Analysis of the Young equation and use of the Kelvin equation in calculating the Gibbs excess

Colloids and Surfaces A: Physicochemical and Engineering Aspects 202 (2002) 33 – 39 www.elsevier.com/locate/colsurfa

Analysis of the Young equation and use of the Kelvin equation in calculating the Gibbs excess Alan E. Johnson * 57 Scarisbrick Road, Rainford, St. Helens, Merseyside, WA11 8JN, UK Received 13 June 2001; accepted 13 September 2001

Abstract Vertical resolution of the forces acting in the Young model [Phil. Trans. Roy. Soc. 84 (1805)] of a drop of liquid on a flat surface leads to an expression for the vertical component of force on the three-phase-line and a quantitative explanation of contact angle hysteresis. Derivation of a modified Kelvin equation has been used to explain the significance of the cubic turning points of the Van der Waals’ equation and the calculation of the Gibbs’ surface excess in a two-phase system. © 2002 Published by Elsevier Science B.V. Keywords: Surface tension; Contact angle; Kelvin equation; Young equation

1. Introduction Boundaries, such as surfaces, lines and points occur where two, three or four immiscible phases Abbre6iations: kLV, force per unit length or energy per unit area of the liquid/vapour interface (mN m − 1); kSV, force per unit length or energy per unit area of the solid/vapour interface (mN m − 1); kSL, force per unit length or energy per unit area of the solid/liquid interface (mN m − 1); YLV, surface concentration at liquid/vapour interface (mol m − 2); YSV, surface concentration at ‘‘ solid/vapour interface (mol m − 2); qE, equilibrium contact angle; qA, advancing contact angle; qR, receding contact angle; „, angular difference from the equilibrium contact angle; z, density of the liquid phase (kg m − 3); M, molecular mass of the vapour or liquid; P , saturated vapour pressure above pressure above a plane liquid surface (Pa); Pr, saturated vapour pressure above a spherically curved meniscus with radius r (Pa); TPL, three-phase-line boundary. * Tel.: +44-174-488-4978. E-mail address: [email protected] (A.E. Johnson).

meet, and at all of these divisions there are molecules with higher energies than those in the bulk phase. The influence of an adjacent phase contributing to any of the above boundaries extends for a finite distance into other boundary phases. Concentration excesses within a surface boundary and the significance of a dividing line between phases are given precise meaning in the technique devised by the Gibbs and in the case of surfaces the magnitude of the excess is calculated from the Gibbs equation. Evidence of excess free energy in the surface and the influence of curvature in the dividing surface is clearly demonstrated by the Kelvin equation, which defines the relationship between droplet radius and equilibrium vapour pressure. In this case, the curved surface of the droplet functions as a potential energy so that less energy is required to convert it to vapour than an identical mass of liquid from a plane surface.

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Equilibrium cases involving three-phase have been analysed by extending the derivation introduced by Kelvin for a two-phase system to a three-phase system based upon the ideal model of a liquid drop on a surface first envisaged by Young.

equation will give rise to error, but it was found that the solution to this problem gives significance to the cubic turning points of a Van der Waals gas.

3. The Kelvin equation 2. Liquid at equilibrium with vapour in a cylinder or capillary The liquid is assumed to be at the equilibrium contact angle. As the case of q = y/2 is trivial the principle cases are those of a concave and a convex meniscus (Fig. 1). The work done dW on the system during reversible transfer of dn moles from a flat surface with equilibrium pressure P to the meniscus with an equilibrium vapour pressure Pr, is: dW =dn·RT·ln

  Pr P

(1)

Since there is no change in the surface area of the meniscus the change in surface energy dW is equal to the product of the increase in wetted area of the capillary and the difference in surface energy between the S/L and S/V surfaces. dW = 2yr·dL(kSL −kSV)

(2)

1000yr 2z dL M

(3)

dn=

From the diagram geometry: r = a cos q

(4)

Curvature is at the very root of the Kelvin equation, but it is unlikely that Kelvin realised before he derived the equation just how extreme the curvature had to be, before its effect became significant. The equation is important since it quantifies the effect of curvature on the vapour pressure of small liquid drops and also the effect of a negative radius explains the phenomenon of capillary condensation, which plays a role in surface chemistry. Ultimately, when the radius becomes very small, failure to take account of the Gibbs excess associated with the drop leads to error. In the following derivation the procedure for drawing the phase boundary between the liquid and vapour is the same as used by Gibbs. For mathematical purposes the excess is considered to be totally on the vapour phase side of the boundary and, therefore, it has a positive value. Consequently, the liquid phase consists entirely of core molecules. Since the shift in vapour pressure predicted by the equation only becomes significant when the radius approaches the nanometre scale this takes the derivation using the radius of a sphere as a continuous variable is close to the limit of its validity for a liquid made up of discrete

Using Eqs. (1)– (4) and the Young equation leads to the Kelvin equation:

 

ln

P MkLV = Pa 500azRT

(5)

There is a sign convention associated with the Kelvin equation, such that convex radii are positive and concave radii are negative. Therefore, Eq. (5) can be obtained by substituting −a in the standard equation which is derived for a sphere. The Kelvin equation is one of the cornerstones of surface science, but under certain circumstances the theory presented below predicts that this

Fig. 1. Filling a capillary under reversible conditions.

A.E. Johnson / Colloids and Surfaces A: Physicochem. Eng. Aspects 202 (2002) 33–39

molecules. In essence the derivation used by Kelvin treats the drop as consisting of bulk molecules and surface energy as a property of an envelope surface which does not have a molecular concentration associated with it. For most of the range of values which the radius can assume, this is a perfectly reasonable approach, but when the radius is very small the molecules close to the surface become a significant fraction of the whole sphere and ultimately the sphere will become unstable as the number of core molecules approaches zero. This effect can be taken care of in a fairly minor modification to the equation, but earlier comments about treating the radius as a continuous variable will still apply. The spherical drop may be considered to be made up of molecules in two energy distributions: (a) core molecules with a lower mean energy; (b) surface molecules with a higher mean energy (The Gibbs excess). When reversible transfer to the drop from a plane surface takes place molecules are supplied to both the core of the drop and to the surface excess. 4000yr 3z N= + 4yr 2GLV 3M

dN 4000yr 2z = + 8yrGLV dr M

  Pr P

(7)

(8)

This is equal to the free energy increase associated with creation of additional surface area dA: dW= kLV·dA 8yr·dr·kLV =dN·RT·ln

  Pr P

 

MkLN Pr = RT·ln 500rz +MGLV P

(9) (10)

(11)

In this equation the pressure Pr tends to a limit as r“0 but in the original Kelvin equation Pr tends to infinity. In the light of Eq. (11) the validity of the derivation of Eq. (5) must be questioned. There is in fact an error in Eq. (5) which is of the same nature as the flaw in the usual derivation of the Kelvin equation, but more subtle. With reference to Fig. 2, YSV is the surface excess on the capillary wall. Reversible transfer of dn moles from a flat surface to the meniscus results in no change in area or curvature of the meniscus but the area of the wall wetted by the liquid increases. Proceeding as previously: r= a cos(y − q)= −a cos q dW= 2yr(kSL − kSV)dL = − 2yrkLV cos q (from the Young equation) dN·RT·ln

Proceeding in an identical manner to Kelvin. Transfer of dN moles of liquid from a plane surface to the drop under reversible conditions requires an input of work dW: dW =dN·RT·ln

Substitute the relationship between the number of moles transferred dN and the change in radius of the drop dr:

(6)

where r, radius of drop; N, total number of moles in the spherical drop. Differentiate Eq. (6) wrt r:

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dN=

 

Pa = − 2yrkLV cos q·dL P

1000yr 2z dL − 2yrGSV·dL M

Hence:

 

RT·ln

Pa MkLV cos q = 500az cos q+MGSV P

(12)

Now since the meniscus is part of a sphere then, Eq. (11) covering the equilibrium of a sphere must also be satisfied which then leads to: MkLV cos q MkLV cos q = 500az cos q+ MGSV 500az +MGLV So the relationship between the excess on the capillary wall YSV, and the excess on the liquid surface YLV is: GLV cos q= GSV

(13)

A.E. Johnson / Colloids and Surfaces A: Physicochem. Eng. Aspects 202 (2002) 33–39

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Fig. 2. Reversible transfer of a liquid from a plane to a convex surface.

4. Use of Van der Waals’ equation P=

RT a − V− b V 2

(14)

&

Reversible transfer of one mole: Vr

PrVr − P V −V = RT



P·dV



V

Vr Vr −b V − − ln V −b Vr −b V −b

− 2a







1 1 − Vr V

(15)

Using Eq. (13) for a quantity dN moles together with Eqs. (8) and (9) now gives:



MkLV Vr V =RT − 500rz + MGLV Vr −b V −b

 

 

− ln

Vr −b V −b

−2a

1 1 − Vr V

(16)

Usually where Van der Waals equation is used, a horizontal straight line is drawn across the pressure/volume curve at the saturated vapour pressure for a plane surface to signify condensa-

tion at constant pressure and the region between the triple point and the critical point of the equation is dismissed as spurious. This is far from truth, since, the Van der Waals equation itself and Eq. (14) constitute simultaneous equations of state covering the situation where a curved surface is in equilibrium with the vapour, and all the points on the cubic curve can be accounted for. Curvature may be of spherical, cylindrical or torroidal nature, the only restriction being that the system is in equilibrium, but for brevity priority will be given to spherical curvature. Van der Waals equation is semi-empirical and incorporates volume, pressure, and temperature in a relationship, which is cubic in V. The equation, was designed to take account of both the forces between molecules and their finite volumes. It covers the gas phase and vapour/liquid mixtures up to the point where condensation to the liquid phase is complete. Where both liquid and vapour exist in equilibrium the equation has three real roots. Two of the roots occur where the substance is entirely in the vapour phase and where condensation to the liquid is complete. The third root occurs part way through the condensation process where the substance exists as a mixture of liquid and vapour. The cubic curves correspond to an

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5. Interpretation of the isotherm

Fig. 3. Significance of Van der Waals isotherm.

equilibrium state when the liquid/vapour interface is curved, and the radius of curvature is given by Eq. (14) for a particular value of volume. Special significance can be attached to the maximum and minimum values of the equation since they occur where r “0 so the Gibbs excess YLV can be calculated from the values of pressure, volume and temperature at these points. Of the two points the maximum is far more important. This follows from the properties of equations since the sum of the roots is equal to the coefficient of V [1] and the difference in volume between the liquid and vapour states is huge so error in the equation will have a bigger effect when the substance is in the liquid state Fig. 3. Substitute r =0 into Eq. (14): kLV V Vr Vr −b =RT − −ln GLV Vr −b V −b V −b



−2a







1 1 − Vr V

Temperature (°C)

Surface tension (mN m−1)

0 50 100 150 200 250 300 350

76.9 66.6 56.4 46.1 35.8 25.5 15.2 5.0



(17)

At point ‘A’ condensation can commence along ‘AD’ at the minimum radius of the concave meniscus. The radius is greater at point ‘N’ and condensation can take place along ‘NM’. The vapour has condensed completely at ‘M’. Line ‘PR’ represents the usual vapour to liquid condensation where the interface is flat. At point ‘H’ the equilibrium interface has a convex radius and reaches its minimum value at point ‘B’. Between ‘B’ and ‘D’ the substance exists as a mixture of liquid and vapour and the equilibrium radius of the interface changes from its minimum convex value at ‘B’ to its minimum concave value at ‘D’. The substance is completely in the liquid state at ‘E’ and the interface has its minimum radius of convex curvature. As pressure is lowered equilibrium can be re-established by increasing the radius of curvature. This procedure can continue until the minimum value of concave curvature is reached at ‘D’. In the table below the molar volume at the pressure maximum for each value are shown for the temperature range 0–350 °C. Isotherms were calculated using constants a and b which are derived by the classic method using the properties of the water at its critical temperature of 374.1 °C. Water vapour, and other real gases do not behave as perfect Van der Waals gases so that a and b are not true constants and there values depend upon temperature. However, the objective here is to give a simple demonstration technique rather than prepare table of very precise values. Molar values for vapour in equilibrium flat-water surface were taken from standard tables [2]. Values for the Gibbs’ excess which show an increase with temperature given in the end column.

Molar volume (max curvature) (ml mol−1) 419 342 284 240 203 172 145 117

Molar volume (flat surface) (ml mol−1) 3 710 000 216 700 30 110 6746 2227 896 394 159

Gibbs excess (mmol m−2) 4.2 4.6 5.0 5.6 6.3 7.4 9.8 34.0

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A.E. Johnson / Colloids and Surfaces A: Physicochem. Eng. Aspects 202 (2002) 33–39

Fig. 4. The equilibrium position and the surface constrained position of the surface tension vectors.

6. The Young equation Young considered a drop of liquid on a plane surface represented an equilibrium and resolved horizontally the three vectors corresponding to the three surface tensions kLV, kSV, and kSL. The solid– vapour and solid– liquid vectors shown in blue were drawn in horizontally opposed positions on the surface, and the liquid– vapour vector shown in red was drawn as a tangent at angle q at the point on the surface where the three phases meet. Young equation: kSV − kSL = kLV cos q

‘O’ then the vectors would take up their equilibrium positions as shown such that the vector equation is satisfied. kLV “ + kSV “ + kSL “ = 0

(19)

Vertical resolution of the vectors in Young’s position produces a resultant, which is shown in green on the right. Clearly the equation to complete the equilibrium can be written down but that alone does not imply physical significance Fig. 4: kLV sin q= kLV sin h+ kSL sin i

(20)

(18)

Young was correct but this is far from obvious, since if he is correct then it follows that his conception of the arrangement of vectors was incomplete. If equilibrium exists then the force vectors should sum to zero and if the forces are resolved in any direction, then they should balance. Even in recent times the equation has been considered invalid for failing this criterion [3,4]. The equilibrium positions of kSV and kSL which make angles h and i, respectively, with the horizontal are shown in red and the three red vectors form a closed triangle which proves that they sum to zero. If the horizontal axis could freely fold at

7. Vector rotation In Fig. 5 equilibrium constrained enced as a given by:

when vector ‘OA’ is turned from its position through an angle q to a position the tangential force expericonsequence of the torque at ‘O’ is

Fq = 2kLV sin

  q q cos 2 2

Hence the work done in rotating the vector through an angle h to its horizontal position on the surface is:

&

Wh = k 2LV

A.E. Johnson / Colloids and Surfaces A: Physicochem. Eng. Aspects 202 (2002) 33–39 h

sin q·dq =k 2LV(1 − cos h)



0

Wh = 2k 2LV sin2

h 2

(21)

When the vectors are constrained in a non-equilibrium position a downward vertical force is developed as a result of torque on the TPL and this opposed by the vertical component of kLV. The torque is the origin of the force components on the right hand side of Eq. (20). So the work of rotation of kSV and kSL from the equilibrium positions to their horizontally opposed positions on the surface is given by:



WSV + WSL =2 k 2SV sin2



 

h i +k 2SL sin2 2 2

This strain energy in the line will be reduced by the line bending and conforming to the roughness of the surface. Rotation of kLV applies torsion and potential energy to the line and rolling begins when this is equal to the work of adhesion. If all the potential energy used in rotation of kSV and kSL is consumed in increasing adhesion to the surface then the limiting values of the advancing and receding angles qA and qR, respectively, can be calculated.



k 2LV sin2





„ h i =k 2SV sin2 +kSL sin2 2 2 2

(22)

When the equation is solved for „ then qA and qR are given by: qA = qE +„,

qR =qE −„

When the solid surface is flat the contact angle has a maximum value and no hysteresis is observed.

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Surface roughness lowers the contact angle and this is explained quantitatively by Wenzel [5] equation. Surface roughness allows the TPL to deform and lose potential energy. This potential energy must be regained before rolling motion can commence which leads to the phenomenon of hysteresis. Numerical example: Let kLV = 60, kSV = 40, kSL = 30 then h= 54.03°, i= 63.25° and „= 12.63°. Hence, q= 80.41° (calculated using the Young equation), qA = (80.41+ 12.63)=93.04°, qR = (80.41− 12.63)= 67.78°. Rough surfaces allow the TPL to release strain energy since the angular features present opportunities for the surface vectors to move towards their equilibrium positions. So the tendency will be for the TPL to release as much strain energy as possible. In practice a minimum energy compromise is reached since the intersecting surfaces also have a tendency to minimise surface energy.

8. Conclusion Certain problems which arise with the Kelvin equation disappear when the derivation takes account of the Gibbs’ surface excess. If Van der Waals equation is used in place of the general gas equation then the cubic curves can be shown to relate to an equilibrium state where the dividing surface is curved. Combining the modified Kelvin equation with Van der Waals equation leads to a novel way of determining the Gibbs surface excess. A solution has been proposed for the long standing problem of the nature of the vertical forces in the derivation of the Young equation.

References

Fig. 5. Work done during vector rotation.

[1] E.D. Eastman, G.K. Rollefson, Physical Chemistry, McGraw-Hill, New York, 1947, p. 280. [2] J.H. Stephens, Kempe’s Engineers Year Book for 1999, Miller Freeman UK Ltd, 1999. [3] J.J. Bikerman, Proc. Second Int. Congress Surface Activity London 111 (1957) 125. [4] A.W. Adamson, Physical Chemistry of Surfaces, Interscience Publishers, New York, 1967, p. 369. [5] R.N. Wenzel, Ind. Eng. Chem. 28 (1936) 988.