Chee 4602010 Lecture 4

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CHEE 460/CHEM 347 Lecture 4 1

Scaling of van der Waals Interactions

Reading assignment: Textbook, sections 10.4b-end of chapter (pp.

477-495)

Recommended reading (optional): Israelachvili, J.,Intermolecular

and Surface Forces 1992, 2nded., Academic Press, Chapters 10

(pp.152-168) & 11 (pp. 176- 192)

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Hamakers Approach

Highlights:

All interactions between bodies and surfaces can be estimated through the

summation of the pairwise combinations of the interactions between theatoms/molecules of which there are composed

For example, for two interacting bodies 1 and 2 with volumes and

densities V1, V2, and r1, r2, respectively, the total van der Waals energy of

interaction per unit surface area can be found through:

(1)=2

216

21

1

// )(

VV

vdWdVdVCW

ll

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Hamakers Approach -Assumptions

It should be stressed that the Hamakers approach is based on the

following assumptions:

Only pairwise interactions are considered, i.e., many-body interactionsare ignored

The interactions are instantaneous

The interacting bodies have uniform densities throughout

The interactions do not alter the original shape of the bodies

The medium is vacuum

The dispersion (London) interactions occur at a single frequency

Permanent dipole and free charge effects are ignored as negligible

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Hamakers Approach Integration

An example of the integrations involved in the calculation of the van

der Waals interaction energy (WvdW) will be given in class for the

case of two semi-infinite blocks (textbook, pp.483-484; see Figure 1)

Figure 1: Interactions between (a) a molecule and a block of material (b) between two blocks of material [1]

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Hamakers Approach Integration result

Considering only pairwise interactions, the resulting van der Waals

interaction between two macroscopic flat surfaces is given by:

(2)

Units: [Energy/unit surface area] Where, l [= m] is the surface-surface distance between the two

interacting blocks and A is the Hamaker constant [=J]

2

//12

)( = ll

AW

vdW

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Hamakers Approach Other Geometries

Figure 2: Summary of formulas for calculating van der Waals interactions between bodies of different geometries. The

equations have been derived on the assumption of pairwise additivity [2]

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Intermolecular vs. Macroscopic Interactions

From the above, the following conclusions can be drawn:

The interaction energy between macroscopic bodies decays much more

slowly with distance (r-1 for spheres and r-2 for surfaces vs. r-6betweenmolecules)

The interactions between macroscopic bodies (spheres, cylinders) are

linear functions of their physical dimension, R (exception: infinite flat

surfaces). Interactions involving molecules with diameter larger than 1.0 nm must

be calculated as macroscopic (or, else, they will be underestimated; see

[2], pg. 160)

The effects of the interaction forces between two (same) individual

molecules and two particles made up of these molecules may be different

(stable vs. unstable suspension; see Fig 3)

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Intermolecular vs. Macroscopic (contd)

Intermolecular and microscopic forces for the same materials can

have different effects, leading to, e.g., stable vs. unstable suspensions

Figure 3: Comparison of interaction potentials and resulting (de)stabilization effects between

molecules and microscopic particles (pg. 154 [2])

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Hamakers Approach -Remarks

This scaling approach is an over-simplification that assumes pairwise

additive interactions (for examples, molecules near the surfaces of two

bodies will screen the interactions between molecules in the bulk of these

two materials)

As we will see soon, a popular approach to calculating the interactions

between two bodies is based on, not molecular parameters, but rather

measurements of bulk or surface material properties (refractive index,

dielectric constants, surface tension/energy, etc.)

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The Derjaguin approximation

The evaluation of two volume integrals is relatively straightforward

for simple geometries, but rather cumbersome (or, computationally

intensive) when complicated geometries are considered The Derjaguin approximation is an approach that can simplify the

calculations in the case we wish to calculate the interaction energy

between two curved bodies

It must be emphasized that the Derjaguin approximation is validonlywhen R1 & R2 >> l (l: surface-to-surface separation)

Here, one will have to integrate the interaction energy per unit area

between two infinite planar surfaces (see semi-infinite blocks) over

the surfaces of the interacting particles

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The Derjaguin approximation (contd)

In the general case, the mathematical expression to be used is of the form:

(3)

The product is a function of the principal radii of the interacting

surfaces

If we assume R1, R2 to be the principal radii of curvature of body 1andR1, R2 the principal radii of curvature of body 2, then:

(4)

llll

dWWo

vdWvdW

= ))((2

)( //21

21

+

+

+=

'

2

'

121

2

'

22

'

11

21

1111sin

1111

RRRRRRRR

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Derjaguin Approximation -Integration

By integrating Equation (4) we get the general expression:

(5)

For two interacting sphere of radius R, we have:

(6)

Similarly, for a sphere of radius R and an infinite flat plate (1/R2

0):

(7)

ll 1

6)(

21AWvdW =

ll

12)(

ARW

vdW

oo =

ll

6)(/

ARW

vdW

o =

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Derjaguin Approximation -Remarks

If one of the spheres is much larger than the other, Eqn. 3 allows the

calculation of the force between a sphere and a flat plate (limiting

case) For two equal spheres, the calculated force is half of that developed

between a sphere and a flat plate

The variation of the calculated force with distance can be totally

different between two interacting flat plates and two spheres (See Fig.4). In other words, two spheres may repel each other while two flat

surfaces (made of the same material) attract!!

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Force vs. Distance: Effect of surface curvature

Figure 4: Force vs. separation between two curved surfaces and two flat surfaces See [2], pg. 164)

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Example 1

Derive and expression for the force vs. distance that develops

between 2 cylinders (R1, R2) crossed at angle .

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References

[1] Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface

Chemistry 1997, 3rd Ed., Marcel Dekker, Inc.

[2] Israelachvili, J.Intermolecular and Surface Forces 1992, 2nd Ed.,Academic Press.

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Chee 4602010 Lecture 4