1

3.4 Contact-Angle Approach to Estimation of Surface Free

Energy Motivation Contact angle approach – Procedural descripti

on Justification of Owen’s equation of adhesion

work Assumptions in previous derivation Some footnotes

2JCheng201211

Motivation

In most cases, it is difficult to obtain data like polarizability, dipole, and ionization energy of a molecule.

Thus, it is desirable to have alternative methods for estimating surface free energies and adhesion work

3JCheng201211

Motivation

Challenge from Solid Surface It’s straightforward to measure L for liquids

because surface area of liquids can be changed under constant P, T, & moles n.

Measurement of S for solids is a serious challenge technically

For the surface area of a solid cannot, in general, be changed without affecting its chemical potential.

Therefore, in changing the area, work needs to be done against the elastic forces in the solids.

In a given experiment involving stretching of solid surfaces, it is often difficult to delineate the effects of bulk and surface mechanics.

4JCheng201211

Contact-Angle Method and Young’s Eq

Contact-Angle Method: What is contact angle?

Young’s equation seems pointing to a possibility of inferring energies involving solids

cosLSLSYoung’s eq.

(See Appendix 5 for derivation of Young’s eq.)

liquid

solid

L

S

SL

5JCheng201211

Historical Account of the Method

Equations Used to Deduce S from

cosLSLS

1221A

12W

Young’s eq. (1805) Energy minimization

WA: adhesion work

SLLSA

SLW

(1)

(2))cos1(W L

ASL (

3)

2 unknowns but 1 equationwhere unknowns: S & SL

known: L; measured:

Dupre’s eq. (1869) Energy conservation

(1) & (2)

1 unknown WSL, 1 equation, but lost individual info on S

& SL

6JCheng201211

Historical Account of the Method

Equations Used to Deduce S from (contd)contd

Owen’s eq. (Owens & Wendt 1969, Kaelble & Uy 1970)

p2

p1

d2

d1

A12 22W (

4) Or Wu’s eq. (Wu 1982)

p2

p1

p2

p1

d2

d1

d2

d1A

12

44W

(5)

7JCheng201211

Historical Account of the Method

Equations Used to Deduce S from (contd)contd

(3) & (4)

(6)

pL

pS

dL

dSL 22)cos1(

Or (3) & (5)

pL

pS

pL

pS

dL

dS

dL

dS

L

44cos1

(7)

8JCheng201211

Contact-Angle Method – Procedure

Data Processing Steps for Calculating S from

Prepare the solid surface of interest Measure the contact angles, i of two or more than

two liquids with well-known ip and i

d on the solid surface

Apply i and the ip and i

d of the test liquid i to either eq. (6) or eq. (7)

Solve the equations, each with i for the two unknowns s

p and sd

Finally, we have s = sp + s

d

9JCheng201211

Surface Tensions of Test Liquids

Water: H2O

ethylene glycol 乙二醇 : C2H4(OH)2

diiodomethane 二碘甲烷 : CH2I2

glycerol 甘油 : C3H5(OH)3

source: http://www.accudynetest.com/surface_tension_table.html, 3/17/2010

test liquid (20oC) d p water 21.8 51 72.8

ethylene glycol 29 19 48

diiodomethane 50.8 0 50.8

glycerol 34 30 64

10

Justification of Owen’s Equation of Adhesion Work

Idea behind the derivation Derivation details

p2

p1

d2

d1

A12 22W

11JCheng201211

Recalls about Adhesion Work

21

21

disp1

21

21

ind1

21

21

orient1

1 r24

C

r24

C

r24

C

disp

12ind12

orient122

12

212

12

VDW1221A

12 CCCr12r12

CW

20

22

21orient

124kT3

uu2C

20

2201

2102ind

124

uuC

212

0

210201disp12

4

h

2

3C

20

41orient

14kT3

u2C

20

2101ind

14

u2C

20

1201disp

14

h

4

3C

If 2112 rrr disp ind, orient, n,CCC n2

n1

n12 an

dhold

we would have

disp2

disp1

ind2

ind1

orient2

orient1

A12 22W

12JCheng201211

20

41orient

14kT3

u2C

20

42orient

24kT3

u2C

orient2

orient1 CC

orient122

0

22

21orient

2orient1 C

4kT3

uu2CC

C12orient is the geometric average of C1

orient and C2

orient!

orient2

orient1

orient12 CCC

13JCheng201211

20

2102ind

14

u2C

20

2201ind

24

u2C

ind2

ind1 CC

20

2201

2102ind

124

uuC

20

2201

2102ind

2ind1

4

uu2CC

Q. When can the geometric average approach be an adequate approximate of C12

ind and C12disp?

=???

212

0

210201disp12

4

h

2

3C

20

1201disp

14

h

4

3C

20

2202disp

24

h

4

3C

disp2

disp1 CC 2

0

210201disp2

disp1

4

h

4

3CC

=???

disp2

disp1

disp12

ind2

ind1

ind12 CCC,CCC

14JCheng201211

2201

2102

uyux2

2012102

2201

2102

ind12

ind2

ind1

2yx

yx

2uu

uu

C

CC

Note

2

1yx

21

21disp12

disp2

disp1

yx

2yx2

C

CC

contd

When x and y are within 2.5 times of each other, we have an approximation error less than 10%

disp2

disp1

disp12

ind2

ind1

ind12 CCC,CCC

(Another better discussion appears in Appendix 6)

15JCheng201211

In summary,

20

22

21orient

2orient1

orient12

4kT3

uu2CCC

ind2

ind12

0

2201

2102ind

12 CC4

uuC

5.2u

u4.0

2201

2102

when

disp2

disp1

212

0

210201disp12 CC

4

h

2

3C

4.2h

h4.0

2

1

when

disp2

disp1

ind2

ind1

orient2

orient1

VDW12 CCCCCCC

disp2

disp1

ind2

ind1

orient2

orient1

VDW12 CCCCCCC

(Inequality for harmonic mean can be found in

Appendix 2.1)

16JCheng201211

Prediction of Adhesion Work

Q. Since VDW coefficients of dissimilar molecules can be reasonably approximated by the geometric average of those of individual molecules

Can the geometric average be applied to adhesion work?

22

disp2

22

21

disp1

21

22

ind2

22

21

ind1

21

22

orient2

22

21

orient1

21

?

212

disp1221

212

ind1221

212

orient1221

212

VDW1221A

12

r12

C

r12

C

r12

C

r12

C

r12

C

r12

C

r12

C

r12

C

r12

C

r12

CW

21

VDW1

21C

1 r12

CW

2

2

VDW2

22C

2 r12

CW

recallin

g

17JCheng201211

Prediction of Adhesion Work

For the geometric average to be a good approximate, one needs only to check if

contd

21

?2

12 rrr

Case 1 r1 = 1, r2 = 2, r12 = (1 + 2)/2

A sufficient condition is

5.24.02

1

Thus, in the following, we assume

212

12 rrr

18JCheng201211

Prediction of Adhesion Work

With contd

disp2

disp1

ind2

ind1

orient2

orient1

22

disp2

22

21

disp1

21

22

ind2

22

21

ind1

21

22

orient2

22

21

orient1

21A

12

222

r12

C

r12

C

r12

C

r12

C

r12

C

r12

CW

Recall we have defined ind1

orient1

p1

disp1

d1 )(2W d

1p1

C1

We can approximate

d2

d1

p2

p1

A12 22W if 5.24.0

ind1

orient1

5.2u

u4.0

2202

2101

&

we have 212

12 rrr

5.24.0ind2

orient2

&

19JCheng201211

Assumptions in Previous Derivation[Cha1996, p. 105]

First of all, The surface free energies (& components) are internal energy function

s; all attempts to derive a relationship between the work of adhesion an

d surface free energies are based on interaction models that ignore entropy.

Plus assumptions in computation of interaction energies

All initial theories of interfacial interactions assumed a pairwise additivity rule for intermolecular potential

The problem is not too severe for higher frequency dispersion interactions.

But for zero frequency interaction, such as dipole-dipole interactions, random orientations of dipoles cancel each other's field, which reduces the interaction energy substantially from the value calculated from the pairwise additivity rule. (To understand, we need to review Lifshitz theory)

Thus, the above derived theory can be safely applied to non-polar molecules but care should be taken when polar molecules are involved.

20JCheng201211

Some Footnotes

The geometric average approach implies that interfacial free energy is always 0

For dispersion energy, harmonic average can be applied when polarizabilities of both molecules are the same

This is probably the reason why Wu’s harmonic average approach can predict adhesion work better when organic polymers, which are generally non-polar, are involved.

However, when polarizabilities are dissimilar, it may be better to still adopt the geometric mean approach

21JCheng201211

References

Chaudhury, M. K., “Interfacial interaction between low-energy surfaces,” Materials Science and Engineering, R16, 97-159, 1996

Woodward, R. P., “Prediction of Adhesion and Wetting from Lewis Acid Base Measurements (as presented at TPOs of Automotive 2000),” from First Ten Angstroms, Inc., 2000