PhD thesis – Simon Lang

Abstract

Miniaturization processes constitute a success story in science and development, which have optimized the efficiency of various technological processes by orders of magni- tude. In many of these processes in micro- and nanotechnology, liquids are present, and since these liquids are confined on the microscopic scale they naturally interact strongly with boundaries and walls. To understand the behavior of such liquids is therefore of extraordinary importance. The motivation of this thesis is to contribute to the understanding of such liquids confined on the molecular scale from the perspective of theoretical physics.

When liquids interact with external walls, there is a complex interplay due to the constraints of the boundary and the intrinsic interactions of the particles. As a con- sequence, the particles on average favor certain positions, other positions are avoided and the structure of such liquids becomes inhomogeneous due to the breaking of spa- tial symmetries. The celebrated Noether-theorem implies a reduction of the conserved quantities of the system, for example the liquid exchanges and transfers momentum with the confining walls. When constructing a theory for the dynamics of such liquids, particular requirements are needed.

For densely-packed liquids, the dynamics of the molecular constituents slow down due to inner friction by orders of magnitude and the liquid can even amorphously vit- rify on macroscopic time scales. The details of this slow dynamics is encoded in the intermediate scattering function, which is directly accessible to scattering experiments or can be reconstructed by recording the trajectories of the particles in molecular dy- namics simulations. The mode-coupling theory of the glass transition (MCT) is a microscopic theory, which can quantitatively explain this peculiar dynamics via self- consistent equations of motion for the intermediate scattering function. The interplay of the slow dynamics close to vitrification and the interaction with boundaries is insuf- ficiently understood so far.

To date, it has been unclear how to generalize MCT to describe densely-compressed liquids confined between two parallel plates. This generalization is presented in this thesis, and allows the investigation of the dynamical behavior of liquids in confined geometry systematically based on the fundamental interactions. The particular chal- lenge for the development of the theory is the identification of the commensurable modes to generalize the intermediate scattering function to a matrix-valued quantity. By numerical solution of the complicated equations, the liquid-glass state diagram in confined geometry is determined for the first time. The theory predicts an intriguing phenomenon: For fixed density, it is possible to melt a glass and re-vitrify again by mere manipulation of the wall separation. Such a behavior is referred to as a reentrant transition and also seen in recent molecular dynamics (MD) simulations for systems of controlled polydispersity, performed by our collaborators (S. Mandal, F. Varnik). In- terestingly, the MCT even predicts that the coexistence of glass states and liquid states can be observed in-situ for slightly tilted wedges. These completely unknown effects demonstrate that intriguing physics accompany the regime of confined liquids.

Simulations of hard spheres in confined geometry reveal that the diffusion coef- ficient along the walls oscillates as a function of the wall separation. To date, these oscillations have not been explained by a dynamical microscopic theory. Here, a the- ory for the incoherent dynamics is developed, from which a microscopic expression for the lateral diffusion coefficient is extracted. The incoherent scattering function encodes all the relevant information on the one-particle dynamics in confined geom- etry. A novel strategy for nonorthogonal projectors is developed to extract the lateral dynamics from the matrix-valued theory, which can be probed readily within experi- ments or simulations recording only the lateral coordinates of the particles.

The microscopic density in confined geometry can vary in time due to two currents parallel and perpendicular to the walls. A similar feature is found for molecular liq- uids, where the temporal variation of the microscopic density occurs due to rotation and translation of the molecular constituents. This property is reflected by a com- plex structure of the corresponding equations of motion in comparison to simple bulk liquids. These equations have not yet been understood well, although the theory for molecules exists since 1997. In the framework of this dissertation a better understand- ing of this class of mode-coupling equations is achieved and it is shown, that many of the neat properties for the conventional MCT can be generalized to this more complex MCT. This is an important contribution providing a solid mathematical framework for the results and predictions for the MCT of confined and molecular liquids.

An intriguing regime is when the fluid is confined on a scale, which only slightly extends the diameter of the molecular constituents. Such a system can be modeled by successively decreasing the wall separation until only small fluctuations in the direc- tion of the walls are feasible. For the understanding of such fluids an important break- through has been achieved here. The major insight to describe extremely confined liquids is that one can use the transversal degrees of freedom as a small perturbation to the lateral degrees of freedom based on microscopic statistical physics without further approximations. For hard spheres, a novel cluster expansion is found, from which the contributions of the transversal degrees of freedom can be systematically calculated. The exact determination of the partition function allows us to establish the connection to the world of thermodynamics, for example the transversal pressure exerted on the walls can be calculated explicitly to leading and next-to-leading order. A new small parameter of the confinement problem is identified. For hard spheres, the cluster ex- pansion allows the calculation of an effective interaction potential, mapping extremely confined hard spheres onto two-dimensional disks with a soft renormalized interaction layer. With these results one can even describe the equilibrium phase behavior by a simple formula in the regime of extreme confinement.

For the characterization of confined fluids, structural correlation functions are of utmost importance. One can use the solution strategy of the expansion with respect to the transversal degrees of freedom to study the convergence properties of the structural quantities. In the most general form, this is performed for the m-particle distribution function, from which the whole hierarchy of static correlation functions can be re- constructed. Corrections to the structural quantities of the 2D reference system are calculated analytically. An interesting insight is that the 2D limit is subtle and the convergence of various correlation functions depends strongly on the specific type of wall interactions.

The results of the dissertation provide a fruitful ground for many new research directions. For the future it will be important to study the time-resolved correlation function from the derived equations of motion of densely-packed liquids. The inter- play of localization and diffusion in confined geometry, the dynamical behavior of extremely confined liquids with small transversal fluctuations or the competition of binary composition with the reentrant scenario constitute new scientific ground and promise intriguing perspectives for the future.