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Analysis of a perturbed Cahn–Hilliard model for Langmuir–Blodgett films

Abstract

An advective Cahn–Hilliard model motivated by thin film formation is studied in this paper. The one-dimensional evolution equation under consideration includes a transport term, whose presence prevents from identifying a gradient flow structure. Existence and uniqueness of solutions, together with continuous dependence on the initial data and an energy equality are proved by combining a minimizing movement scheme with a fixed point argument. Finally, it is shown that, when the contribution of the transport term is small, the equation possesses a global attractor and converges, as the transport term tends to zero, to a purely diffusive Cahn–Hilliard equation.

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References

  1. 1.

    Abels, H., Depner, D., Garcke, H.: On an incompressible Navier–Stokes/Cahn–Hilliard system with degenerate mobility. Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 1175–1190 (2013)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Abels, H., Röger, M.: Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 2403–2424 (2009)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Akagi, G.: Global attractors for doubly nonlinear evolution equations with non-monotone perturbations. J. Differ. Equ. 250, 1850–1875 (2011)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ambrosio, L.: Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19(5), 191–246 (1995)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser, New York (2000)

    MATH  Google Scholar 

  6. 6.

    Barbu, V.: Analysis and Control of Nonlinear Infinite-Dimensional Systems. Mathematics in Science and Engineering, vol. 190. Academic Press Inc, Boston (1993)

    MATH  Google Scholar 

  7. 7.

    Blodgett, K.B.: Films built by depositing successive monomolecular layers on a solid surface. J. Am. Chem. Soc. 57(6), 1007–1022 (1935)

    Article  Google Scholar 

  8. 8.

    Boyer, F.: Nonhomogeneous Cahn–Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 225–259 (2001)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cahn, J.W., Hilliard, J.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

    Article  Google Scholar 

  10. 10.

    Chen, X.D., Lenhert, S., Hirtz, M., Lu, N., Fuchs, H., Chi, L.F.: Langmuir–Blodgett patterning: a bottom-up way to build mesostructures over large areas. Acc. Chem. Res. 40, 393–401 (2007)

    Article  Google Scholar 

  11. 11.

    Cherfils, L., Miranville, A., Zelik, S.: The Cahn–Hilliard equation with logarithmic potentials. Milan J. Math. 79, 561–596 (2011)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Colli, P., Krejc̆í, P., Rocca, E., Sprekels, J.: Nonlinear evolution inclusions arising from phase change models. Czechoslovak Math. J. 57(132), 1067–1098 (2007)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Colli, P., Gilardi, G., Sprekels, J.: On an application of Tikhonov?s fixed point theorem to a nonlocal Cahn–Hilliard type system modeling phase separation. J. Differ. Equ. 260, 7940–7964 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Colli, P., Gilardi, G., Sprekels, J.: On a Cahn–Hilliard system with convection and dynamic boundary conditions. Ann. Mat. Pura Appl. 197, 1445–1475 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Della Porta, F., Grasselli, M.: Convective nonlocal Cahn–Hilliard equations with reaction terms. Discret. Contin. Dyn. Syst. Ser. B 20, 1529–1553 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Della Porta, F., Grasselli, M.: On the nonlocal Cahn–Hilliard–Brinkman and Cahn–Hilliard–Hele–Shaw systems. Commun. Pure Appl. Anal. 15, 299–317 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Eden, A., Kalantarov, V.K., Zelik, S.V.: Global solvability and blow up for the convective Cahn–Hilliard equations with concave potentials. J. Math. Phys. 54, 041502 (2013)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Ei, S.-I.: The effect of nonlocal convection on reaction–diffusion equations. Hiroshima Math. J. 17, 281–307 (1987)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: \(L^p\) Spaces. Springer, New York (2007)

    MATH  Google Scholar 

  20. 20.

    Frank, F., Liu, C., Alpak, F.O., Riviere, B.: A finite volume/discontinuous Galerkin method for the advective Cahn–Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging. Comput. Geosci. 22, 543–563 (2018)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Frigeri, S., Grasselli, M.: Global and trajectory attractors for a nonlocal Cahn–Hilliard–Navier–Stokes system. J. Dyn. Differ. Equ. 24, 827–856 (2012)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Frigeri, S., Grasselli, M., Rocca, E.: A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility. Nonlinearity 28, 1257–1293 (2015)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Gal, C.G., Grasselli, M.: Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 401–436 (2010)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Gal, C.G., Grasselli, M.: Longtime behavior of nonlocal Cahn–Hilliard equations. Discret. Contin. Dyn. Syst. 34, 145–179 (2014)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Grasselli, M., Miranville, A., Rossi, R., Schimperna, G.: Analysis of the Cahn–Hilliard equation with a chemical potential dependent mobility. Commun. Partial Differ. Equ. 36, 1193–1238 (2011)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Amer. Math. Soc, Providence, RI (1988)

    MATH  Google Scholar 

  27. 27.

    Ignat, L.I., Rossi, J.D.: A nonlocal convection–diffusion equation. J. Funct. Anal. 251, 399–437 (2007)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Iuorio, A., Melchionna, S.: Long-time behavior of a nonlocal Cahn–Hilliard equation with reaction. Discret. Contin. Dyn. Syst. 38, 3765–3788 (2018)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Köpf, M.H., Gurevich, S.V., Friedrich, R., Thiele, U.: Substrate-mediated pattern formation in monolayer transfer: a reduced model. New J. Phys. 142, 023016 (2012)

    Article  Google Scholar 

  30. 30.

    Köpf, M.H., Gurevich, S.V., Friedrich, R., Chi, L.: Pattern formation in monolayer transfer systems with substrate-mediated condensation. Langmuir 26(13), 10444–10447 (2010)

    Article  Google Scholar 

  31. 31.

    Köpf, M.H., Thiele, U.: Emergence of the bifurcation structure of a Langmuir–Blodgett transfer model. Nonlinearity 27(11), 2711–2734 (2014)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Ladyzhenskaya, O.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  33. 33.

    Langmuir, I.: The constitution and fundamental properties of solids and liquids. II. Liquids. 1. J. Am. Chem. Soc. 39(9), 1848–1906 (1917)

    Article  Google Scholar 

  34. 34.

    Leoni, G.: A First Course in Sobolev Spaces, 2nd edn. American Mathematical Society, Providence, RI (2017)

    Book  Google Scholar 

  35. 35.

    Liu, W., Bertozzi, A., Kolokolnikov, T.: Diffuse interface surface tension models in an expanding flow. Commun. Math. Sci. 10, 387–418 (2012)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Liu, J., Dedé, L., Evans, J.A., Borden, M.J., Hughes, T.J.R.: Isogeometric analysis of the advective Cahn–Hilliard equation: spinodal decomposition under shear flow. J. Comput. Phys. 242, 321–350 (2013)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Miranville, A., Schimperna, G.: On a doubly nonlinear Cahn–Hilliard–Gurtin system. Discret. Contin. Dyn. Syst. Ser. B 14, 675–697 (2010)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Novick-Cohen, A.: The Cahn–Hilliard equation. Handbook of differential equations: evolutionary equations. Vol. IV, 201–228, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008)

    Chapter  Google Scholar 

  39. 39.

    Ó Náraigh, L.: The role of advection in phase-separating binary liquids. Ph.D. thesis, (2008)

  40. 40.

    Oliveira Jr., O.L.: Langmuir–Blodgett films—properties and possible applications. Braz. J. Phys. 22(2), 60–69 (1992)

    Google Scholar 

  41. 41.

    Purrucker, O., Förtig, A., Lüdtke, K., Jordan, R., Tanaka, M.: Confinement of transmembrane cell receptors in tunable stripe micropatterns. J. Am. Chem. Soc. 127, 1258–1264 (2005)

    Article  Google Scholar 

  42. 42.

    Rocca, E., Sprekels, J.: Optimal distributed control of a nonlocal convective Cahn–Hilliard equation by the velocity in three dimensions. SIAM J. Control Optim. 53, 1654–1680 (2015)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Roberts, G. (ed.): Langmuir–Blodgett Films. Springer, New York (1990)

    Google Scholar 

  44. 44.

    Schimperna, G.: Global attractors for Cahn–Hilliard equations with nonconstant mobility. Nonlinearity 20, 2365–2387 (2007)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Watson, S.J., Otto, F., Rubinstein, B.Y., Davis, S.H.: Coarsening dynamics of the convective Cahn–Hilliard equation. Phys. D 178, 127–148 (2003)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Wilczek, M., Gurevich, S.V.: Locking of periodic patterns in Cahn–Hilliard models for Langmuir–Blodgett transfer. Phys. Rev. E 90, 042926 (2014)

    Article  Google Scholar 

  47. 47.

    Zasadzinski, J.A., Viswanathan, R., Madsen, L., Garnaes, J., Schwartz, D.K.: Langmuir–Blodgett films. Science 263, 1726–1733 (1994)

    Article  Google Scholar 

  48. 48.

    Zhu, J., Wilczek, M., Hirtz, M., Hao, J., Wang, W., Fuchs, H., Gurevich, S.V., Chi, L.: Branch suppression and orientation control of Langmuir–Blodgett patterning on prestructured surfaces. Adv. Mater. Interfaces 3, 1600478 (2016)

    Article  Google Scholar 

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Acknowledgements

The authors thank the hospitality of the departments of mathematics of the universities of Heidelberg and Vienna, of SISSA, and of the E. Schrödinger Institute in Vienna, where this research was developed. The authors are members of the GNAMPA group of INdAM. This work was partially supported by the 2015 GNAMPA project Fenomeni Critici nella Meccanica dei Materiali: un Approccio Variazionale. E.D. acknowledges the support of the Austrian Science Fund (FWF) project P 27052 and of the SFB project F65 Taming complexity in partial differential systems. The research of M.M. was partially supported by grant FCT-UTA_CMU/MAT/0005/2009 Thin Structures, Homogenization, and Multiphase Problems, by the ERC Advanced grant Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture (Grant agreement no.: 290888), by the ERC Starting grant High-Dimensional Sparse Optimal Control (Grant agreement no.: 306274), and by the DFG Project Identifikation von Energien durch Beobachtung der zeitlichen Entwicklung von Systemen (FO 767/7). Finally, the authors thank Stefano Melchionna for fruitful discussions.

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Appendix A

Appendix A

We collect here a few results of technical nature which are needed throughout the paper. The following lemma will be instrumental in the proof of Proposition 4.4.

Lemma A.1

Let \(q\in L^2(0,T;V)\) and let \(\tau :=\frac{T}{N}\) for \(N\in {\mathbb {N}}\). Then, as \(\tau \rightarrow 0\),

$$\begin{aligned} \frac{1}{\tau }\sum _{k=1}^N \chi _{((k-1)\tau ,k\tau )}(t) \int _{(k-1)\tau }^{k\tau } q'(s)\,\mathrm ds \rightarrow q'(t) \qquad \text {weakly in } L^2(0,T;L^2(0,L)). \end{aligned}$$
(A.1)

Proof

We set

$$\begin{aligned} g_\tau (x,t):=\frac{1}{\tau }\sum _{k=1}^N \chi _{((k-1)\tau ,k\tau )}(t) \int _{(k-1)\tau }^{k\tau } \bigl ( q'(x,t)-q'(x,s) \bigr )\,\mathrm ds. \end{aligned}$$

We can estimate

$$\begin{aligned} |g_\tau (x,t)|^2&= \frac{1}{\tau ^2} \sum _{k=1}^N \chi _{((k-1)\tau ,k\tau )}(t) \bigg | \int _{(k-1)\tau }^{k\tau } \bigl ( q'(x,t)-q'(x,s) \bigr )\,\mathrm ds \bigg |^2 \\&\leqslant \frac{1}{\tau } \sum _{k=1}^N \chi _{((k-1)\tau ,k\tau )}(t) \int _{(k-1)\tau }^{k\tau } |q'(x,t)-q'(x,s)|^2 \,\mathrm ds \\&\leqslant 2|q'(x,t)|^2 + \frac{2}{\tau } \sum _{k=1}^N \chi _{((k-1)\tau ,k\tau )}(t) \int _{(k-1)\tau }^{k\tau } |q'(x,s)|^2 \,\mathrm ds\,, \end{aligned}$$

so that by integrating on \((0,L)\times (0,T)\) we obtain

$$\begin{aligned} \int _0^T\int _0^L |g_\tau (x,t)|^2\,\mathrm dx\mathrm dt \leqslant 4\int _0^T\int _0^L |q'(x,t)|^2\,\mathrm dx \mathrm dt =:M <\infty . \end{aligned}$$

This implies that, up to subsequences, \(g_\tau \rightharpoonup g\) weakly in \(L^2((0,L)\times (0,T))\).

We shall now show that \(g=0\). The estimate

$$\begin{aligned} |g_\tau (x,t)| \leqslant \frac{1}{\tau } \int _{t-\tau }^{t+\tau } \big | q'(x,t)-q'(x,s) \big | \,\mathrm ds \end{aligned}$$

implies that \(\Vert g_\tau (t)\Vert _{L^2(0,L)}\rightarrow 0\) for almost every \(t\in (0,T)\) by [34, Theorem 8.19]; in turn, given \(\varepsilon >0\), by Egorov’s Theorem we can find a set \(A_\varepsilon \subset (0,T)\) with measure smaller than \(\varepsilon \) such that \(\Vert g_\tau (t)\Vert _{L^2(0,L)}\rightarrow 0\) uniformly on \((0,T)\setminus A_\varepsilon \). Therefore for any test function \(\varphi \in L^2((0,L)\times (0,T))\) we have

$$\begin{aligned} \limsup _{\tau \rightarrow 0}\bigg |\int _0^T\int _0^L g_\tau (x,t)\varphi (x,t)\,\mathrm dx\mathrm dt \bigg |&\leqslant \limsup _{\tau \rightarrow 0} \int _0^T \Vert g_\tau (t)\Vert _{L^2(0,L)} \Vert \varphi (t)\Vert _{L^2(0,L)} \,\mathrm dt\\&= \limsup _{\tau \rightarrow 0} \int _{A_\varepsilon } \Vert g_\tau (t)\Vert _{L^2(0,L)} \Vert \varphi (t)\Vert _{L^2(0,L)} \,\mathrm dt\\&\leqslant M^\frac{1}{2}\biggl ( \int _{A_\varepsilon } \Vert \varphi (t)\Vert _{L^2(0,L)}^2\,\mathrm dt \biggr )^\frac{1}{2}, \end{aligned}$$

and the right-hand side can be made arbitrarily small as \(\varepsilon \rightarrow 0\). This shows that \(g_\tau \rightharpoonup 0\) weakly in \(L^2((0,L)\times (0,T))\) and (A.1) holds. \(\square \)

We discuss in the following two lemmas the dependence on t of the function \(z_{\varphi (t)}\) introduced in (2.10), when \(\varphi \) depends on t.

Lemma A.2

Let \(q\in L^2(0,T;V')\) and let \(z_{q(t)}\) be defined by (2.10). Then the map \(t\mapsto z_{q(t)}\) belongs to \(L^2(0,T;V)\).

Proof

For every \(\varphi \in V'\) the map

$$\begin{aligned} t \mapsto \left\langle \varphi ,z_{q(t)}\right\rangle _{V',V} {\mathop {=}\limits ^{(2.10)}} \int _0^L z_\varphi '(x)z_{q(t)}'(x)\,\mathrm dx = \left\langle q(t),z_\varphi \right\rangle _{V',V} \end{aligned}$$

is measurable thanks to the assumption \(q\in L^2(0,T;V')\). Hence the map \(t\mapsto z_{q(t)}\) is weakly measurable from (0, T) to V and in turn strongly measurable, by Pettis Theorem (see [34, Theorem 8.3]) and the separability of V. Moreover, by (2.10) we have

$$\begin{aligned} \int _0^T \left||z_{q(t)}\right||^2_V\,\mathrm dt = \int _0^T \left||q(t)\right||_{V'}^2 <\infty \end{aligned}$$

since \(q\in L^2(0,T;V')\), which shows that the map \(t\mapsto z_{q(t)}\) is Bochner integrable and belongs to \(L^2(0,T;V)\). \(\square \)

Notice that, if \(q\in L^2(0,T;L^2(0,L))\), then by (2.15) we have \(q\in L^2(0,T;V')\). Hence we can apply the previous lemma to deduce that also in this case \(z_{q(t)}\in L^2(0,T;V)\).

Lemma A.3

Let \(u\in H^1(0,T;V')\) and let \(z_{u(t)}\) be defined by (2.10). Then the map \(\psi (t) :=z_{u(t)}\) is in \(H^1(0,T;V)\), and

$$\begin{aligned} {\dot{\psi }}(t) = z_{{\dot{u}}(t)}. \end{aligned}$$
(A.2)

Proof

We first observe that \(\psi \in L^2(0,T;V)\), thanks to Lemma A.2. Similarly, the map \(t\mapsto z_{{\dot{u}}(t)}\) is in \(L^2(0,T;V)\), thanks to the same lemma applied to \(q={\dot{u}}\in L^2(0,T;V')\).

We now show equality (A.2), which will complete the proof of the lemma. By definition of weak derivative, (A.2) is equivalent to show

$$\begin{aligned} \int _0^T z_{{\dot{u}}(t)}\varphi (t)\,\mathrm dt = - \int _0^T z_{u(t)}{\dot{\varphi }}(t)\,\mathrm dt \end{aligned}$$
(A.3)

for every \(\varphi \in C^1_{\mathrm c}(0,T)\), where (A.3) is an equality between elements of V. For every \(\eta \in V\) we have

$$\begin{aligned} \begin{aligned} \Bigl (\int _0^T z_{u(t)}{\dot{\varphi }}(t)\,\mathrm dt \,,\, \eta \Bigr )_V&= \int _0^T \bigl (z_{u(t)}{\dot{\varphi }}(t), \eta \bigr )_V\,\mathrm dt = \int _0^T \int _0^L z'_{u(t)}(x){\dot{\varphi }}(t)\eta '(x) \,\mathrm dx \mathrm dt \\&{\mathop {=}\limits ^{(2.10)}} \int _0^T{\dot{\varphi }}(t) \left\langle u(t),\eta \right\rangle _{V',V}\,\mathrm dt = \left\langle \int _0^T u(t){\dot{\varphi }}(t)\,\mathrm dt,\eta \right\rangle _{V',V} \\&= - \left\langle \int _0^T {\dot{u}}(t)\varphi (t)\,\mathrm dt,\eta \right\rangle _{V',V} = -\int _0^T \varphi (t) \left\langle {\dot{u}}(t),\eta \right\rangle _{V',V}\,\mathrm dt \\&{\mathop {=}\limits ^{(2.10)}} -\int _0^T \int _{0}^L \varphi (t) z'_{{\dot{u}}(t)}(x)\eta '(x) \,\mathrm dx \mathrm dt = -\int _0^T \bigl ( \varphi (t) z_{{\dot{u}}(t)}, \eta \bigr )_V \mathrm dt \\&= - \Bigl (\int _0^T \varphi (t)z_{{\dot{u}}(t)}\,\mathrm dt \,,\, \eta \Bigr )_V \,. \end{aligned} \end{aligned}$$

In the previous chain of equalities we used several times the property that the Bochner integral commutes with linear operators, see for instance [34, Theorem 8.13]. This proves claim (A.3). \(\square \)

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Bonacini, M., Davoli, E. & Morandotti, M. Analysis of a perturbed Cahn–Hilliard model for Langmuir–Blodgett films. Nonlinear Differ. Equ. Appl. 26, 36 (2019). https://doi.org/10.1007/s00030-019-0583-5

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Keywords

  • Evolution equations
  • Cahn–Hilliard equation
  • Langmuir–Blodgett transfer
  • Minimizing movements
  • Fixed point theorem
  • Thin films
  • Global attractor

Mathematics Subject Classification

  • 35K35
  • 49J40
  • 37L30
  • 74K35