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doi: 10.3934/dcdss.2022035
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## Undulated bilayer interfaces in the planar functionalized Cahn-Hilliard equation

 1 Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, USA 2 Department of Mathematics, Ohio University, Morton Hall 321, 1 Ohio University, Athens OH 45701, USA

* Corresponding author: Qiliang Wu

Received  April 2021 Revised  January 2022 Early access February 2022

Fund Project: K. Promislow acknowledges support from NSF-DMS grant 1813203. Q. Wu acknowledges support from NSF-DMS grant 1815079

Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, [8]. Undulated bilayers are characterized by oscillatory perturbations of the bilayer width, which decay on a spatial length scale that is long compared to the bilayer width. We mimic defects within the functionalized Cahn-Hillard free energy by introducing spatially localized inhomogeneities within its parameters. For length parameter $\varepsilon\ll1$, we show that this induces undulated bilayer solutions whose width perturbations decay on an $O\!\left( \varepsilon^{-1/2}\right)$ inner length scale that is long in comparison to the $O(1)$ scale that characterizes the bilayer width.

Citation: Keith Promislow, Qiliang Wu. Undulated bilayer interfaces in the planar functionalized Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022035
##### References:
 [1] R. Choksi, Partial Differential Equations: A First Course, American Mathematical Society, Providence, RI, 2021. [2] A. Christlieb, N. Kraitzman and K. Promislow, Competition and complexity in amphiphilic polymer morphology, Phys. D, 400 (2019), 132144, 20 pp. doi: 10.1016/j.physd.2019.06.010. [3] A. Doelman, G. Hayrapetyan, K. Promislow and B. Wetton, Meander and pearling of single-curvature bilayer interfaces in the functionalized Cahn–Hilliard equation, SIAM J. Math. Anal., 46 (2014), 3640-3677.  doi: 10.1137/13092705X. [4] G. Faye and A. Scheel, Center manifolds without a phase space, Trans. Amer. Math. Soc., 370 (2018), 5843-5885. [5] G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119. [6] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. [7] S. Jain and F. Bates, On the origins of morphological complexity in block copolymer surfactants, Science, 300 (2003), 460-464. [8] S. Jain and F. Bates, Consequences of nonergodicity in aqueous binary PEO-PB micellar dispersions, Macromolecules, 37 (2004), 1511-1523. [9] A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3 (1967), 546-570. [10] M. Matsen and M. Schick, Stable and unstable phases of a diblock copolymer melt, Phys. Rev. Lett., 72 (1994), 2660-2663. [11] V. Pliss, A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR Ser. Mat., 28 (1964), 1297–1324 (Russian). [12] K. Promislow and B. Wetton, PEM fuel cells: A mathematical overview, SIAM J. Appl. Math., 70 (2009), 369-409.  doi: 10.1137/080720802. [13] K. Promislow and Q. Wu, Existence of pearled patterns in the planar functionalized Cahn-Hilliard equation, J. Differential Equations, 259 (2015), 3298-3343.  doi: 10.1016/j.jde.2015.04.022. [14] K. Promislow and L. Yang, Existence of compressible bilayers in the functionalized Cahn–Hilliard equation, SIAM J. Appl. Dyn. Syst., 13 (2014), 629-657.  doi: 10.1137/130931060. [15] M. Teubner and R. Strey, Origin of scattering peaks in microemulsions, J. Chem. Phys., 87 (1987), 3195-3200. [16] M. Teubner and R. Strey, Fluctuating interfaces in microemulsions and sponge phases, Phys. Rev. E, 50 (1994), 1325-1335. [17] A. Vanderbauwhede and G. Iooss, Center Manifold Theory in Infinite Dimensions, Dynamics Reported: Expositions in dynamical Systems, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 1, Springer, Berlin, 1992,125–163.,

show all references

##### References:
 [1] R. Choksi, Partial Differential Equations: A First Course, American Mathematical Society, Providence, RI, 2021. [2] A. Christlieb, N. Kraitzman and K. Promislow, Competition and complexity in amphiphilic polymer morphology, Phys. D, 400 (2019), 132144, 20 pp. doi: 10.1016/j.physd.2019.06.010. [3] A. Doelman, G. Hayrapetyan, K. Promislow and B. Wetton, Meander and pearling of single-curvature bilayer interfaces in the functionalized Cahn–Hilliard equation, SIAM J. Math. Anal., 46 (2014), 3640-3677.  doi: 10.1137/13092705X. [4] G. Faye and A. Scheel, Center manifolds without a phase space, Trans. Amer. Math. Soc., 370 (2018), 5843-5885. [5] G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119. [6] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. [7] S. Jain and F. Bates, On the origins of morphological complexity in block copolymer surfactants, Science, 300 (2003), 460-464. [8] S. Jain and F. Bates, Consequences of nonergodicity in aqueous binary PEO-PB micellar dispersions, Macromolecules, 37 (2004), 1511-1523. [9] A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3 (1967), 546-570. [10] M. Matsen and M. Schick, Stable and unstable phases of a diblock copolymer melt, Phys. Rev. Lett., 72 (1994), 2660-2663. [11] V. Pliss, A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR Ser. Mat., 28 (1964), 1297–1324 (Russian). [12] K. Promislow and B. Wetton, PEM fuel cells: A mathematical overview, SIAM J. Appl. Math., 70 (2009), 369-409.  doi: 10.1137/080720802. [13] K. Promislow and Q. Wu, Existence of pearled patterns in the planar functionalized Cahn-Hilliard equation, J. Differential Equations, 259 (2015), 3298-3343.  doi: 10.1016/j.jde.2015.04.022. [14] K. Promislow and L. Yang, Existence of compressible bilayers in the functionalized Cahn–Hilliard equation, SIAM J. Appl. Dyn. Syst., 13 (2014), 629-657.  doi: 10.1137/130931060. [15] M. Teubner and R. Strey, Origin of scattering peaks in microemulsions, J. Chem. Phys., 87 (1987), 3195-3200. [16] M. Teubner and R. Strey, Fluctuating interfaces in microemulsions and sponge phases, Phys. Rev. E, 50 (1994), 1325-1335. [17] A. Vanderbauwhede and G. Iooss, Center Manifold Theory in Infinite Dimensions, Dynamics Reported: Expositions in dynamical Systems, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 1, Springer, Berlin, 1992,125–163.,
Cryo-TEM images of blends of amphiphilic diblock polymer in water. A mixture of diblocks with hydrophobic/hydrophillic chain lengths of 170/110 and 46/58, respectively. This mixture produces visible undulations behind the endcaps, see the red boxes outlining the structures labeled '2' and '3'. Reprinted with permission from Figures 7 & 8 of [8]
The operator $\mathbb{L}( \varepsilon)$ admits two purely imaginary eigenvalues (blue crosses) with algebraic multiplicity 2 when $\varepsilon = 0.$ Given $\varepsilon>0$, for $\alpha_0>0$ they remain purely imaginary while for $\alpha_0<0$ they split into four geometrically simple modes (red crosses)
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