Computer-Simulationen zu Strukturen und Phasenumwandlungen in Modell-Kolloiden

Computer-Simulationen zu Strukturen und

Phasenumwandlungen

in Modell-Kolloiden

Strukturen und Phasenumwandlungen in Modell-Kolloiden

Dissertation

zur Erlangung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.)

an der Universit¨at Konstanz,

Mathematisch-Naturwissenschaftliche Sektion, Fachbereich Physik,

vorgelegt von Peter Henseler

Tag der m¨undlichen Pr¨ufung:

17. Juli 2008

Referent: Prof. Dr. Peter Nielaba Referent: Prof. Dr. Paul Leiderer

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/6143/

List of Figures v

List of Tables xi

1 General Introduction: Computer Simulation of Soft Matter Systems 1

1.1 Colloidal Systems as Model Systems in Experiment and Simulation . . . 1

1.2 Outline of the Thesis . . . 3

Part A Elastic Properties of Colloidal Crystals

3.2 Basic Equations . . . 11

3.2.1 Strains, Stresses and Elastic Constants . . . 11

3.2.2 Fourier Space . . . 15

3.2.3 Stress-Strain Relations . . . 16

3.2.4 Compatibility Relations for the Strains . . . 17

3.3 Crystal Symmetry Properties . . . 18

3.3.1 Isotropic Systems . . . 19

3.3.2 Two-Dimensional Structures . . . 19

3.3.3 Cubic Crystals . . . 21

3.3.4 Elasticity Tensors for Some Lower Symmetries . . . 26

3.4 Methods for Determination of Elastic Constants . . . 26

3.4.1 Block Analysis of the Sengupta Method . . . 27

3.4.2 Parinello-Rahman Method . . . 31

3.4.3 Kantor Method . . . 35

3.4.4 Free Volume Approximation . . . 37

3.4.5 Displacement Fluctuations . . . 38

3.4.6 Analysis of Lattice Vibrations . . . 39

3.5 Symbols Used for the Theory of Elasticity . . . 40

4 Simulation Details 41 4.1 Monte-Carlo Simulations . . . 41

4.2 General Remarks on the Simulation Algorithm . . . 45

4.3 Data Analysis . . . 47

4.3.1 Numerical Computation of the Lagrangian Strains . . . 47

4.3.2 Simulation Details of the Block Analysis Method . . . 49

4.3.3 Order Parameters . . . 50

5 Elastic Constants From Microscopic Strain-Strain Fluctuations 53 5.1 Linear Harmonic Chain . . . 53

5.2 Hexagonal Systems in 2D . . . 53

5.3 Cubic Crystals . . . 55

5.3.1 Elastic Constants of Hard-Sphere Crystals . . . 55

5.3.2 Elastic Constants of Lennard-Jones Crystals . . . 60

5.4 Anisotropy of Strain Fluctuations . . . 64

6 Conclusions and Outlook of Part A 71

Part B Colloidal Transport in Micro-Channels

8.2 Dimensionless Interaction Strength . . . 83

8.3 Experimental and Simulation Setup of Haghgooie . . . 84

9 Brownian Motion 85 9.1 Mathematical Description . . . 87

9.1.1 Stochastic Processes . . . 87

9.1.2 Langevin Equation . . . 88

9.1.3 Time Correlation Functions and Time Scales . . . 92

9.1.4 Wiener Process . . . 96

9.1.5 Fokker-Planck Equation . . . 98

9.1.6 Overdamped Limit . . . 101

9.2 Brownian Dynamics Simulation . . . 103

9.2.1 Conventional Brownian Dynamics . . . 103

9.2.2 Stochastic Runge-Kutta Method . . . 104

9.2.3 Smart Monte-Carlo Method . . . 105

9.2.4 Gaussian Random Numbers . . . 106

9.3 Hydrodynamic Interactions . . . 107

9.3.1 Stokesian Dynamics . . . 108

9.3.2 Dissipative Particle Dynamics and Further Developments . . . 110

9.4 Algorithm . . . 112

9.4.1 Particle Pair Potentials . . . 113

9.4.2 Reduced Units . . . 115

9.4.3 Boundary Conditions . . . 116

10 Equilibrium Properties of the Channels 121

10.1 Influence of the Confinement . . . 121

10.2 Layer Order Parameter . . . 124

10.3 Phase Diagram of the Laterally Confined Dipolar System . . . 126

10.4 Confined Three-Dimensional Systems . . . 133

10.5 Single-File Diffusion . . . 134

11 Transport Behavior of Colloids in Microchannels 137 11.1 Layer Reduction . . . 137

11.1.1 Density Gradient along the Channel . . . 139

11.1.2 Drift Velocity . . . 141

11.1.3 Example Particle Trajectories . . . 141

11.1.4 Defect Removal . . . 144

11.1.5 Connection between the Layer Transition and the Density Gradient . . . 145

11.1.6 Comparison with Experiments . . . 148

11.1.7 Oscillatory Behavior of the Layer Transition . . . 150

11.1.8 Influence of the Particle Interaction Range . . . 152

11.1.9 Three-Dimensional Channels . . . 153

11.2 Influence of Boundary Conditions . . . 155

11.2.1 Fixed Edge Boundary Particles . . . 155

11.2.2 Influence of the Boundary Conditions in Flow Direction . . . 157

11.3 Transversal Barriers . . . 161

11.3.1 Single Line Barrier . . . 162

11.3.2 Two Line Barriers . . . 165

12 Counterflow 173 12.1 Simulation Results in 2D . . . 173

12.2 Simulation Results in 3D . . . 183

13 Junctions and Particle Mixing Behavior 187 13.1 Symmetric Junction Geometry . . . 188

13.2 Asymmetric Junction Geometry . . . 195

14 Conclusions and Outlook of Part B 199 15 Zusammenfassung 205

Part C Appendix

C1.2 Program Compilation . . . 216

C1.3 Example Start Parameter Files . . . 217

C2 Literature Values for Elastic Constants 219

C3 Elastic Constants of a Harmonic Simple Cubic Crystal 221

Bibliography 225

Danksagung 239

2.1 Configuration snapshots of particles in two planar cross sections of a cubic crystal

as obtained by confocal fluorescence microscopy. . . 9

3.1 Cauchy stress tensor components acting on a body element. . . 13

3.2 Three different types of body deformation. . . 23

3.3 Plot of the finite size scaling function for 3D systems. . . 30

3.4 Periodically repeated simulation box in the Parinello-Rahman method . . . 31

3.5 The pressure and the independent elastic constants for a cubic system of hard spheres of volume fractionρ/ρ₀=0.8 determined with the Kantor method. . . 37

4.1 The nearest neighbor cluster of a fcc-crystal. . . 47

5.1 Finite size scaling analysis of a linear harmonic chain. . . 54

5.2 Block size dependent strain fluctuations for a 2D triangular system particles with the inverse 12^thpower potential. . . 54

5.3 Representative strain histograms ofhε₁₁−ε₂₂i^Lb of a hard core system with fcc- lattice symmetry. . . 55

5.4 Block size dependent strain fluctuations of a cubic crystal with hard-core inter- action which are evaluated for cubic blocks. . . 56

5.5 Elastic constants for a cubic crystal with hard-core interaction evaluated for spher- ical blocks. . . 57

5.6 Results for the elastic constants of a cubic crystal with hard-core interaction com- pared to the literature values. . . 58

5.7 Elastic constants evaluated by using the Kantor method. . . 59

5.8 Elastic constants for a cubic crystal with Lennard-Jones particle interaction. . . 61

5.9 Comparison of the elastic constants for a Lennard-Jones crystal of a fcc-structure with the results of Cowley. . . 61

5.10 Representative plot for the system size dependency of the block size dependent strain fluctuations. . . 62

5.11 Distribution of the variableD²(t)of equation (4.17) for a hard-core crystal with fcc symmetry. . . 63

5.12 Projections of the different non-zero strain-strain correlations of a fcc crystal with Lennard-Jones pair interaction onto a crystal symmetry plane. . . 69

5.12 Figure5.12continued. . . 70

7.1 Biological ion channel. . . 77

7.2 AtomicPtnanowire. . . 78

8.1 SEM images of the channel geometry in the experiment of K¨oppl and Erbe . . . 81

8.2 Sketch of the video microscope setup . . . 83

9.1 Realization of a Brownian trajectory . . . 85

9.2 Different coarse-graining levels of a colloidal suspension. . . 86

9.3 Time dependency of the velocity autocorrelation and mean squared displacement for a ’free’ Brownian particle . . . 94

9.4 Colloidal time scales . . . 95

9.5 Realization of the fluctuating Gaussian white noiseδF(t)and the corresponding realization ofW(t). . . 96

9.6 Different ways of coarse-graining of the solvent in a simulation of a colloidal suspension. . . 110

9.7 Scheme of the channel and the reservoir setup. . . 118

10.1 Typical snapshots of equilibrated configurations and their averaged density pro- files across the channel. . . 122

10.2 Full density profile transverse to the walls forLy=20σfor the solid state. . . . 123

10.3 Comparison of the interaction ranges of the screened Coulomb potential (YHC) for a selection of values forκ_Dand the dipolar pair potential. . . 123

10.4 Full density profiles transverse to the walls forLy=8σof systems with screened Coulomb interaction. . . 124

10.5 Comparison of the local layer oder parameter and the local orientational order parameter. . . 125

10.6 Snapshot of a partition of an equilibrium defect configuration. . . 125

10.7 The layer order parameter as a function of the channel width. . . 127

10.8 The bulk defect concentration as a function of the channel width. . . 127

10.9 Time evolution of the defect concentration. . . 128

10.10 Density profiles transverse to the walls forLy=9σandLy=10σin dependence ofΓ. . . 129

10.11 Global layer order parameters as a function of the interaction strength for a chan- nel of widthLy=9σ. . . 129

10.12 Order parameters in dependency of the dimensionless interaction strength for a selection of channel widths. . . 130

10.13 Superimposed particle positions in equilibrium for two channel widths. . . 132

10.14 Time dependency of kinetic and potential energy during the equilibration process. 133 10.15 Snapshot of an equilibrium configuration in 3D confined between two plates. . . 134

10.16 Single file diffusion behavior for a channel setup withLy=0.5σ. . . 135

10.17 Comparison of the mean-square displacement parallel and perpendicular to the confining channel walls for the two channel widthsL_y=9σ andL_y=10σ. . . . 136

11.1 Full channel snapshot from simulation for a channel with ideal hard walls . . . . 137

11.2 Snapshots of the stationary non-equilibrium configurations around the layer tran- sition . . . 138

11.3 Stationary non-equilibrium density profiles along the channel for several values of slopeα . . . 139 11.4 Average particle drift velocity as a function of the inclination. . . 140 11.5 Drift velocity histograms of the particles in different channel regions. . . 140 11.6 Example particle trajectories of the particles crossing the layer reduction zone . 142 11.7 Superimposed configuration snapshots for the full channel under influence of an

external driving field. . . 143 11.8 Removal of a defect after the layer transition region . . . 144 11.9 Local lattice constants, particle density and layer order parameters. . . 145 11.10 Local lattice constantd_y as a function of the local particle densityρ for various

inclinationsα. . . 146 11.11 Result of the stretching analysis of static channel configurations of a channel with

the widthL_y=10 and dipolar pair interaction . . . 147 11.12 Equilibrated configuration snapshot of a funnel geometry. . . 148 11.13 Local lattice constantsdxanddyand local particle densityρ(x)in the experiment

of K¨oppl. . . 148 11.14 Snapshots of defect configurations obtained from a Delaunay triangulation of the

particles moving in the channel. . . 149 11.15 Movement of thex-position of layer transition for the transitions 8→7 layers

and 7→6 layers. . . 150 11.16 Movement of thex-position of layer transition for the transitions 8→7 for an

inclinationα=0.2^◦and otherwise identical parameters as in the previous figure. 151 11.17 Movement of the x-position of layer transition for the transitions 8→7 in the

experiment of K¨oppl and Erbe. . . 152 11.18 Local lattice constants, particle density and layer order parameters for a system

with screened Coulomb interaction. . . 153 11.19 Configuration snapshots showing the occurrence of the layer reduction of a 3D

system with YHC pair interaction. . . 154 11.20 Contour plots of the potential energy surface in the presence of fixed dipolar point

particles on the wall. . . 155 11.21 Snapshots of a driven dipolar system with fixed boundary particles at two differ-

ent times. . . 156 11.22 Schematic image of a microchannel setup with periodic entropic channel walls. . 156 11.23 Density profiles for a selection of inclinationsα of a system with the driving

force applied only within a partition of the channel. . . 158 11.24 Stationary non-equilibrium situations of systems where the driving force is ap-

plied only within part of the channel region for a selection of inclinations. . . . 159 11.25 Superimposed configurations of systems with screened Coulomb pair interaction

for a selection of inverse screening lengths . . . 160 11.26 Density profiles for a selection of Debye screening lengths of a YHC system with

the driving force applied only within a partition of the channel. . . 161 11.27 Snapshots of a channel with a barrier of repulsive optical traps taken from simu-

lation and experiment . . . 162 11.28 Time evolution of the change of the local density in front of the barrier. . . 164

11.29 Local lattice constants and density for a dipolar system with barrier atx=500σ. 164 11.30 Sketch of a magnetoblockade setup. . . 165 11.31 Channel snapshot of a channel with two barriers of repulsive optical traps of

strength ˜k=−6 at the positionsx=300σ andx=400σ. . . 166 11.32 (a)Local density along the channel in non-equilibrium steady state of a channel

with two barriers atx=300σ andx=400σ for three trapping strengths ˜k. (b) Time dependency of the average particle densitynbetween the two barriers. . . 166 11.33 Snapshots of the magnetoblockade effect for a channel of widthL_y=2σ. . . 167 11.34 Superimposed snapshots near the two transversal barriers for a selection of three

barrier strengths ˜k. . . 168 11.35 Flow rate across the second barrier of a magnetoblockade as function of the sim-

ulation time for a selection of trap strengths ˜k. . . 169 11.36 Average particle numberN_Island and corresponding particle flux across two-line

barriers. . . 170 11.37 Average particle numberNIslandand particle flux across two-line barriers as func-

tion of the inclination for fixed barrier strength. . . 171 12.1 Snapshots and lane formation order parameter for a system of counterflowing

particles confined by two wall of separationLy=100σ. . . 174 12.2 Typical simulation snapshots of counterflowing particles for different driving

forces. . . 176 12.3 Time evolution of buildup and breakage of an intermediate jammed state of coun-

terflowing particles. . . 177 12.4 Particle trajectories and time-dependence of the velocities for three typical parti-

cles moving in positivex-direction. . . 178 12.5 Metastability of the lanes. . . 179 12.6 Snapshots of systems withL_y=12.5σ andL_y=20σ andn=0.4σ⁻². . . 180 12.7 Global lane formation order parameter ΦLF(t) for a selection of driving force

strengths of a YHC binary system withV₀=200 andκ_D=4σ⁻¹. . . 181 12.8 Snapshots of a system with counterflow of a binary mixture of particles with

screened Coulomb interaction. . . 181 12.9 Velocity histograms for a counterflow system of two different particle species

species. . . 182 12.10 Typical simulation snapshots for counterflow in 3D of particles with YHC pair

interaction. . . 184 13.1 Sketch of the general channel geometry of a junction as being realized in the

simulation code. . . 187 13.2 Superimposed snapshots of a symmetric junction . . . 189 13.3 Line density histograms(a)forx∈[385.5,400]σ and(b)forx∈[400,550]σ of

the system of figure13.2. . . 189 13.4 Local density histograms along the channel for the system of figure13.2. . . . 190 13.5 Mixing behavior of a symmetric junction . . . 191 13.6 Schematic graph of the flow of particles along the outer walls and along the inner

walls near the mixing region. . . 192

13.7 Mixing behavior of a symmetric junction . . . 193 13.8 The mixing order parameter as function of the applied magnetic field strength. . 194 13.9 Superimposed snapshots for three different interaction strengths to highlight the

change in the mixing behavior. . . 194 13.10 Superimposed snapshots for an asymmetric junction. . . 195 13.11 Local density histograms along the channel for the system of figure13.10. . . . 196 13.12 Detail of a snapshot for an asymmetric junction. . . 196 13.13 Superimposed snapshots for an asymmetric junction. . . 197 13.14 Line density histograms(a)forx∈[80,360]σ of the particles in input channel 2

and(b)forx∈[420,700]σ in the joint output channel 3 of the system shown in figure13.13. . . 198 13.15 Local density histograms along the channel for the system shown in figure13.13. 198 C1.1 File and subdirectory listing of the directorysrc/phd src/ . . . 213 C1.2 File and subdirectory listing of the directorysrc/phd src/(continued) . . . . 214 C1.3 File and subdirectory listing of the directorysrc/BD/ . . . 215 C1.4 File and subdirectory listing of the directorysrc/TOOLBOX/. . . 216 C1.5 Example output of a successful compilation process of the executable program

BD2D. . . 216 C1.6 Example start parameter file of the executableBD2D. . . 217 C1.7 Example start parameter files of the executableEC3D. . . 218 C3.1 Particle coordinates and stains of the of the [100]-symmetry plane of a simple

cubic lattice. . . 221

4.1 Q4andQ6for various perfect crystal lattices. . . 51 8.1 Particle properties of the DynabeadsR used in the experiment. . . 82 9.1 Fundamental quantities and the corresponding derived quantities in reduced units. 116 9.2 Choices for the energy scaleεfor different particle pair potentials. . . 116 C2.1 Isothermal elastic constants of a fcc hard sphere solid. . . 219 C2.2 Isothermal elastic constants for a Lennard Jones fcc crystal. . . 220 C3.1 The components of the displacement vectors for the nearest and next nearest

neighbors of the central atom 1 of a simple cubic lattice. . . 222

Simulation of Soft Matter Systems

1.1 Colloidal Systems as Model Systems in Experiment and Simulation

This thesis deals with so-calledsoft (condensed) matter systems[Gen92,Wit99,L¨ow01,Cha00, Bar03,Jon04], which is a convenient term for materials in states of matter that are neither crys- talline solids nor simple liquids. Especially, we concentrate on colloidal dispersions [Rus89, Eva99,Ham00] (sometimes also calledcomplex fluids) which are solutions of mesoscopic solid particles with a stable (i.e. non-fluctuating) shape and dimensions in the range of 1 nm to 10µm embedded in a molecular fluid solvent.¹ It is typical for soft matter systems, that they contain multiple length scales. Many colloidal dispersions are ubiquitous in our everyday life and play a key role in industrial processes – examples include: clay, mist, smoke, blood, milk, mayonnaise, paint, ink, lubricants, and pastes. All these disparate materials have in common a length scale in- termediate between atomic sizes and macroscopic scales. Irregular thermal fluctuations (Brownian motion) are important on typical time scales of milliseconds in the description of such systems.

Generally, a colloidal particle is defined by its dynamic behavior, which restricts its length and time scales [Fre02a,Dho96].

Apart from numerous practical applications colloidal particles in a suspension provide versatile model systems for studying basic questions of condensed matter physics. In this field of physics the basic aim is to understand the collective behavior of large assemblies of particles in terms of the interactions between their components. Hereby, the detailed knowledge of the “internal”

properties of a colloidal particle,i.e.the discrete atomic nature of matter is not so relevant. Thus, colloids can be looked at as “giant molecules” where the detailed microscopic arrangement of its constituents is ignored. Such a process is called coarse graining. Therefore, colloidal sys- tems can provide insight into classical properties of many-body quantum systems. Generally, colloidal suspensions serve as model systems for the study of stochastic many-body effects, be- cause their behavior is governed by the laws of (classical) statistical mechanics. Additionally, colloidal suspensions under the influence of an external force field are an ideal testing ground for the formulation of a stochastic theory of non-equilibrium systems.

Due to their intrinsic length and time scales in the micrometer and millisecond range colloidal suspensions are readily accessible to experimental measurements by means of light scattering, in- ternal reflection microscopy, and video microscopy [Coc96,Hab02], respectively. The method of confocal video microscopy [Coc96,Pra07] makes it possible to directly observe the particle tra- jectories of three-dimensional systems in real-space. Colloidal systems allow for a wide variation

1Other prominent soft matter systems are: polymer melts or solutions, foams, and liquid crystals.

of relevant parameters. In particular the particle density can easily be varied by several orders of magnitude. Also the size, the shape, and the interaction between the colloidal particles can be designed by appropriate manufacturing processes. Additionally, the interaction energy between the particles can be tuned from values much smaller to values much larger than the thermal energy k_BT. For instance, the interaction range of the electrostatic repulsion of like-charged colloids can be tuned between very short range and long range by simply changing the salt concentration of the suspension. Another common type of pair interaction is the magnetic dipole-dipole interaction of so-called superparamagnetic colloids. Density matching of the colloidal particles and the sus- pension makes it possible to study the dynamics of the colloids without the influence of gravity.

Suspensions of mesoscopic colloidal particles typically show a very rich phase behavior, making them excellent realizations of classical statistical models.

The accessibility of colloidal systems in real-space enables the direct comparison of experiment and simulation under equilibrium and non-equilibrium conditions at various levels of complexity.

The complexity of the problem increases with increasing deviation from equilibrium conditions.

Additionally the complexity of the system itself can be increased by going from mono-disperse systems to multicomponent systems (e.g. binary systems) of spherical particles and even further by going to mixtures of spheres and disks (or plates). Exciting new physical phenomena occur in the presence of a confinement or under the influence of external fields.

The applications and benefits of computer simulations are manifold as can been seen from the ref- erences [Nie02,Fer06a,Fer06b]: In contrast to purely analytical models, numerical simulations allow us to follow the dynamics of complex many-body model systems without making reference to unnecessary strong approximations. Simulations can be used in an efficient and systematic manner to isolate and to investigate the influence of microstructure, composition, geometry, and external perturbation over a wide range of parameters. Experimental realizations are usually re- stricted to a specific setup and to a smaller range of parameters. For example, changes of compo- sition, geometry, or dimension of space require very different experimental methods, whereas the numerical realizations are much more straightforward. Furthermore, simulations serve as a good testing ground for theoretical models, because they provide an undistorted and more rigorous en- vironment than is experimentally accessible. The simplest colloidal model system consists of a suspension of hard spheres (discs),i.e. solid particles which do not overlap. Such systems show purely entropic phase transitions and nowadays serve as “standard model” for computer simula- tions of colloidal systems, since their phase behavior is rigorously analyzed and well understood.

The experimental realization of such a hard-core system can be achieved only in an approximate manner. Finally, simulations provide more detailed and direct information on the dynamics and the structure of particles than experimental measurements.

This thesis presents the results obtained by numerical analysis of two different model systems, which are both realized at the University of Konstanz by the two experimental physics groups of Prof. G. Maret and Prof. P. Leiderer.

1.2 Outline of the Thesis

We present simulation results and analytical calculations for two distinct topics, which are con- cerned with colloidal systems in equilibrium and in non-equilibrium, respectively. Therefore, the thesis is divided into two main parts: Part AandPart Brepresenting these two distinct projects.

These parts are followed by the Appendix inPart C, which contains additional information with respect to parts A and B.

Part A provides a methodical investigation of the determination of elastic properties of three- dimensional crystal structures. Part B explores a variety of colloidal particle transport phenom- ena through two- and three-dimensional microchannels, and is directly connected to experimental studies. Several unexpected phenomena found in the simulation will be presented, directly com- pared to experimental results, and predictions for further interesting effects will be made.

A brief guide of the three parts A, B, and C is given in the following:

Chapter 1 is a general introduction into the field of computer simulation of soft matter systems.

Part A – Elastic Properties of Colloidal Crystals

Part A deals with the elastic properties of two- (2D) and three-dimensional (3D) colloidal crystals in equilibrium and comprises the chapters 2 to 6.

Chapter 2 gives a motivation why the elastic properties of colloidal crystals remain an interesting subject for research nowadays.

Chapter 3 shortly introduces, after a comment on the notation in section3.1, the basic equations of the theory of elasticity (section3.2). In section3.3we will discuss the symmetry properties of some basic two-dimensional and three-dimensional crystal structures and their implications for the response of the elastic system. In section3.4we will introduce different methods which may be used for the calculation of elastic constants by computer simulation. At the end of the chapter a list is given of the most important symbols being used in section3.5.

Chapter 4 gives the details of the simulation algorithm used in this thesis. After a brief reminder of the Monte-Carlo simulation method in section4.1 we make some general state- ments on the implementation and features of our self-written algorithm (section4.2).

Section4.3is devoted to the data analysis of the simulations. It contains the descrip- tion of a variety of methods for the calculation of strains from configuration data, a comment on the explicit implementation of the Sengupta method in the simulation algorithm, and finally an introduction of order parameters which can be used for the characterization of the system state.

Chapter 5 presents the results obtained in our simulation runs. In section 5.1we examine the linear harmonic chain and in section 5.2 the two-dimensional hexagonal lattice as test cases. Section5.3discusses the elastic constants of cubic crystals in presence of either a hard-core or of a Lennard-Jones pair interaction potential. The differences between the values obtained by our simulation method and known literature values

are related to the anisotropy of the crystals as being shown in section5.4.

Chapter 6 summarizes the results of Part A and gives an outlook on possible future develop- ments.

Part B – Colloidal Transport in Micro-Channels

Part B describes the driven transport behavior of colloidal particles through micro-channels mainly in two dimensions.

Chapter 7 highlights our motivation of studying driven colloidal suspensions and makes con- nections to other systems showing similar phenomena.

Chapter 8 introduces in section8.1the experimental setup of M. K¨oppl, A. Erbe and P. Leiderer, which is the experimental complement to our computer simulation. In section 8.2 follows the definition of the dimensionless interaction strength. Details on a similar experimental setup of R. Haghgooie are given in section8.3.

Chapter 9 gives an overview of the main features of the mathematical description of Brownian motion in section 9.1. Various possibilities of realization of a Brownian dynamics simulation are described is section 9.2. In section 9.3 we comment on the effects and possible implementation of hydrodynamic interactions. Details of our simulation algorithm are given in section9.4.

Chapter 10 includes in section10.1a discussion of the confinement induced equilibrium proper- ties of our system. We compare these results with the findings in the literature. In section10.2we introduce the so-called layer order parameter which we developed to analyze the system in equilibrium as well as in non-equilibrium. This chapter also includes a (equilibrium) phase-diagram of our laterally confined dipolar system in section10.3, where we summarize the influences of the channel width and of the inter- action strength. Section10.4deals with confined three-dimensional systems. Finally, we close the chapter with simulation results of single-file diffusion, which occurs for very small channel widths where particles cannot pass each other (section10.5).

Chapter 11 is devoted to the so-called layer-reduction of a driven non-equilibrium system under lateral confinement. In section11.1we extensively discuss the different physical phe- nomena which are important for the complete understanding of the layer-reduction.

These include the density gradient along the channel, its connection to the layer tran- sition, the particle drift velocity, and the particle trajectories, the removal of defects, the analysis of the oscillatory behavior of the layer transition position, the importance of the particle interaction range, and the comparison with the experimental results.

Also, remarks on the layer transition in three dimensional microchannels are made. In section11.2we analyze the influence of the boundary conditions and in section11.3 the influence of additional parabolic traps of finite range on the transport behavior.

Chapter 12 extends the unidirectional particle transport in a micro-channel system to the situation of particles driven in opposite directions under the influence of a confinement. Here we discuss the phenomenon of so-called lane-formation in 2D- (section 12.1) and 3D-systems (section12.2) under confinement.

Chapter 13 deals with geometrical modifications of the channel setup. In particular, we take a closer look at the situation of a junction with two input channels which merge into a single outflow channel. We discuss the particle mixing behavior and the conditions under which vortices can appear. Section13.1is about symmetric intersections, and section13.2deals with asymmetric ones.

Chapter 14 gives a review of the main conclusions of part B being obtained by simulations of equilibrium and non-equilibrium systems and the comparison with the experimental results. Finally, we make some proposals for interesting directions of future research.

A general summary of the thesis in German language is given inChapter 15.

Part C – Appendix

The appendix comprises additional information of parts A and B concerning the simulation algo- rithm and some reference results from literature. It consists of the following three chapters:

Chapter C1 makes some detailed remarks on the simulation source code and its compilation pro- cess. Also, we list some examples of start parameter files, which have been used for our simulation runs.

Chapter C2 summarizes known literature values of the elastic constants in 3D for the hard-core and for the Lennard-Jones pair interaction potential.

Chapter C3 contains an analytic calculation of the elastic constants of a cubic crystal with har- monic pair interaction of its nearest and next nearest neighbors.

Elastic Properties of Colloidal Crystals

Abstract We perform classical (Metropolis) Monte Carlo simulations in the canonical en- semble to calculate isothermal elastic constants of crystal phases of hard-sphere and of Lennard-Jones systems. The elastic constants are determined in a sin- gle simulation run from microscopic fluctuations of the instantaneous local La- grangian strain tensor and by the use of a finite-size scaling theory. This ap- proach is a generalization of the method of Senguptaet al.[Sen00b] from two to three dimensions. We will discuss the importance of finite-size effects, of non-localities of the strain fluctuations, and of the anisotropy of the underlying reference lattice by comparing our results to simulation results from literature.

Part A of the thesis is devoted to the calculation of elastic constants of three-dimensional crystals by the evaluation of the instantaneous particle positions. These positions can be determined either by microscopic simulation procedures as Monte Carlo or molecular dynamics simulations or by direct observations in experiments where colloidal particle trajectories are recorded. In the experi- ment the so-calledconfocal fluorescence microscope[Pra07] allows for the recognition of particle positions in three dimensions. Such an experiment was done, for example by D. Reinke [Rei06f]

of the group of Prof. G. Maret at the University of Konstanz. They used fluorescent and sterically stabilized perspex beads of diameter 1.66µm which were suspended in a density and refraction index matched solution. With the help of a confocal microscope they were able to record the par- ticle trajectories with a high precision. The obtained data were used for the determination of the elastic properties of colloidal crystals.¹

x y

x z

Figure 2.1: Configuration snapshots of particles in two planar cross sections of a face-centered cu- bic crystal as obtained by confocal fluorescence microscopy after a first step of image processing. The image is taken from the thesis of Reinke [Rei06f].

The classical theory of elasticity describes the response behavior of deformable elastic bodies under the influence of external forces, i.e. bodies which recover their original shape when the forces causing the deformation are removed. Already in the year 1678 Robert Hooke (1635–1703) formulated the famous law, that the deformation (measured by the strain) of an elastic body is proportional to the applied stress. The proportionality factors, which are material specific values, are the so-called elastic constants. Hooke’s law was the beginning of the theory ofcontinuum elasticity.

Although elasticity is a rather old subject with a well formulated theory for macroscopic sys-

1Generally, the elastic constants are proportional to the inverse volume of the basic crystal cell. Colloids are by a factor of 10000 larger than atoms. Therefore, the elastic constants of colloidal crystals are of about 12 orders of magnitude smaller than those of atomic crystal lattices. This explains why colloidal crystals are so-calledsoft materials.

tems (see for example [Bor68,Ash76,Wal70,Lan91]), there still remain many open questions in the context of elasticity of micro- and nano-systems [Gol02,Del05], of elasticity of amorphous complex or granular systems [Bar06], of martensitic [Bha07] and two-dimensional phase transi- tions [Sen00a,Bin02,Zah03] and in the context of crack formation [Sch04]. Such questions in- clude for example [Bar06]: What are the appropriate measures of stresses and strains at the atomic length scale? Can the constitutive laws of macroscopic elastic theory be applied to microscopic systems, and if so up to what length scale? Are the elastic constants of nano-systems identical to those of the bulk system? Which statements can be made about the elastic/ plastic transition at small length scales? So, there is a strong interest in developing successful “coarse-graining”

schemes,i.e.methods to build up a description of a macroscopic system from the knowledge of its microscopic variables, such as particle positions and velocities, together with a detailed knowledge of the interatomic potentials.

Generally, elastic constants are important because they provide a link between the mechanical and the dynamical behavior of crystals. Additionally, they provide also a means to probe the interatomic forces of (experimental) systems, which usually are not fully known a-priori. For example non-central forces have been identified in systems of charged colloids by the violation of the Cauchy symmetry relation of elastic constants of a cubic crystal [Rei07c]. Furthermore, one may ask: How does the presence of quenched impurities modify the elastic response behavior of a crystal lattice? For metallic alloys, like steel, it is well known, that the presence of impurities significantly enhances the material stiffness.

The determination of the response of the system to an imposed external stress was first addressed by simulations by the pioneering work of Parinello and Rahman [Par81,Par82, Ray84]. In the following, a lot of different computer simulation methods have been developed, which we will review and compare to our method. The latter method is a generalization of the method proposed by Sengupta [Sen00b] for two-dimensional systems to three dimensions. It is based on the analysis of the finite-scaling of (microscopic) strain fluctuations for different subsystem sizes. The main advantage of this method is, that it does not make any direct reference to the particle pair-potential, and thus it is applicable to any crystalline system where the particle trajectories are known. Model systems of particles interacting via singular potentials, as for example the hard-core potential, can be analyzed by this method.

3.1 Notation

The theory of elasticity is a rather old subject of physics and thus many different notations exist.

This fact sometimes makes it difficult to capture the underlying physical phenomenon. As a help to the reader all important symbols used are tabulated in section3.5.

Bold letters indicate vectors and underlined symbols are assigned to tensors which refer to a fixed Cartesian system(x1,x2,x3). Roman indices refer to Cartesian components of vectors or tensors, Greek subscripts mark vectors or tensors in Voigt notation, and Greek superscripts refer to the particle numbers.

Unless otherwise specified, the Einstein sum convention is used, where two repeated indices (Greek or Roman) are implicitly summed over. By convention, the following writing is used:

∂A

∂r

i j≡∂A_i

∂rj and

∂²B

∂r∂s

i jkl≡ ∂²B_{i j}

∂rk∂sl.

The superscript^T indicates the transpose matrix and⁻¹ its inverse. The identity matrix is written as 1, and its components asδi j.

Vectors and tensors in Fourier space are denoted by a tilde on top of the symbol, e.g. ˜u_i.

3.2 Basic Equations

This section will present the basic relations of thermoelasticity in an outline form. The summary of the following subsection is based upon the following books on elasticity [Lan91,Wal70,Wal72, Wal02,Lov72,PG00]. More detailed information can be found therein.

3.2.1 Strains, Stresses and Elastic Constants

Assume a system in an initial equilibrium state where the particles are located at positions R.

Under the influence of an applied stress the initial configuration changes to the final position r.

For each particle an instantaneousdisplacement vectoru(R,t)is defined having the components ui(R,t) =ri(t)−Ri. (3.1) Forhomogeneous strain,i.e.constant strain throughout the material, the transformation fromRto ris linear and can be written as

ri(R) =αi jRj, (3.2)

where the elements of the transformation gradient tensor αi j ≡∂ri/∂Rj are constants. Equiva- lently the transformation can be characterized by the displacement gradientsu_{i j}≡∂u_i/∂R_j, which are related to the transformation gradients by

α_{i j}=δ_{i j}+u_{i j}. (3.3)

Thus for homogeneous strain the u_{i j} are also constants. The volumeV(r) after deformation is related to the original reference volumeV(R)by

det[αi j] = V(r)

V(R). (3.4)

Elastic theory describes the deformation of any configuration from a reference configuration in terms of a strain tensor. Pairs of particles in the initial (reference) configuration and in the final configuration are separated by∆Rand∆rrespectively. For constant strainsεi jover the region∆R, the relative positions of all particles in configuration{r}are completely specified by the reference configuration{R}and the strainsε_{i j}

|∆r|²− |∆R|²=2ε_{i j}∆R_i∆R_j. (3.5) Here the so-calledLagrangian strain tensor¹at reference positionRis defined as

(ε)_{i j}≡ε_{i j}=1 2

∂u_i

∂R_j+∂uj

∂R_i + ∂u_i

∂R_k

∂uj

∂R_k

. (3.6)

This tensor obviously is symmetric (ε_ij=ε_ji). Its’ significance is based on the fact that it is not limited to small deformations. Even for a finite strain it completely determines the deformation of the body. But in most cases the deformationsu_i are small and the non-linear term which is of second order, can be neglected:

ε_{i j} =1 2

∂u_i

∂R_j +∂uj

∂R_i

. (3.7)

In this work, only the case of small deformations is considered. So it will be totally safe to work with the linearized Lagrangian strain tensor (3.7). Inserting (3.5) in (3.6) it follows that the Lagrangian strains are related to the transformation gradients by the relationship

ε_{i j}=1

2 α_kiα_{k j}−δ_{i j}

. (3.8)

In contrast to a characterization of the deformation in terms ofα_{i j} oru_{i j}, the Lagrangian strains contain no information about the rotations of the solid body.

Inversion of the relation (3.8) results in the power series αi j=δi j+εi j−1

2εkiεk j+... (3.9)

From Eq. (3.4) the volume ratio of the body in the starting (V0) and the final configuration (V) is V

V₀ =det[α_{i j}] =1+ε_ii+... (3.10)

σ

σ

σ

y z

x

σ

σ σ

σ

31

σ

23 22

σ

Figure 3.1: Components of the Cauchy stress tensorσ_{i j}acting on a body element.

which gives the connection between the traceε_iiof the strain tensor and the relative volume change.

On deformation of an elastic body the particles are moved from their equilibrium position. The internal forces which are calledstressestry to bring the body back to its equilibrium situation. The forces f_i acting on the surface of a crystal can be written as divergences of the applied Cauchy stress tensorσ_ik[Lan91,Cha00],

f_i=∂σ_ik

∂R_k. (3.11)

Positive stresses are directed outward from the crystal surfaces as depicted in figure3.1. In equi- librium the forces due to internal strains have to compensate each other in every partial volume of the deformed body,i.e. f_i=0. So the equilibriumcondition for mechanical stabilityis

∂σ_ik

∂Rk =0. (3.12)

The configurational dependency of the state functions is given by the relative positions of all particles only. Thus for essentially constant strains within the range of the interaction the internal energyU and the free energyF depend on the actual configuration{r}only through{R}andε according to equation (3.5): U=U({r},S) =U({R},ε,S) andF =F({r},T) =F({R},ε,T).

The thermodynamics of a deformation of a solid body can be described by the differentials of the state functions²

dU=dU({R},ε,S) =T dS+δW=T dS+V0τ_ik^Sdε_ik (3.13) or equivalently by the Helmholtz free energy differential

dF=dF({r},T) =dF({R},ε,T) =−SdT+V₀τ_ik^Tdε_ik, (3.14) sinceF=U−TSwhich is the result of the Legendre transformation fromStoT . The variableV₀ denotes the volume of the (non-deformed) reference system,Sis the entropy, andδW=−V₀τ_ikdε_ik is the work due to the change of the strain tensor. τ_ik^S and τ_ik^T are the adiabatic or isothermal

1The components of the strain tensor are functions of the coordinates and thus not independent of each other, because the six different componentsε_{i j}are expressed by derivatives of just three independent functions, the components of the displacement vectoru(cf. sub-section (3.2.4)).

2For hydrostatic compression the stress tensor isτ_{i j}=−pδ_{i j}, wherepis the pressure normal to the body interfaces.

Soτ_ikdu_ik=−pdu_ii=−pdVand thus equation (3.13) reduces to the usual formdU=T dS−pdV.

stress tensor respectively. It is also assumed that at every instant of the deformation process the interior of the body is in equilibrium according to the external conditions. From equation (3.14) we therefore get for the (isothermal)thermodynamic stress tensor(tension) evaluated at the initial configuration{R}

τ_{i j}^T= 1 V0

∂F

∂εi j

T, (3.15)

where the derivative is evaluated at constant temperature keeping all other strain components ε_kl6=εi j fixed.^3,4Analogous relations also hold for the adiabatic situation which are not explicitly written down.

For small homogeneous deformations, as it is certainly the case for particles oscillating about the initial equilibrium configuration{R}, the Helmholtz free energy can be expanded according to

F({R},ε,T) =F({R},0,T) +V₀τ_{i j}^Tε_{i j}+1

2V₀C_{i jkl}^T ε_{i j}ε_kl+O(ε³) (3.16) Here the fourth rank tensorC_{i jkl}^T is thetensor of (isothermal) elastic constants.

The isothermal or adiabatic elastic constants can also be obtained from the second derivative of the strain energyW which refers either to the free energy or to the internal energy

C_{i jkl} = ∂τ_{i j}

∂εkl = 1 V0

∂²W

∂εi j∂εkl

(3.17) This scheme can be extended to define higher order elastic constants like

D_{i jklmn}= 1 V₀

∂³W

∂ε_{i j}∂ε_kl∂ε_mn

. (3.18)

Due to the symmetry of the strain tensor and due to the permutability of partial derivatives, the elasticity tensor is symmetric under interchange of the first two indices, the last two indices and the first and last index pair,i.e.

C_ijkl=C_jikl, C_ijkl=C_ijlk and C_ijkl=C_klij. (3.19) Instead of maximal 81 independent values, there exists a maximum of 21 independent elastic constants. This number can be drastically reduced taking into account the point group symmetry of the crystal equilibrium configuration (cf. Section3.3). In general, the number of independent elastic constants decreases as the point group symmetry of the crystal increases. Cubic crystals have three independent elastic constants. Hexagonally ordered two-dimensional crystalline solids and isotropic solids have two independent elastic constants.

3Note that the expressionτ_{i j}^T=1/V₀ ∂F/∂ε_{i j}

Thas to be understood asdF=τ_{i j}du_{i j}, where all elements withi6=j of the symmetric tensordu_{i j}occur twice. Thus the differentiation of the free energyFreturns values forτ_{i j}which have a value twice as high fori6= jcompared to the evaluation from expressionτ_{i j}^T=C_{i jkl}^T ε_kl. [Lan91]

4When the stress components are evaluated at the final configuration{r}starting from an arbitrary initial configuration {R}, then [Wal70]

τ_{i j}^T= 1 V₀α_ikα_jl

∂F

∂ε_kl

T.

Because of the symmetry requirements of equation (3.19) there exist only six independent pairs of subscriptsij. These can be numbered from 1 to 6 according to the following scheme, which was first introduced by Voigt [Voi10]:

i j = 11 22 33 32 or 23 31 or 13 21 or 12

α = 1 2 3 4 5 6 (3.20)

The elastic constants inVoigt’s notationtake the formC_αβ ≡C_{i jkl}whereα andβ take the values 1 to 6. From the third equality of equation (3.19) we haveC_αβ =C_βα.

If the strain tensor depends on the locationR, the free energy is a functional of the strain tensor F[ε(R)] = 1

V₀ Z

dRF({R},ε(R),T). (3.21)

Further important thermodynamic functions, useful in this context are the pressureP, the (isother- mal)bulk modulus B^T, and thecompressibilityκ^T being defined as [Wal72]

P = − ∂F

∂V

T (3.22)

B^T = 1

κ^T =−V₀ ∂P

∂V

T. (3.23)

The two constantsB^T andκ^T are a measure of the substances’ resistance to volume changes.

3.2.2 Fourier Space

Sometimes it is helpful to work in Fourier space instead of the position space, because in Fourier space the differential equations become algebraic equations. The atomic displacements for aN particle system transform as

˜

ui(k) = 1 N

∑

α=1(x^α_i −X_i^α)e^−ik·hr^αi, (3.24) and the Lagrangian strains in Fourier space read⁵

ε˜_{i j}(k) =i

2[k_iu˜_j(k) +k_ju˜_i(k)]. (3.25)

5Fourier transform of the strain tensor written explicitly:

ε˜_{i j}(k) =

√N V₀

Z

d³Re^−ik·Rε_{i j}(R) =

√N V₀

Z

d³Re^−ik·R1 2

∂u_i(R)

∂R_j +∂u_j(R)

∂R_i

=

√N 2V₀

Zd³R ∂

∂R_j

e^−ik·Ru_i(R)

+ik_ju_i(R)e^−ik·R+ ∂

∂R_i

e^−ik·Ru_j(R)

+ik_iu_j(R)e^−ik·R

= i

2

k_iu˜_j(k) +k_ju˜_i(k) .

In the third step the first and the third summand vanish, because the displacement vector is zero on the box edges.

In general, spatial partial derivatives in position space ∂/∂Ri change to factors of ki in Fourier space. The equilibrium condition for mechanical stability of equation (3.12) is

k_jσ˜_{i j}=0. (3.26)

So the elastic free energy∆F per particle for an elastically homogeneous and isothermal system of constant particle numberNin absence of external stress (σ=0) transforms as:⁶

∆F = 1 2

V

NC_{i jkl}

∑

α=1ε_{i j}(hr^αi)ε_kl(hr^αi) (3.27)

= 1 2

V

NC_{i jkl}

∑

α=1

∑

k

∑

h

ε˜i j(h)ε˜_kl(k)e^i(h+k)·hr^αi (3.28)

= V

2C_{i jkl}

∑

k k_jk_lu˜_i(k)u˜_k(−k). (3.29) 3.2.3 Stress-Strain Relations

For small deformations the stresses depend linearly on the strains [Wal70,Wal72]

τ_{i j}=B_{i jkl}ε_kl, (3.30)

with thestress-strain coefficients(δ_ikis the Kronecker delta symbol) B_{i jkl}=1

2 σ_ilδ_jk+σ_jlδ_ik+σ_ikδ_jl+σ_jkδ_il−2σ_{i j}δ_kl

+C_{i jkl}. (3.31)

This is the generalized version of the well knownHooke’s law. The stressesσ in equation (3.31) are those existing prior to the imposition of the strain,i.e.the stresses in the initial state. For sys- tems under no external stress (σ=0) the stress-strain coefficients reduce to the elastic constants,

6Side calculation to get from equation (3.28) to (3.29):

∆F = 1

2 V NC_{i jkl}

∑

α=1

∑

k

∑

h

ε˜_{i j}(h)˜ε_kl(k)e^i(h+k)·hr^αi (3.25)= 1

2 V NC_{i jkl}

∑

k

∑

h

i

2 h_iu˜_j(h) +h_ju˜_i(h)i

2(k_ku˜_l(k) +k_lu˜_k(k))

∑

α=1e^i(h+k)·hr^αi

= −V

8C_{i jkl}

∑

k

∑

h h_iu˜_j(h) +h_ju˜_i(h)

(k_ku˜_l(k) +k_lu˜_k(k))δ_h,−k

= V

8C_{i jkl}

∑

k k_iu˜_j(−k) +k_ju˜_i(−k)

(k_ku˜_l(k) +k_lu˜_k(k))

= V

8C_{i jkl}

∑

k k_ik_ku˜_j(−k)u˜_l(k) +k_ik_lu˜_j(−k)u˜_k(k) +k_jk_ku˜_i(−k)u˜_l(k) +k_jk_lu˜_i(−k)u˜_k(k)

= V

2C_{i jkl}

∑

kk_jk_lu˜_i(k)u˜_k(−k)

In the last step the symmetry properties of the tensor of elasticity are exploited and use is made of the possibility that indices occurring twice can be renamed arbitrarily. The following representation of theδ-function is used:

δ_k,k⁰= 1 N

∑

α=1e^i(k−k0)·r^α.

i.e. B_{i jkl}=C_{i jkl}. The adiabatic and isothermal stress-strain coefficients, which sometimes are also calledBirch moduliobey the relations

B^S_{i jkl}= ∂τi j

∂ε_kl

S and B^T_{i jkl}=

∂τi j

∂ε_kl

T. (3.32)

The superscriptsS andT will be omitted in the following, because all relations hold for the adi- abatic and for the isothermal case. From the expression (3.31) it is clear that the stress-strain coefficients B_{i jkl} are symmetric in the first and the in last pair of indices. So, they can also be written in Voigt notation, but do not satisfy all Voigt symmetry relations, becauseB_αβ 6=B_βα. Thecompliance tensor S_αβ is defined as the inverse of the elastic tensorB_αβ,i.e.

S_αβB_βγ=δαγ. (3.33)

By multiplication of Hooke’s law byS_αβ one obtains in Voigt notation

S_γατ_α ^(3.30)= S_γαB_αβε_β^(3.33)= δ_γβε_β =ε_γ (3.34)

⇒S_αβ = ∂εα

∂τ_β, (3.35)

which obviously is the inverse of (3.32). Equation (3.34) is analogous to the paramagnetic case, where a fieldH induces a magnetizationM=ξHand the compliance tensorS plays the role of the magnetic susceptibilityξ =∂M/∂H.

For isotropic pressure Pthe stress tensor is diagonal and can be written as⁷ τ_{i j} =−Pδ_{i j}. This reduces the relation (3.31) to

B_{i jkl}=−P δ_jlδ_ik+δ_ilδ_jk−δ_{i j}δ_kl

+C_{i jkl}. (3.36)

3.2.4 Compatibility Relations for the Strains

The components of the Lagrangian strain tensor(ε)_{i j}≡εi j are defined by equation (3.7) as partial derivatives of the displacement field with respect to some given reference lattice. Different strain tensor components cannot be arbitrary functions of the coordinates x, y, and z. They obey the followingcompatibility relations

O×(O×ε)^T =0, (3.37)

which were first derived by Saint Venant in 1890. These relations express the physical fact, that all material of a body before and after the deformation is continuous and connected. Inside of the body there do not occur dislocations or penetrations of body material due to deformation.

The compatibility relations can be obtained as mathematical identities after execution of the sec-

7The sign difference between the stress tensor components and the pressure is due to the sign convention: Positive stress on a body is exerted outward whereas positive pressure acts in the opposite direction.