THE SURFACE TENSION OF PURE SUBSTANCES

1. THE SURFACE TENSION OF PURE SUBSTANCES INTRODUCTION

2. Introduction • Surface tension  is the contractile force which always exists in the boundary between two phases at equilibrium • Its actually the analysis of the physical phenomena involving surface tension which interests us

3. Our topics primarily concern on • Surface tension as a force • Surface tension as surface free energy • Surface tension and the shape of mobile interfaces • Surface tension and capillarity • Surface tension and intermolecular forces

4. Surface Tension As A Force: The Wilhelmy Plate • The surface of a liquid appears to be stretched by the liquid it encloses • Example of this are: • the beading of water drops on certain surfaces; • the climbing of most liquids in glass capillaries • The force acts on the surface and operates perpendicular and inward from the boundaries of the surface, tending to decrease the area of the surface

6. Noted • Equation above defines the units of surface tension to be those of force per length or dynes per centimeter in the cgs system • The apparatus shown resembles a two-dimensional cylinder/piston arrangement, so its analogous to a two dimensional pressure • A gas in the frictionless, three-dimensional equivalent to the apparatus of the figure would tend to expand spontaneously. For a film however the direction of spontaneous change is contraction

7. A quantity that is closely related to surface tension is the contact angle , defined as the angle (measured in the liquid) that is formed at the junction of three phases, as shown in figure 6.1b • Although the surface tension is a property of two phases which form the interface,  requires that three phases be specified for its characterization

8. The Wilhelmy Plate • Figure 6.2 The Wilhelmy plate method for measuring . In (a) the base of the plate does not extend below the horizontal liquid surface. In (b) the plate is partially submerged to buoyancy must be considered a b

9. Figure 6.2 represent a thin vertical plate suspended at a liquid surface from the arm of tarred balance • The manifestation of surface tension and contact angle in this situation is the entrainment of a meniscus around the perimeter of the suspended plate • Assuming the apparatus is balanced before the liquid surface is raised to the contact position, the imbalance that occurs on contact is due to the weight of the entrained meniscus • Since the meniscus is held up by the tension on the liquid surface, the weight measured by the apparatus can be analyzed to yield a value for 

10. The observed weight of the meniscus w, must equal the upward force provided by the surface w = 2(l+t) cos  •  is the contact angle, l and t are the length and thickness of the plate. Because of the difficulties in measuring , the Wilhelmy plate method is most frequently used for system in which  = 0 so w = 2(l + t) • Since the thickness of the plate used is generally negligible compared to their length (t <<< l) equation may approximated: w = 2l 

11. Surface Tension As Surface Excess Free Energy • The work done on the system of figure 6.1 is given by Work = F dx = 2l dx =  dA • This supplies a second definition of surface tension, it equals the work per unit area required to produce new surface • If the quantity w’ is defined to be the work done by the system when its area is changed, then equation becomes w’ = -γdA according to the first law • dE = q - w in which w is the work done by the system and q is the heat absorbed by the system. It relates to Gibbs free energy by following equation : dG = TdS – pdV - wnon-pV + pdV + Vdp – TdS – SdT for a constant temperature, constant pressure and reversible process dG = -wnon-pV

12. That is dG equals the maximum non-pressure/volume work derivable from such a process since maximum work is associated with reversible process • We already seen that changes in surface area entail non-pV work, therefore we identify w’ as wnon-pV and write dG = γdA • Even better in view of the stipulations we write γ = (G/A)T,p • This relationship identifies the surface tension as the increment in Gibbs free energy per unit increment in area

13. The Laplace Equation

14. Our attention will focus on those specific surfaces with most readily allow the experimental determination of  • The shape assumed by a meniscus in a cylindrical capillary and the shape assumed by a drop resting on a planar surface (called a sessile drop) are most useful in this regard • Figure 6.4 may be regarded as a portion of the surface of either of these cases • As can be seen the curve represent the profile of a sessile drop; inverted, the solid portion represent the profile of a meniscus • The actual surfaces are generated by rotating these profiles around the axis of symmetry

16. Because the symmetry of the surface, both values R must be equal at the apex of the drop • The value of the radius of curvature at this location is symbolized b, therefore, at the apex (subscript 0) • Next, let us calculate the pressure at point S. At S the value of p equals the difference between the pressure at S in each of the phases • These may be expressed relative to the pressure at the reference plane through the apex (subscript 0) as follows

17. In phase A: pA = (pA)0 + Agz • In phase B: pB = (pB)0 + Bgz • Therefore, p at S equals (p)S = pA – pB = (pA)0 – (pB)0 + (A - B)gz = (p)0 + gz Where  = A - B and we can write it

19. Notes • If A > B,  will be positive and the drop will be oblate in shape since the weight of the fluid tends to flatten the surface • If A < B, a prolate drop is formed since the larger buoyant force leads to a surface with much greater vertical elongation. In this case  is negative • A  value of zero correspond to a spherical drop and in a gravitational field is expected only when p = 0 • Positive values of  correspond to a sessile drops of liquid in gaseous environment • Negative  values correspond to sessile bubbles extending into a liquid

21. Notes • The previous statement imply that the drop is resting on a supporting surface • If instead the drop is suspended from a support (called pendant drops or bubbles), g becomes negative, and it is the liquid drop that will have the prolate ( < 0) shape and the gas bubbles the oblate ( >0) shape

23. Measuring Surface Tension: Sessile Drops • The Bashfort and Adams tables provide an alternate way of evaluating  by observing the profile of a sessile drop of the liquid under investigation • Once  known for a particular profile, the Bashfort Adams tables may be used further to evaluate b • For the appropriate  value, the value of x/b at  = 90o is read from the tables. This gives the maximum radius of the drop in units of b • From the photographic image of the drop, this radius may be measured since the magnification of the photograph is known • Comparing the actual maximum radius with the value of (x/b)90 permits the evaluation of b

24. The figure can use for example of the procedure described • Theoretically its shown to correspond to a  value of 10,0 then b is evaluated as follows • The value of (x/b)90 for  = 10 is found to be 0,60808 from the tables • Assume the radius of the actual drops is 0,500 cm at its widest point • Item (1) and (2) describe the same point; therefore b = 0,500/0,60808 = 0,822 cm • Assuming  to be 1,00 g cm-1 and taking g = 980 cm s-2 gives for 

25. Measuring Surface Tension: Capillary Rise • A simple relationship between the height of capillary rise, capillary radius, contact angle and surface tension can derived 2R cos  = R2h g (48) • Its difficult to obtain reproducible result unless  = 0o, so the equation simplifies to (49) • The cluster constant 2/(g) is defined as the capillary constant and is given the symbol a2; • (50)

26. The apex of the curved surface is identified as the point from which h is measured. As we have seen before, both radii of curvature are equal to b at this point • At the apex of the meniscus, the equilibrium force balance leads to the result (51) (52)

27. Equation (48) is valid only when R = b, that is for a hemispherical meniscus. • In general this is not the case and b is not readily meaured so we have not yet arrived at a practical method of evaluating γfrom the height of capillary rise. Again the tables of Bashfort and Adams provide the necessary information • For liquid to make an angle of 0o with the supporting walls, the walls must be tangent to the profile of the surface at its widest point • Accodingly (x/b)90o in the Bashfort and Adams tables must correspond to R/b. since the radius of the capillary is measurable, this information permits the determination of b for a meniscus in which θ = 0 • However there is a catch. Use of the Bashfort and Adams tables depends on knowing the shape factor β. It is not feasible to match the profile of a meniscus with theoritical contours, so we must find a way of circumventing the problem

28. The procedure calls for using successive approximation to evaluate β. Like any iterative procedure, some initial values are fed into a computational loop and recycle until no further change results from additional cycles of calculation • In this instance, initial estimates of a and b (a1 and b1) are combined with Eqs. (46) and (50) to yield a first approximation to β (β1) • The value of (x/b)90o for β1 is read or interpolated from the tables • This value and R are used to generate a second approximation to b (b2). By Eq.(52) a second approximation of a (a2) is also obtained and –starting from a2 and b2 – a second round of calculation is conducted. • The following table shows an example of this procedure

29. It is sometimes troublesome to find a starting point for these iterative calculations. The following estimates are helpful for the capillary rise problem: • From Eqs (49) and (50) a1 Rh • Treating the menicsus as hemisphere b1  R • The initial value of table 6.3 assuming R = 0.25 cm and h = 0.40 cm

30. Measuring Contact Angle • The experimental methods used to evaluate θ are not particularly difficult, but the result obtained may be quite confusing • The situation is best introduced by refering figure right-below which shows a sessile drop on a tilted plane • It is conventional to call the larger value the advancing angle θa and the smaller one the receding angle θr • With the sessile drop, the advancing angle is observed when the drop is emerging from a syringe or pipet at the solid surface • The receding angle is obtained by removing liquid from the drop

31. Weight of a meniscus in a Wilhelmy plate experiment versus depth of immersion of the plate. In (a) both advancing and receding contact angles are equal. In (b) a > r

32. Schematic energy diagram for metastable states corresponding to different contact angles • The general requirement for hysteresis is the existence of a large number of metastable states which differ slightly in energy and are separated from each other by small energy barriers • The metastable states are generaly attributed to either the roughness of the solid surface or its chemical heterogeneity, or both.

33. Cross section of a sessile drop resting on a surface containing a set of concentric grooves. For both profiles, the contact angle is identical microscopically, although macroscopically different

34. Kelvin Equation • Another result of pressure difference is the effect it has on the free energy of the material possessing the curved surface • Suppose we consider the process of transferring molecules of a liquid from a bulk phase with a vast horizontal surface to a small spherical drop of radius r • Assuming the liquid to be incompressible and the vapor to be ideal, ∆G for the process of increasing the pressure from po to po + ∆p is as follows:

36. The Kelvin equation enables us to evaluate the actual pressure above a spherical surface and not just the pressure difference across the interface, as was the case with the Laplace equation • Using the surface tension of water at 20oC, 72,8 ergs cm-2, the ratio p/po is seen to be • Or 1,0011; 1,0184; 1,1139; and 2,9404 for drops of radius 10-4, 10-5, 10-6 and 10-7 cm respectively. • Thus for a small drops the vapor pressure may be considerably larger than for flat surfaces

37. The Kelvin equation may also be applied to the equilibrium solubility of a solid in a liquid • In this case the ratio p/po in equation is replaced by the ratio a/ao where ao is the activity of dissolved solute in equilibrium with flat surface and a is the analogous quantity for a spherical surface • For an ionic compound having the general formula MmXn the activity of a dilute solution is related to the molar solubility A as follows:

38. The equation provides a thermodynamically valid way to determine SL, for example the value of SL for the SrSO4-water surface has been found to be 85 ergs cm-2 and for NaCl-alcohol surface to be 171 ergs cm-2 by this method • The increase in solubility of small particles and using it as a means of evaluating SL is fraught with difficulties: • The difference in solubility between small particles and larger one will probably differ by less than 10% • Solid particles are not likely to be uniform spheres even if the sample is carefully fractionated • The radius of curvature of sharp points or protuberances on the particles has a larger effect on the solubility of irregular particles than the equivalent radius of the particles themselves.

39. The Young Equation • Suppose a drop of liquid is placed on a perfectly smooth solid surface and these phases are allowed to come to equilibrium with the surrounding vapor phase • Viewing the surface tension as forces acting along the perimeter of the drop enables us to write equation which describes the equilibrium force balance LV cos  = SV - SL

40. First Objections • Real solid surfaces may be quite different from the idealized one in this derivation • Real solid surface are apt to be rough and even chemically heterogeneous • If a surface is rough a correction factor r is traditionally introduced as weighting factor for cos , where r > 1 • The factor cos  enters equation by projecting LV onto the solid surface • If the solid is rough a larger area will be overshadowed by the projection than if the surface were smooth • Young’s equation becomes rLV cos  = SV - SL • A surface may also be chemically heterogeneous. Assuming for simplicity that the surface is divided into fractions f1 and f2 of chemical types 1 and 2 we may write LV cos  = f1(S1V - S1L) + f2(S2V - S2L) • Where f1 + f2 = 1

41. Second Objection • The issue of whether the surface is in a true state of thermodynamically equilibrium, it may be argued that the liquid surface exerts a force perpendicular to the solid surface, LV sin  • On deformable solids a ridge is produced at the perimeter of a drop; on harder solids the stress is not sufficient to cause deformation of the surface • Is it correct to assume that a surface under this stress is thermodynamically the same as the idealized surface which is free from stress? • The stress component is absent only when  = 0 in which case the liquid spreads freely over the surface and the concept of the sessile drop becomes meaningless

42. Notes • We must assume that SV and S may be different • Let us consider what occurs when the vapor of a volatile liquid is added to an evacuated sample of a non volatile solid • This closely related to the observation that the interface between a solution and another phase will differ from the corresponding interface for the pure solvent due to the adsorption of solute from solution • For now we may anticipate a result to note that adsorption always leads to decrease in , therefore:

43. Figure shows relationship between terms write at the right hand side, it also suggest that the shape of the drop might be quite different in equilibrium and non equilibrium situations depending on the magnitude of e • There are several concepts which will assist us in anticipating the range of e values: • Spontaneously occurring processes are characterized by negative values of ∆G • Surface tension is the surface excess free energy; therefore the lowering of  with adsorption is consistent with the fact that adsorption occurs spontaneously • Surfaces which initially posses the higher free energies have the most to gain in terms of decreasing the free energy of their surfaces by adsorption • A surface energy value in the neighborhood of 100 ergs cm-2 is generally considered the cutoff value between ‘high energy’ and ‘low energy’ surfaces • The equation must be corrected to give

44. ADHESION AND COHESION • Figure illustrates the origin of surface tension at the molecular level

45. In (a) which applies to a pure liquid, the process consists of producing two new interfaces, each of unit cross section, therefore for the separation process: ∆G = 2A = WAA • The quantity WAA is known as the work of cohesion since it equals the work required to pull a column of liquid A apart • It measures the attraction between the molecules of the two portion

46. G = WAB = final - initial = A + B - AB • This quantity is known as the work of adhesion and measures the attraction between the two different phases • The work of adhesion between a solid and a liquid phase may be define analog: WSL = S + LV - SL • By means of previous equation S may eliminated to gives WSL = SV + e + LV - SL • Finally Young’s equation may be used to eliminate the difference: WSL = LV(1 + cos ) + e

47. e  0 where the equality holds in the absence of adsorption • High energy surface bind enough adsorbed molecules to make e significant, example of these are metals, metal oxides, metal sulfides and other inorganic salts, silica, and glass • On the other hand e is negligible for a solid which possesses a low energy surface, most of organic compounds, including organic polymers are in this criteria.

48. The difference between the work of adhesion and the work of cohesion of two substances defines as quantity known as the spreading coefficient of B on A, SB/A: SB/A = WAB – WBB • If WAB > WBB the A-B interaction is sufficiently strong to promote the wetting of A by B (positive spreading). Conversely no wetting occurs if WBB > WAB since the work required to overcome the attraction between two molecules B is not compensated by the attraction between A and B (negative spreading). SB/A = A - B - AB = A – (B + AB)

49. The Dispersion Component of Surface Tension 1  repulsion 2  attraction 3  specific Interaction 4  resultant

50. Interface between two phases A A A A A A Bulk phase A A A A A A