W. Richard Bowen and Nidal Hilal 4
W. Richard Bowen and Nidal Hilal 4
W. Richard Bowen and Nidal Hilal 4
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8.3 ENd-gRAFTEd POLyMER CHAINS 235<br />
with a spherical probe: F(D) � exp(�2�D/L 0), where F(D) is the force<br />
exerted on the probe by the brush at a separation D between the probe<br />
surface <strong>and</strong> the solid substrate <strong>and</strong> L 0 is the brush thickness.<br />
We can directly check this equation on our data. We fitted several force<br />
profiles with the above equation <strong>and</strong> we obtained a brush thickness in the<br />
range of 100 nm, which is of the same order (albeit a bit larger) of the distance<br />
D, where we observe in force profiles the initial increase of the force,<br />
signifying the extent (thickness) of the brush away from the surface. The<br />
discrepancy could arise from the fact that since our probe is small, a large<br />
proportion of the chains lie near the edge <strong>and</strong> therefore are expected to<br />
bend than to compress. A more appropriate theory must take into account<br />
the spherical shape of the tip <strong>and</strong> the lateral bending of the chains.<br />
The second type of force profile is shown in Figure 8.9(a). The force<br />
profile during probe-tip approach is essentially the same: gradual increase<br />
of the loading force due to gradual compression of the brush. However,<br />
during the reverse force profile as the tip moves away from the surface,<br />
we observe some long-range attractive forces. The full length of one chain<br />
should be around 120 nm (calculated using the number-average molecular<br />
weight M n), which is consistent with the stretching events occurring up<br />
to approximately 200 nm. The magnitude of the attractive forces is in the<br />
range of 0.1–0.5 nN. Forces in this range can be attributed to physisorption<br />
of several monomers to a surface. Since we observe a gradual increase in<br />
the first derivative of these forces, we attribute them on parts of chains,<br />
individual chains <strong>and</strong>/or small clusters of chains that were adhered<br />
non-specifically to the tip <strong>and</strong> subsequently stretched during the reverse<br />
movement. The continuous lines in Figure 8.9 correspond to the entropic<br />
force for a single polymer chain using the freely jointed chain (FJC) model<br />
with full length approximately equal to the distance between the contact<br />
F (nN)<br />
1.5<br />
1.0<br />
0.5<br />
0.0<br />
Forward<br />
Reverse<br />
Theory<br />
–0.5<br />
50<br />
0 50 100<br />
D (nm)<br />
150<br />
(a) (b)<br />
D (nm)<br />
80<br />
70<br />
60<br />
Non-linear elasticity<br />
0.00 0.05 0.10 0.15<br />
F (nN)<br />
FIgurE 8.9 (a) Second type of force curve observed over a polymer brush. The continuous<br />
lines correspond to the theoretical prediction for stretching of individual chains or<br />
parts of a chain. (b) Zoom of one of the long-range attractive force curves.