Eq. (13) and imposing h = hC, h = 0 yields the solution

hðrÞ ¼ r sin u

2H ¼ 

In preparation for solving the AYL, we first solve the Young– Laplace equation (Eq. (12) without disjoining pressure)

ð10Þ

The two-dimensional slice is similar to the ‘‘pancake” geometry described by Brochard-Wyart et al. [22] (also de Gennes et al. [23]) and is the two-dimensional analog of a sessile puddle for which the curvature in a direction normal to the plane of the figure is of order

2

cos h ¼ 1 

hC

2L2C

þ

hhC L2C

2



h

2L2C

ð14Þ

Introducing the macroscopic static contact angle, ho, as the boundary condition at the solid surface for solutions of the Y–L equation, Eq. (14) becomes

Fig. 3. Angle of inclination of the vapour/liquid interface near the static contact line (for n-heptane on PTFE) as a function of the logarithm of film thickness. The solid line is the analytical solution of the Young–Laplace equation; the dashed line is the analytical solution of the AYL equation, terminated at the outer edge of the molecular region.

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M. Elena Diaz et al. / Journal of Colloid and Interface Science 348 (2010) 232–239 2

cos ho ¼ 1 

hC

ð15Þ

2L2C

Returning to the AYL, we can now integrate Eq. (12), impose the condition h = hC, h = 0, and make use of Eq. (15) to give cos h as a function of h, hC, hm and ho

! 2 h h 2ð1  cos ho Þ cos h  cos ho ¼ 2  2 þ  hC 2h2C 2h 2hC 2

hm

2

hm

ð16Þ

Eq. (16) is the analytical solution of the AYL, that gives the shape of the interface throughout the capillary and transition regions down to the edge of the molecular region. Graphs of inclination angle, h, against the logarithm of film thickness for solutions to both the YL and AYL equations, given by Eqs. (14) and (16) respectively, are shown in Fig. 3 for heptane on PTFE for which ho = 21o. The two curves are almost identical, as expected, for h > 108m. For h < 108 m the Y–L solution increases monotonically to 21o at h = 0, whereas the AYL solution is seen to reach a maximum value and then begin to fall with the computations terminated at the outer edge of the molecular region, h ffi 109 m. This observed behaviour in the case of heptane is found to be typical of the n-alkanes on PTFE and dispersive liquids on both PDMS and PE. The behaviour of the AYL solution is clearly illustrated in Fig. 4 where the logarithm of [ho  h ] is plotted against the logarithm of h(z). The graph has a minimum close to 108 m where the difference between ho and h is of O(105 radians). We shall investigate this minimum more closely in Section 4. A further observation of the AYL solution in Figs. 3 and 4 is that the angle h remains close to 21o (within half of one degree) over a range extending from close to one nanometer to a few micrometers from the contact line; again this will be considered further in Section 4. Fig. 5 shows the film thickness profile for heptane on PTFE in which the logarithm of film thickness is plotted as a function of the logarithm of distance, z, along the solid surface. This follows from a straightforward numerical integration of dh/dz = tan h, with cos h given as a function of h by Eq. (16). Integration of Eq. (16) is

done as an initial value procedure starting at the top of the drop. The second boundary condition relating the capillary shape to the solid/liquid properties, is found as the adsorbed film thickness when the angle of inclination of the interface is zero. With the log– log graph having a slope of 0.996, the liquid film is effectively a wedge extending down to the molecular region. The solid line (dotted line) represent film thickness profiles for the YL (AYL) equations, respectively and appear undistinguishable until z falls below 108 m. As the molecular region is approached and entered, the two profiles are seen to diverge as the slope of the YL profile approaches tan(21o) whilst that of the AYL continues to fall. This is examined further in Section 5. 4. Analysis of transition region The transition region connects the molecular region to the capillary region, i.e. it is a central region that links a domain where molecular interactions have a major effect (in generating disjoining pressure and interfacial free energies) to one in which they have no effect. In the light of Fig. 3 we define the transition region as a liquid wedge in which the effect of curvature is negligible so that the angle of inclination at each point of the interface is close to the static contact angle in the sense that

jho  hj < d

ð17Þ

where d is a small parameter yet to be specified. 4.1. Position of null curvature (ht, ht) A characteristic film thickness for the transition region is ht, the film thickness at the position of null curvature where the slope angle of the vapour–liquid interface is ht. Here we shall show that ht is simply related to the other two characteristic lengths, hm and hC whilst ht is closely related to the macroscopic contact angle ho. At the position of null curvature, hydrostatic and disjoining pressure have equal and opposite effects such that, via Eq. (12)

Fig. 4. Logarithm of [ho  h] as a function of the logarithm of h(z) where h is given by the AYL equation for n-heptane on PTFE. Also shown is the extent of the transition region hL < h < hU where |ho  h| < e.

M. Elena Diaz et al. / Journal of Colloid and Interface Science 348 (2010) 232–239

237

Fig. 5. Logarithm of liquid film thickness (film profile) as a function of the logarithm of the distance to the apparent contact line for n-heptane on PTFE. Shown also is film thickness; ht, and the upper and lower ends, (hu, and hL), of the transition region. The solid line is the analytical solution of the Young–Laplace equation and the dashed line is the analytical solution of the AYL equation, terminated at the outer edge of the molecular region.

2

hm 3 ht

¼

hC L2C

  ht 1 hC

 ht 2ð1  cos ho Þ þ smaller terms hC    2 3 hm 2 ho ¼ 6e2 sin ffi 2 ht 2

cos ht  cos ho ¼

In practice ht is several orders of magnitude smaller than hC; and so provided

ht =hC  1

ð19Þ

hm L2C hC

ð21Þ

It then follows from Eqs. (20), (15)

ðiÞ

  hm ho ¼ 2e sin ht 2

ðiiÞ

ht ¼ e2 hc

ðiiiÞ

  hm ho ¼ 2e3 sin hc 2

þ

ð23Þ

  ho 2

ð24Þ

ð20Þ

It is convenient to introduce a small, system parameter, e, defined by the relation

ht =hc ¼ e2

2 2ht

When interpreting expression (24) there are two points to note:

2

3

hm

Expanding cos ht in a Taylor series about ho gives

ho  ht ffi 3e2 tan

then

ht ¼



2

ð18Þ

ð22Þ

So for a given liquid–solid combination where hm, hc, and ho are known, Eq. (22(iii)) determines e and either Eq. (22(i)) or (22(ii)) gives ht. For dispersive liquids and the n-alkanes on various substrates with static contact angles in the range of 7–50°, Table 1 shows that typically 1011 m < hm < 1010 m;108 m < ht < 107 0:25emm;hC  103 m, thus confirming the validity of condition (19) and that e is a small parameter, 103  e  102 . In order to gain some insight into the significance of ht, the angle of inclination at the point of null curvature, (h, h) = (ht, ht) is substituted into Eq. (16) to give

(i) Since ho > ht and ht is the largest value of the slope angle h, then ho > h for all h. Hence there is no point on the vapour– liquid interface where the angle of inclination h is identically equal to the static contact angle ho. (ii) The difference between ho and ht is of O(e2) and so negligibly small. Hence to this order of approximation, ho, the macroscopic static contact angle, can be identified as the angle of inclination of the vapour–liquid interface at the point of null curvature.

ho ffi ht

ð25Þ

4.2. Extent of the transition region The transition region is bounded by upper and lower film thicknesses hU and hL, where the interface has angles of inclinations hU and hL, respectively. Over the range hL < h < hU, the variation in the inclination of the vapour/liquid interface from its maximum value ht and also from ho [in view of Eq. (24)] is assumed to be ‘‘small” so that condition (17) remains valid, although the small parameter d has yet to be specified. Sufficient conditions for Eq. (17) to hold are

jho  hU j < d and jho  hL j < d

ð26; 27Þ

and therefore the aims of this section are: (i) to derive expressions for hU and hL that will ensure the validity of conditions (26) and (27)

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M. Elena Diaz et al. / Journal of Colloid and Interface Science 348 (2010) 232–239

and (ii) to show that an appropriate value for d is d = e where e is given by Eq. (21).

now clear from the above expression that an appropriate choice for d is

4.2.1. Upper limit, hU, (h = hU) The upper limit of the transition region will be located within the capillary region where the Young–Laplace equation holds and the angle of inclination h is related to ho and h via Eqs. (14) and (15)

d¼e

cos h  cos ho ¼

"

h hC

 

 2 # 1 h 2ð1  cos ho Þ 2 hC

ð28Þ

On substituting, h = hU, h = hU, and expanding cos hU in a Taylor series expansion about ho, Eq. (28) gives

"    2 # hU hU ð1  cos h0 Þ ho  hU ’ 2  sin ho hc hc

ð29Þ

Hence |ho  hU| < d implies that for hU/hC << 1

hU d < cotðho =2Þ hC 2

ð30Þ

4.2.2. Lower limit, hL, (h = hL) Eq. (16) gives the solution of the curvature (AYL) equation that is valid outside the molecular region. Near h = hL, only the first and third terms on the right hand side of Eq. (16) are significant since

hL =hC < ht =hC ffi Oðe2 Þ  1

ð31Þ

and the first will dominate provided

ðhL =ht Þ3  1

ð32Þ

Hence the form of Eq. (16) valid near h = hL, is given by 2

cos h  cos ho ffi

hm 2

2h

ð33Þ

Substituting h = hL and h = hL, and expanding cos hL about cos ho yields 2

ðho  hL Þ ffi

hm

1 2 hL 2 sin h0

ð34Þ

Hence |ho  hL| < d is satisfied provided

 2 hm < d2 sin ho hL

ð35Þ

So to summarize, all angles of inclination within the transition region will satisfy |h  ho| < d, provided

   2 hU d hm < cotðho =2Þ; < d2 sin ho 2 hC hL

ð36Þ

since then gVL(hL)/r > 0.995; interfacial tension is within 0.5% of its constant value, r, for all the solid/liquid combinations in Table 1. The boundaries of the transition region are now given by

hU ¼ hC

e 2

hm hL ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin ho e

cotðho =2Þ;

ð39Þ

Table 1 gives the full range of data for the n-alkanes on PTFE and for dispersive liquids on PDMS and PE. Note the following points. 4.2.3.1. Characteristic lengths. The molecular length, hm, is approximately 1010 m for n-alkanes on PTFE and 1.7.1011–1010 m for dispersive liquids on PE and PDMS. The transition, ht, and capillary, hc, lengths exhibit a factor 3 and factor 7 variation respectively across the whole range of liquids. For the case of small contact angles (bromobenzene and tert-butyl naphthalene on PE) we can observe the behaviour hc ! 0 as h0 ! 0 in accordance with hc ! ho Lc which follows from Eq. (15). 4.2.3.2. Small parameters. For the analysis in Section 4 to remain valid, the two ratios hL/hC and (hL/ht)3 are required to be much smaller than one. Table 1 shows that over the range of available data, hL =hC  5:106 andðhL =ht Þ3  103 , which in turn, ensure that the theoretical predictions for hL and hU, will be in close agreement with those obtained by numerical computations using Eqs. (16), (26) and (27), see Table 1. In fact the error is less than 1% except for tert-butyl naphthalene (<1.75%). A further observation is that, for all liquid/solid combinations in Table 1, hL =hU < 103 showing that the transition region extends over 3–4 orders of magnitude in film thickness. This is clearly illustrated in Fig. 4 for heptane on PTFE for which e = 7.12  103, hU = 1.2  105 m and hL = 1.17  109 m. 5. Molecular region and the fully augmented Young–Laplace equation Earlier we observed from Figs. 3 and 5 that film thickness profiles for the YL and AYL equations begin to diverge as the molecular region is approached (h < 2.109 m). Once the molecular region is entered and the effect of interfacial tension varying with disjoining pressure is taken into account, the curvature equation is the fully augmented Young–Laplace equation (FAYL), given by Eq. (3). With disjoining pressure and interfacial tension given by Eqs. (7) and (8), Eq. (3) curvature given by Eqs. (3) and (10) reduces to 2

d cos h hm ¼ 2 2 dh hðh  hm =2Þ

where

 3   hL hU  1; and 1 ht hC

ð38Þ

ð37Þ

4.2.3. Choice of d There are three factors influencing the choice of d which, in turn, will determine the extent of the transition region. First, d must be larger than O(e2), otherwise the only points satisfying |ho  h| < d will be in a small range centered on the null curvature point, see Eq. (24). Second, d must be sufficiently small so that the h values are close to ho. Thirdly, the choice must give rise to a value for hL, that is at the edge of the molecular region or beyond where interfacial free energy, gVL, given by Eq. (8) has effectively 2 attained its constant value, r. Therefore, by substituting h ¼ 2 2 hL ¼ hm =2d sin ho into Eq. (8) gives gVL(hL) = r(1  d sin ho). It is

ð40Þ

where the effect of hydrostatic pressure is now negligible compared to that of disjoining pressure. Integrating Eq. (40) using partial fractions gives 2

cos h ¼  ln 1 

1 hm 2 h2

! þC

ð41Þ

where the constant, C, is determined via the condition h = hL when 2 2 h = hL, C ¼ cos hL þ lnð1  hm =2hL Þ which via Eq. (34) becomes

 4 hm C ¼ cos ho þ O hL

ð42Þ

Hence to O(hm/hL)4, the solution for h = h(h, hm, ho) is given by

M. Elena Diaz et al. / Journal of Colloid and Interface Science 348 (2010) 232–239

" cos h ¼ cos ho  ln 1 

# 2

1 hm 2 h2

ð43Þ

Near the outer edge of the molecular region, ðhm =hÞ2  1, and so expression (43) becomes 2

cos h  cos ho ffi

1 hm 2 h2

ð44Þ

which is identical with Eq. (33), the solution of the AYL equation near h = hL. This confirms that the inner solution given by Eq. (43) runs smoothly into the outer solution given by Eq. (16). It follows from Eq. (43) that dh/dh > 0 and, therefore starting at the outer edge of the molecular region, h will continue to decrease as h decreases. Finally there are two points to note concerning the relevance and function of this analytical solution. First, there is a special case, hm = 0, for which Eq. (43) implies that h = ho for all h in the molecular region whilst Eq. (6) gives an equation for ho in terms of the Hamaker constants ½6

G½6 ðho Þ ¼

ASL

½6

ALL

239

is shown to have no precise location on the vapour–liquid interface, the angle of inclination of the interface at the position of null curvature is the closest with the difference of O(e2). Furthermore, all angles of inclination within the transition region satisfy |ho  h| < e, and therefore, to O(e), the macroscopic static contact angle can be identified as the angle of inclination of the wedgeshaped transition region. It is the presence of this transition region extending from a few micrometers to a few nanometers of the solid surface, that provides an accurate means of measuring contact angles directly from an image of the interface. The procedure involves matching a digitized image of the vapour–liquid interface to a solution of the Y–L equation [11] and extending this solution until it intersects the tangent to the solid surface. Clearly this method removes the uncertainty of how much we should amplify a digitized image and how close to the solid surface we should measure the contact angle. By measuring the contact angle anywhere within the transition region, one can ensure a precision better than e = 8.8  103 radians (approximately one half of one degree) which is substantially better than our current ability to measure experimental contact angles.

ð45Þ

This is precisely the same equation, derived independently by Miller and Ruckenstein [16] and Jameson and del Cerro [17] yet, somewhat surprisingly, none of these authors seemed to appreciate its physical significance. If hm = 0 and ho is given by Eq. (45), it follows from Eq. (5) that P 0 and dg VL =dh 0, i.e. disjoining pressure is everywhere zero and interfacial tension is constant along the whole of the interface. Clearly this corresponds to a unique configuration in which the effect of the solid–liquid molecular interactions exactly nullifies the effect of the liquid–liquid interactions so as to generate no disjoining pressure. In practice, hm = 0 refers to a configuration in which the vapour–liquid interface is described by the Y–L equation and includes a wedge of angle ho extending down to the solid substrate in the absence of either a liquid film or an adsorbed layer. Second, the solution given by Eq. (43) could be extended further into the molecular region so as to connect with either a thin film or an adsorbed layer and hence, once again, relate the static contact angle, ho, to the Hamaker constants. The physical significance of the cases, hm = 0 and hm – 0 were presented in a companion paper on hysteresis during the measurement of static contact angles [26].

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

[21]

6. Conclusions Analytical solutions are presented for the shape of the vapour– liquid interface within and outside the molecular region. These solutions facilitate the identification of, and measurement of the static contact angle. Although the macroscopic static contact angle

[22] [23] [24] [25] [26]

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