W. Richard Bowen and Nidal Hilal 4

8.3 ENd-gRAFTEd POLyMER CHAINS 235
with a spherical probe: F(D) � exp(�2�D/L 0), where F(D) is the force
exerted on the probe by the brush at a separation D between the probe
surface and the solid substrate and L 0 is the brush thickness.
We can directly check this equation on our data. We fitted several force
profiles with the above equation and we obtained a brush thickness in the
range of 100 nm, which is of the same order (albeit a bit larger) of the distance
D, where we observe in force profiles the initial increase of the force,
signifying the extent (thickness) of the brush away from the surface. The
discrepancy could arise from the fact that since our probe is small, a large
proportion of the chains lie near the edge and therefore are expected to
bend than to compress. A more appropriate theory must take into account
the spherical shape of the tip and the lateral bending of the chains.
The second type of force profile is shown in Figure 8.9(a). The force
profile during probe-tip approach is essentially the same: gradual increase
of the loading force due to gradual compression of the brush. However,
during the reverse force profile as the tip moves away from the surface,
we observe some long-range attractive forces. The full length of one chain
should be around 120 nm (calculated using the number-average molecular
weight M n), which is consistent with the stretching events occurring up
to approximately 200 nm. The magnitude of the attractive forces is in the
range of 0.1–0.5 nN. Forces in this range can be attributed to physisorption
of several monomers to a surface. Since we observe a gradual increase in
the first derivative of these forces, we attribute them on parts of chains,
individual chains and/or small clusters of chains that were adhered
non-specifically to the tip and subsequently stretched during the reverse
movement. The continuous lines in Figure 8.9 correspond to the entropic
force for a single polymer chain using the freely jointed chain (FJC) model
with full length approximately equal to the distance between the contact
F (nN)
1.5
1.0
0.5
0.0
Forward
Reverse
Theory
–0.5
50
0 50 100
D (nm)
150
(a) (b)
D (nm)
80
70
60
Non-linear elasticity
0.00 0.05 0.10 0.15
F (nN)
FIgurE 8.9 (a) Second type of force curve observed over a polymer brush. The continuous
lines correspond to the theoretical prediction for stretching of individual chains or
parts of a chain. (b) Zoom of one of the long-range attractive force curves.