Abstract
Methods to derive an interface concentration profile in a two component system are discussed on the basis of squared gradient theories. Starting point is the description of soft matter systems, where the correlation length of fluctuations becomes the relevant length scale. A phase diagram which contains bulk and interface phase transitions is used as a road map to the involved phenomena. The Landau theory and the Flory-Huggins theory as a typical representative of soft matter mean field theories are outlined as motivations for the squared gradient approach. A brief discussion of bulk properties forms the basis for the discussion of the interface profile of a two component system and the wetting behavior of this system at a substrate.
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Notes
- 1.
Strobl uses in his book the term free enthalpy instead of free energy, which is employed here. In order to describe experiments theoretically, the free enthalpy would be more appropriate, since experiments are in most cases performed at constant pressure, not at constant volume. However, the Flory-Huggins theory contains the volume V as explicit parameter, not the pressure P. Therefore, it is formally a free energy, not a free enthalpy. Since the volume change of polymer blends upon mixing or heating is usually negligible, there is basically no difference between the free energy and the free enthalpy.
- 2.
The formal difference between the total free energy density in (8.3) and the free energy increment to a constant background is of minor importance, as any calculation is based on derivatives of the free energy, where any constant shift of the energy scale cancels out.
- 3.
For a finite scattering volume the fluctuations form a Fourier series, not a Fourier integral transformation.
- 4.
For data gained from a real experiment, the measured intensity needs to be corrected by subtracting the background scattering of the solvent.
- 5.
- 6.
An excellent discussion of thermal excitation and time correlation function as described by a Langevin equation is found in the book of Doi and Edwards [19].
- 7.
In this step, a \(\phi \) dependence of the elastic constant \(\kappa \) would lead to an additional term, which does not fit to the following manipulations in a simple way.
- 8.
For a neutral substrate with \(\gamma _{\mathrm{{AS}}}=\gamma _{\mathrm{{BS}}}\) and thus no preferential adsorption, the substrate would have no effect on the sample. This boring case does not need further discussion.
- 9.
The delta function is also briefly discussed in the appendix.
- 10.
- 11.
The mechanism of such a continuous transition is different from the discussion of Bonn and Ross [4]. They investigate conditions for the contact composition which could lead to a continuous transition based on a graphical method eqivalent to (8.55) with an additional, \(\phi \) dependent term on the right side. Based on a discussion of this slope of the right side, they identify conditions for the contact composition where the wetting transition becomes continuous. They find continuous transitions at a higher temperature than \(T_{\mathrm{{pw}}}\). In the discussion here, in contrast, the reason for continuous wetting is a calculated value \(|\cos (\varTheta _{\mathrm{{a}}})|>1\) at \(T_{\mathrm{{pw}}}\), which leads to continuous wetting at lower temperature than \(T_{\mathrm{{pw}}}\). The discussion of Bonn and Ross does not include a test of the magnitude of \(|\cos (\varTheta _{\mathrm{{a}}})|\) resulting from their derived boundary values.
- 12.
Due to the linearity of (8.59), \(\phi '\) can simply replace \(\phi \) in all derivatives, e.g. \(\frac{\mathrm{{d}}\phi }{{\mathrm{d}}z}=\frac{\mathrm{{d}}\phi '}{{\mathrm{d}}z}\) or \(\frac{\mathrm{{d}}\omega }{{\mathrm{d}}\phi }=\frac{\mathrm{{d}}\omega }{{\mathrm{d}}\phi '}\). In order to apply the equations in Sect. 8.5.1 with \(\phi '\) instead of \(\phi \), one could write (8.43) with a new function \(\tilde{\omega }(\phi ')=\omega (\phi _{\mathrm{{c}}}+\phi ')\), and repeat all derivations of Sect. 8.5.1 with \(\tilde{\omega }(\phi ')\) instead of \(\omega (\phi )\). In order to keep the notation traceable, we use the same symbol and write sloppily \(\omega (\phi ')\) for \(\tilde{\omega }(\phi ')\).
- 13.
For a quick and dirty check, one can plot (8.93) with example values \(\phi _{\mathrm{{a}}}'=-1\), \(\phi _{\mathrm{{b}}}'=+1\), and \(\phi _0''=-1\).
- 14.
For the integration of the profile, use a partial fraction decomposition of (8.103)
$$\begin{aligned} \frac{2(\vartheta +1)Z_{\mathrm{{red}}}}{Z_{\mathrm{{red}}}^2-Z_{\mathrm{{red}}}-\vartheta /4}=\frac{\sqrt{\vartheta +1}(1+\sqrt{\vartheta +1})}{Z_{\mathrm{{red}}}-\frac{1}{2}-\frac{1}{2}\sqrt{\vartheta +1}}-\frac{\sqrt{\vartheta +1}(1-\sqrt{\vartheta +1})}{Z_{\mathrm{{red}}}-\frac{1}{2}+\frac{1}{2}\sqrt{\vartheta +1}}{.} \end{aligned}$$.
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Acknowledgments
The author thanks Helgard Sigel for her hospitality during summer 2014 in Markdorf, when the first part of this contribution was written.
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Sigel, R. (2016). Interfaces of Binary Mixtures. In: Lang, P., Liu, Y. (eds) Soft Matter at Aqueous Interfaces. Lecture Notes in Physics, vol 917. Springer, Cham. https://doi.org/10.1007/978-3-319-24502-7_8
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