Brownian motion: elements of colloid dynamics

187 Pages·2018·3.136 MB·English

by  Philipse, Albert P

Undergraduate Lecture Notes in Physics
Albert P. Philipse
Brownian Motion
Elements of Colloid Dynamics
Undergraduate Lecture Notes in Physics
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Albert P. Philipse
Brownian Motion
Elements of Colloid Dynamics
123
Albert P. Philipse
Van ’tHoff Laboratory forPhysical
andColloidChemistry, Debye Institute
forNano-Materials Science
UtrechtUniversity
Utrecht, The Netherlands
ISSN 2192-4791 ISSN 2192-4805 (electronic)
Undergraduate Lecture Notesin Physics
ISBN978-3-319-98052-2 ISBN978-3-319-98053-9 (eBook)
https://doi.org/10.1007/978-3-319-98053-9
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Preface
Thisbooknucleatedasasetofundergraduatelecturenotes.Lookingforadditional
suitable textbook material on Brownian motion, this was often found either too
briefandqualitativelyortooextensivewithmoremathematicsthandesirableforan
introductory course. This book aims to be somewhere in between and intends to
provideatreatmentofBrownianmotiononalevelappropriateforbachelorstudents
of physics, chemistry, soft matter, and the life sciences.
One very appealing aspect of Brownian motion, as this book also illustrates, is
that the subject connects a broad variety of topics, including thermal physics,
hydrodynamics, reaction kinetics, fluctuation phenomena, statistical thermody-
namics, osmosis, and colloid science. For basic courses on any of these topics,
I hope this book will offer useful and motivating study material.
IwouldliketoacknowledgethemanyinsightfuldiscussionswithProf.Agienus
Vrij on colloids, Brownian motion, and osmotic pressure. Maria Bellantone
(SpringerUK)isthankedforthepleasantcollaboration—andforencouragingmeto
finally complete this book. Maria Uit de Bulten-Weerensteijn and Yvette Roman
arethankedfortheirhelpinthepreparationofthebook.SamiaOuhadjjiandBonny
Kuipers are acknowledged for proofreading. The anonymous referees and Ute
Heuser (Springer Physics) have offered helpful comments. Any remaining typos,
mistakes or unclarities are, of course, the author’s sole responsibility.
Utrecht, The Netherlands Albert P. Philipse
2018
vii
Contents
1 A First Round of Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 A Restless Realm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Stokes-Einstein Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 The Particle Quartet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 A Feverish Sphinx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Through a Small Grain of Glass . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Molecular Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Molecular Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Colloids Are Molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Kinetic Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 The Basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Free Volumes and Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Pressure from Ideal Thermal Particles . . . . . . . . . . . . . . . . . . . 27
3.4 Velocity Distributions and Energy Equipartition . . . . . . . . . . . . 31
3.5 Soft Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 A Tale of Ten Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Brownian Versus Ballistic Motion . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Mass-Related Time Scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 The Diffusive Regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Continuity, Gradients and Fick’s Diffusion Laws . . . . . . . . . . . . . . 61
5.1 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Constitutive Equations and Fick’s Laws. . . . . . . . . . . . . . . . . . 64
ix
x Contents
5.3 Stationary Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Diffusion in a Dilute Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6 Brownian Displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.1 Einstein for Chemists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 Translational Diffusion Coefficient from Equilibrium. . . . . . . . . 77
6.3 Quadratic Displacements via Einstein’s Diffusion Approach . . . 80
6.4 Brownian Motion from Newtonian Mechanics . . . . . . . . . . . . . 82
6.5 Angular Displacements from a Diffusion Equation . . . . . . . . . . 85
6.6 The Rotational Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . 88
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7 Fluid Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.1 Fluid Velocity Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2 The Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3 Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.4 On Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8 Flow Past Spheres and Simple Geometries . . . . . . . . . . . . . . . . . . . 105
8.1 Slits and Tubes—and Darcy’s Law . . . . . . . . . . . . . . . . . . . . . 105
8.2 Friction Factor of a Rotating Sphere . . . . . . . . . . . . . . . . . . . . 109
8.3 The Translational Friction Factor. . . . . . . . . . . . . . . . . . . . . . . 113
8.4 Stick, Slip and the Lotus Sphere . . . . . . . . . . . . . . . . . . . . . . . 118
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9 Encounters of the Brownian Kind. . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.1 Diffusion Versus Convection. . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.2 Brownian Motion Towards a Spherical Absorber . . . . . . . . . . . 123
9.3 Diffusional Sphere Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.4 Birth and Growth of Brownian Clusters . . . . . . . . . . . . . . . . . . 127
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
10 Random Walks in External Fields . . . . . . . . . . . . . . . . . . . . . . . . . 133
10.1 One-Dimensional Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
10.2 Radial Brownian Motion and Colloidal Stability. . . . . . . . . . . . 136
10.3 Brownian Motion in a Shear Flow. . . . . . . . . . . . . . . . . . . . . . 138
10.4 Brownian Magnets in a Magnetic Field . . . . . . . . . . . . . . . . . . 139
10.5 Gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
11 Brownian Particles and Van ’t Hoff’s Law. . . . . . . . . . . . . . . . . . . 147
11.1 Thermodynamics of Dilute Solutions . . . . . . . . . . . . . . . . . . . . 148
11.2 Osmotic Pressure Gauged via the Solvent . . . . . . . . . . . . . . . . 150
Contents xi
11.3 Osmotic Pressure from Brownian Motion; Vrij’s Statistical
Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Appendix A: Moments, Fluctuations and Gaussian Integrals ..... .... 155
Appendix B: Summary Vector Calculus. .... .... .... .... ..... .... 159
Appendix C: Answers to Selected Exercises .. .... .... .... ..... .... 165
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 175