Molecular Transport Phenomena Lecture notes Dr. Pouyan Boukany & Dr. Valeria Garbin Department of Chemical Engineering Faculty of Applied Sciences Delft University of Technology Delft, the Netherlands August 25, 2021 Chapter 1 Introduction 1.1 Course motivation These are the lecture notes for the course Molecular Transport Phenomena, an introductory course in our MSc program of Chemical Engineering. The course and these lecture notes have been developed over the years by several teachers: Harry van den Akker, Rob Mudde, Michiel Kreutzer, Volkert van Steijn, Laura Rossi; and most recently Pouyan Boukany and Valeria Garbin. The course is designed to stimulate students to explore the interaction between a molecular view, based on the undergraduate courses in physical chemistry, quantum mechanics and thermodynamics, and a continuum mechanics view, that they have been exposed to in engineering courses on transport phenomena and applied physics. We feel that these subjects are often taught without making connections. For instance, most transport classes jump immediately to Fick’s law when discussing diffusion, leaving the microscopic origin based on random walks aside. Especially for students that are well trained in chemistry but lacking in engineering, such an approach fails to recognize, or connect to, their “home ground”, the body of science that they have mastered. The last page of this chapter features a paper, which connects the microscopic view point with phenomena observed macroscopically and acts as an appetizer to this course. In this course, we will consider transport processes in bulk fluids and, for example, explain Fickian diffusion as a random walk of molecules. So far, you might expect that all diffusive processes can be described with Fick’s laws. You will learn that Fick’s law is not at all wrong, but a more generic description is needed to describe the diffusive processes in which multiple species are involved. This generic description is known as Maxwell-Stefan diffusion, and also here we will explain the foundations by considering the interactions between the different species. In addition to bulk processes, we will also consider processes at interfaces. Examples include the motion of fluids due to solid-liquid interactions at the surface of charged particles, and the statics and hydrodynamics of fluid-fluid interfaces. 1.2 Course learning objectives At the end of this course, a student should be able to: 3 4 CHAPTER 1. INTRODUCTION 1. Reproduce the basic laws of transport phenomena (including, but not limited to: Fick’s laws of diffusion, conservations laws for mass, species, momentum, and energy, the Hagen-Poisseuille equation for steady laminar pipe flow); solve problems involving the transport of mass, species, momentum, and energy, including setting up the problem using a shell balance, deriving the corresponding differential equation(s), formulating of the initial and boundary conditions, and solving basic ordinary differential equations. 2. Use simple microscopic models (examples: kinetic gas theory, random walk, Langevin’s theory) to derive expressions for quantities observed at the macro scale, such as pressure, diffusion coefficients, interfacial tension; estimate the order of magnitude of these macroscopic quantities using these expressions. 3. Explain the microscopic framework of multi-component diffusion and solve simple quasi-static one-dimensional problems using the Maxwell-Stefan equations, without and with additional forces (such as wall friction and electric fields) 4. Describe the motion of isolated and non-isolated charged particles using a molecular model; solve basic problems that involve electroosmotic and electrophoretic flows. 5. Explain interfacial tension as a force, pressure, energy and reproduce basic laws on capillarity (e.g. Laplace law, Young’s law). Solve the shape of static interfaces (e.g. based on energy minimization). Solve simple hydrodynamic problems in which the motion of interfaces is driven by gravity and pressure, and opposed by viscous friction. 1.3 Course assessment method The assessment for this course is designed to ensure that you stay on top of the material throughout the course and do not delay studying until the exam. The assessment includes both a formative component (assignments) and a summative component (final exam). The two take-home assignments will enable students to apply the concepts learned in the lectures and working classes to an unfamiliar problem. Each assignment can count towards 10% of the final grade. A final exam based on problem solving is meant to test a deeper understanding of the course material. Specific exam-level exercises will be provided in order to prepare well for the final exam. The final exam, in principle, counts for 80% of the final grade. In case the resulting course grade based on assignments and the exam is lower than the grade of the exam itself, then the exam counts for 100% of the final grade. Students who do not secure a passing mark for the course, based on the assignments and the final exam, can do a resit in period 3. In this case, the points obtained for the assignments will not be used to calculate the final resit grade, and the resit alone counts for 100% of the final course grade. Note for Academic Year 2021/22: the exam and the resit are planned to be on-campus but the exact format of assessment may change due to Covid-19 regulations. 1.4. COURSE SCHEDULE 1.4 5 Course schedule A detailed schedule of the course can be found on Brightspace. Updates to the schedule will be posted on Brightspace; students will be notified ahead of time via Brightspace announcements if there are changes to the schedule. 1.5 Levelling expectations Here a brief set of pointers to help align expectations between teachers, teaching assistants (TAs) and students. • TAs are available to answer questions, to help solving tutorial exercises and to provide clarifications on the study material when necessary. TAs (and lecturers) are however not required, neither should be expected, to know the solutions of ALL molecular transport phenomena exercises available on the internet and in books, unless these were part of specific assignments. Students should feel free to ask questions about all material, however help could also be given in the form of thinking along with the lecturer or TA rather than obtaining a direct solution to the problem. • Since this course covers a number of connected topics, developed from different fields of research, we will sometimes need to handle different terminologies for the same physical quantities. There is for example no universal agreement what symbol to use for what physical quantity, and different symbols will be used by different people. At the graduate level, and therefore also in this course, students are expected to be flexible enough to handle different terminology. Understanding, rather than memorizing, important equations will be of great help when dealing with these points. • The lectures and slides are given to support the material provided in the lecture notes or the book chapters you will read regarding a specific topic. Students should therefore not expect a 1:1 repetition of the lecture notes/book material in the lectures, but rather a supplementary tool to understand the topic. • Practicing solving old exam questions that are given by the lectures helps students test their skills and knowledge level; however students should not expect the questions of current exams to be a simple repetition of old exam questions. We expect students to be able to apply the learned theory and skills to a problem thet haven’t seen before. ESSAY NATURE|Vol 446|22 March 2007 Frontier at your fingertips Putting the pieces together The Hitchhiker’s Guide to the Galaxy famously features a supercomputer, Deep Thought, that after millions of years spent calculating “the answer to the ultimate question of life, the Universe and everything”, reveals it to be 42. Douglas Adams’s cruel parody of reductionism holds a certain sway in physics today. Our 42 is Schroedinger’s many-body equation: a set of relations whose complexity balloons so rapidly that we cannot trace its full consequences up to macroscopic scales. All is well with this equation, provided we want to understand the workings of isolated atoms or molecules up to sizes of about a nanometre. But between the nanometre and the micrometre wonderful things start to occur that severely challenge our understanding. Physicists have borrowed the term ‘emergence’ from evolutionary biology to describe these phenomena, which are driven by the collective behaviour of matter. Take, for instance, the pressure of a gas — a cooperative property of large numbers of particles that is not anticipated from the behaviour of one particle alone. Although Newton’s laws of motion account for it, it wasn’t until more than a century after Newton that James Clerk Maxwell developed the statistical description of atoms necessary for understanding pressure. The potential for quantum matter to develop emergent properties is far more startling. Atoms of niobium and gold, individually similar, combine to form crystals that, kept cold, show dramatically different properties. Electrons roam free across gold crystals, forming the conducting fluid that gives gold its lustrous metallic properties. Up to about 30 nanometres, there is little difference between gold and niobium. It’s beyond this point that the electrons in niobium start binding together into the coupled electrons known as ‘Cooper pairs’. By the time we reach the micrometre scale, these pairs have congregated in their billions to form a single quantum state, transforming the crystal into an entirely new metallic state — that of a superconductor, which conducts without resistance, excludes magnetic fields and has the ability to levitate magnets. Superconductivity is only the start. In assemblies of softer, organic molecules, a tenth of a micrometre is big enough for the emergence of life. Self-sustaining microbes little more than 200 nanometres in size have recently been discovered. Although we understand the principles that govern the superconductor, we have not yet grasped those that govern the emergence of life on roughly the same spatial scale. In fact, we are quite some distance from this goal, but it is recognized as the far edge of a frontier that will link biology and physics. Condensed-matter physicists have taken another cue from evolution, and believe that a key to understanding more complex forms of collective behaviour in matter lies in competition not between species, but between different forms of order. For example, high-temperature superconductors — materials that develop superconductivity at liquid-nitrogen temperatures — form in the presence of a competition between insulating magnetic behaviour and conducting metallic behaviour. Multi-ferroic materials, which couple magnetic with electric polarization, are found to develop when magnetism competes with lattice-distorting instabilities. A related idea is ‘criticality’ — the concept that the root of new order lies at the point of instability between one phase and another. So, at a critical point, the noisy fluctuations of the emergent order engulf a material, transforming it into a state of matter that, like a Jackson Pollock painting, is correlated and self-similar on all scales. Classical critical points are driven by thermal noise, but today we are particularly interested in ‘quantum phase transitions’ involving quantum noise: jigglings that result from Heisenberg’s uncertainty principle. Unlike its thermal counterpart, quantum noise leads to diverging correlations that spread out not just in space, but also in time. Even though quantum phase transitions occur at absolute zero, we’re finding that critical quantum fluctuations have a profound effect at finite temperatures. For example, ‘quantum critical metals’ develop a strange, almost linear temperature dependence and a marked predisposition towards developing superconductivity. The space-time aspect of quantum phase transitions gives them a cosmological flavour and there do seem to be many links, physical and mathematical, with current interests in string theory and cosmology. Another fascinating thread here is that like life, these inanimate transformations involve the growth of processes that are correlated and self-sustaining in time. Some believe that emergence implies an abandonment of reductionism in favour of a more hierarchical structure of science, with disconnected principles developing at each level. Perhaps. But in almost every branch of physics, from string theory to condensed-matter physics, we find examples of collective, emergent behaviour that share common principles. For example, the mechanism that causes a superconductor to weaken and expel magnetic fields from its interior is also responsible for the weak nuclear force — which plays a central role in making the Sun shine. Superconductors exposed general principles that were used to account for the weak nuclear force. To me, this suggests that emergence does not spell the end for reductionism, but rather indicates that it be realigned to embrace collective behaviour as an integral part of our Universe. As we unravel nature by breaking it into its basic components, avoiding the problem of ‘42’ means we also need to seek the principles that govern collective behaviour. Those include statistical mechanics and the laws of evolution, certainly, but the new reductionism that we need to make the leap into the realm between nano and micro will surely demand a new set of principles linking these two extremes. ■ Piers Coleman is in the Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA. FURTHER READING Anderson, P. W. Science 177, 393 (1972). Laughlin, R. B. A Different Universe (Basic Books, 2005). Davis, J. C. http://musicofthequantum.rutgers.edu (2005). Coleman P. & Schofield, A. J. Nature 433, 226–229 (2005). For other essays in this series, see http:// nature.com/nature/focus/arts/connections/ index.html CONNECTIONS Piers Coleman J. KAPUSTA/IMAGES.COM Between the nano- and micrometre scales, the collective behaviour of matter can give rise to startling emergent properties that hint at the nexus between biology and physics. 379
