INSTITUTE OF CHEMICAL TECHNOLOGY, PRAGUE

Faculty of Chemical Engineering

Department of Physical Chemistry

Dissertation

Author: Mgr. Stanislav Pařez

Supervisor: Consultant:

RNDr. Milan Předota, Ph.D.

Study program: Chemistry

Field of study: Physical Chemistry

Prague 2013

COMPUTER SIMULATIONS OF SOLID-LIQUID INTERFACES

VYSOKÁ ŠKOLA CHEMICKO-TECHNOLOGICKÁ V PRAZE

Fakulta chemicko-inženýrská

Ústav fyzikální chemie

Disertační práce

Autor: Mgr. Stanislav Pařez

Školitel: Konzultant:

RNDr. Milan Předota, Ph.D.

Studijní program: Chemie

Studijní obor: Fyzikální chemie

Praha 2013

POČÍTA ČOVÉ SIMULACE ROZHRANÍ PEVNÝCH LÁTEK A KAPALIN

This dissertation was written at the E. Hála Laboratory of Thermodynamics of Institute of Chemical Process Fundamentals of Academy of Sciences of the Czech Republic in Prague

from Oct 2009 to Mar 2013.

I hereby declare that this thesis is my own work. Where other sources of information have been used, they have been acknowledged and referenced in the list of used literature and other sources. I have been informed that the rights and obligations implied by Act No. 121/2000 Coll. on Copyright, Rights Related to Copyright and on the Amendment of Certain Laws (Copyright Act) apply to my work. In particular, I am aware of the fact that the Institute of Chemical Technology in Prague has the right to sign a license agreement for use of this work as school work under §60 paragraph 1 of the Copyright Act. I have also been informed that in the case that this work will be used by myself or that a license will be granted for its usage by another entity, the Institute of Chemical Technology in Prague is entitled to require from me a reasonable contribution to cover the costs incurred in the creation of the work, according to the circumstances up to the full amount. I agree to the publication of my work in accordance with Act No. 111/1998 Coll. on Higher Education and the amendment of related laws (Higher Education Act).

In Prague on 29th March, 2013

It has been a great honour for me to work under supervision of Milan Předota. His enthusiasm for science has motivated me to actively contribute to the issues concerned in this thesis. I particularly appreciate his ability to look at complex problems in a simple way, which has inspired me to test my own hypothesis about solutions. I am also very grateful to co-authors of the attached papers, Ivo Nezbeda and Jadran Vrabec. Numerous discussions with both of them have raised my awareness about interesting issues in the field of molecular simulation and science in general.

SUMMARY The behavior of fluids at solid-liquid interfaces is studied by means of molecular simulation. The introduction of the surface forces leads to interesting surface-driven changes in fluid properties, which normally take place on a scale of a few molecular lengths and depend markedly on the position with respect to the surface. In this thesis, interfacial properties of aqueous solutions are studied for two geometries of a solid: (i) a spherical particle of various sizes and (ii) a planar surface.

In the first case, effect of particle size (curvature) on the structure of interfacial water was investigated. Changes in radial density profile as well as orientations of water molecules relative to the surface were traced. An effective coarse-grained interaction potential between a water molecule and the spherical particle was derived, which smears out the atomistic nature of the spherical solute, to shorten computational time and to investigate the effect of size systematically.

For the planar surface, extensive calculations were performed to study shear viscosity of aqueous mixtures and dielectric properties of water in the interfacial region. Here, rutile (α-TiO2) (110) surface was considered, which is also frequently probed by experimentalists for its chemical stability. Profiles of viscosity and relative permittivity, as a function of the distance from the surface, were examined. The results show significant deviations of the properties from their bulk values. Viscosity was studied for pure water and its mixture with methanol over the whole composition range. Extension of calculations to mixtures was enabled by generalization of the standard non-equilibrium molecular dynamics method that is designed for pure fluids. Further investigations on transport properties of mixtures were carried out addressing mutual diffusion in the ternary mixture water + methanol + ethanol for bulk homogenous system. Fick diffusion coefficients were predicted for the ternary mixture, where no experimental data had been available. Dielectric properties were calculated from a response of water molecules to a homogeneous electric field applied in the direction normal to the surface. Results provide molecular-level understanding which is in contrast to classical picture. Here, simulation data could be used to facilitate the development of theoretical models of an interface, such as the electric double layer.

SOUHRN Pomocí molekulárních simulací byly studovány jevy na rozhraní pevných látek a kapalných roztoků. Interakce s pevnou látkou ovlivňuje zásadním způsobem řadu fyzikálně-chemických vlastností kapaliny nacházející se v blízkosti několika molekulárních délek od rozhraní. Především jsem se zaměřil na popsání a vysvětlení vlastností kapalin na rozhraní v závislosti na vzdálenosti od (i) sférické koloidní částice různých velikostí a (ii) rovinného povrchu.

V prvním případě jsem studoval vliv velikosti (nebo křivosti) částice na strukturu vody v blízkosti rozhraní. Za tímto účelem jsem sledoval změny v radiálním hustotním profilu a v orientaci molekul vody vzhledem k povrchu. Pro snížení výpočetní náročnosti byl odvozen a aplikován efektivní potenciál, který popisuje interakci molekuly vody se sférickou částicí jako celkem zanedbáním její atomární struktury.

V případě rovinného povrchu jsem provedl detailní studie zaměřené na viskozitu vodných roztoků a na dielektrické vlastnosti vody. Jako pevný povrch byl modelován (110) krystalografický povrch rutilu (α-TiO2), jenž je díky své chemické stabilitě často zkoumán i v experimentech. Výrazně odlišné chování viskozity a relativní permitivity u rozhraní než v homogenní fázi je demonstrováno na výsledných prostorově proměnlivých profilech obou veličin. Viskozita byla studována pro čistou vodu i pro vodný roztok methanolu v celém koncentračním rozmezí. Abych mohl určovat viskozitu směsí, zobecnil jsem standardní metodu v rámci nerovnovážné molekulární dynamiky, která byla navržena pouze pro čistou látku. Studium transportních vlastností směsí bylo završeno výpočtem bulkových hodnot vzájemné difúze u ternární směsi voda + methanol + ethanol. Simulace zde vedly k předpovědi hodnot Fickových difúzních koeficientů, které dosud nebyly známé. Dielektrické vlastnosti byly určovány pomocí odezvy molekul vody na působení homogenního elektrického pole kolmo k rozhraní. Simulační data v tomto případě poskytují fundamentální vhled do chování dipólů na rozhraní z hlediska mikroskopické struktury. Překvapivé výsledky, které jsou v rozporu s tradiční představou, mohou posloužit dalšímu vývoji klasických modelů rozhraní jako je model elektrické dvojvrstvy.

CONTENTS

1. INTRODUCTION 1

2. STATE OF THE ART 4

2.1 Methodology ........................................................................................... 4

2.2 Application of Simulations ...................................................................... 7

2.2.1 Interfacial Properties of Aqueous Solutions at Planar Rutile Surface............................................................................................. 7

2.2.2 Effect of Size of a Spherical Colloid Particle on Structure of Interfacial Water.............................................................................. 9

2.2.3 Mutual Diffusion in Multicomponent Liquids .................................. 9

3. GOALS OF THE THESIS 10

4. THEORETICAL BACKGROUND 11

4.1 Molecular Simulation Methods ............................................................. 11

4.1.1 Transport Properties ........................................................................ 11

4.1.2 Dielectric Properties ........................................................................ 13

4.2 Molecular Models ................................................................................. 15

4.3 Simulation Details ................................................................................. 17

5. RESULTS AND DISCUSSION 18

5.1 Interfacial Properties of Aqueous Solutions at Planar Rutile Surface .. 18

5.1.1 Viscosity of Mixtures by Non-equilibrium Molecular Dynamics .. 18 5.1.2 Viscosity in the Bulk Region .......................................................... 19

5.1.3 Interfacial Viscosity ........................................................................ 20

5.1.4 Relative Permittivity by Non-equilibrium simulation..................... 23 5.1.5 Dielectric Properties of Interfacial Water Layers ........................... 23

5.2 Effect of size of a spherical colloid particle on structure of interfacial water ................................................................................... 30

5.2.1 Coarse-grained Interaction Potential ............................................... 30

5.2.2 Structure of Interfacial Water.......................................................... 31

5.3 Mutual Diffusion in Multicomponent Liquids ...................................... 34

5.3.1 Thermodynamic Factor ................................................................... 34

5.3.2 Fick Diffusion Coefficients ............................................................. 35

6. CONCLUSION 38

7. REFERENCES 40

8. APPENDICES 43

8.1 APPENDIX A ....................................................................................... 44

8.2 APPENDIX B ....................................................................................... 57

8.3 APPENDIX C ....................................................................................... 81

8.4 APPENDIX D ..................................................................................... 103

9. PUBLICATIONS 123

1

1. INTRODUCTION

Phenomena at solid-liquid interfaces play an important role in physics, chemistry, biology, engineering, electrochemistry, geophysics as well as daily life. Among numerous examples, let us mention heterogeneous or photo- catalysis, function of biological membranes, electrochemical cells, corrosion and environment preserving applications. The specific behavior of liquids, or generally fluids, is caused by the interaction with the solid phase. The introduction of the surface forces gives rise to new phenomena that are quite unexpected on the basis of our knowledge acquired from bulk systems. These include new kinds of phase transitions, e.g. capillary condensation, layering, wetting, as well as altering dynamical and transport properties of fluids, e.g. space dependence of diffusivity and viscosity.

The above mentioned phenomena mostly take place on very small scale - typically within 20 Å from the surface. Because of this dimension, interfacial fluid properties are very difficult to study from a purely experimental perspective, making molecular simulations an attractive alternative, and in some cases the only feasible tool. Nowadays, molecular simulations have become a common tool for supply of experimental data and verifying theoretical predictions, improving understanding of studied processes from the molecular-level perspective. With improvement of molecular models, molecular simulations have achieved the role of virtual experiments. Focusing on solid-liquid interfaces, progresses in simulation methodology, which have increased the number of determinable quantities, and development of computer technology allowed substitution of original works on structure of fluid near a smooth planar surface by detailed studies of various properties at fully atomistic solids of arbitrary shapes.

Despite all the achievements in simulations of confined fluids, numerous problems are yet to be solved. This thesis is mainly devoted to development and test of simulation methods with respect to their applicability to solid-liquid interfaces. Among most of simulation methods, equilibrium statistical ensemble averages are used to calculate fluid properties [1], including transport coefficients that are by definition related to perturbations taking the system out of equilibrium. These methods require infinite, or possibly very large, systems to suppress finite size effects. In contrast, the interface is restricted in one dimension to a few molecular lengths. Thus, application of such methods on interface is often not plausible and can lead to significant deviations. An alternative approach, tested here, is to use methods in which properties of molecular liquid are averaged locally to yield observables known from the theory of continuum. E.g., shear viscosity can be determined on the basis of the classical fluid dynamics [2] from study of laminar flow, where properties of the liquid may vary with position of the given lamina from the surface. Similarly, local dielectric properties can be derived from the theory of dielectrics [3] as a response of the system to an external field. This approach, following the theory of continuum, ultimately leads in non-equilibrium simulations. The local properties as a function of the distance from the surface are used to describe the interfacial region, while results from bulk phase, i.e. where the influence of the solid is negligible, are compared to those from equilibrium method as well as experiment in order to assess the method and to see the effect of the magnitude of the driving force (generating the flow or the dielectric response) on the studied properties.

One of the most important challenges in chemistry is proper description of mixtures. To this point, simulation methods should take into account the effect of composition on the

2

studied properties. However, several techniques are available for pure fluids only. This drawback will be resolved here for shear viscosity determination by non-equilibrium molecular dynamics simulation (NEMD). The issue of transport properties of mixtures has been studied further, even slightly beyond the scope of the dissertation theme. To be specific, a consistent simulation method was tested for determination of mutual diffusion coefficients in multicomponent homogeneous liquid, i.e. a system without a solid phase. In this method, Maxwell-Stefan (MS) and Fick diffusion coefficients are calculated based on given molecular model, unlike the common method that requires the use of experimental vapor-liquid equilibrium (VLE) data.

The results of this thesis point to the challenge of accurate and consistent description of interfacial liquid properties. Although the advancement of experimental techniques, nowadays capable of probing interfacial region with a resolution of a few nanometers, has resulted in more detailed understanding of interfacial properties [39], in the nanoscale rather crude approximations are often adopted during the interpretation of the experimental data due to a lack of detailed information. In the interpretation of experimental measurements, a model of the interface as a sequence of several discrete layers of constant or linearly varying parameters (e.g. dielectric constant, capacitance, ionic strength, mobility) is frequently applied [41,6], or local inhomogeneities and microscopic details are ignored at all and bulk values are used. In contrast, many computer studies indicate that the structure of the interface is much more complex [40,9-11].

Present results continue previous computer studies providing further insight into the experimental observations and model picture of a solid-liquid interface, and extend the range of applications of molecular simulations to systems that were previously inaccessible. It is convenient to divide the results into three sections based on four original papers published in, or submitted to, peer-review international journals:

Interfacial properties of aqueous solutions at planar rutile surface

[A] Pařez, S.; Předota, M. Determination of Distance-dependent Viscosity of Mixtures in Parallel Slabs using Non-equilibrium Molecular Dynamics. Phys. Chem. Chem. Phys. 2012, 14, 3640.

[B] Pařez, S.; Předota, M. Dielectric properties of water at solid-liquid interfaces. Submitted to J. Phys. Chem. C

Effect of size of a spherical colloid particle on structure of interfacial water

[C] Předota, M.; Nezbeda, I.; Pařez, S. Coarse-grained potential for interaction with a spherical colloidal particle and planar wall. Collect. Czech. Chem. Commun. 2010, 75, 527.

Mutual diffusion in multicomponent liquids

[D] Pařez, S.; Guevara-Carrion, G.; Hasse, H.; Vrabec, J. Mutual diffusion in the ternary mixture water + methanol + ethanol and its binary subsystems. Phys. Chem. Chem. Phys. 2013, 15, 3985.

This thesis is organized as follows. In section 2, the state-of-the-art of the addressed

issues is overviewed and motivation for present studies is outlined. In section 3, the goals of the thesis are clearly summarized. The simulation methods, employed molecular models and

3

technical simulation details are introduced in section 4. In section 5, results are presented and discussed. Finally, conclusions are drawn. The full texts of the author's original papers [A-D], which the thesis is based on, are attached in the Appendices.

4

2. STATE OF THE ART

2.1 Methodology

Molecular simulations provide us with a chain of representative configurations of molecules of the studied system, including their mutual interaction. In molecular dynamics (MD), the configurations form a time sequence corresponding to the evolution of the system, while in Monte Carlo (MC) method, the configurations are generated randomly, but their occurrences in the chain are proportional to their Boltzmann factors [8,1]. The natural way to determine thermodynamic quantities from a simulation is to employ principles of statistical thermodynamics to calculate averages over the chain of configurations. To obtain reasonable estimates for thermodynamic values, the simulated system has to be sufficiently large. This requirement is not only to achieve sufficient statistical accuracy of the results, but also because of neglecting the surface effects, so that the average value was independent of the system size. For example, if a bulk system is divided into two halves, calculated thermodynamic properties in both parts are the same as in the original system provided that the mutual interaction between the parts is much smaller than energy of each part. This is, however, not the case of interfaces. As the thickness of an interfacial layer of liquid is only a few molecular lengths, though the layer might be large in the other two dimensions and contain virtually infinite number of molecules, the interaction with its neighborhood cannot be neglected due to large surface/volume ratio.

Several properties of liquid are defined in terms of a response of a system to a perturbation. For example, diffusion coefficient relates a molar flux to a concentration gradient, shear viscosity couples shear stress and velocity gradient [1,2], and relative permittivity (a.k.a static dielectric constant) describes the polarization response to an applied electric field [3]. Starting from the relation between perturbation and response, two simulation approaches follow. Either, the perturbation is explicitly introduced in the simulation by external fields closely mimicking the experimental setup. This approach is referred to as the non-equilibrium approach. Alternatively, the effect of perturbation is accounted for in the distribution function of states, expansion of which in powers of perturbation strength is considered. By keeping only linear terms of the ensemble average for the response, and comparison with the corresponding continuum relation, the resulting expression for the studied quantity can be identified as an equilibrium ensemble average [1,8]. Clearly, this approach, known as linear response theory, holds only for small perturbations of the real system under which the response is proportional to its driving force. Here, the approach is referred to as the equilibrium approach, since it can be computed from a simulation of an equilibrium ensemble without applying external fields.

The equilibrium expression for some properties can be written as a single molecule function. In other words, these properties can be obtained by tracing the behavior of a single molecule, while averaging the value over the entire system “only” increases its statistical accuracy. Self-diffusivity is an example of such single-molecule property, cf. eq. (6). In this case, the property can be assigned to a certain layer or bin, if the molecule resides the bin for sufficiently long time, e.g. until convergence of the velocity auto-correlation function integral is reached when considering self-diffusivity. This approach has been employed to calculate a

5

profile of self-diffusion coefficient of water at the rutile surface [9], where the results conformed to the hydrogen-bonding (HB) analysis showing the small diffusivity at interfacial layers as a result of dense HB structure, and explained the differences between the component of diffusivity perpendicular and parallel to the surface. On the other hand, properties like viscosity or relative permittivity are related to collective behavior of molecules. Their equilibrium expressions contain virial, cf. eqs. (3)-(4), resp. total dipole moment fluctuations, cf. eq. (13); quantities that depend on correlation between different molecules. If we divide a system into two parts and calculate, for example, the mean square of the total dipole momentum, the sum of the contributions from both parts differs from calculation from the entire system by correlations between dipoles from different parts. Although these correlations may be small for sufficiently large subsystems in the bulk phase, they cannot be presumed to be small in such inhomogeneous and thin region which the interface presents.

The above mentioned problems in application of equilibrium methods on interfaces led us to considering non-equilibrium methods. Since non-equilibrium approach follows closely the continuum description of liquid, the governing equations can be derived for local properties. This perfectly fits the description of interfaces, where properties of liquid vary with the position with respect to the surface due to spatial inhomogeneities. In paper [A], shear viscosity was determined by NEMD using Poiseuille flow [1] in a slab. The approach had been implemented before, mostly for atomic fluids based on the Lennard-Jones potential or its modifications [14-18]. Consequently, pure water was studied in Ref. [20] and, in a detail, in Ref. [9]. Our present results extend the application of the methodology to more complex systems (aqueous alcohols) and, in particular, to mixtures. Although a method for determination of viscosity of mixtures is readily available in the framework of equilibrium molecular dynamics (EMD) [1,24], only the NEMD method allows calculation of viscosity in the inhomogeneous region formed at a solid surface, which, however, was designed for pure fluids only. Therefore, its generalization for mixtures was necessary for studying interfacial profile of viscosity of solutions. A very few attempts have been made to calculate interfacial viscosity of a mixture in the literature [18,19]. However, authors employed intuitive formulas that approximate the expression derived in [A] for a low solute concentration range and/or for situations when all components drift with similar velocities. In paper [A], we give an expression that is valid for an arbitrary composition of a mixture with an arbitrary number of components. Provided that the flow is laminar, the choice of external forces acting on each component, and generating the flow, is arbitrary too. Therefore, the method can be applied even for situations in which the drift velocities of individual components are not similar, such as in the case of electroosmosis, where oppositely charged particles move in opposite directions.

In paper [B], a non-equilibrium approach to determine interfacial dielectric properties is introduced, based on the polarization response of molecular liquid to an applied electric field. Derived from the theory of dielectrics, the method describes dielectric properties of liquid by polarization vector (volume density of dipole momentum). Again, the local description is achieved by averaging molecular dipoles over a close neighborhood of a given point to yield local values of polarization. This approach has been used to evaluate relative permittivity for homogeneous liquids [21], for which it yields consistent results with the equilibrium simulation [22] using the total dipole moment fluctuation formula [4,8] derived from the linear response theory. To study the interfacial region, the fluctuation formula applied on bins has been employed in most works [12,13], although its use is not plausible. Besides

6

neglecting correlations between neighboring bins, the formula is derived for homogenous isotropic systems, and the fluctuation of the total dipole moment of the given bin contains a spurious contribution from the fluctuation of the number of particles in the bin. In Ref. [23], a part of these problems was resolved and a much more consistent fluctuation formula was derived for the slab geometry. The results were favorably compared to the non-equilibrium approach for a very simple model of a polar fluid at a graphite wall modeled by 9-3 Lennard-Jones (LJ) potential. However, the role of fluctuations of the dipole moment of a layer due to molecules entering/leaving the layer is not addressed, and the theory does not cover the situation met by real liquids when a non-zero polarization exists at zero external field due to the interaction with the surface.

Methodology advance in treatment of mixtures is again addressed in paper [D], where mutual diffusion in multicomponent mixtures is studied by means of equilibrium simulation. A consistent approach for calculation of Fick diffusion coefficients on the basis of a given molecular model is introduced challenging the classical simulation approach. Both approaches determine Fick diffusion coefficients in two steps: first, MS diffusion coefficients are calculated using standard Green-Kubo expression [29]. Second, the thermodynamic factor is evaluated serving as a conversion factor between both sets of mutual diffusion coefficients. The difference between the two simulation approaches lies in the parameterization of an excess Gibbs energy GE model used in the evaluation of the thermodynamic factor. In the classical approach [30-33], the GE model is regressed to experimental VLE or excess enthalpy data, while in the present approach, the same model is regressed to composition dependence of chemical potentials that were sampled by MC simulation. The latter approach was preferred, because the classical approach (i) is highly sensitive to the underlying GE model or particular experimental data set used [34] and (ii) is limited to the range of substances and thermodynamic conditions for which the experimental data exist. This drawback has mostly disqualified the classical approach from application to ternary and higher multicomponent mixtures where the experimental VLE data are scarce [35]. The approach presented in paper [D] allows prediction of Fick diffusion coefficients in an arbitrary multicomponent mixture.

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2.2 Application of Simulations

Determination of structural and dynamical properties of liquid phase in contact with solid surfaces has progressed substantially in recent years due to new experimental techniques which can probe the interfacial region with a typical resolution of the order of a few nanometers. These techniques include temperature programmed desorption, surface titration, high resolution electron energy loss spectroscopy, X-ray and neutron scatterings [39]. Quasielastic neutron scattering is well suited for studying the mobility of water molecules and has been applied to a great variety of aqueous solutions, both organic and inorganic. The exceptionally large incoherent neutron cross-section of hydrogen compared to other elements makes neutron scattering an attractive technique to study the dynamics of interfacial water [40]. The elastic signal is predominantly due to scattering from immobile (on the time scale of the measurement) water species, while the inelastic signal originates from the mobile hydrogens. Both the translational and rotational modes of motion are reflected in the neutron scattering spectra.

Despite the advancement of these experimental techniques, their resolution is still insufficient to access variation of properties on molecular scale, which is however essential to describe the complexity of the solid-liquid interface. As a consequence, experimental data are combined with theoretical models that account for distance dependence of liquid properties approximately, usually based on relations valid in bulk or using bulk values and ignoring interfacial effects at all. The aim of this thesis is to provide valuable pieces of information about interfacial phenomena for proper interpretation of experimental data and development of theoretical models by means of molecular simulations.

2.2.1 Interfacial Properties of Aqueous Solutions at Planar Rutile Surface Most of the results in this thesis are obtained for aqueous solution at planar (110) rutile nonhydroxylated surface. Rutile is a chemically stable metal oxide showing complex behavior at the interface due to the interplay between long-range electrostatic forces, van der Waals interactions of molecules and ions with the solution and the surface, and the specific effects in the inhomogeneous aqueous phase at the interface, resulting in the formation of an electric double layer (EDL). For these properties, the rutile-aqueous interfaces are frequently studied by both molecular simulations [9-11] and experiment [42,43,39].

The rutile surface in contact with aqueous solutions has been studied extensively in our group. The structure of water and ions at the interface from our MD simulations agrees very well with experimental data obtained by X-ray standing wave and X-ray reflectivity (e.g., crystal truncation rod) analysis of surface structures at ambient conditions [9-11,42,43]. In the ongoing research, diffusivity [11], shear viscosity [11] and hydrogen bonding structure [44] of pure water in the interfacial region were studied. The results showed that interfacial properties can significantly differ from their values in bulk, which in turn questions using the bulk values in theories where local values are requisite. This problem was pointed out in Ref. [11] concerning remarkable disagreement between the structural results of our MD simulations and X-ray structural measurements on one hand, and, on the other hand, the experimental electrophoretic data interpreted by electrokinetic theory in terms of a “shear plane” at a considerable distance (~several nm) from the surface. The interpretation of the electrophoretic

8

data relies on the Helmholtz-Smoluchowski equation [45] rεε

ηµζ0

= , where shear viscosity

η and relative permittivity εr should refer to the values at the Stern plane (3-4 Å from the surface). Because of the lack of the availability of space-dependent values at the interface, the bulk values are routinely used.

Motivated by the need for detailed profiles of local values, interfacial viscosity and permittivity were investigated in papers [A] and [B] respectively. Viscosity was consistently determined for water + methanol system, extending the range of molecular simulation application to mixtures and highly polar hydrogen-bonding liquids, as discussed above in the Methodology chapter. Determination of relative permittivity of the interface, and dielectric properties in general, is an important step to test theoretical models describing an interface in terms of EDL, which are used in interpretation of experimental surface titration data [5,6,44]. The EDL models, surveyed e.g. in Ref. [5], are modifications of the basic Stern (or Gouy-Chapman-Stern) model [46,7], which describes the electric double layer as two layers: (i) the Stern layer (or the Helmholtz layer), referring to the compact layer of immobile ions adsorbed to the surface, and (ii) the diffuse layer where the ions are mobile and their concentration profile follows the solution of Poisson-Boltzmann equation [45]. The role of water is that of continuum with dielectric constant εr,i filling the layer i. The model is parameterized by values of capacitances of the layers which are fitted to experimental data from the surface titration. Values of permittivity do not follow directly from EDL model parameters, but can be linked to information on positions of the Stern or diffuse layer from X-ray diffraction and other techniques according to the equation iiri dC /0, εε= , where Ci and di are capacitance and

thickness of the layer respectively. The resulting values of permittivity corresponding to the basic Stern model is a low value of ~6 (metal electrodes) or ~40 (metal hydroxides) [5] in the Stern layer, and only slightly reduced value relative to the bulk value in the diffuse layer. However, the values of permittivity are sensitive to the particular model applied [5,6]. Knowledge of relative permittivity εr,i of water from molecular simulation would thus enable to assess various models and verify their applicability to study interfacial phenomena.

In the literature, most of the works on interfacial permittivity address refinement of the EDL models by introducing distance- or field-dependent permittivity [7,47,48] in order to solve the Poisson-Boltzmann equation more accurately. The resulting picture is permittivity monotonously decreasing over the Stern layer towards the surface, as a consequence of a strong preferential orientation of water dipoles inside the Stern layer due to the strong electric field generated by the surface and adsorbed ions. However, the relations that describe the variance of permittivity take the same form as for homogeneous bulk systems, thus completely ignoring molecular nature of water in the interfacial region exhibiting strong inhomogeneities in density, dipole orientation, charge distribution and related quantities. These effects might contribute to the electrostatic energy by an amount comparable to, or even exceeding, the ionic contribution [9]. As argued by Bockris and Reddy [49], the EDL picture has often been “ionic centric” rather than “water centric.”

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2.2.2 Effect of Size of a Spherical Colloid Particle on Structure of Interfacial Water

While papers [A] and [B] investigate properties of liquid at a planar surface, this issue addresses the role of geometry, i.e. curvature, for the surface for structure of interfacial liquid. The knowledge of the structure (both spatial and orientational) of fluids at solid surfaces is of great importance from both the technological and scientific points of view. The structure of water around large biomolecules or membranes affects their biological function. Environmental applications are also of exceptional interest since some surfaces in contact with contaminated solutions are capable of adsorbing hazardous materials which then can be transported elsewhere for further processing [39]. In these examples, the curvature of the solid has an impact on its functionality. Hence, understanding the effect of curvature is essential for design and optimal performance of novel devices.

The structure of the fluid is affected by two main factors: the intermolecular interactions and the geometry of the surface. The interaction of a fluid molecule with the surface is given, in general, by its interaction with all atoms/molecules of the surface as the sum of the direct site–site pair contributions (provided the pair additivity of the interaction holds). However, in some cases as, e.g., when the solid atoms are relatively small and closely packed, or when it is the presence of the surface itself which is more important than the specific individual atom–atom interactions, the surface can be treated as structureless and the sum can be replaced by an effective interaction. In paper [C], such effective or “coarse-grained” potential is derived for interaction between a point particle and a spherical particle of an arbitrary size for an inverse power law form of the site–site interaction. The coarse-grained potentials between interacting bodies have been studied for a number of geometries, though often restricting tor–6 dispersion term only. These include a potential between two spherical homogeneous particles [50–53], a point particle and a spherical surface (motivated by study of fullerene C60–C96 buckyballs) [54,55], a point particle and an infinitely long cylinder or fiber (developed to study an interaction of aerosol particles with fibrous filters) [56,57], or two torus-shaped colloidal particles [58].

2.2.3 Mutual Diffusion in Multicomponent Liquids Mutual diffusion is of high importance for chemical engineering, being the rate determining step in many technological processes, such as distillation, absorption or extraction. Generally, diffusion coefficients are determined by experimental techniques, theoretical or empirical approaches and molecular simulations. However, the experimental techniques are mostly designed for binary mixtures and their use for multicomponent mixtures is hindered by serious complications [25,26]. The theoretical and empirical approaches (see for example Ref. [27-29]) often fail in predictive applications, especially when highly polar and hydrogen-bonding liquids are considered, because they relate mutual diffusion coefficients to one-component properties or simplify the interaction between unlike molecules. In this context, molecular simulation offers a promising alternative. In paper [D], a simulation technique was applied on the ternary system water + methanol + ethanol for which no experimental or other data had previously existed. Although other alternative purely simulation approaches have been introduced, mostly based on Kirkwood-Buff theory, such as Ref. [36-38], they have been tested for simpler systems than the strongly non-ideal aqueous alcohol mixture.

10

3. GOALS OF THE THESIS

The general aim of this thesis is development of molecular simulation methods and their application on solid-liquid interfaces. It covers both methodology advances and application of molecular simulations to explore interesting behavior of confined liquid at the interface. Specifically, the goals in the methodology part have been:

1. development and test of existing methods with respect to their applicability on liquid at solid interfaces. In particular, the use of non-equilibrium methods was investigated.

2. generalization of simulation techniques for mixtures. Non-equilibrium technique for determination of shear viscosity, and equilibrium technique for determination of Fick diffusion coefficients in homogeneous systems are concerned.

The application part encompasses:

1. determination of profiles of the studied properties, as a function of the distance from the surface. This includes structural properties, e.g. radial density and orientational structure, and relative permittivity of interfacial layers, as well as dynamic properties, e.g. viscosity and diffusivity. Results are discussed with respect to theoretical models used for interpretation of experimental data.

2. examination of the effect of geometry of the solid on structural properties of interfacial solvent.

3. prediction of mutual diffusion coefficients for systems where other methods fail.

11

4. THEORETICAL BACKGROUND

4.1 Molecular Simulation Methods

4.1.1 Transport Properties Transport properties are derived from the classical fluid dynamics [2]. Within the non-equilibrium simulation approach, the flow of liquid generated by the driving force of the perturbation is directly simulated. In the equilibrium approach, a relationship is established between a transport coefficient and the time integral of the correlation function of the corresponding microscopic flux in a system under equilibrium (Green-Kubo formalism) [1,8]. Alternatively, instead of evaluating the time integral of the correlation function, the associated Einstein relation [1] can be employed. For transport properties, MD is preferable to MC because of availability of molecular velocities.

For shear viscosity η, the non-equilibrium approach is based on the relation [2,1]

γη

xzP−= , (1)

where Pxz is shear stress (the off-diagonal component of the external pressure tensor) and γ is shear rate. To study viscosity of liquid in the vicinity of a surface, the NEMD technique based on Poiseuille flow in the slab geometry [1,14] is particularly appealing. In this case, shear is generated by a homogeneous force field Fx applied in the direction parallel with the surfaces (along the x axis) leading to the dependence of the quantities in eq. (1) on the distance from the surface (measured by the coordinate z centered in the middle of the slab)

( ) ( )∫ ′′=z

xxz zdzFzP0

ρ ,

( ) ( )z

zvz

x

∂∂

=γ ,

(2)

where ρ(z) and vx(z) are number density and drift velocity respectively. Regarding the equilibrium approach, the Green-Kubo relation for viscosity reads [1]

∫∞

=0

)0()( xzxz PtPdtV

βη , (3)

where β = 1/kBT denotes the Boltzmann factor, V stands for the volume, and the value of shear stress Pxz(t) at time t can be calculated from EMD simulation as

∑∑∑∑∑= ≠ = == ∂

∂−=

N

i

N

ij

k

a

l

bz

ab

ijxij

N

i

zi

xii

xz

r

urvvmP

1 1 11 2

1, (4)

in which the first term is momentum flux, the second term is virial, and the indices a and b count the interaction sites while the indices i and j denote the molecules of the system.

12

Similar pair of relations can be obtained for the self-diffusion coefficient Dself. The continuum transport equation founding the non-equilibrium approach relates the molar flux J to the gradient of concentration c [1,8]

cD ∇−= selfJ , (5)

while the equilibrium expression is based on individual molecule velocity v auto-correlation function [1,8]

)0()(3

1

0

self vv tdtD ∫∞

= . (6)

In liquid mixtures, diffusion is rigorously described by Fick's law or MS theory [28]. Fick's law relates the molar flux to the set of independent gradients of concentrations

∑∑−

=

−

=

∇−=∇−=1

1

1

1

n

jjij

n

jjiji xDcD ρJ , i = 1,..,n-1 , (7)

where indices i and j distinguish between components of the total n-component mixture, ρ is the molar density and xj stands for the mole fraction of component j. In MS theory, the driving force of diffusion is the gradient of the chemical potential µi, which is assumed to be balanced by a friction force that is proportional to the mutual velocity ui - uj between the components

∑=≠ ∆

−−=∇

n

ij ij

jiji

x

1

)( uuµβ , i = 1,..,n . (8)

Both theories describe diffusion independently by a set of either Fick diffusion coefficients Dij or MS diffusion coefficients ∆ij. Clearly, a relation exists between both sets of diffusion coefficients [28]

ΓBD 1−= , (9)

in which all the three symbols represent (n-1)×(n-1) matrices. D is the matrix of Fick diffusion coefficients Dij, the elements of the matrix B are the following functions of MS diffusion coefficients

∑=≠ ∆

+∆

=n

ij ij

j

in

iii

xxB

1

,

∆−

∆−=

inijiij xB

11, (10)

and the thermodynamic factor ΓΓΓΓ is defined by

jkxpTj

iiijij x

x

≠∂

∂+=Γ

,,

lnγδ , (11)

where δij is the Kronecker delta and γi is the activity coefficient of component i. Hence, the MS diffusion coefficients can be transformed to the Fick ones and vice versa if the thermodynamic factor is known. However, there is a fundamental difference between both sets of coefficients. Only the Fick diffusion coefficients can be measured experimentally since they are defined using directly observable quantity, such as concentration or mole fraction, whereas in the definition of MS diffusion coefficients, chemical potentials appear. On the

13

other hand, MS coefficients can be conveniently obtained from EMD simulation using the Green-Kubo expression based on the net velocity correlation function [29]

( )ijij LfB =−1 , ∑∑∫==

∞

=ji

N

llj

N

kkiij tdt

NL

1,

1,

0

)()0(3

1vv , (12)

where N is the total number of molecules, Ni is the number of molecules of component i, vi,k(t) denotes the velocity of molecule k of component i at time t, and the exact functional form f of the elements of B-1 matrix, cf. eq. (9) , depends on the total number of components in the mixture. Note that the convergence of time integral in eq. (12) is slowed down compared to the case of self-diffusion coefficient, cf. eq. (6), where the correlation function reflects single-molecule property and thus can be averaged over all molecules of the given component.

4.1.2 Dielectric Properties Similar to transport properties, the equilibrium and the non-equilibrium approaches can be distinguished for determination of dielectric properties in the framework of the theory of dielectrics [3].

The equilibrium approach derived for homogeneous isotropic systems using linear response theory relates relative permittivity εr to fluctuations of the total dipole moment M as

( )22

0r

rr )12)(1(MM −=

+−Vε

βε

εε, (13)

where ε0 is the permittivity of vacuum. This equation is known as Kirkwood fluctuation formula.

The non-equilibrium expression for permittivity follows from the constitutive relation [3]

ε rε 0E = ε 0E + P , (14)

where E is the total electric field, which can be decomposed as the sum of the external field Eext and the polarization field Ep due to polarized liquid

E = Eext + Ep . (15)

The polarization field typically acts opposite the external field, so that the total electric field is screened on filling the volume with dielectrics proportionally to its permittivity. For the slab geometry and external homogeneous electric field acting in the direction normal to the surfaces, polarization field has non-zero only its z-component, normal to the surface,

zEE pp ≡ . Its value can be derived from electrostatics of a charge distribution ρq using the

Gauss’s law [3]

( ) zdzzEz

q ′′= ∫∞−

)(1

0p ρ

ε, (16)

or alternatively from the point dipole approximation by averaging molecular dipoles to yield the local value of polarization vector P. Having the polarization vector, the electrostatic potential φp and the corresponding polarization field are given by [3]

14

( ) [ ] z'z

z'zz'Pdz'

z'zy'x'

z'zz'Pdy'dx'dz'z

LL

−−=

−++

−= ∫∫∫∫∞

∞−

∞

∞−

)(2)(

))((4

02/3222

0

p0 πϕπε ,

( )0

p

)(

εzP

zE −= ,

(17)

where L is the thickness of the slab and zPP ≡ .

15

4.2 Molecular Models

Throughout this work, rigid, non-polarizable molecular models of united-atom type were used. These simple models account for the intermolecular interactions, including HB, by a set of LJ sites and point charges which may or may not coincide with the LJ site positions. The potential energy uij between two molecules i and j can thus be written as

( ) ∑∑= =

+

−

=

k

a

l

b ijab

ba

ijab

ab

ijab

ababijabij r

qq

rrru

1 1 0

612

44

πεσσε , (18)

where a denotes an interaction site on molecule i, b a site on molecule j, and k and l are the numbers of sites on molecules i and j, respectively. The LJ size and energy parameters are σab and εab, while qa and qb denote the point charges.

For water, the TIP4P/2005 model [59] or SPC/E model [60] were used. They consist of one LJ site and three point charges. The TIP4P/2005 model was appreciated to be the most successful among the rigid non-polarizable water models with respect to various structural and transport properties [61]. The SPC/E model has been the most widely used model among simulation studies of water, including our works [9-11]. The molecular models for methanol and ethanol were taken from Refs. [62,63]. They consist of two (methanol) or three (ethanol) LJ sites and three point charges each. These models proved to perform well not only for pure substances, but also for their mutual mixture or aqueous solutions regarding both static and dynamic properties [24,64]. Finally, carbon model comprises only LJ parameters that were taken from Ref. [65]. This model is commonly used to describe the interaction of carbon atoms in various morphologies.

The rutile wall, introduced in a detail in Refs. [9,66], represents a realistic, fully atomistic model of the (110) surface. The wall consists of four TiO layers, where the two deepest maintain the strictly periodic bulk crystal structure while the other two are ab initio relaxed [66]. The surface is terminated with rows of bridging oxygens protruding out of the interface layer towards the aqueous phase, see Fig. 1. The interaction parameters between the wall and water are based on the ab initio derived potentials [9,66]. These potentials describe the Ti-O(water) in terms of Buckingham potential while the O(wall)-O(water) interaction is given by LJ potential. The results of this thesis were obtained for fixed wall atoms in their potential minimizing positions since no significant effects of the surface flexibility on the studied properties were observed.

To define a molecular model for a multicomponent mixture on the basis of pairwise additive pure substance models, only the unlike interactions have to be specified. In the case of point charges, this can straightforwardly be done using the laws of electrostatics. However, for the unlike LJ parameters, there is no physically sound approach so that combining rules have to be employed for predictions. Here, the interactions between unlike LJ sites of two molecules were determined by the simple Lorentz–Berthelot combining rule

16

2bbaa

ab

σσσ += , and

bbaaab εεε = .

(19)

The mixture data presented below are therefore strictly predictive, because not a single experimental data point on mixture properties was considered in the model parameterization.

17

4.3 Simulation Details

In this subsection, the main technical details are overviewed to outline the simulation techniques employed and simulation parameters used. The explicit values or full description are sometimes omitted as each system requires specific simulation details for optimal modeling. For more detailed information, the reader is referred to Simulation Details sections of the original papers [A-D] presented in Appendices.

Interfacial properties at planar rutile surface were investigated by MD simulation in the slab geometry using an own parallel simulation tool developed by the author from uncompleted supervisor’s code. The simulations were performed in the canonic (NVT) ensemble with temperature maintained by Nosé-Hoover thermostat [8]. The unit simulation cell was a rectangular prism with a base parallel to the surface of an area 35.5×39.0 Å2. These lengths were chosen to match 12×6 replicas of the unit rutile (110) cell. The length of the simulation cell in the normal direction was adjusted so as to match the desired density of solution in the bulk region. The number of molecules, density and lengths of the unit cell resulted in the thickness of the slab of about 70 Å. Since the inhomogeneous interfacial region extends typically 15 Å from the surface, the remaining space occupied by the bulk phase covered the distance of about 40 Å. Although the simulated system is 2D-periodic in the plane of surface vectors, periodic boundary conditions were applied on the unit simulation cell in all three directions as long-range electrostatic interactions are faster evaluated in 3D-periodic systems. The Ewald summation technique using the dipole correction term by Yeh and Berkowitz [67] was employed with sufficient spaces of vacuum between periodic replicas in z-direction ensuring an excellent agreement with the results from the rigorous two dimensional Ewald summation [8,67].

To study the effect of size of a spherical colloid particle on structure of interfacial water, MD simulations were utilized in GROMACS [68]. Here, Berendsen thermostat and barostat were applied to keep average temperature 300 K and pressure 1 bar. Particle-mesh Ewald summation was used to treat the long-range electrostatic interactions.

Prediction of mutual diffusion in multicomponent homogeneous liquids was done by MD as well as MC simulations with the program ms2 [69]. The simulations were performed in the NVT ensemble. Electrostatic long-range corrections were considered by the reaction field technique with conducting boundary conditions (εRF = ∞). MC simulations were employed to sample chemical potentials. Because the system under study is an associating liquid with a high density, Widom’s test molecule insertion [8] is inappropriate. Here, the gradual insertion method [70] was used, where instead of inserting a complete test molecule, a fluctuating molecule is introduced that appears in different states of coupling with the other molecules.

Generally, the number of molecules and the cut-off radius were chosen so as to suppress finite size effects, which the transport properties may be particularly sensitive to [24]. The production as well as equilibrium runs were carried out over large number of steps of the order 106 or higher. The statistical uncertainties of the predicted values were estimated with the block averaging method [71]. In MD simulations, the fifth-order Gear predictor-corrector integrator [8] or the Verlet algorithm [8] for simulations performed in GROMACS were used to propagate the trajectories with the time step of 0.9-1 fs.

18

5. RESULTS AND DISCUSSION

5.1 Interfacial Properties of Aqueous Solutions at Planar Rutile Surface

Interfacial behavior of viscosity of aqueous methanol solution and relative permittivity of pure water were investigated at ambient conditions. The following results were obtained using TIP4P/2005 water model and nonhydroxylated version of our rutile surface model, depicted in Fig. 1 and described in Molecular Models chapter of the Theoretical Background section.

aqueoussolution

bulk TiO2

bridgingoxygen

Fig. 1: Structure of the neutral nonhydroxylated rutile (110) surface. The Ti atoms are gray while O atoms are red.

5.1.1 Viscosity of Mixtures by Non-equilibrium Molecular Dynamics To generalize the existing NEMD method based on calculation of density and drift velocity profiles, cf. eqs. (1) and (2), the formulas for shear stress and shear rate have to be modified to reflect all-component properties. Starting from the equation of motion for the liquid in steady state [2], which establishes balance between friction and the external force

∑=

+∂

∂−=n

ii

xi

xz

Fz

P

1

0 ρ , (20)

where xiF is the external force acting on a mole of component i with molar density ρi, it turns

out that shear stress of a mixture is the sum of all component contributions

( ) ( )∑ ∫=

′′=n

i

z

ix

ixz zdzFzP

1 0

ρ . (21)

19

The generalization of shear rate formula (2) requires substitution of the drift velocity xv as in

a mixture each component can drift with a different velocity xiv . When the equation for the

steady flow of a viscous liquid [2] is written in terms of additive quantities, i.e. momentum density and mass density, the expression for the shear rate reads

( ) ( )z

zv

zM

zvzM

zz

x

n

iii

xi

n

iii

∂∂

=

∂∂=

∑

∑

=

= COM

1

1

)(

)()(

ρ

ργ , (22)

where Mi is the molar mass of component i. Hence, the shear viscosity of a mixture can be calculated in the same fashion as in the case of a pure fluid provided the shear stress is a sum of all components’ shear stresses and the streaming velocity of the given layer of the fluid is

characterized by the center of mass (COM) velocity of the layer ( )zvxCOM . While the latter

result might seem intuitive, the preceding works on inhomogeneous viscosity [18,19] considered only the solvent velocity and differences between streaming velocities of individual components were omitted. Although both points can be subtle, namely (i) for low salt concentration the solvent contribution by far dominates that of ions and (ii) in many situations the streaming velocities of individual components are similar and therefore close to the streaming velocity of COM of the layer, there are examples in which it is essential to calculate viscosity using generally valid eq. (22) for shear rate instead of a solvent velocity derivative or other one component formula. As the particular example of considerably different streaming velocities of individual components are electrokinetic or electroosmotic experiments. However, even applying constant acceleration on all components, as e.g. in the gravity driven flow, does not generally guarantee equal streaming velocities of all components, owing to unequal mobilities affected by different strengths of interactions and shape of molecules. Moreover, in interfacial region, different interactions with the surface (e.g. in aqueous solution of ions) can lead to different streaming velocities of ions strongly interacting with the surface and less interacting water or vice versa.

The new NEMD method yields viscosity for the given set of external forces xiF . To

verify that the value of viscosity is independent of the particular combination of forces used (within the linear regime), three independent simulations were carried out for a binary mixture, in which the external force was applied to one component, to the other, or to both of them, respectively. The resulting profiles of distance-dependent viscosity and related quantities is shown in Fig. 2 for water + methanol mixture at methanol mole fraction xMeOH = 0.31. Although a different velocity, and thus shear rate profiles developed in each simulation, and the same holds for shear stress, the profiles of viscosity are the same within their statistical uncertainties, including the interfacial region.

5.1.2 Viscosity in the Bulk Region Viscosity of homogenous isotropic systems can be calculated by both NEMD, cf. eqs. (1),(21),(22), and EMD, cf. eqs. (3),(4). In Fig. 3, the bulk values of viscosity following from the present non-equilibrium method are compared to the results of the equilibrium method [64] as well as to experimental data [72-74] for water + methanol mixture. The present results

20

show a perfect agreement with the equilibrium method, and, thanks to the performance of molecular models used, they also reproduce the experimental data very well. It should be pointed out that the EMD data were obtained using the same molecular models, so a true comparison of two simulation approaches is given in Fig. 3.

-30 -20 -10 0 10 20 30

v xCO

M [m

/s]

0

5

10

15

20

25

30

350 10 20 30

z [Å]-30 -20 -10 0 10 20 30

γ [1

010 s

-1]

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.00 10 20 30

z [Å]-30 -20 -10 0 10 20 30

Pxz

[106 P

a]

-30

-20

-10

0

10

20

30

z´ [Å]0 10 20 30

z [Å]

z´ [Å] z´ [Å]a) b)

d)c) z´ [Å]0 10 20 30η

[10-4

Pa.

s]

5

10

15

20

25

30

35

40

z [Å]-30 -20 -10 0 10 20 30

Fig. 2: (a) Center of mass streaming velocity, (b) shear rate, (c) shear stress, and (d) resulting viscosity profiles for water + methanol mixture for mole fraction xMeOH = 0.31. Results of three independent simulations for three different protocols of external forces per molecule are shown: Fx

w = 5.52×10-13 N, Fx

MeOH = 0 (red); Fxw = 0, Fx

MeOH = 9.32×10-13 N (blue), and Fxw = 1.38×10-13 N, Fx

MeOH = 2.07×10-13 N (green). The z-coordinate is centered in the middle of the slab. The coordinate z' = Lz/2 - |z|, which gives the distance from a surface, is given on the top of the graphs.

5.1.3 Interfacial Viscosity In the preceding paragraph, viscosity in the bulk region was discussed to provide the comparison of our results to the equilibrium approach and experimental measurements. Nevertheless, the key advantage of the present non-equilibrium method is that it enables prediction of distance-dependent viscosity throughout the whole slab, including interfacial regions close to the surfaces where strong viscosity variation can be expected due to fluid–surface interactions and inhomogeneous structure. Consequently, behavior of viscosity in the interfacial region depends on specific details of surface-liquid interactions and geometry (smoothness) of the surface.

21

xMeOH

0.0 0.2 0.4 0.6 0.8 1.0

η [1

0-4 P

a.s]

6

8

10

12

14

16

18 our NEMD simulationEMD simulation, Guevarra et al.exp. data, Mikhail et al.exp. data, Isdale et al.exp. data, Kubota et al.

Fig. 3: Composition dependence of shear viscosity of water + methanol mixture. Present NEMD simulation results from the bulk region (full squares) are compared to EMD simulation data (open squares) and to experimental data (triangles).

ρ [g

/cm

3 ]

0

1

2

η [1

0-4 P

a.s]

0

5

10

15

20

25

30

35

z´ [Å]0 5 10 15 20 25

density of waterdensity of MeOHstreaming velocitynumerical viscosityshear

v xCO

M [m

/s]

γ [1

010 s

-1]

Fig. 4: Detailed view of the properties of the interfacial region as functions of the distance from the rutile wall for mole fraction xMeOH = 0.31. The values of center of mass streaming velocity (red) and numerical viscosity (green) are shown in corresponding units on the left axis. Densities of water and methanol (black), and shear rate (blue) calculated as numerical derivative of streaming velocity are given by the right axis.

22

Fig. 4 reveals interfacial profiles of the properties relevant for viscosity calculation in a detail. As can be seen from the density profiles, surface-liquid interactions cause strong ordering of the liquid forming an inhomogeneous region, which typically extends up to 15 Å from the surface. In this region significant changes of dynamic properties of the liquid take place. The streaming velocity profile is not parabolic as in the bulk region but passes through an inflection point identified by the extreme of shear rate. Closer to the surface, absolute value of shear rate decreases while shear stress monotonically increases in agreement with eq. (21). This behavior combined leads, according to eq. (1), to a viscosity increase by several times of its bulk value. The streaming velocity and accordingly the derived properties (shear rate, viscosity) were not considered closer to the surface than the distance of the second liquid layer because of poor statistics resulting from lack of molecules in a gap between first and second liquid layers. However, the streaming velocity of the first layer is zero in the case of rutile surfaces, indicating strong adsorption, no-slip boundary conditions and huge or virtually infinite viscosity of the first layer.

z [Å]

0 5 10 15 20

η [1

0-4 P

a.s]

0

20

40

60

80

100

120

140 waterxMeOH = 0.3

xMeOH = 0.6

xMeOH = 0.8

methanol

z' [Å]

Fig. 5: Left: Profiles of shear viscosity in the interfacial region as functions of the distance from the rutile wall for selected mole fractions of methanol xMeOH. Right: Numerical density profiles of water (─) and methanol (− −) for selected mole fractions of methanol xMeOH.

The composition trend of viscosity in the interfacial region is different than in bulk.

While viscosity in the bulk reaches maximum for xMeOH ≈ 0.3 (Fig. 3), its increase in the interfacial region is largest for xMeOH ≈ 0.6 - 0.8, as indicated by Fig. 5. We were able to link this effect to number of hydrogen bonds (HB) between adjacent layers. The most rapid increase of viscosity leading to deviation from the bulk-like composition dependence takes place within ~5 Å from the surface, which corresponds to the region of the first two liquid layers. We understand this viscosity increase to be related to the number of HBs formed between the first and the second liquid layers. As the external force is given per molecule, number of HBs between the first and the second layer per molecule in the second layer was

0.00

0.05

0.10ρ [Å

-3]

0.00

0.05

0.10

0.15

pure SPC/E

xMeOH = 0.3

xMeOH = 0.6

pure MeOH0.00

0.05

0.10

0 2 4 6 8 10 12

0.00

0.05

0.10

0.15

0.20

z´ [Å]

pure TIP4P/2005

pure MeOH

23

analyzed, denoted as HBn 21− . The composition dependence of HBn 21− was found to be in qualitative

agreement with the composition dependence of viscosity, having minimum for pure water (0.8 HB/molecule), raising towards the maximum at xMeOH = 0.6 - 0.8 (1.1 HB/molecule) and decreasing for higher xMeOH (0.9 HB/molecule for pure methanol). Similar characteristics

were gathered concerning the number of bonds between the second and the third layers HBn 32− .

The composition dependence of HBn 32− is qualitatively the same as the bulk one. This is also in

agreement with Fig. 5, where the viscosity composition trend follows bulk behavior (with a maximum around xMeOH = 0.3) for all distances further the break-event point about 5 Å from the surface, which can be identified as the beginning of the third layer.

Fig. 5 also shows that the increase of viscosity is limited to a region extending over the distance of ~15 Å from the surface, similar to the case of density variations. This implies that significant changes affecting flow dynamics of confined liquid can be expected for flows in channels or pores of width of a few nanometers. In other words, flow of liquid in nanospace should not be described by experimental values of transport coefficients measured in the bulk phase.

5.1.4 Relative Permittivity by Non-equilibrium simulation The non-equilibrium method for determination of relative permittivity, introduced by eq. (14), becomes more subtle when a system with an interface is considered. Because of a wall-induced polarization, see Fig. 6, which exists even at vanishing external electric field, polarization is no more proportional to the external field. To describe the response of liquid to the changes induced by the external field, the values of polarization and field in eq. (14) were replaced by their changes ∆P and ∆E with respect to the reference values at zero external field

εrε0∆E = ε0∆E + ∆P . (23)

By construction, ∆P as well as ∆E = Eext + ∆Ep vanish for vanishing external field. The modified constitutive relation (23) thus enables comparison of permittivity in the bulk and at an interface for the same Eext. Note that outside the interfacial region the wall-induced polarization is zero as the bulk structure is recovered and eq. (23) reduces to the standard expression (14).

Considering the relation (17) between polarization and field, eq. (23) implies the following expression for permittivity

( )zSFzEE

E

zE

Ez

−=

∆+=

∆=

1

1

)()()(

pext

extextrε , (24)

where we introduced the term screening factor extp EESF ∆−≡ , a dimensionless factor

describing the portion of external field that is screened by the polarization response.

5.1.5 Dielectric Properties of Interfacial Water Layers MD simulations of water slab between parallel rutile walls were performed under a strong external electric field applied in the direction normal to the surfaces. Particular attention was paid to the polarization of interfacial layers aiming to estimate their relative permittivity.

24

Ep(

z) [V

/Å] 5

0

-5

Ep(

z) [V

/Å]

-10

-5

0

5

z [Å]0 2 4 6 8 10

Ep(

z) [V

/Å]

-20

-15

-10

-5

0

5

L2 L3L1

Eext = -1 V/Å

Eext = 0 V/Å

Eext = 1 V/Å

Fig. 6: Left: Snapshots from simulations of water at rutile wall: a lateral look at a block of water molecules above 6×3 replicas of the unit rutile (110) cell for the external electric field -1 (top), 0 (center) and 1 (bottom) V/Å. The preferential orientation of molecules in interfacial water layers results in non-zero polarization even when no external field is applied. Right: Polarization field as a function of the distance z from the surface. The results based on simple atomistic electrostatics (− −), cf. eq. (16), are compared to the ones from the molecular theory of dielectrics (─), cf. eq. (17). The vertical lines indicate boundaries of the first three molecular layers L1-L3.

First, possibility of calculation of permittivity on a molecular scale was tested, as the

definition of permittivity (14) or (23) relies on the point dipole approximation for molecular electric field, accuracy of which might become poor when the volume is restricted to a molecular length in one dimension. To this end, polarization field Ep calculated by the dielectric expression (17) (molecular approach), i.e. based on the point dipole approximation, was compared to the result of the electrostatic expression (16) (atomistic approach). In the first approach the polarization field of the given bin stems from the molecules with center of mass in the bin while in the latter approach the charged sites located in the bin matter,

- 1 V/Å

0 V/Å

+ 1 V/Å

25

regardless the position of the molecules which they belong to. Fig. 6 (right) compares the two approaches for the external electric fields -1, 0 and 1 V/Å. The results differ at the interface, particularly in the first two water layers (L1 and L2). Hence, the field inside a layer of a molecular thickness is not accurately described by the field of point dipoles. This should not, however, lead us to disqualification of the dielectric expression (17), because the studied dielectric properties descend from the same theory and thus their values can be determined to the same order of approximation. The difference between both approaches gives us an estimate for the principal inaccuracy of calculation of dielectric properties in the molecular layers. On the other hand, such inaccuracy is also present in the models of a solid-liquid interface that have served as a motivation for the present calculations. In addition, the difference between both approaches is rather quantitative, and it might be further alleviated by averaging of the polarization field over the layer and subtraction of zero field values since in calculation of permittivity the change of the mean polarization field with the external field strength appears. Bearing in mind the limitation of calculation of interfacial polarization and derived properties, we proceeded further.

To assess relative permittivity, the screening factor extp- EESF ∆≡ is of the crucial

importance since it compares the change of the polarization field to the change of external field, i.e. the ability of matter to weaken the external field. SF = 0 if the matter does not react to the external electric field at all (εr = 1) and SF = 1 if the matter is able to cancel the external field completely (εr = ∞). The values of the screening factor based on the mean polarization fields in the layers L1-L3 and in the bulk are plotted in Fig. 7 over the range of external fields. An immediate striking result of this figure is that some values are larger than 1. We will refer to this phenomenon as overcompensation since it occurs when the magnitude of the polarization field response exceeds the magnitude of the external field, so the change of the total electric field is oriented oppositely to the external field. An ultimate consequence is a negative value of permittivity defined by eq. (24).

To show that such unexpected dielectric behavior, unknown from our experience from the bulk phase, is not only an effect of the dipole approximation discussed above, results from both electrostatic (atomistic) and dielectric (molecular) approaches are reported in Fig. 7 (left). The differences in polarization profiles, shown in Fig. 6, remain evident on values of the screening factor in Fig. 7 (left). Nevertheless, qualitative outcomes of both approaches are the same: they yield the same statement about the presence (or absence) of the overcompensation for the physically interesting range of small external fields. In the following, only results of the dielectric (molecular) approach are discussed because of its simple assignment of polarization to the layers.

The dependence of the screening factor on the external field is replotted with a uniform vertical scale in Fig. 7 (right), together with averaged cosine of the angle between the water

dipole and the surface normal µ

µ

µµ

izzz L)(

= , where µ z(z) is the z-component of individual

water dipole moment and µ = 0.48 eÅ is its magnitude (for TIP4P/2005 model). It is instructive to plot these two quantities together because, in the present approach, dielectric properties are derived from the polarization response which is governed by changes in dipole orientation. Considering the relation

26

iiizzzPE zL LL

00p )()(

1)(

1 µρεε

−=−= , (25)

and very small variations of densityi

zL

)(ρρ = with external field, the value of the screening

factor at given Eext is thus proportional to the slope of the secant line between the value of average cosine at Eext and 0. Hence, the value of the screening factor can be viewed as originated by two factors: local dipole susceptibility, i.e. the response of individual dipole orientation to the field, and local density. The role of density is straightforward – if each water dipole changed its orientation by the same amount as in the bulk, the polarization response and thus the screening factor of a layer would be enhanced proportionally to the density. This proportionality holds in L1 for small fields, where the slope of cosine dependence is the same

as in bulk and thus 8.1bulk

L1

bulk

L1 ≈=ρρ

SF

SF. However, this proportionality does not work for

other layers and for stronger external fields. This makes clear that dielectric properties at the interface cannot be understood as those of homogeneous continuum with increased density, and therefore, local dipole susceptibility is the crucial factor.

Again, this dipole susceptibility is related to the slope of the external field dependence of dipole cosine, see Fig. 7 (right). The truly linear dependence of dipole cosine, corresponding to the constant SF, is inherently present at small external fields for bulk phase. Closer to the interface, surface interactions affect orientation of water dipoles, producing a nonzero polarization (µz /µ ≠ 0) at zero external field and altering the dipole cosine differently than in bulk. Particularly, interfacial forces hinder alignment of dipoles of L1 in the direction of the zero-field polarization (along positive z-axis), whereas they are less resistant to reorientation in the opposite direction. As a result, the increase in dipole cosine is smaller than linear for positive external fields and, equivalently, SF decreases. Different behavior is found in L2. Here, the dipoles are very reluctant to reorientation, which is identified with the low slope of the cosine dependence. As a result, the magnitude of the screening factor is only about half of the bulk value (for small fields), despite the enhanced density. Influence of rutile wall interaction persists even in L3 and continues the series of structured layers with oppositely oriented dipole moments: µz /µ = 0.76, -0.52, 0.05 in L1, L2, L3 respectively at Eext = 0. The slope of the cosine dependence in L3 for small external fields is slightly higher than the corresponding value in bulk. Consequently, the value of SF is larger, causing a weak overcompensation.

We repeated the present study for a smooth graphite wall modeled by 9-3 LJ potential to see the effect of different water-wall interaction. The qualitative outcome, including the overcompensation in L1, was the same, though the screening factor was smoother and closer to the value of 1 in the interfacial layers. The results of analogous study of a simple polar liquid interacting with graphite wall are given in article [23]. The overcompensation of the external field is observed in L1 even in that case.

27

Eext [V/Å]-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

− ∆E

p / E

ext

0.98

0.97

− ∆E

p / E

ext

1.05

1.00

0.95

− ∆E

p / E

ext

1.4

1.0

0.6

0.2

− ∆E

p / E

ext

2.2

1.8

1.4

1.0

0.6

L2

L1

bulk

L3

<µz>

/ µ

-0.5

0.0

0.5

− ∆E

p / E

ext

2.0

1.5

1.0

0.5

− ∆E

p / E

ext

2.0

1.5

1.0

0.5

<µz>

/ µ

-0.5

0.0

0.5

− ∆E

p / E

ext

2.0

1.5

1.0

0.5

<µz>

/ µ

-0.5

0.0

0.5

1.0

L2

L1

Eext [V/Å]

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

<µz>

/ µ

-1.0

-0.5

0.0

0.5

− ∆E

p / E

ext

2.0

1.5

1.0

0.5

bulk

Eext [V/Å]

L3

Fig. 7: Left: Dielectric screening in terms of the screening factor (SF) produced by water in the first three molecular layers (L1-L3) and in bulk. The results based on simple atomistic electrostatics (○), cf. eq. (16), are compared to the ones from the molecular theory of dielectrics (●), cf. eq. (17), that were fitted by polynomial (···) to interpolate the value at Eext = 0 V/Å. Right: Screening factor (●, left vertical axes) and average cosine of the angle between water dipole and the surface normal (□, right vertical axes) in the first three molecular layers (L1-L3) and in the bulk.

The values of the screening factor can be directly transformed to relative permittivity

through eq. (24). However, the values for small external electric field strengths that are available in common experiments are in our calculations corrupted by large uncertainties due to the factor of 1/Eext. Therefore, the dependence of SF on the external field strength was interpolated, as shown in Fig. 7 (left), and the relative permittivity was estimated from the

28

value of the interpolation function at vanishing external field. The resulting permittivity is listed in Tab. 1. Permittivity was also calculated by equilibrium approach using the total dipole moment fluctuation of the given layer, according to eq. (13). As can be seen from Tab. 1, non-equilibrium and equilibrium approaches yield completely different values. We understand this as a consequence of implausible application of the equilibrium formula on interfaces. The Kirkwood fluctuation formula is derived (i) for homogeneous isotropic fluid assuming that the studied system is symmetrically surrounded by dielectric medium [8], and (ii) for an NVT ensemble - however, in an interface, the number of molecules contributing to the total dipole moment of the given layer varies with time due to the molecules crossing the layer. When applied on homogeneous environment of bulk region, the fluctuation formula yields permittivity in agreement with non-equilibrium approach as well as data from the literature [61] for the TIP4P/2005 model: εr = 58 ± 3.

zmin [Å] zmax [Å] SF εr εrKirkwood

L1 2.11 2.98 1.78 (1) -1.28 (2) 33 (2) L2 2.98 4.04 0.48 (1) 1.92 (6) 59 (6) L3 4.04 8.47 1.015 (5) -69 (23) 49 (2) bulk 15 L-15 0.9825 (6) 57 (2) 58 (2)

Table 1: Relative permittivity of water in the first three molecular layers L1-L3 and in bulk for vanishing external field. Positions of the layers are given by interval (zmin, zmax) along the z-axis. Permittivity from the present non-equilibrium approach is compared to that from the equilibrium Kirkwood fluctuation formula (13). The numbers in parentheses indicate the statistical uncertainty in the last digits.

Our results of the distance-dependent permittivity at a solid-liquid interface are in

contrast with the common model picture [5-7], where relative permittivity of the interfacial water is presumed to monotonically increase from as low value as ~6 (depending on the nature of the surface; metal or metal-oxide surfaces are commonly considered) [5] at the first hydration layer to the bulk value reached in the diffuse layer. Apart from particular values, the common feature of the models is that they account for dielectric properties of the interface using the same relations as for homogeneous systems. E.g., permittivity of the interfacial water is believed to be lower than in bulk due to strong local electric field radiated by the surface and ions which lower permittivity according to a relation derived for homogeneous continuum, such as the one by Booth [75,7]

−−+=bE

bEbE

nnE rr

1)coth(

3))0(()( 22 εε , βµ)2(

6

73 2 += nb , (26)

where n is refractive index. This feature is also present in the equilibrium simulation approach based on the dipole moment fluctuation. Here, the decrease of permittivity is linked to

alignment of dipoles in the external field, in accordance with lowering of 22 MM − in

homogeneous isotropic NVT ensemble due to saturation of the total dipole moment M

towards its maximum value. In contrast, our results exhibit significant differences between the bulk and interfacial layers leading to the response of water molecules in the interfacial layers following different pattern than in the bulk (see Fig. 7). Particularly, it even depends on the

29

orientation of the external field relative the surface and might lead to the overcompensation of the external field by the polarization response, i.e. phenomena that do not occur in the bulk at any field strength. Overall, Fig. 7 presents a simulation evidence that interfacial permittivity cannot be described by the same value nor the same field dependence as in bulk.

The dielectric properties were attributed to two factors: local dipole susceptibility, i.e. ability of dipoles to change their orientations on applying the external field, and local density; contrary the common picture where relative orientation of the dipoles vs. the field plays the central role while dipole susceptibility and density are presumed to be the same as in bulk. Our findings are supported by Ref. [76] addressing rotational diffusivity of interfacial water layers in contact with rutile surface. It was found that one component of rotational diffusivity of first layer is exceptionally high, while rotational diffusivity of second layer is reduced compared to other layers as well as bulk. These findings were explained by the average number of HBs per water molecule being 2.08, 4.21, 3.82, and 3.70 for L1, L2, L3, and bulk, respectively. In this light it is not surprising that L2 is most reluctant to reorientation due to external field, a fact documented by its low screening factor. On the contrary, L1 forms fewer bonds, which are moreover weaker (longer) and therefore this layer can strongly respond to the external field, which causes a strong and even overcompensating screening factor, further boosted up by high density.

30

5.2 Effect of size of a spherical colloid particle on structure of interfacial water

SPC/E water in contact with spherical graphite particle of various sizes was studied in order to examine the effect of geometry of the surface on structural properties of interfacial liquid. To this end, a set of MD simulations was carried out spanning the entire range of sphere sizes - from a point particle, through a particle of a few nanometers in the diameter, up to the limiting case of an infinitely large sphere, which corresponds to a planar interface.

5.2.1 Coarse-grained Interaction Potential The interaction between a water molecule and the whole spherical particle was modeled by a single coarse-grained potential. The potential is derived by summing all site-site contributions related to the interaction sites inside the spherical particle that are assumed to be distributed homogeneously over the volume of the sphere. This derivation was done for a general site-site interaction of the form u(x) = x-n, where x is the distance between site on the water molecule and the given site in the spherical particle. The resulting coarse-grained potential is given by

=−= ∫V

udDrU )(),( xrx

( ) ( )( )

( ) ( )( )

−−−+−

−−−+

−=

−−−−

n

DrDr

rn

DrDr

n

nnnn

3

2/2/

4

2/2/

2

2 3344πρ,

(27)

where r is the distance between the center of the sphere and the site on the water molecule, D is the diameter of the sphere and ρ = 112.9125 nm-3 is the number density of interacting sites (atoms) within the sphere corresponding to mass density of graphite 2.25 g/cm-3. The formula (27) covers the two most interesting cases of Coulombic ~1/x and LJ

−

612

4xx

σσε interactions. Because an uncharged particle was simulated, the following

results are based on the site-site potential of the LJ form. The application of the coarse-grained potential in our simulations has significantly

shortened the CPU time, which becomes particularly extensive when the diameter of the spherical particle exceeds nanometer size. In addition, it enables a systematic study of the effect of the particle size on the properties of the interface without the need for taking into account specific surface interactions, e.g. special surface groups, charged atoms, etc. The resulting profile of the coarse-grained potential based on LJ interaction is plotted in Fig. 8 for a set of particle sizes as a function of the distance from the surface, z = r - D/2. Here, σ denotes the LJ parameter of SPC/E water. The smooth transition of the potential from a point-like particle to the planar wall, represented by LJ 9-3 potential, is evident.

31

z/σ1.0 1.5 2.0 2.5

U/(ερσ3)

-2

0

2

4

6

8

D = 0.5σD = 2σD = 5σD = 25σD = 100σplanar wall

Fig. 8: Dependence of the interaction with a spherical LJ particle of diameter D as a function of the distance z from the surface.

5.2.2 Structure of Interfacial Water The radial distribution functions between the center of the sphere and the oxygen atom of a water molecule for differently sized spheres, Fig. 9, show the obvious effect of the sphere size. Smaller sphere allows closer approach of water towards its surface as fewer surface atoms participate in the repulsion compared to the planar wall due to larger curvature of the surface. Interesting phenomenon is the height of the first peak which is not monotonous, but exhibits maximum in the 2σ - 5σ solute size range, which must be linked with the most convenient packing and/or hydrogen bonding structure of water molecules at these surfaces.

z=rSO - D/2 [nm]0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

gSO

0

1

2

D = 0.5σD = 1σD = 2σD = 5σD = 10σD = infinity

IaIb

Fig. 9: Sphere-oxygen radial distribution functions for differently sized spheres as a function of distance surface of the sphere. The density profile at planar wall is included (black).

32

Fig. 9 also indicates the split of the first hydration shell into two subshells, Ia and Ib, for which the angular distribution of water molecules was analyzed in detail. The angular distribution of water molecules in the vicinity of the solute can be unambiguously described by angular bivariate plots [77], 2D-histograms describing the orientation of the water molecule relative to the normal of the surface (center of sphere – oxygen vector) in terms of cosϑ, where ϑ is the vector between the surface normal and the dipole vector of the molecule, and angle φ, formed by the projection of the surface normal to the plane perpendicular to the molecule’s dipole moment. The bivariate plots for differently sizes spheres and the planar wall, Fig. 10, show first of all that orientation changes are evident in the Ia subshell, while the structure of subshell Ib does not change significantly. With the help of numbering selected orientations of water molecules relative to the solute depicted in Fig. 11, and their location on bivariate plots, the effect of increasing solute’s size can be identified as decreasing population of the orientation I with two hydrogens and one lone-pair site straddling the sphere, and orientation II with one hydrogen site and two lone-pair sites straddling the sphere. At the same time, orientation III with the molecular plane of water parallel to the surface becomes predominantly populated. The transition from the largest sphere studied, D=10 σ, towards the planar wall leads to even further reorientation of water molecules, with increased number of configurations V and VI, both pointing with the hydrogen site or the lone-pair site directly toward the surface and thus sacrificing the hydrogen bond of this site.

φ [d

eg]

0

30

60

90Ib

φ [d

eg]

0

30

60

90

cos ϑ-1.0 -0.5 0.0 0.5 1.0

φ [d

eg]

0

30

60

90

cos ϑ-1.0 -0.5 0.0 0.5 1.0

φ [d

eg]

0

30

60

90

cos ϑ−1.0 −0.5 0.0 0.5 1.0

φ [d

eg]

0

30

60

90

cos ϑ−1.0 −0.5 0.0 0.5 1.0

Ia

I

II

III

IV

V

VI

I

II

I

II

III III

D=0.5 σσσσ

D=1 σσσσ

D=2 σσσσ

D=10 σσσσ

planar wall

φ [d

eg]

0

30

60

90 0

1 2

III

D=5 σσσσ IbIa

II

Fig. 10: Bivariate plots of angular distribution of water molecules around differently sized spheres and planar wall. The reorientation of water molecules on increasing the size of the solute is mostly restricted to the subshell Ia, a narrow layer inside the first hydration shell.

33

I III V

I III

III V

II IV VI

II IV

IV VI

Fig. 11: Illustration of the identified orientations I–VI of the water molecules relative to the solute. The grey and purple sticks indicate the O–H bonds and the lone pair directions, respectively. The reference vector pointing from the water oxygen to the solute is also shown. The curved arrows show the rotation that transforms the given orientation to the indicated one.

34

5.3 Mutual Diffusion in Multicomponent Liquids

5.3.1 Thermodynamic Factor Mutual diffusion is investigated by means of molecular simulation for homogeneous liquid mixtures containing water + methanol + ethanol at ambient conditions. To calculate experimentally observable Fick diffusion coefficients, MS diffusion coefficients need to be converted using the thermodynamic factor, according to eqs. (9)-(11). The critical step of this

scheme is the calculation of the thermodynamic factor

jkxpTj

iiijij x

x

≠∂

∂+=Γ

,,

lnγδ .

Composition derivatives of activity coefficients γi appearing in its definition are commonly obtained from a regression of a thermodynamic model of excess Gibbs energy GE to experimental VLE data. We avoided this approach for (i) its high sensitivity to the underlying GE model, and (ii) the need of experimental data for desired substances and conditions. In the present approach, the GE model is regressed directly to the composition dependence of chemical potentials that were sampled by MC simulation. With respect to the following results, the Wilson model [34]

( ) ∑ ∑∑= ==

Λ−=≡n

i

n

jijji

n

iiiB

E xxxTkG1 11

lnlnγx , 1=Λ ii , (28)

with adjustable parameters Λij, i ≠ j, was found to favorably describe the composition variance of chemical potentials of the systems studied.

methanol0.0 0.2 0.4 0.6 0.8 1.0

ethanol

0.0

0.2

0.4

0.6

0.8

1.0

water

0.0

0.2

0.4

0.6

0.8

1.0

x2 / mol mol-1

x3 / mol mol-1x1 / mol mol-1

Fig. 12: Left: Compositions of the ternary mixture water(1) + methanol(2) + ethanol(3) for which the chemical potentials (×) and transport properties (○) were calculated. Right: Chemical potential of water (red), methanol (blue) and ethanol (green) in their ternary mixture as a function of mole fractions. The plot is based on the ternary Wilson model (28) that was fitted to the present simulation results.

35

5.3.2 Fick Diffusion Coefficients MS diffusion coefficients and chemical potentials were calculated for various compositions of the ternary mixture as depicted in Fig. 12 (left). Chemical potentials of all components were fitted by the ternary Wilson model, eq. (28) with n = 3. The resulting composition dependence of the chemical potentials for optimal parameters of the model is plotted in Fig. 12 (right). Since this picture shows chemical potentials diminished by the ideal mixing term lnxi, the composition dependence demonstrates strong non-ideality of the system. Having the Wilson model parameters, the MS diffusion coefficients sampled by EMD, cf. eq. (12), were transformed in the Fick diffusion coefficients listed in Table 6 of Paper [D]. To our best knowledge, these are the first results ever reported in literature for this ternary system.

Nevertheless, to verify our approach, we give the comparison with experimental data and the common simulation approach for the binary subsystems, where literature data are available. Fig. 13 (left) shows simulation results for chemical potentials in the binary mixtures and performance of the Wilson model that was fitted to chemical potentials of the ternary mixture (see Fig. 12). In Fig. 13 (right), the thermodynamic factor following from this fit is compared to the result of the common simulation approach, in which the Wilson model was regressed to experimental VLE data [78]. Clearly, both approaches give different thermodynamic factors, except for the ideally behaving mixture methanol + ethanol. Fig. 14 shows both types of mutual diffusion coefficients: MS diffusion coefficient (left) and Fick diffusion coefficient (right). The Fick diffusion coefficients from both simulation approaches, as obtained by multiplying the MS diffusion coefficient (left) with the corresponding thermodynamic factor from Fig. 13, are compared to experimental values [79,D]. Present simulation method yields a smoother composition dependence of the Fick diffusion coefficient for the aqueous binary subsystems, which qualitatively better reproduces experimental data than the common method based on VLE data. Although both methods yield comparable deviations from the experimental data points, the dependence resulting from the common approach is flatter and exhibit a broader minimum, which is most pronounced for the case water + methanol. On the other hand, both sets of the simulation results show a systematic deviation at the alcohol end of the composition range and a shift of the composition minimum, mainly caused by deviations of pure component properties. Considering that simple molecular models were used that were fitted to static properties only and the Lorentz–Berthelot combining rule was employed to describe the interaction between unlike LJ sites, the prediction quality with respect to mutual diffusion is nonetheless remarkable. For the mixture methanol + ethanol, both approaches predict the same values on account of ideality of this mixture, cf. corresponding composition dependence of chemical potentials and thermodynamic factor in Fig. 13.

Overall, the present simulation approach relying on direct calculation of chemical potentials is more successful in prediction of the composition dependence of Fick diffusion coefficient. At the same time, the approach is more plausible, as its results are consistently derived from the given molecular model, and it can be straightforwardly applied on ternary or higher multicomponent mixtures unlike the common approach limited by existence of experimental data at desired conditions.

36

0.0 0.2 0.4 0.6 0.8 1.0

µ i / k B

T

-12

-10

µ i / k

BT

-14

-12

-10

-8

xi / mol mol-1

µ i / k B

T

-16

-14

-12

-10

-8

methanol + ethanol

water + methanol

water + ethanol

Γ

0.0

0.2

0.4

0.6

0.8

1.0

xi / mol mol-10.0 0.2 0.4 0.6 0.8 1.0

Γ

0.0

0.2

0.4

0.6

0.8

1.0

Γ

0.0

0.2

0.4

0.6

0.8

1.0

methanol + ethanol

water + methanol

water + ethanol

methanol + ethanol

Fig. 13: Left: Chemical potentials of the binary subsystems water + methanol (top), water + ethanol (center) and methanol + ethanol (bottom) as functions of the mole fraction of the second respective component. The chemical potentials of water (■), methanol (●) and ethanol (▲) are denoted by full symbols. The corresponding open symbols denote the values after subtraction of ideal mixing term. The solid line (−) represents the ternary Wilson model. Right: Thermodynamic factor from the ternary Wilson model fitted to present simulation results for the chemical potentials (−) is compared to the Wilson model based on experimental VLE data reported by Hall et al. [78] at 298.15 K (– –).

37

D /

10-1

0 m2 s-1

0

5

10

15

20

xi / mol mol-10.0 0.2 0.4 0.6 0.8 1.0

∆ / 1

0-10 m

2 s-1

0

5

10

15

20

∆ / 1

0-10 m

2 s-1

0

5

10

15

20∆

/ 10-1

0 m2 s

-1

0

5

10

15

20

methanol + ethanol

water + methanol

water + ethanol

xi / mol mol-10.0 0.2 0.4 0.6 0.8 1.0

D /

10-1

0 m2 s-1

0

5

10

15

20

D /

10-1

0 m2 s

-1

0

5

10

15

20

methanol + ethanol

water + methanol

water + ethanol

Fig. 14: Left: Maxwell–Stefan diffusion coefficient of the binary subsystems water + methanol (top), water + ethanol (center) and methanol + ethanol (bottom) as a function of the mole fraction of the second respective component. Right: Fick diffusion coefficient. Present simulation results (○) are compared to simulation results using the thermodynamic factor from experimental VLE data (triangles) and to experimental data [79,D] (-■-). The straight lines serve as a guide to the eye.

38

6. CONCLUSION

This text summarizes results achieved during author's doctoral study, which led to four original papers [A-D]. The general aim of the conducted research has been to provide a molecular-level insight into phenomena that give rise peculiar properties of solid-liquid interfaces. Molecular simulations play a key role in this aim as they supply pieces of information that are inaccessible in experiments and allow testing predictions of theories, which necessarily simplify the concept of liquid, often not taking into account molecular nature of solvent. Our results, on the other hand, prove that molecular description is crucial when addressing solid-liquid interfaces. Another goal has appeared in due course of my work; namely tackling simulation methods for prediction of properties of liquid mixtures. To this end, a technique for determination of (distance dependent) shear viscosity of mixtures was developed and a technique for calculation of Fick diffusion coefficients was successfully tested.

Most of the presented results address interfacial behavior of aqueous solution at planar surface made of rutile, a prototypical metal-oxide frequently studied by experiments as well as simulations. Significant surface-driven changes in liquid properties relative to their bulk values were found. These properties, in particular shear viscosity and relative permittivity, were calculated by non-equilibrium methods to be able to capture their distance dependence over the interfacial region.

The profile of viscosity showed rapid increase towards the surface on account of attractiveness of rutile surface leading to substantial changes of liquid flow in slabs of thickness of a few nanometers or less. In our case of water + methanol mixture, the increase of viscosity in the contact layer can be more than 10 times the bulk value. The interfacial viscosity profiles for various methanol concentrations were interpreted using the structural data and information on hydrogen bonding. The viscosity data around second and third fluid layers were found to correlate with hydrogen bonding, i.e. more hydrogen bonds per molecule leads to steeper increase of local viscosity.

The interfacial dielectric properties were found to be strongly inhomogeneous, challenging the common picture of relative permittivity monotonously decreasing to a low, but positive value at the contact layer. In contrast, our results exhibit strong polarization response of the contact layer leading to the overcompensation of the external field, and strong differences between response of bulk and interfacial layers, demonstrated by non-symmetrical field dependence of the interfacial properties. Such behavior is a direct consequence of molecular nature of solvent manifested by density and orientational response variations among interfacial layers due to the interaction with the surface and local hydrogen bonding structure, and resulting in dielectric response that strongly depends on the distance from the surface and follows different pattern than in bulk. Therefore, our results evidence that interfacial dielectric properties cannot be described by the same field or density dependence which holds in bulk.

Properties of liquid in contact with solid are affected not only by parameters of their mutual interaction, but also by the geometry of the surface. To assess the latter effect, changes in structure of water at a spherical graphite particle of various sizes were investigated. Results led to identifying the size of the sphere for which the largest packing fraction of the first

39

hydration layer occurs, and representative orientations of water molecules relative to the surface. This work contributes to studies on solute-driven changes in the structure of the solvent. In addition, a coarse-grained potential for the interaction of a point particle or point interaction site with the entire spherical particle was introduced, intended to describe interaction with homogeneous sphere in situation when the steric van der Waals interaction is dominant.

Finally, mutual diffusion coefficients were predicted for the strongly non-ideal ternary mixture water + methanol + ethanol by a consistent simulation approach and were reported for the first time. The presented ternary diffusion data should facilitate the development of aggregated predictive models for diffusion coefficients of polar and hydrogen-bonding systems. The resulting Fick diffusion coefficients in binary subsystems were compared to experimental measurements as well as results from the common simulation approach, which requires experimental vapor–liquid equilibrium data. The present approach was found to be applicable to multicomponent mixtures, and at the same time to be more successful than the common approach, which is mostly limited to binary mixtures.

40

7. REFERENCES

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8. APPENDICES

Full text of the papers [A-D] are attached.

[A] Pařez, S.; Předota, M. Determination of Distance-dependent Viscosity of Mixtures in Parallel Slabs using Non-equilibrium Molecular Dynamics. Phys. Chem. Chem. Phys. 2012, 14, 3640.

[B] Pařez, S.; Předota, M. Dielectric properties of water at solid-liquid interfaces. Submitted to J. Phys. Chem. C

[C] Předota, M.; Nezbeda, I.; Pařez, S. Coarse-grained potential for interaction with a spherical colloidal particle and planar wall. Collect. Czech. Chem. Commun. 2010, 75, 527.

[D] Pařez, S.; Guevara-Carrion, G.; Hasse, H.; Vrabec, J. Mutual diffusion in the ternary mixture water + methanol + ethanol and its binary subsystems. Phys. Chem. Chem. Phys. 2013, 15, 3985.

44

8.1 APPENDIX A

57

8.2 APPENDIX B

58

81

8.3 APPENDIX C

82

103

8.4 APPENDIX D

104

123

9. PUBLICATIONS

1. Předota, M.; Nezbeda, I.; Pařez, S. Coarse-grained potential for interaction with a spherical colloidal particle and planar wall. Collect. Czech. Chem. Commun. 2010, 75, 527.

2. Pařez, S.; Předota, M. Determination of Distance-dependent Viscosity of Mixtures in Parallel Slabs using Non-equilibrium Molecular Dynamics. Phys. Chem. Chem. Phys. 2012, 14, 3640.

3. Pařez, S.; Guevara-Carrion, G.; Hasse, H.; Vrabec, J. Mutual diffusion in the ternary mixture water + methanol + ethanol and its binary subsystems. Phys. Chem. Chem. Phys. 2013, 15, 3985.