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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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CHEE 460/CHEM 347 Lecture 4 8
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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CHEE 460/CHEM 347 Lecture 4 10
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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CHEE 460/CHEM 347 Lecture 4 11
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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