Soft Matter at Aqueous Interfaces by Peter Lang & Yi Liu

Author:Peter Lang & Yi Liu
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

9.1.4 Scaling and Blobs

Let us now consider the example of a chain in a slit in more detail. Since only the scaling variable can control the physics we can write

(9.10)

where denotes the (average) extension of the chain in the lateral direction and R 0 denotes again the extension of the free chain. The function of the scaling variable, , is called cross-over or scaling function. For , (very wide slits) we have , and . More interesting is the opposite case where the chain is squeezed in a narrow silt, see Fig. 9.3. In this case the chain is two-dimensional for and we expect: using Eqs. (9.8, 9.9). But how can we smoothly change from the behavior to This is only possible if the cross-over function displays a power-law behavior in the limit 1 of the general form with an unknown exponent m. Substituting this into Eq. (9.10) we arrive at: with the solution . We can conclude that the real chain in good solvent expands laterally due to compression according to

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