The electric double layer put to work thermal physics at electrochemical interfaces PhD thesis, Utrecht University, April 2017 Cover design by Mathijs Janssen ISBN: 978-90-393-6761-2 The electric double layer put to work thermal physics at electrochemical interfaces De dubbellaag aan het werk gezet thermische fysica aan elektrochemische grensvlakken (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof.dr. G.J. van der Zwaan, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op woensdag 19 april 2017 des middags te 2.30 uur door Mathijs Adriaan Janssen geboren op 28 maart 1989 te Uithoorn Promotor: Prof.dr. R.H.H.G. van Roij Dit proefschrift werd mogelijk gemaakt met financiële steun van een NWO-VICI subsidie. Contents Publications vii 1 Introduction 1 1.1 Thermodynamics in a nutshell . .1 1.2 A capacitor cycle . .3 1.3 The electric double layer... .4 1.4 ...put to work . .6 1.5 Outline . .8 1.A Appendix: The potential of blue energy devices . .9 2 Electric double layer theory 11 2.1 Setup . 11 2.2 Density functional theory . 14 2.3 Classic EDL models . 18 2.4 Discussion: ionic correlations . 22 2.A Appendix: Density functional theory . 23 3 Capacitive mixing and deionization 25 3.1 Introduction . 25 3.2 Setup . 27 3.3 Theory . 29 3.4 Concentration profiles . 31 3.5 Charging and capacitance . 33 3.6 Cyclic processes . 37 3.7 Discussion and conclusion . 41 3.A Appendix: The excess free energy functional . 43 4 Boosting capacitive blue-energy and desalination devices with waste heat 45 4.1 Introduction . 45 4.2 Setup and capacitive behavior . 47 4.3 Boosting the work output of CAPMIX . 48 4.4 Adiabatic heating by charging . 50 4.5 Capacitive deionization . 51 4.6 Discussion . 53 4.7 Conclusion . 54 5 Thermal response in adiabatically charged electric double layer capacitors 55 5.1 Introduction . 55 vi Contents 5.2 Setup . 57 5.3 Thermodynamics . 57 5.4 Dynamics . 59 5.5 Results . 60 5.6 Conclusion . 62 5.A Appendix: Derivation heat equation . 63 5.B Appendix: Analytical approximation to the adiabatic temperature rise . 66 5.C Appendix: Excess correlations . 67 6 Capacitive thermal energy extraction 69 6.1 Introduction . 69 6.2 Thermodynamic description . 71 6.3 Experimental system . 74 6.4 Theoretical model . 78 6.5 Discussion and conclusion . 81 6.A Appendix: Comparison with thermoelectric devices . 83 6.B Appendix: Thermocapacitive Carnot cycles . 84 7 Heat of electric double layer formation in porous carbon electrodes 87 7.1 Introduction . 87 7.2 Theory for heat flow and production during EDL buildup . 89 7.3 Experimental methods . 93 7.4 Temperature-dependent experiments of EDL formation . 95 7.5 Discussion and conclusion . 100 8 Harvesting vibrational energy with liquid-bridged electrodes 103 8.1 Introduction . 103 8.2 Setup . 106 8.3 Thermodynamics . 106 8.4 Conclusion . 114 8.A Appendix: Traditional RC-circuit . 115 8.B Appendix: Dimensionless equations . 116 8.C Appendix: Static limit of square wave driving . 117 Bibliography 126 Samenvatting 127 Acknowledgements 133 CV 135 vii Publications Next to unpublished results, this thesis is based on the following work: 1. Mathijs Janssen, Andreas H¨arteland Ren´evan Roij, “Boosting Capacitive Blue-Energy and Desalination Devices with Waste Heat”, Physical Review Letters 113, 268501 (2014), (Chapter 4). 2. Andreas H¨artel,Mathijs Janssen, Sela Samin, and Ren´evan Roij, “Fundamental measure theory for the electric double layer: implications for blue-energy harvesting and water desalination”, Journal of Physics: Condensed Matter 27, 194129 (2015), (Chapter 3). 3. Andreas H¨artel, Mathijs Janssen, Daniel Weingarth, Volker Presser, and Ren´evan Roij, “Heat-to-current conversion of low-grade heat from a thermocapacitive cycle by superca- pacitors”, Energy & Environmental Science 8, 2396–2401 (2015), (Chapter 6). 4. Mathijs Janssen, Ben Werkhoven, and Ren´evan Roij, “Harvesting vibrational energy with liquid-bridged electrodes: thermodynamics in mechanically and electrically driven RC-circuits”, RSC Advances 6, 20485–20491 (2016), (Chapter 8). 5. Mathijs Janssen and Ren´evan Roij, “Reversible Heating in Electric Double Layer Capac- itors”, Physical Review Letters 118, 096001 (2017), (Chapter 5). 6. Mathijs Janssen, Elian Griffioen, Ren´evan Roij, Maarten Biesheuvel, Ben Ern´e, “Heat of electric double layer formation in porous carbon electrodes”, in preparation, (Chapter 7). Popular publications: 7. Mathijs Janssen, “(B)lauwe energie”, Nederlands Tijdschrift voor Natuurkunde, 82-3, 64- 67 (2016), (Samenvatting). English translation: “Blue and lukewarm energy”, Dutch Journal of Physics, 1, (2016). 1 Introduction 1.1 Thermodynamics in a nutshell The formulation and development of thermodynamics was driven by the practical need to optimize steam-engine performance. With the depletion of fossil fuels and ever-increasing world population, the construction of new types of engines to harvest energy from sustainable sources, as well as a fundamental exploration of the parameters governing their performance, is as relevant today as it was two centuries ago during the early days of thermodynamics. It is worthwhile to recap a few of the key notions of thermodynamics, since the motivation behind large parts of this thesis builds on analogous arguments and reasoning. Traditional heat engines consist of a gas at temperature T and pressure p that is enclosed in a container that prevents particles from leaking out, but does allow heat flow through its walls. While mostly rigid, on one side a piston can move frictionlessly to adjust the volume V of the gas (see Fig. 1.1(a), inset). The engine is alternately brought into thermal contact with two heat reservoirs at high and low temperature (TH and TL, respectively) during a com- pression/expansion cycle. The heat engine works by intercepting the heat flow that would spontaneously arise if the two heat reservoirs were brought into contact with each other, and performs it in a controlled manner. Consider the canonical example of the Stirling cycle as in- dicated in Fig. 1.1(a). The cycle consists of a compression phase at TL, a heating step towards TH by bringing the cylinder in contact with the high-temperature reservoir, followed by an expansion stroke at TH, and a final cooling step towards TL to bring the engine into its original position. As the piston moves over an infinitesimal distance, the gas performs an infinitesimal amount of work δW = pdV on its environment. This product of the conjugate thermodynamic variables p and V is of dimension energy, typically measured in units of joule. Integrated over a full expansion/compression cycle this yields an amount of work I I W = δW = p(V, T )dV. (1.1) We see that the work delivered by the engine during a cycle is exactly the surface enclosed by the cycle in the pressure/volume representation of Fig. 1.1(a). To draw explicit cycles and to find explicit expressions for the work performed, we need to know the relationship p(V, T ) between the pressure, volume and temperature. A milestone stone of 19th century physics is the formulation of the ideal-gas equation of state by Clapeyron (1834) which is a combination of Boyle’s law (1662), Charles’s law (∼ 1780), and Avogadro’s law (1811). Written in “modern” statistical mechanics language it reads NkBT p(V, T ) = , (1.2) V 2 1 Introduction (a) (b) +Q e x p d C an si on -Q ) 2 a t h heating igh te (V) m Ψ (N/m per p a work ture co work B m cooling p re discharging at low capacitance ssio n at low te mperature A charging at high capacitance 0 V (m3) 0 Q (C) Figure 1.1: (a) A Stirling compression-heating-expansion-cooling cycle of a heat engine, plotted in the p-V plane of conjugate thermodynamic variables. (b) Ψ-Q plot of the charging cycle ABCA of a plate capacitor. with N the (fixed) number of particles and kB Boltzmann’s constant. We now see that the engine provides net mechanical work because the compression phase at low temperature costs less energy than the amount of energy harvested during the expansion phase at high temperature. But where does the harvested mechanical energy come from? The first law of thermodynamics, dU = δQ − δW, (1.3) states that one can change the internal energy U of a system, either by adding heat δQ to the system, or by letting the system perform work δW on its environment. With this first law and H dU = 0, true for any state function integrated over a closed path, we can express the work H equivalently as W = δQ ≡ QH − QL, where QH and QL are the heat flows from the hot and cold reservoirs, respectively; a portion of the heat flow out of the hot reservoir is converted into mechanical work. The discussed Stirling cycle contains isochoric (dV = 0) heating (cooling) steps, in which a heat flow into (out of) the engine causes a temperature change. Obviously, the efficiency η ≡ W/QH of the engine suffers from these isochoric steps since they do not add to the performed work, but do add to the total heat flow. Is it possible to change the temperature of the gas without heat flowing? To answer this question it is convenient to introduce the concept of entropy, denoted by S. While work and heat are not state functions, indicated with δ for inexact differentials, it was realized by Clausius that the combination dS = δQ/T is. From this definition we see that no heat is exchanged along adiabatic (dS = 0) paths.