Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, [
Citation: |
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]
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)
[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