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

Keith Promislow 1, and  Qiliang Wu 2,,

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.

Keywords: Functionalized Cahn-Hilliard, defects, dumb-bell, spatial dynamics, invariant manifold.
Mathematics Subject Classification: Primary: 35B36, 35Q92, 37L10.
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.,

Figure 1.  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]
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Figure 2.  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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