Studentische Forschung an der Fakultät für Mathematik


Discrete-to-Contiuum Limit for a 2D Model of "Water Dislocation"

Bachelorarbeit von Leonard Kreutz, SS 2013
Betreuer: Prof. Dr. Marco Cicalese

Overview: In this article, I present the content of my bachelor's thesis which I wrote in the summer term of 2013 and I give a short outlook on how I will continue my studies on this topic. I started studying mathematics in the fall of 2010. Right from the very beginning I knew that I wanted to focus on Analysis and later on work in this field. In the summer of 2012 I applied to the TopMath program, since I knew that, by being part of it, I would have the opportunity to quickly deepen my studies in Analysis. I chose Prof. Cicalese as mentor because of his research interest in Analysis and applications.

In the fall of 2013 I attended a study group organized by Prof. Cicalese and Prof. Friesecke. These meetings where occasions of informal discussions about future research directions in applied analysis. In one of the meetings, Prof. Cicalese presented the discrete-to-continuum analysis of water models as an interesting and challenging research field to explore. A few days later we decided to choose this topic as the subject of my bachelor thesis.

Introduction: In my bachelor thesis, I provide a rigorous discrete-to-continuum analysis of the two dimensional Bell-Lavis energy model for water in the low density case. Studying the \( \Gamma \)-limit of proper scalings of the energy as the number of water molecules increases at constant density, I show the emergence of the so called "water dislocations", special microscopic arrangements of the water molecules driven by the presence of interfacial energy barriers between incompatible ground states. The complete description of the geometry of the microscopic ground states is accomplished by computing an explicit formula for the energy density of the asymptotic energy functional.

lattice1.jpg
Fig. 1: Generic configuration on the lattice.
The setting of the problem is the following: One is given a set \( \Omega \subset \mathbb{R}^n \) with a sufficiently regular boundary, and a triangular lattice \( \mathcal{L}_n \) with lattice spacing \( \frac{1}{n} \), \( n \in \mathbb{N} \). Each point of the lattice \( \mathcal{L}_n(\Omega)=\mathcal{L}_n \cap \Omega \) may be occupied by a water molecule that is modeled as a material point with three bonding arms at \( 120^\circ \). The arms represent the direction of the H-bonds of the water molecule. The strength of the interaction between pairs of nearest neighbours depends on whether the arms of the particles are facing each other or not. A generic configuration is shown in Figure 1.

The Bell-Lavis energy I consider is usually written in the following form: $$E_n^{BL}(\eta,\tau)~=~-\sum_{i,j \in \mathcal{L}_n(\Omega)}\frac{1}{n^2}\eta_i\eta_j(\epsilon_{vdw}+\epsilon_{hb}\tau_i^{ij}\tau_j^{ji})-\mu\sum_{i\in \mathcal{L}_n(\Omega)}\frac{1}{n^2}\eta_i,$$ where \( \epsilon_{vdw}>0 \),\( \epsilon_{hb}>0 \) and \( \mu>0 \) are the strengths of the Van-der-Waals forces, hydrogen bonds and the chemical potential, respectively. Here \( \eta: \mathcal{L}_n(\Omega) \to \{1,0\} \) is the occupational variable and \( \eta_{i} \) takes values \( 1 \) or \( 0 \) depending on whether or not the site \( i \) is occupied. The variable \( \tau : \mathcal{L}_n(\Omega) \times \mathcal{L}_n(\Omega) \to \{0,1\} \) is the orientational variable and the product \( \tau_{i}^{ij}\tau_{j}^{ji} \) is \( 1 \) or \( 0 \) depending on whether the H-bonds of the molecules sitting at \( i \) and \( j \) are facing each other or not. As a consequence, the Bell-Lavis energy favors the formation of clusters of bonded molecules.

Performing a change of variables one can pass from the Bell-Lavis model to a Blume-Emery-Griffith (spin-\( 1 \)) model as follows. Introducing the variable \( s: \mathcal{L}_n(\Omega) \to \{\pm1,0\}\) defined as $$s_a=s(a)=\begin{cases} 1&\text{if at}~a~\text{the particle is facing right}\,,\\-1&\text{if at}~a~\text{the particle is facing left}\,,\\0&\text{if there is no particle at}~a\,, \end{cases}$$ one may rewrite the energy as $${E}_n(s)= \begin{cases} \underset{a \in \mathcal{L}_n(\Omega)}{\sum}\frac{1}{n^2}\big(f(s_{a+\frac{v_2}{n}},s_a,s_{a+\frac{v_1}{n}})\\ \hspace{8mm}+f(s_{a+\frac{v_1}{n}},s_{a+\frac{1}{n}(v_1-v_2)},s_a)\big), & s:\mathcal{L}_n(\Omega) \to \{\pm1,0\},\\+\infty, &\text{otherwise,}\end{cases}$$ for some \( f:\mathbb{R}^3 \to \mathbb{R} \).

lattice2.jpg
Fig. 2: Ground state configurations.
In terms of the new variable \( s \) the asymptotic analysis of the model as \( n\to \infty \) is significantly simplified. In particular I can study the phenomenon of phase transitions in the system, which in the present case translates into showing the presence (or not) of (interfacial) energy barriers between two different ground states. To this aim I first observe that without any constraint on the density of particles no phase transition occurs. This means that there is no stability of minimal states in the sense that different ground states can be mixed without paying any additional energy. In fact, when no density constraint is assumed, the two ground-state cells look like in Fig. 2 (top row).

lattice3.jpg
Fig. 3: Coexistence of the two phases.
One can see that these two ground states can be distinguished in terms of the average value of the variable \( s \) in each triangle of the lattice. This turns out to be \( -1/3 \) (two particles in state \( -1 \) and one in state \( 1 \)) for the configuration on the left and \( 1/3 \) (two particles in state \( 1 \) and one in state \( -1 \)) for the one on the right. These two average values of the order parameter \( s \) may be seen as two different phases of the system. In Figure 3, I show how these two phases can coexist without increasing the total energy of the system. In fact one should notice that across the red line, which ideally divides the two phases, each cell is still in the minimal energy state.

The construction above suggests that some constraint on the density of the system could modify the microstructure of the ground state enough to make different ground states incompatible and lead to the emergence of phase transitions. In my thesis I discuss the case of the so called (local) low density constraint, i.e. in each triangle at most two particles are allowed. Up to translation, in this case the ground states look like in Figure 2 (bottom).

lattice4.jpg
Fig. 4: Water dislocations.
This time when two different ground states coexist in the system an interfacial energy appears. In fact if two different phases meet the cells of the lattice at the boundary of the phases (grey cells in Fig. 4) are not in a minimal state, so that the overall energy is increased of an amount proportional to the length of the common boundary. This interfacial energy accounts for the loss of translation invariance in the regular pattern of the H-bonds. The so generated 'defects' are also known as water dislocations (see Figure 4).

The above observations can be made rigorous computing the \( \Gamma \)-limit as \( n\to \infty \) of the functionals \( E_n \) as well as of the scaled functionals $$E_n^{(1)}(s)=\frac{E_n(s)-\min E_n}{\frac{1}{n}}.$$ I prove that, for some \( g:\mathbb{R}\to \mathbb{R} \), the \( \Gamma \)-limit \( E \) of the functionals \( E_n \) (with respect to the \( L^{\infty} \) weak-\(\star\) convergence) has the following integral structure $$E(u)=\int_\Omega g(u(x))dx.$$ With this result at my disposal, and restricting the analysis to special \( \Omega \) in order to avoid boundary layer effects, I compute the phase transition energy \( E^{(1)} \) as the \( \Gamma \)-limit, with respect to the \( L^{1}(\Omega) \) convergence, of the scaled functionals \( E_{n}^{(1)} \) obtaining $$E^{(1)}(u)=\int_{S(u)}\varphi(u^+,u^-,\nu)d\mathcal{H}^1.$$ Here the energy stays finite on those functions \( u\in BV(\Omega) \) taking only a finite number of values (each of them identifies one possible phase of the system). In this setting \( S(u) \) denotes the jump set of \( u \), while the energy density \( \varphi \) depends on the traces \( u^{\pm} \) of \( u \) at \( S(u) \) as well as on the normal \( \nu \) to \( S(u) \). In other words, this limit energy says that the macroscopic phases may be described as sets of finite perimeter and that whenever two of these phases have a common boundary, this carries an energy proportional to its length. The precise form of \( \varphi \), that I have computed, depends on the optimal microscopic structure of the phases and reflects the symmetries of the lattice. In the context of the Bell-Lavis theory this is the energy of water dislocations.

Outlook: The results I have obtained raise new questions. Below I list some of the themes I am planning to address during my PhD studies:
  • energy driven motion of water dislocation microstructures,
  • energy scaling and optimal microstructures under mild density constraints,
  • analysis of stochastic instead of triangular lattice water models.

Zur vollständigen Bachelorarbeit von Leonard Kreutz hier.


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Steckbrief des Autors

portait-kreutz.JPG

  • Leonard Kreutz
  • Bachelorstudium Mathematik an der TUM von WS 2010/11 bis SS 2012
  • Bachelor-Elitestudiengang TopMath an der TUM von WS 2012/13 bis SS 2013
  • B.Sc. in Mathematik mit Nebenfach Informatik und Schwerpunkt Angewandte Analysis, SS 2013
  • nach dem Bachelor: Promotionsphase im Studienprogramm TopMath
  • Forschungsinteressen: Angewandte Analysis, besonders im Gebiet der Variationsrechnung
 

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