Nucleation of Gas Hydrates [1st ed.] 9783030518738, 9783030518745

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Nobuo Maeda

Nucleation of Gas Hydrates

Nucleation of Gas Hydrates

Nobuo Maeda

Nucleation of Gas Hydrates

123

Nobuo Maeda Department of Civil and Environmental Engineering, School of Mining and Petroleum Engineering University of Alberta Edmonton, AB, Canada

ISBN 978-3-030-51873-8 ISBN 978-3-030-51874-5 https://doi.org/10.1007/978-3-030-51874-5

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© Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Hiroshi Maeda who showed me the way.

Preface

Nucleation is intrinsically an interfacial phenomenon. Classical nucleation theory is based on nucleation work of cluster formation that arises from the interfacial free energy. This feature is compounded in the nucleation of gas hydrates due to their unique attributes. It is therefore not obvious how best to introduce an enormously complex subject of nucleation of gas hydrates. The basic approach I ended up adopting in writing this monograph was the principle of Occam’s razor—keep it as simple as possible but not simpler. Abinitio considerations were employed where appropriate to minimize the number of underlying assumptions on which our subjects are built. Also, effort was made to describe relevant concepts as simple as possible so that the readers would be able to intuitively grasp the essence of the conceptual ideas but not so simple that the descriptions would become inaccurate. Richard Feynman once remarked that he would consider that he has understood an equation when he could guess its answer without actually solving it. The first two chapters are of general utility and applicable to nucleation phenomena of any system. Chapter 1 introduces classical nucleation theory and its underlying conceptual ideas. Chapter 2 details the two leading experimental techniques for investigations of nucleation rates, which should also be of general utility and applicability to nucleation studies of any system. Nucleation rates are central to nucleation phenomena of a given system and so is the case for gas hydrates. In fact, a primary difficulty in the investigation of clathrate hydrate nucleation has been the inability of the researchers to determine and compare nucleation rates of clathrate hydrates across various systems of different scales and complexities, which in turn has been limiting the ability of the researchers to study the nucleation process itself. Chapter 3 introduces selected physical properties of gas hydrates that are relevant to nucleation. Chapter 4 then introduces interfacial gaseous states. Heterogeneous nucleation of gas hydrates requires not only a solid substrate that could lower the nucleation work and the activation barrier but also ample supplies of guest gases in its vicinity. This unique attribute of gas hydrate systems renders interfacial gaseous states the key concept in the heterogeneous nucleation of gas hydrates. Finally, Chapter 5 introduces nucleation of ice and nucleation of gas hydrates. Since this monograph is meant to introduce the complex subject of nucleation of gas hydrates, vii

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Preface

we only cover its simplest aspects. For example, we only concern ourselves with nucleation of incipient gas hydrates, that is, nucleation of gas hydrate from a hydrate-free state, and do not touch any secondary nucleation, that is, nucleation of a second type of gas hydrate in the presence of a hydrate phase. A short tea break topic is inserted at the end of each chapter. An appendix is added at the end of the monograph that lists nucleation rate data of clathrate hydrates of some guest gases. Edmonton, Canada

Nobuo Maeda

Contents

1 Nucleation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Phase Equilibria and Phase Behavior . . . . . . . . . . . . . . . . . 1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Clausius–Clapeyron Equation . . . . . . . . . . . . . . . . . 1.1.3 The Principle of Detailed Balance . . . . . . . . . . . . . . 1.2 Classical Nucleation Theory . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Source of Fluctuations . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Nucleation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Scaling Laws for Nucleation Rates . . . . . . . . . . . . . 1.2.4 Nucleation Work of Homogeneous Nucleation and Heterogeneous Nucleation . . . . . . . . . . . . . . . . . . . 1.3 Homogeneous Nucleation of Solutes from Supersaturated Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Nucleation of Two-Dimensional Solid Layers . . . . . . . . . . 1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Estimation of the Contact Angle of Alkane Melt on Frozen Alkane Monolayers . . . . . . . . . . . . . . . . . . . 1.4.3 Latent Heat of Surface Fusion . . . . . . . . . . . . . . . . 1.4.4 Nucleation of Two-Dimensional (2D) Solid Layers . 1.5 Tea Time Break: Nucleation in Capillary Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Experimental Methods for Determination of Nucleation Rates 2.1 Constant Temperature Method . . . . . . . . . . . . . . . . . . . . . . 2.2 Linear Cooling Ramp Method . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Analysis in the Subcooling Domain . . . . . . . . . . . . 2.2.3 Analysis in the Time Domain . . . . . . . . . . . . . . . . .

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2.2.4 Protocol of Converting an Experimentally Measured Survival Curve to a Nucleation Curve . . . . . . . . . . . 2.2.5 The Impact of Experimental Cooling Rate . . . . . . . . 2.3 Tea Time Break: The Fourier Transform and the Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Gas Hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Physical Properties of Gas Hydrates . . . . . . . . . . . . . . . . . 3.1.1 Crystal Structures of Clathrate Hydrates . . . . . . . . . 3.1.2 Hydration Numbers . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Electromagnetic Properties . . . . . . . . . . . . . . . . . . . 3.2 Thermodynamic Aspects of Gas Hydrates . . . . . . . . . . . . . 3.2.1 Phase Diagrams of Clathrate Hydrates at Relatively Low Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Phase Diagrams of Clathrate Hydrates at Higher Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Phase Equilibrium Computation of Clathrate Hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Gas Hydrate Inhibitors . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Thermodynamic Hydrate Inhibitors (THI) . . . . . . . . 3.3.2 Kinetic Hydrate Inhibitors (KHI) . . . . . . . . . . . . . . 3.3.3 Anti-Agglomerates (AA) . . . . . . . . . . . . . . . . . . . . 3.3.4 Anti-freeze Proteins . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Tea Time Break: Similarities of Biological Systems to the Flow Assurance Challenges of Oil and Natural Gas Pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Interfacial Gaseous States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Disjoining Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Phenomenological Descriptions of Disjoining Pressure in a Single-Component System . . . . . . . . . . . . . . . . . . 4.1.3 The Origin of the Disjoining Pressure . . . . . . . . . . . . . 4.1.4 Van der Waals Forces . . . . . . . . . . . . . . . . . . . . . . . .

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4.2 Interfacial Gaseous Layers . . . . . . . . . . . . . 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . 4.2.2 Experimental Observations . . . . . . . . 4.2.3 Thermodynamic Considerations . . . . 4.3 Is the Surface of Gas Hydrates Dry? . . . . . . 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . 4.3.2 Thermodynamic Basis of Pre-melting 4.4 Tea Time Break: Disjoining Pressure and Polywater . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Nucleation of Gas Hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Nucleation of Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Physical Properties of H2O . . . . . . . . . . . . . . . . . . . . 5.1.3 Nucleation of Ice from Bulk Liquid Water . . . . . . . . 5.1.4 Nucleation of Ice from Water Vapor . . . . . . . . . . . . . 5.2 Nucleation Rate of Gas Hydrates . . . . . . . . . . . . . . . . . . . . . 5.2.1 Unique Attributes of Gas Hydrate Nucleation . . . . . . 5.2.2 Historical Perspective of Gas Hydrate Nucleation . . . 5.2.3 Empirical Approach . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Scaling Laws for the Nucleation Rates of Gas Hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Theoretical Front and the Applicability of Classical Nucleation Theory to Gas Hydrate Systems . . . . . . . . 5.2.6 Nucleation Pathways . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Memory Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Major Attributes of the Memory Effect and Proposed Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Remaining Loose Ends . . . . . . . . . . . . . . . . . . . . . . 5.4 Effects of Electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Locations of Ions in a Salt Solution . . . . . . . . . . . . . 5.4.3 Gibbs Adsorption Isotherm . . . . . . . . . . . . . . . . . . . . 5.4.4 Concentration Gradient of Guest Gases Near a Salt Solution–Guest Gas Interface . . . . . . . . . . . . . . . . . . 5.4.5 Salting-Out Effect of Ions . . . . . . . . . . . . . . . . . . . . . 5.4.6 Experimental Observations . . . . . . . . . . . . . . . . . . . . 5.5 Concluding Remarks: The End of the Beginning . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix: Nucleation Rate Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Chapter 1

Nucleation Theory

1.1 Phase Equilibria and Phase Behavior 1.1.1 Introduction There are excellent textbooks on phase equilibria and phase diagrams, like [1], so here we only cover the essence of phase behavior that is necessary for an understanding of nucleation of gas hydrates or clathrate hydrates. We note at this stage that we regard the terms “gas hydrates” and “clathrate hydrates” as interchangeable throughout this book. A good starting point of the topic would be the Gibbs phase rule [1]. Conditions for coexistence of macroscopic phases can be set out in terms of intensive thermodynamic parameters such as pressure, temperature, and composition, in accordance with the Gibbs phase rule [1]. F =C − P +2

(1.1.1)

where F is the number of degrees of freedom, C is the number of components, and P is the number of phases in the system. Therefore, given the pressure and the temperature (F = 2) of a one-component system (C = 1), the phase state of the system is completely fixed (P = 1). Below, we consider a one-component system (C = 1). The van der Waals equation of state is the first of a series of cubic equations of state that accounted for phase transitions. There have been a number of variants to the cubic equations of state since the time of van der Waals, such as Redlich–Kwong equation of state, Soave–Redlich–Kwong equation of state, Zudkevitch–Joffe–Redlich–Kwong equation of state, Peng–Robinson equation of state, and so on [1]. The van der Waals equation of state will suffice for our purpose here. A remarkable feature of the van der Waals equation of state is that adding the intermolecular attractions and repulsions to the ideal gas law leads to the presence of a critical point and phase transitions. It is quite mind-bending that van der Waals © Springer Nature Switzerland AG 2020 N. Maeda, Nucleation of Gas Hydrates, https://doi.org/10.1007/978-3-030-51874-5_1

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1 Nucleation Theory

T > Tc Pressure

Fig. 1.1 A schematic illustration of how V varies with P at a constant T. Three representative isotherms for each case of T > T c , T = T c and T < T c are shown

C T = Tc T < Tc

Volume

came up with his famous equation at a time when the existence of molecules was still in doubt. For one mole of gas, the van der Waals equation of state is P = RT /(V − b) − a/V 2

(1.1.2)

There are three independent parameters in the van der Waals equation of state: pressure (P), molar volume (V ), and temperature (T ). R is the gas constant, and a and b are called the attraction parameter and the repulsion parameter, respectively. Both of these corrections are to the volume of the gas. Equation (1.1.2) shows that V approaches a limiting value, b, at high pressures (P → ∞). This limiting value is due to the finite size of molecules (volume occupied by the molecules). The a/V 2 term reduces the system pressure (the actual pressure of van der Waals gas is smaller than the pressure of an ideal gas by a/V 2 ). A term that reduces P represents intermolecular attraction (repulsion would increase the pressure). We can select T to be a constant and see how V varies with P at a constant T (i.e., draw an isotherm) in a P–V diagram. Depending on the choice of T, isotherms can take different shapes. V monotonically decreases with increasing P for a sufficiently large choice of T. Then, an inflection point appears on the isotherm at a certain value of T. The isotherm has a minimum and a maximum below this choice of T. The schematic of these three representative isotherms is shown in Fig. (1.1). The T at which the isotherm has an inflection point, C (critical point), is what we call a critical temperature, T c . Importantly, both coefficients of the van der Waals equation of state can be uniquely determined if we know the physical parameters (P, T, ρ where ρ is the density) at the critical point, Pc , V c , and ρ c [1]. It is therefore sometimes more convenient to express these parameters in reduced properties (relative to the respective values at the critical point) [1]. Tr ≡ T /Tc ,

Pr ≡ P/Pc , Vr ≡ V /Vc , ρr ≡ ρ/ρc

(1.1.3)

Then, the constants a and b in the van der Waals equation are given in terms of Pc and T c as a = 27R2 T 2c /64Pc and b = RT c /8Pc . These expressions follow from

1.1 Phase Equilibria and Phase Behavior

3

Fig. 1.2 A schematic illustration of an isotherm in a P–V diagram for T < T c that is expressed in reduced properties

Pr

a

e f

b

g d

Vr the mathematical conditions of an inflection point of the isotherm for T = T c , that is, ∂P/∂V = ∂ 2 P/∂V 2 = 0 at the critical point. Then, from Eq. (1.1.3), the van der Waals equation of state (Eq. 1.1.2) can be simplified in terms of T r , Pr , and V r to 

 Pr + 3/Vr2 (3Vr − 1) = 8Tr

(1.1.4)

There is no phase transition above T c because V monotonically decreases with increasing P. Below T c , in contrast, there is a section in the isotherm in which V increases with increasing P (the section between points d and e in Fig. (1.2)). A cubic equation of state, like the van der Waals equation of state, generally has three roots. However, not all of them are physically real. The situation is analogous to finding the length of a side of a square when its area is given: solving a quadratic equation yields two roots, one positive and the other negative, but only the positive root is physically real. Similarly, for the van der Waals equation, that the system volume expands with pressurization (∂V /∂P > 0) is unphysical and this section of the isotherm (between d and e) cannot materialize in reality. Instead, the limit of metastability (spinodal) is reached, and the system will separate into two phases of liquid (a–b) and gas (f –g). The position of the horizontal dashed straight line through b and f is such that the two areas enclosed by the van der Waals curve and the horizontal straight line are equal.

1.1.2 Clausius–Clapeyron Equation A phase boundary defines regions of the thermodynamically stable phases in a phase diagram. To construct a phase diagram, therefore, one needs to know how a phase boundary varies with a change in pressure or temperature.

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The Clausius–Clapeyron equation can be used to find the relationship between pressure and temperature along phase boundaries [2, 3]. Feynman had a unique way of explaining key thermodynamic concepts in his lecture series (44–46), including the Clausius–Clapeyron equation in series (45) [2]. His illustrations of the Carnot cycle, the efficiency of an ideal engine, and the ratchet model are among the most elegant explanations I have come across on these topics, and the readers are recommended to read the Feynman lecture series for an alternative derivation. Here, we follow the approach adopted by Reif [3]. The general expression for the Gibbs free energy is [3, 4] G ≡ U − T S + PV

(1.1.5)

where G is the Gibbs free energy, U is the internal energy, T is the system temperature, S is the system entropy, P is the system pressure, and V is the system volume. The Gibbs free energy of a system is also equal to the total of the chemical potential of all the molecules in the system. G = μN

(1.1.6)

where μ is the chemical potential and N is the number of the molecules in the system. Now we consider two phases of a closed single-component system, (1) and (2), that have the chemical potential of μ1 and μ2 , respectively. If we denote the number of the molecules in each phase as N 1 and N 2 , then the total Gibbs free energy of the system is G = μ1 N1 + μ2 N2

(1.1.7)

Since the system is closed, the total number of molecules, N 1 + N 2 , is a constant. Then, dG = μ1 dN1 + μ2 dN2 = μ1 dN1 + μ2 d(constant − N1 ) = μ1 dN1 − μ2 dN1 = (μ1 − μ2 )dN1

(1.1.8)

For the two phases to be in equilibrium, dG = 0, and hence, (μ1 − μ2 ) = 0 or μ1 = μ2

(1.1.9)

In other words, the transfer of a molecule from one of the two phases to the other does not change the system free energy when μ1 = μ2 . The physical meaning of a phase boundary is almost clear now. Since G = G(P, T ) [3, 4], if P and T were such that μ1 > μ2 then the minimum value of G in Eq. (1.1.7) would be achieved if all the molecules transformed from Phase (1) to Phase (2). Likewise, if P and T were such that μ1 < μ2 then the minimum value of G in Eq. (1.1.7) would be achieved if all the molecules transformed from Phase (2) to Phase (1). Thus, the two phases could only coexist in equilibrium when μ1 = μ2 because a transformation of a number of molecules from one phase to the other

1.1 Phase Equilibria and Phase Behavior

5

does not alter G at such combinations of P and T that satisfies the condition μ1 = μ2 . In other words, the locus of points where P and T satisfy the condition μ1 = μ2 represents the phase boundary between the two phases (1) and (2). On one side of such phase boundary, μ1 > μ2 and Phase (2) is the thermodynamically stable phase, whereas μ1 < μ2 on the other side of such phase boundary and Phase (1) is the thermodynamically stable phase. Is there a way to construct a phase diagram without knowing the functional form of μ(P, T )? By definition, μ1 is equal to μ2 on the phase boundary, so μ1 (P, T ) = μ2 (P, T )

(1.1.10)

Meanwhile, since dG = V dP − SdT [3, 4], dμ1 = dG 1 /N1 = (V1 /N1 ) · dP − (S1 /N1 ) · dT = v1 dP − s1 dT

(1.1.11)

dμ2 = dG 2 /N2 = (V2 /N2 ) · dP − (S2 /N2 ) · dT = v2 dP − s2 dT

(1.1.12)

where v is the molecular volume and s is the entropy per molecule. From Eq. (1.1.10), dμ1 = dμ2 as long as the change is along the phase boundary. Thus, v1 dP − s1 dT = v2 dP − s2 dT

(1.1.13)

(v1 − v2 )dP = (s1 − s2 )dT

(1.1.14)

(dP/dT ) = (s1 − s2 )/(v1 − v2 ) = NA (s1 − s2 )/NA (v1 − v2 )

(1.1.15)

where N A is the Avogadro number. Thus, Eq. (1.1.15) states that the local slope (dP/dT ) on a phase boundary is equal to the change in the molar entropy divided by the change in the molar volume between the two phases (1) and (2) at that P and T: (dP/dT ) = Smolar /Vmolar

(1.1.16)

How can one find S molar ? A great property of the first-order phase transition is that the transformation takes place at a constant temperature T and the change in the entropy is the latent heat of the transition between the two phases (1) and (2), L 12 , divided by that temperature: (dP/dT ) = L 12 /(T Vmolar )

(1.1.17)

As can be seen from Eq. (1.1.17), the greater the change in V molar (as in a liquid– gas transition) the smaller the slope, dP/dT, in the phase diagram. When the change in V molar is negative (as in the ice–liquid water transition), the slope, dP/dT, in the phase diagram is also negative.

6

1 Nucleation Theory

1.1.3 The Principle of Detailed Balance In the framework of thermodynamics up until now, there has been no concept of time so far. That two phases (1) and (2) can coexist in thermodynamic equilibrium when μ1 (P, T ) = μ2 (P, T ) does not mean complete absence of transformation of molecules between the two phases. Rather, there are equal numbers of molecules transforming in one direction as the other. The principle of detailed balance states that each elementary process should be balanced by its reverse process in equilibrium. Consider a situation in which a single-component liquid is introduced to a closed system that is maintained at a constant temperature. The amount of liquid is plentiful and assumes that the space above the liquid in the closed system at time = 0 is a vacuum. How quickly will the liquid evaporate? After a long time, after the system has attained an equilibrium, the space above the liquid will be totally saturated with the vapor. Since the system is maintained at a constant temperature, any cooling due to the latent heat of evaporation will be promptly replenished by the heat reservoir, and the vapor will attain an equilibrium vapor pressure of the liquid at that temperature. The principle of detailed balance states that, after the system has achieved an equilibrium, the elementary rate of evaporation of a molecule is equal to the elementary rate of condensation of the molecule. For the system as a whole, the rate of evaporation is constant with time after time = 0 because the amount of the liquid is plentiful. The rate of condensation, in contrast, is zero at the beginning (time = 0) because the upper space has started from a vacuum. Even though the elementary rate of evaporation of a molecule is equal to the elementary rate of condensation of the molecule, since there is no molecule in the vacuum to condense from at time = 0, the total rate of condensation is zero. The system-wide rate of condensation then gradually increases with time as the upper space is progressively populated with the vapor molecules. Eventually, the system-wide rate of condensation equals the system-wide rate of evaporation at which point an equilibrium is established. The great value of the principle of detailed balance comes from the realization that the elementary rate of evaporation of a molecule can be estimated from the elementary rate of condensation of a molecule in equilibrium and vice versa. For example, consider two liquids, A and B. Liquid A has a much greater equilibrium vapor pressure than Liquid B at a certain temperature. Which liquid takes longer to attain equilibrium if each liquid is introduced to a pre-evacuated chamber of the same temperature at the same time? That the equilibrium vapor pressure of A is higher than B means that both the rate of evaporation and the rate of condensation of the molecule of A are greater than those of the molecule B. Therefore, Liquid A will attain equilibrium first.

1.1 Phase Equilibria and Phase Behavior

7

Another example is laser. It is known that a laser requires at least three, preferably four, energy levels (electronic states) to create a population inversion that must precede before any stimulated emission can be induced. Why is it impossible to create a population inversion with only two energy levels? The principle of detailed balance states that the rate of transfer of an electron from one energy level to the other is the same. At the beginning of an excitation process, the vast majority of the electrons are in the lower energy level. As the pumping progresses, more and more electrons are transferred to the upper energy level. However, the rate of the downward transfer increases as the upper energy level is increasingly more populated. The rate of the downward transfer matches that of the upward transfer by the time the population of both levels become equal, so it is impossible to populate the upper level more than the lower level—i.e., impossible to attain a population inversion. Thus, at least three and typically four energy levels are required in a laser, as shown in Fig. (1.3). In a three-state laser (the left panel), the stimulated emission occurs between Level 2 and Level 1. The electrons that have been excited from Level 1 to Level 3 quickly relax (lose some energy) to a slightly lower energy Level 2, which preferably has a long lifetime. As Level 3 is promptly evacuated by such relaxation, electrons from the bottom state (Level 1) can continuously be excited to Level 3. Then, a population inversion can be established between Level 2 and Level 1. In a four-state laser (the right panel), a fourth level, termed Level 2, can be used for a stimulated emission from Level 3 to Level 2. The electrons that have been excited from Level 1 to Level 4 quickly relax (lose some energy) to a slightly lower energy Level 3, which preferably has a long lifetime. Since Level 2 is hardly populated at the beginning, a population inversion between Level 3 and Level 2 can be established more easily than in a three-state laser (although at a cost of the narrower energy gap between Level 3 and Level 2 than the energy gap between Level 2 and Level 1 of the three-state laser). After the stimulated emission, the electrons relax from Level 2, which preferably has a short lifetime, back to Level 1. Level 3

Level 4

Level 2

Level 3

Laser emission

Laser emission

Level 2 Level 1

Level 1

Fig. 1.3 A schematic illustration of energy levels (electronic energy states) of a three-state laser (left) and a four-state laser (right)

8

1 Nucleation Theory

1.2 Classical Nucleation Theory The definition of the term “classical nucleation theory” is not clear or universal in the literature. In a broad sense, classical nucleation theory may refer to a theoretical framework that allows the formation of spatially non-uniform patches of different densities (e.g., clusters of a thermodynamically stable phase within a metastable parent phase), as opposed to a simultaneous and spatially uniform transition of macroscopic phases (the density of the metastable parent phase uniformly and simultaneously changes to that of the thermodynamically stable phase). In a narrow sense, the term may refer to a particular aspect of said broad theoretical framework. There are excellent textbooks on classical nucleation theory, such as Kashchiev’s book [5], so here we only cover the essence of classical nucleation theory that is necessary for an understanding of nucleation of clathrate hydrates. The readers are also referred to Feynman [2] and Callen [4] for thermodynamics and Reif [3] for statistical mechanics.

1.2.1 Source of Fluctuations According to the theory of canonical ensemble of classical statistical mechanics, the probability that one finds a particle in a certain energy state is described by the Boltzmann distribution [3]: P = A exp(−E/kT )

(1.2.1)

where P is the probability of finding a particle in an energy state, E, A is a constant, k is the Boltzmann constant, and T is the absolute temperature. An important conclusion of the Boltzmann distribution is that there is always a finite (non-zero) probability that a particle can occupy a very high energy state, no matter how unstable. In other words, if one waits long enough, one will sooner or later encounter a moment such an event occurs. This fluctuation is central to nucleation phenomena. But, physically, where did such fluctuations come from? A salient point is that a transition of an elementary particle (photon, electron, etc.) between two energy states can occur not only between two real energy states but also between a real state and a virtual one. The uncertainty principle of quantum mechanics states that the lifetime of such a virtual state is inversely proportional to the energy gap between the virtual state and the nearest real state. In other words, a particle can assume virtually any energy state, real or virtual, but it becomes progressively less likely to find the particle in a virtual state that is far away from a real state. This is the source of fluctuation embedded at the heart of quantum mechanics. In addition to the non-zero temperatures on earth, high energy cosmic radiation and environmental background radiation could provide the necessary energy to surmount an activation barrier for nucleation.

1.2 Classical Nucleation Theory

9

In a system that contains multiple particles, the Gibbs free energy, G, takes the place of E in Eq. (1.2.1) when the presence of one particle is not independent of the presence of the others. Still, the essence remains the same: the theory of grand canonical ensemble states that such particles follow a free energy probability distribution of the Boltzmann form [3]: P = A exp(−G/kT )

(1.2.2)

where P is the probability of finding the system that contains multiple particles in a free energy state G, A is a constant, k is the Boltzmann constant, and T is the absolute temperature. Consider a one-component system at the melting point of the component. The solid phase and the liquid phase can coexist at the melting point. If one heats the solid at the melting point, it will melt due to the presence of the pre-melting layer (this is the topic of Chap. 4). If one cools the liquid at the melting point, however, it does not freeze immediately. Instead of releasing the latent heat at the melting point, the liquid can cool beyond the melting point as if nothing has happened. This remarkable attribute of a liquid to cool beyond the melting point (subcooling or supercooling) suggests that the free energy of the system does not remain constant at a constant temperature but somehow increases during the freezing process. This increase in the free energy during the freezing process at a given temperature is called an activation barrier or activation free energy barrier. Where did such an activation barrier come from? Kashchiev showed that the probability that a phase transition proceeds through a spatially uniform change in density throughout a system is vanishingly small [5]. Instead, a spatially non-uniform pathway is much less taxing [5]. This spatially non-uniform pathway is the basis of classical nucleation theory, which allows the formation of not only the monomers but also clusters of various sizes in a supposedly spatially uniform metastable parent phase. Given that the presence of such clusters contradicts the spatial uniformity of a phase (definition of a phase in Sect. 1.1), the formation of such entities is considered transient in nature. Classical nucleation theory further postulates that, given the transient nature of such clusters, the population of such clusters follows the Boltzmann distribution of the form Eq. (1.2.2). In classical nucleation theory, the formation of a cluster will create an interface around it and it costs interfacial free energy to do so. This free energy cost is proportional to the interfacial area. We may denote this free energy cost as Ginterfacial . In addition, as two molecules come together, the system as a whole will gain intermolecular potential energy [6] but lose entropy [3]. The same consideration applies when multiple molecules come together. These effects are proportional to the mass or the volume of the cluster. We may then combine the two effects in the form of Gbulk . Then G activation = G interfacial + G bulk

(1.2.3)

10

1 Nucleation Theory

Fig. 1.4 A schematic illustration of how Gactivation (r), the blue curve, varies with the size of the cluster at a constant temperature below the melting point (for which Gbulk < 0)

G(r)interfacial ΔG(r*)activation

G(r)bulk radius

ΔG(r)activation r*

Gbulk can be positive, negative, or zero depending on whether the system is above, below, or at the melting point, respectively. In contrast, Ginterfacial varies with the temperature but always stays positive. How Gactivation varies with the size of the cluster depends on the intermolecular potential energy profiles and the shape of the cluster, and as such precise descriptions are highly complex. Here, we only note the general trend. Figure (1.4) shows a schematic picture of how Gactivation could vary with the size of the cluster at a constant temperature below the melting point (for which Gbulk < 0). The interfacial area scales with the square of the radius, r, of the cluster, whereas its volume scales with the cube of the radius. Since they are each proportional to Ginterfacial and Gbulk , respectively, there will be a maximum in Gactivation somewhere along the line. For a cluster smaller than such a critical size, r*, for which Gactivation increases with r, its growth is unfavorable and hence does not proceed. In contrast, once a cluster finds a way to exceed such critical size, r*, then the cluster can keep growing to macroscopically detectable sizes that will eventually consume all the metastable phase. We may denote such a maximum, G(r*)activation , as G ∗activation . This is the activation barrier that needs to be overcome for a nucleation event to materialize. The shapes of both G(r)bulk and G(r)interfacial shown in Fig. (1.4) vary with temperature, and hence so does the shape of G(r)activation . It is clear, then, that a liquid-to-solid phase transition cannot proceed at or above the melting point, because G(r)activation monotonically increases with the cluster size and consequently G ∗activation , which corresponds to the maximum of G(r)activation , diverges to infinity. Thus, a phase transition can only occur when G(r)bulk is negative (subcooled) and by a sufficient extent that can offset the positive G(r)interfacial term. This excess Gbulk required for the phase transition is called the driving force. The driving force is defined as the free energy difference between the metastable state and the thermodynamically stable state, and increases with the system subcooling. G driving_force ≡ G metastable − G equilibrium

(1.2.4)

Let us now consider how the three curves shown in Fig. (1.4) may vary with the system subcooling. The G(r)bulk curve moves lower, while the G(r)interfacial curve moves slightly higher (the interfacial free energy generally decreases with heating

1.2 Classical Nucleation Theory

11

and approaches zero toward the critical point) and consequently the G(r)activation curve moves lower as the system cools. Then the maximum of the G(r)activation curve, G ∗activation , also diminishes with the system subcooling. If the driving force becomes so large that G ∗activation < 0, then the formation of clusters of any sizes will become favorable and, as such, the phase transition will proceed with certainty. Here, G(r)activation < 0 for all r and its maximum is found at r = 0 where G ∗activation = G(0)activation . We also note G(r)activation → −∞ as r → ∞. In reality, though, the phase transition is likely to proceed before the driving force becomes so large. The reason is the probabilistic nature of the Boltzmann distribution.   P = A exp −G ∗activation /kT

(1.2.5)

where A is a constant. The probability of finding the system above G ∗activation increases with the system subcooling because G ∗activation decreases with the system subcooling. At some point before G ∗activation reaches zero, the likelihood of finding a sufficiently large cluster of r > r* becomes high enough for the phase transition to proceed from there. Another important attribute of the Boltzmann distribution is that the size distribution of such clusters becomes sharper as the system cools. This is the essence of nucleation and the source of stochasticity in nucleation phenomena.

1.2.2 Nucleation Rate The physical meaning of Arrhenius’s law is almost clear now. The probability distribution of a grand canonical ensemble is the Boltzmann distribution in which the probability of finding a system in a free energy state G is proportional to exp(−G/kT ). For nucleation, the probability of finding a cluster that can surmount a free energy gap diminishes exponentially with the size of the free energy gap. In classical nucleation theory, the free energy gap to be surmounted is given by the activation free energy barrier that arises from the interfacial free energy cost of cluster formation, G ∗activation . Thus,   J ≡ A exp −G ∗activation /kT

(1.2.6)

where J is called nucleation rate [7] and A is a constant. The constant A may include the frequency of attachment of molecules and the concentration of potential nucleation sites. As can be seen from Eq. (1.2.6), the nucleation rate is of central importance to nucleation phenomena of any system that defines the rate at which critically sized nuclei form [5]. Nucleation rate is thus equivalent to nucleation probability density per unit time. Since such nucleation probability density depends on the system size, it is customary to normalize the nucleation rate to unit system size of a suitable measure. An important note: an underlying assumption is that the formation of clusters will not affect the chemical potential of the surrounding monomer molecules more than

12

1 Nucleation Theory

the usual entropic component of the free energy. However, formation of a cluster will deplete the monomer molecules from its surroundings and as such if the monomers are not replenished from the rest of the system in a timely manner, the local concentration and hence the chemical potential of the monomer molecules could temporarily fall in the vicinity of the cluster due to this depletion. This effect is generally negligible in a single-component system but may not necessarily be so in nucleation of clathrate hydrates as we will discuss in Chap. 5. Another important note to make at this point is that a basic premise of classical nucleation theory is that a metastable system has no memory of the past. It means that the nucleation probability density of a system only depends on the driving force at the moment of interest, regardless of its history. It follows that the nucleation rate becomes constant (time-invariant) when the driving force is constant. Since this constant driving force condition is the simplest case and forms the basis for further analysis, we will spend some time in explanation of this setting. First, we define the survival probability at a constant subcooling temperature as F(t), which represents the survival probability of a liquid that diminishes with time, t. F(t) is a function of time and has a numerical value at any given moment but, unlike the nucleation rate, is not a probability density. F(t) is a probability, as opposed to a probability density, and directly measurable for a given system: Experimentally, if one can repeat an identical measurement 100 times and if only one of the 100 measurements has experienced nucleation at a particular time, then the survival probability at that time can be interpreted as 99%. Strictly speaking, no two measurements can be identical; at the very least either the time or the space in which said two measurements are carried out must be different (never mind that the universe itself is expanding at a rapid rate). In reality, more factors other than just the time or the space are likely different, so the concept of being “identical” is only valid in a thought experiment. Nevertheless, we phrase two very similar measurements as “identical” in this book, and in the world in fact, which incidentally highlights the intrinsically approximate nature of Modern Physics. We define the nucleation probability density (i.e., nucleation rate) at time t as p, which, as we saw above, is time-invariant under the constant driving force condition. The nucleation probability between t and a short time dt later, (t + dt), can be expressed in terms of the nucleation rate (nucleation probability density), p, as p · dt. Ab initio considerations show that the survival probability at (t + dt) must be the survival probability at time t multiplied by the probability that nucleation does not occur in the subsequent duration dt. Mathematically, this probability can be expressed as [1 − p · dt]. Thus, F(t + dt) = F(t) · [1 − p · dt]

(1.2.7)

[F(t + dt) − F(t)]/F(t) = − p · dt

(1.2.8)

Then,

1.2 Classical Nucleation Theory

13

It follows that d(ln F(t))/dt = − p

(1.2.9)

Equation (1.2.9) shows that the nucleation probability density, p, at a given moment t is given by the negative of the derivative of lnF with respect to t at that moment. Solving the differential equation (Eq. 1.2.9) while neglecting the constant of integration yields F(t) = exp(− p · t)

(1.2.10)

If we define the most probable (expected) survival time, , as the time when it is equally likely that a randomly selected sample has nucleated or not (experimentally, when half of the samples are found to have nucleated), then < t >≡ ln 2/ p(t) ≈ 0.7/ p(t)

(1.2.11)

The validity of Eq. (1.2.11) can be verified by substituting t = ln2/p(t) into F(t) = exp[−p · t], which will yield F(t) = 2−1 = 0.5. This most probable (expected) survival time is what induction time in the relevant literature is referring to, which is the time elapsed from the attainment of subcooling of interest to the eventual moment of nucleation. We neglected the constant of integration when we solved the differential equation (Eq. 1.2.9). While the nucleation rate is physically equivalent to the nucleation probability density, we still have to account for the constant of integration for these two properties to match. For now, let us employ a different symbol, k, to express the nucleation rate so that we can distinguish the nucleation rate from the nucleation probability density, p. The nucleation rate, k, can be defined as the inverse of the average (most probable) induction time: k ≡< t >−1

(1.2.12)

k = p(t)/ ln 2 ≈ p(t)/0.7

(1.2.13)

Then, from Eq. (1.2.11),

Thus, strictly speaking, the nucleation rate, k, differs from the nucleation probability density, p, by a constant, which has a numerical value of ln2. Substituting Eq. (1.2.13) into Eq. (1.2.10) reveals an important attribute of a nucleation rate under a constant driving force condition: the survival probability of the system, F(t), diminishes exponentially with time as e−ckt , where c is a constant (c = ln2 in our definition) [5, 8–12]. This exponential distribution of induction times can be verified from the expected fact that the most probable induction time, , should coincide with the time at which the integrated area of F from 0 to becomes equal to the

14

1 Nucleation Theory

integrated area of F from to infinity:

 t 0

t

∞

0

t

Fdt =

∞ t

Fdt:

∫ exp(−ckt)dt = ∫ exp(−ckt)dt

(1.2.14)

Solving Eq. (1.2.14), one will yield exp(−ck < t >) = 0.5

(1.2.15)

A salient point to make here is that these attributes do not depend on the nature of the system or specific mechanisms of nucleation involved in a given system, as our ab initio derivation shows [8]. A nucleation event generally involves multiple steps. Even for the simplest case of freezing of a single-component liquid, a nucleation process must at the very least involve formation of transient clusters of various sizes that, when time-averaged, have the Boltzmann distribution [5]. The formation of a critically sized nucleus (sometimes called “supernucleus”) could consist of a certain number of molecules (or other units of building blocks), n*. A number of different paths exist that a cluster of n*-mers can form from a given ensemble under a set of initial conditions; one path might be progressive cascading of monomers, dimers, trimers, etc., that eventually form a small population of clusters of (n* − 1)-mers and then n*-mers. In some other cases, such a path might involve a larger leap or two in between. But the salient point is that, regardless of the nature of such specific nucleation pathways or the complexity of the system, a nucleation event can only be realized when a continuous path forms from the beginning to the end (i.e., all the pieces of the puzzle are in place). And the nucleation rate represents the rate at which such occurrence is realized, over a whole nucleation pathway, in a unit system size. We note at this stage that another basic premise of classical nucleation theory is that a typically large driving force (excess chemical potential) that is required to overcome an activation barrier would lead to an irreversible overgrowth of the nuclei to macroscopic and experimentally detectable sizes once the activation barrier has been surmounted [5]. In other words, the differential in the chemical potential between the driving force and the thermodynamically stable phase is so large that any countering factor like the release of the latent heat would be insufficient to deter the growth of the supernucleus. If this assumption holds, then it becomes possible to experimentally determine nucleation rates from measurable nucleation probability density per unit system size per unit time once such detection delay can be accounted for [9]. This second approximation becomes less accurate as the driving force becomes small, and the nucleation rate low; the smaller the driving force more wrong we will be. An experimental determination of a nucleation probability density per unit system size and per unit time requires a suitable measure of unit system size. The traditional wisdom has been that a suitable measure of the system size for homogeneous nucleation is the system volume because the number of potential nucleation sites is

1.2 Classical Nucleation Theory

15

expected to be proportional to the system volume [5]. Likewise, it has been traditionally considered that a suitable measure of the system size for heterogeneous nucleation is the area of a surface or an interface that is responsible for providing the heterogeneous nucleation sites.

1.2.3 Scaling Laws for Nucleation Rates We now consider how nucleation rates might scale with the system size. Once again, we start from ab initio considerations and apply this line of thought to a thought experiment. Suppose we have two identical droplets and denote that the nucleation probability per unit time of one of them is P. The same probability also applies to the other droplet which is supposed to be identical. Then the nucleation probability per unit time in either one or both of the two droplets becomes one minus the probability that neither droplet nucleates in the unit time: 1 − (1 − P)2 = 2P − P 2

(1.2.16)

The same consideration can be extended to an ensemble of n identical droplets. The nucleation probability per unit time in any of the n droplets is 1 − (1 − P)n

(1.2.17)

This relationship can be further generalized to a domain V that consists of multiple subdomains of dV each. Here, V is a measure of the system size that is not necessarily volume. The relationship between the nucleation probability per unit time of the parent domain V, which we may denote as P(V ), and the nucleation probability per unit time of a subdomain dV, which we may denote as P(dV ), is P(V ) = 1 − [1 − P(dV )](V /dV )

(1.2.18)

It can be readily verified from Eq. (1.2.18) that P(0) = 1 − 1 = 0 for an infinitesimally small system size and P(∞) = 1 − 0 = 1 for an infinitely large system, as expected. Below, we solve Eq. (1.2.18) to first derive the scaling law with respect to space (as opposed to time). We start from the simplest case of a first-order approximation. For a first-order approximation, we will only leave the first term of the Taylor expansion and neglect all the higher order terms. Then P(V ) = 1 − [1 − P(dV )]V /dV ≈ 1 − [1 − (V /dV ) · P(dV )] = (V /dV ) · P(dV ) (1.2.19) P(V )/V ≈ P(dV )/dV

(1.2.20)

16

1 Nucleation Theory

It can be seen that the first-order approximation led to a conclusion that the nucleation probability density per unit time scales linearly with the system size, whatever the measure of the system size might be. This is an expected conclusion and has indeed been traditionally done in the relevant literature; normalize the experimentally measured nucleation probability density per unit time to the unit system size of a relevant measure. Then, under this first-order approximation, the expected induction time (time at which it is equally likely that a given sample has nucleated or not) scales with the system size as follows. Noting that P(V ) is for unit time (i.e., P(V ) equals the nucleation rate), the survival probability F(t) for each of the two otherwise identical systems except for the different sizes of V 1 and V 2 is F1 (t) = e−ct P1

(1.2.21)

F2 (t) = e−ct P2 = e−ct (V2 /V1 )P1

(1.2.22)

Equation (1.2.22) shows that the second system behaves as if it has an effective nucleation probability density of (V 2 /V 1 )P1 . Therefore, one can deduce the nucleation probability density of a system of interest (P2 of the system size of V 2 ) if one can measure the nucleation probability density of a similar system (P1 of a presumably more experimentally convenient system of size V 1 ). We defined above the nucleation rate, k, so that τ ≡ 1/k gives the expected induction time (time at which it is equally likely that a given sample has nucleated or not) [8]. Then, the expected induction time of System 2, τ 2 , in terms of the experimentally measurable expected induction time of System 1, τ 1 , becomes τ2 = 1/[(V2 /V1 )P1 ] = (V1 /V2 ) · τ1

(1.2.23)

Thus, under this first-order approximation, the expected induction time of the second system is merely the expected induction time of the first system (which presumably is experimentally more convenient to measure) multiplied by the relative ratio of the two system sizes of interest. We can apply essentially the same logic to the time domain. We may consider how the nucleation probability of a unit size of a sample scales with time under a constant driving force condition. Cumulative nucleation probability of a unit size over a duration t, P(t), may be subdivided to smaller durations of dt that add up to t. Then, analogous to Eq. (1.2.18), P(t) = 1 − [1 − P(dt)](t/dt)

(1.2.24)

It can be readily verified from Eq. (1.2.24) that P(0) = 1–1 = 0 for an infinitesimally short time and P(∞) = 1 − 0 = 1 for an infinitely long time, as expected. It is important to note that P(t) is no longer for unit time and therefore no longer equals the nucleation rate.

1.2 Classical Nucleation Theory

17

Now we may consider full solutions. We may treat the system size, V, and time, t, independently: dP(V, t) = (∂ P/∂ V ) · dV + (∂ P/∂t) · dt

(1.2.25)

Then, Eq. (1.2.18) for unit time and Eq. (1.2.24) for unit system size yield the general form P(V, t) = 1 − [1 − P(dV, dt)](V /dV ).(t/dt)

(1.2.26)

We selected the pre-factor of the nucleation rate to be the inverse of the expected induction time, τ, at which it is equally likely that the system has nucleated or not, or equivalently, the survival probability becomes 50% [8]. A pertinent question is: how can one find τ 2 that renders P(V 2 , t 2 ) = F(V 2 , τ 2 ) = 0.5 when the three other variables of V 1 , V 2 , and τ 1 are known. We may consider two systems, Systems 1 and 2, that have the system sizes of V 1 and V 2 , respectively, and apply Eq. (1.2.26) to each of the two systems. Then P(V2 , t2 ) = 1 − [1 − P(dV, dt)](V2 /dV )·(t2 /dt)

(1.2.27)

for the first system and P(V1 , t1 ) = 1 − [1 − P(dV, dt)](V1 /dV )·(t1 /dt)

(1.2.28)

for the second system. Taking the natural logarithm of both sides of Eq. (1.2.27) yields ln[1 − P(V2 , t2 )] = (V2 /dV ) · (t2 /dt) · ln[1 − P(dV, dt)]

(1.2.29)

Likewise, taking the natural logarithm of both sides of Eq. (1.2.28) yields ln[1 − P(V1 , t1 )] = (V1 /dV ) · (t1 /dt) · ln[1 − P(dV, dt)]

(1.2.30)

Dividing Eq. (1.2.29) by Eq. (1.2.30) then yields ln[1 − P(V2 , t2 )]/ ln[1 − P(V1 , t1 )] = (V2 /V1 ) · (t2 /t1 )

(1.2.31)

We defined the most probable or expected induction time of System 1, τ 1 , so that P(V 1 , τ 1 ) becomes P(V 1 , τ 1 ) = F(V 1 , τ 1 ) = 0.5. Therefore ln[1 − P(V2 , t2 )] = −(V2 /V1 ) · (t2 /τ1 ) · ln 2 = −Ct2

(1.2.32)

where C = (V 2 /V 1 )·(ln 2/τ 1 ). Solving Eq. (1.2.32) yields P(V2 , t2 ) = 1 − exp(−Ct2 )

(1.2.33)

18

1 Nucleation Theory

Thus, the nucleation probability of System 2 when the three variables of V 1 , V 2 , and τ 1 are known is given as a function of time by Eq. (1.2.33). In short, when one cannot directly measure the nucleation probability of a system of interest but knows the size of the system, and if one can measure the time evolution of another system of a different scale, then one can indirectly deduce the nucleation probability of the first system. Finally, a simpler equation can be derived when one is only interested in the most probable or expected induction time of System 2, τ 2 , and not in its full-time profile. Substituting P(V 2 , τ 2 ) = F(V 2 , τ 2 ) = 0.5 into Eq. (1.2.33) yields τ2 = (V1 /V2 ) · τ1

(1.2.34)

This is the same result as Eq. (1.2.23) obtained from the first-order approximation. In the end, we found that nucleation rate scales proportionately, both with respect to time and with respect to system size. These conclusions might have been intuitively expected before all these calculations. Nevertheless, it is reassuring to know that these conclusions can be deduced from ab initio calculations that do not depend on any system-specific knowledge or assumptions. Still, the use of Eq. (1.2.33) or Eq. (1.2.34) depends on one’s ability to specify the appropriate system sizes that can be used for comparison, in addition to one’s ability to measure the most probable induction time in a relevant system of a different scale. How one can identify an appropriate measure of the system size of a given system is another matter that we will deal with later, and the matter will be deferred to Chap. 5 for clathrate hydrates.

1.2.4 Nucleation Work of Homogeneous Nucleation and Heterogeneous Nucleation The nucleation work (and the activation barrier) of homogeneous nucleation and the nucleation work (and the activation barrier) of heterogeneous nucleation are related to each other. Here, we only consider a representative case of a spherical cap-shaped nucleus on a flat foreign substrate (Fig. 1.5). The question is, how the presence of a foreign surface impacts the nucleation work and the activation barrier? The activation barrier due to the interfacial free energy for a spherical cap-shaped nucleus consisting of n molecules on a flat surface of a foreign substrate is given by Eq. (3.60) of Ref [5]: γheterogeneous =  1/3 aγn 2/3

(1.2.35)

Ψ = (1/4)(2 + cos θ )(1 − cos θ )2

(1.2.36)

where

1.2 Classical Nucleation Theory Fig. 1.5 Schematic illustration of a spherical cap-shaped nucleus of the thermodynamically stable phase on a flat foreign substrate in the metastable parent phase

19

Metastable parent phase

θ

Thermodynamically stable phase

Foreign substrate

Here a = (36πv20 )1/3 , γ is the specific surface free energy (per unit area) between the metastable parent phase and the thermodynamically stable phase, v0 is the volume of a molecule of interest, and θ is the contact angle of a spherical cap-shaped nucleus that forms on the substrate in the metastable parent phase [5]. In contrast, the activation barrier due to the interfacial free energy for the homogeneous nucleation of a sphere consisting of n molecules, γ homogeneous , is aγ n2/3 from Eq. (3.20) of Ref [5]. From these, 1/3  γheterogeneous /γhomogeneous = Ψ 1/3 = (1/4)(2 + cos θ )(1 − cos θ )2

(1.2.37)

Since 0 ≤ θ ≤ π, Ψ ≤ 1 and the heterogeneous nucleation is favored over the homogeneous nucleation when compared for the same size of nucleus of n. Alternatively, we can compare the maximum nucleation work (that corresponds to G ∗activation in Fig. (1.4)) required for each case. The heterogeneous nucleation work, W heterogeneous (n), for a spherical cap-shaped nucleus on a flat substrate varies with the nucleus size, n, as [5] Wheterogeneous (n) = −nμ +  1/3 aγn 2/3

(1.2.38)

We can use the calculus of variations to find the maximum and dW heterogeneous (n)/dn = 0 yields n ∗heterogeneous = (3μ/2aγ )−3 Ψ where n ∗heterogeneous is the nucleus size for which the heterogeneous nucleation work becomes the maximum. The corresponding maximum nucleation work, W heterogeneous (n ∗heterogeneous ), is      Wheterogeneous n ∗heterogeneous = 4 a 3 γ3 /27 /μ2

(1.2.39)

In contrast, the nucleation work for a spherical nucleus in a homogeneous metastable parent phase varies with the nucleation size as Whomogeneous (n) = −nμ + aγn 2/3

(1.2.40)

20

1 Nucleation Theory

The maximum nucleation work will be realized when n ∗heterogeneous = (3μ/2aγ )−3 , which is larger than n ∗heterogeneous . The corresponding maximum nucleation work, W homogeneous (n ∗heterogeneous ), is      Whomogeneous n ∗homogeneous = 4 a 3 γ3 /27 1/μ2

(1.2.41)

From Eq. (1.2.39) and Eq. (1.2.41),     Wheterogeneous n ∗heterogeneous /Whomogeneous n ∗heterogeneous = 

(1.2.42)

Equation (1.2.36) shows that Ψ → 0 as θ → 0 and Ψ → 1 as θ → π. Likewise, Eq. (1.2.42) shows that W heterogeneous (n ∗heterogeneous ) → 0 as θ → 0 and W heterogeneous (n ∗heterogeneous ) → W homogeneous (n ∗heterogeneous ) as θ → π. In addition, Ψ 1/3 = 0 when Ψ = 0 and Ψ 1/3 = 1 when Ψ = 1. Therefore, from Eq. (1.2.37), γ heterogeneous → 0 as θ → 0 and γ heterogeneous → γ homogeneous as θ → π. Thus, both γ heterogeneous and W heterogeneous approach zero when the thermodynamically stable phase completely “wets” the substrate (no nucleation process is required to overcome an activation barrier when θ = 0). Conversely, both γ heterogeneous and W heterogeneous approach those of homogeneous nucleation, γ homogeneous and W homogeneous , respectively, when the solid substrate does not at all contribute to the reduction of the effective interfacial energy (when θ = π ). A complete wetting case might be realized when the underlying substrate and the emerging thermodynamically stable phase have perfect lattice matching, which gives rise to an epitaxial growth of the thermodynamically stable phase on the substrate. Equation (1.2.37) thus shows that homogeneous nucleation always results in a higher activation energy barrier than heterogeneous nucleation and, as such, heterogeneous nucleation is always energetically preferable whenever a suitable solid is available.

1.3 Homogeneous Nucleation of Solutes from Supersaturated Solutions Clathrate hydrates by definition consist of multiple components, as we will see in Chap. 3. Thus, precipitation of solutes from a supersaturated solution is a closer analogue to nucleation of clathrate hydrates than freezing of a singlecomponent system like ice. To this end, we briefly examine nucleation involved in the precipitation of solutes from a supersaturated solution. Homogeneous nucleation is uncommon in nature, as shown by Eq. (1.2.42). However, there is at least one example in which homogeneous nucleation is commonplace: spontaneous emulsification. Both spontaneous emulsification of oil (solutes) in supersaturated water (solvent) and spontaneous emulsification of water (solutes) in supersaturated oil (solvent) occur.

1.3 Homogeneous Nucleation of Solutes from Supersaturated Solutions

21

When water is added to a Greek liquor, Ouzo, a milky emulsion spontaneously forms. This phenomenon is known as the “Ouzo effect” and is an example of spontaneous emulsification. Ouzo can be described as a dilute oil solution in ethanol as a first approximation (there are myriad of other minor components that add flavor). The physical mechanism of spontaneous emulsification is fairly straightforward: for three components of oil, ethanol, and water, ethanol is fully miscible with each of oil and water, whereas oil and water are sparingly soluble with each other. Thus, when a small volume of oil-in-ethanol solution is added to a large volume of water, ethanol diffuses through water and the solubility of the oil in the increasingly diluted ethanol-in-water solution progressively falls. Eventually, the solubility limit of the oil is reached and the oil precipitates out of the supersaturated solution [13]. Essentially, the same mechanism is at work when a small volume of water-in-ethanol solution is added to a large volume of oil: ethanol diffuses through oil and the solubility of the water in the increasingly diluted ethanol-in-oil solution progressively falls. Eventually, the solubility limit of water is reached and the water precipitates out of the supersaturated solution. Thus, spontaneous emulsification offers a convenient way of dispersing oil in water, or dispersing water in oil, without the use of surfactants or vigorous stirring [13]. These oil droplets in water (or water droplets in oil), once formed, can be somewhat stabilized due to the presence of ethanol. Presumably, ethanol, being amphiphilic (ethanol is known to act as co-surfactants), adsorbs to the oil–water interface. Like all kinetically stabilized emulsions, they will eventually phase separate. Below we only consider the nucleation involved in the formation of oil-in-water emulsions. It has traditionally been considered that only three components are in play in spontaneous emulsification. However, this turned out to be not quite true. Ubiquitous dissolved atmospheric gases, mainly nitrogen, are usually present during the spontaneous emulsification. This is a fact that has largely been overlooked but there is a good reason to believe that the presence of such ubiquitous dissolved atmospheric gases can affect the spontaneous emulsification. First of all, the amount of ubiquitous dissolved atmospheric gases is substantial under atmospheric pressure, as might be expected from Henry’s law—their concentration in water at the standard temperature and pressure is of the order of 1 mM [14]. This concentration becomes higher in the liquids that are involved in the spontaneous emulsification process: oil, ethanol, and their solutions. Then, for a low concentration of oil, for example, 10 μM, there would be 100 times more atmospheric gas molecules than the oil molecules. A typical mechanical vacuum pump can reduce the amount of dissolved atmospheric gases from about 1 mM to about 100 nM. This reduction corresponds to, in relative terms, from about 10000% to about 1% of nitrogen molecules per oil molecule. If such large amounts of any (unintended) substance were present in a sample, they would surely have been treated as impurities. However, no such due diligence has been exercised for ubiquitous dissolved atmospheric gases. Against this backdrop, Sowa et al. investigated the effect of such ubiquitous atmospheric gases on the homogeneous nucleation in spontaneous emulsification [15]. It turned out, perhaps surprisingly, there was a significant difference between the

22

1 Nucleation Theory

spontaneous emulsifications in the presence and in the absence of dissolved atmospheric gases, in which the nucleation became more homogeneous after the removal of dissolved atmospheric gases. In short, removal of dissolved atmospheric gases resulted in larger numbers of smaller and more uniformly sized droplets [15]. The larger numbers of smaller and more uniformly sized droplets after degassing suggest that the nucleation of the oil droplets after degassing became more homogeneous in nature, which underscores our point that ubiquitous atmospheric gases should be regarded as impurities that offer potential heterogeneous nucleation sites for spontaneous emulsification. Removal of any impurities that could act as heterogeneous nucleation sites is expected to allow further diffusion of the oil-in-ethanol solution into the aqueous phase before the oil precipitates out of the aqueous phase. Such enhanced diffusion of the oil-in-ethanol solution not only would distribute the oil over a larger space in the aqueous phase but also would help attain a higher supersaturation of the oil in the aqueous solution. For a sufficiently large supersaturation of the oil, the driving force for precipitation may approach that of the homogeneous nucleation. In this situation, the Boltzmann distribution of the clusters of the oil molecules may show that the driving force is so large that the probability of finding a critically sized cluster is significant. Then, any small fluctuations in the concentration of the oil molecules that can perturb the Boltzmann distribution may trigger the homogeneous nucleation. An important consequence of formation of a nucleus is depletion of the oil monomers from its surroundings (because they are used up for the formation of said nucleus). Then, the next nearest homogeneous nucleation can only occur some distance away from an existing nucleus where the concentration of the oil molecules has not depleted. The end result is that homogeneous nucleation results in the formation of larger numbers of smaller oil droplets that are spatially more uniformly distributed compared to emulsions formed by heterogeneous nucleation. In short, the spontaneous emulsification before degassing is still heterogeneous nucleation, with the dissolved gases acting as heterogeneous nucleation sites, and the spontaneous emulsification after degassing is closer to “truly” homogeneous nucleation as a result of reaching the spinodal limit. Thus, an important conclusion of this section is that “truly” homogeneous nucleation might never occur in a natural system given that even ubiquitous dissolved atmospheric gases (mainly nitrogen gas) can make a measurable difference to homogeneous nucleation of oil droplets in water. In other words, homogeneous nucleation rates of a given system reported in the literature could be overestimated by a significant margin, i.e., “true” homogeneous nucleation would require a greater supersaturation and “true” nucleation rates would be lower than what has been reported for a given supersaturation. Other than the general point made above, the implications of these results to the nucleation of clathrate hydrates are fairly clear. In a multi-component system, nucleation of solutes will deplete solute monomers from the surrounding solution, which will in turn lower the chemical potential of the solute monomers. It follows that homogeneous nucleation of the solutes or its crystal growth cannot proceed without timely replenishment of the solute monomers to the depleted region of the system.

1.4 Nucleation of Two-Dimensional Solid Layers

23

1.4 Nucleation of Two-Dimensional Solid Layers 1.4.1 Introduction As we will see in Chap. 5, clathrate hydrate typically forms as thin films at an aqueous–guest gas interface due to the typically low solubility of the guest gas in water. When the only water present in a system is a droplet floated in the background of the guest gas, clathrate hydrate forms a rind at the surface of the water droplet. This is an example of nucleation of a two-dimensional crystal. Truly two-dimensional crystals are rare in nature but do exist. Straight-chain hydrocarbons (normal alkanes) and straight-chain alcohols of intermediate chain lengths exhibit a rare phenomenon known as surface freezing. Briefly, when a normal alkane solid is heated from below the bulk melting point (T m ), the top monolayer of the solid surface remains frozen up to the surface freezing point (T sf ), which is a few Kelvins higher than T m (Fig. 1.6). This frozen surface monolayer has the same structure as what is known as the rotator phase, the thermodynamically stable solid phase of the bulk of the hydrocarbon below T m [16, 17], in which molecules have rotational freedom, while the long-range positional (translational) order is preserved. The solidified (rotator) surface layer remains monomolecular thick as the temperature is raised from T m to T sf [16, 17]. A differential thermal analysis (DTA) study detected a peak that has arisen from surface freezing of n-pentacontane (C50 H102 ) and n-tetratetracontane Fig. 1.6 The top monolayer of the solid surface of normal alkane remains frozen up to the surface freezing point that is a few degrees above the melting point of the bulk. This frozen surface layer has the same structure as the bulk rotator phase, in which molecules have rotational freedom while the long-range positional (translational) order is fixed

24

1 Nucleation Theory

(C44 H90 ) and found the enthalpy of the surface freezing transition to be 142 J/g, which is close to that of bulk freezing [18]. It has been known that long-chain liquid n-alkanes exhibit little bulk subcooling when cooled from a high temperature before they begin to crystallize. It has been suggested that surface freezing was responsible for this apparent absence of subcoolings of n-alkanes [19, 20]. The frozen surface monolayer has a similar structure and density to those of the bulk rotator phase and forms a few Kelvins above T m . Then, no further energy barrier would be required when the bulk rotator phase epitaxially grows from the already-present surface rotator phase as the system eventually reaches T m . Although even-number alkanes form a bulk crystalline (triclinic) phase, not a rotator phase, directly from the liquid phase at T m , the specific interfacial free energy between the bulk rotator and the bulk crystalline phases of n-alkanes appears very small [19]. It was reported that bulk crystallization of n-hexadecane indeed proceeded via a transient bulk rotator phase [20]. A remarkable feature of surface freezing is that such monomolecular thick two-dimensional solid layers exist above the melting point in equilibrium, and its thickness remains constant as it is heated from T m to T sf [16, 17]. We refer the readers to a comprehensive review article [21] for details of experimental findings on surface freezing of normal alkanes. Another important consequence of surface freezing is the absence of pre-melting of long-chain n-alkanes below the bulk melting point, as noted in 1949 by Bradley and Shellard [22]. As we will see in Chap. 4, the presence of pre-melting is of central importance to the presence or absence of activation barrier and to nucleation of ice. In short, both pre-melting of a solid surface and surface freezing of a liquid surface are of central importance to nucleation of condensed phases. Below, we briefly cover the theoretical background of surface freezing [21].

1.4.2 Estimation of the Contact Angle of Alkane Melt on Frozen Alkane Monolayers The central equation to set the criterion for surface freezing in terms of the specific interfacial free energy is the Young equation [23]: γ1v cos θ = γsv − γs1

(1.4.1)

where the subscripts l, v, and s refer to liquid, vapor, and solid phases, respectively. Formation of a solid-like monolayer must lower the free energy of the system for surface freezing to materialize. Let us assume for the moment that the specific interfacial free energy of either side of the frozen monolayer can be expressed in terms of the macroscopic (semi-infinite media) specific interfacial free energy. Then, γ1v > γsv + γs1

(1.4.2)

1.4 Nucleation of Two-Dimensional Solid Layers

25

Fig. 1.7 The surface phase transition is marked by a distinct change in the temperature dependence of the surface tension. The small surface area of a bubble can be subcooled to below T sf . Image reproduced from Ref. [25] with permission from The American Physical Society

Combined with the Young equation (noting that the specific interfacial free energy terms must not be negative and hence the direction of the inequality does not change by a multiplication or a division of both sides by γ ), cos θ = (γsv − γs1 )/γ1v < (γsv − γsl )/(γsv + γsl ) = 1 − 2γsl /(γsv + γs1 ) < 1 (1.4.3) Equation (1.4.3) shows that the contact angle of the melt on the frozen surface must be finite (non-zero) unless γ sl is zero. This shows that surface freezing is associated with partial wetting of the solid surface by its own melt at T m [23, 24]. We will see later in Chap. 4 a somewhat similar relationship between ice and liquid water. The liquid–vapor surface tension of a subcooled n-octadecane surface below T sf is given by a linear extrapolation of the linear temperature dependence from higher temperatures, which has a negative slope that is characteristic of a common liquid [25] (Fig. (1.7)). The position of T sf is marked by a sharp change in the slope of the surface tension with respect to temperature, in that ∂γ /∂T < 0 above T sf and ∂γ /∂T > 0 below T sf . Interestingly, the small surface area of a small bubble, when cooled, could be subcooled to below T sf until the eventual phase transition (nucleation) brought its surface tension in line with that of a large surface probed by the Wilhelmy plate method, which is assumed to be the thermodynamically stable phase, as indicated by vertical arrows. A schematic picture that illustrates the essence is shown in Fig. (1.8). It was shown that the nucleation for the surface freezing transition can occur either above or below T m , as long as the temperature was below T sf [25]. Thus, in principle, the surface freezing transition can occur at exactly T m , where the solid and the liquid bulk phases can coexist, for which our analysis will be greatly simplified. If we can

26

1 Nucleation Theory

Fig. 1.8 A schematic picture that illustrates the essence of Fig. (1.7). The gap γ between the subcooled γ lv and the frozen (γ sv + γ sl ) gives the driving force for surface freezing. Image adapted from Fig. 2 of Reference [21], with permission from World Scientific Publishing Co

γ

γlv Δγ

γsv + γsl Tm

Tsf

T

assume that the surface tension measured for the subcooled surface at the melting point yields the liquid–vapor specific interfacial free energy at that temperature and that for the frozen surface is given by the sum of the solid–liquid and the solid– vapor specific interfacial free energy terms, then based on our previous data shown in Fig. (1.7) [25]: γ1v ≈ 26.8 mJ/m2

(1.4.4)

γsv + γsl ≈ 26 mJ/m2

(1.4.5)

Using the Young equation once again yields γ1v cos θ = γsv − γsl ≤ γsv + γs1

(1.4.6)

By substituting Eq. (1.4.4) and Eq. (1.4.5) into Eq. (1.4.6), we will find θ > 14°. This lower bound in the contact angle is in good agreement with the measured values of θ for n-alkane droplets on silica (≈10°) [26] and on mica (≈16°) [27] substrates when the adsorbed surface layer is in the frozen state. Given the thickness of the films (≈3 nm) in these studies, which is well within the range of the surface forces exerted by the underlying substrates, the agreement is surprisingly good. A salient point here is that the use of macroscopic specific interfacial free energy values between two semi-infinite media for either side of a frozen layer as thin as a single molecule is a surprisingly reasonable approximation, which gives confidence in us later approximating either side of a quasi-liquid layer in terms of the specific interfacial free energy between two semi-infinite media.

1.4.3 Latent Heat of Surface Fusion The surface excess entropy is given by the temperature derivative of the specific surface free energy [23], which is negative and large in the absolute value for normal alkanes below T sf [16, 25, 28–30]. In contrast, the surface excess entropy of normal

1.4 Nucleation of Two-Dimensional Solid Layers

27

alkanes above T sf is positive and small in the absolute values, as expected of any ordinary liquid. The negative surface excess entropy between the melting point and the surface freezing point shows that the surface is more ordered than the bulk, consistent with the “frozen” state of the monolayer and the disordered state of the underlying liquid bed. For n-octadecane, for example, the surface excess entropy below the T sf , S surface can be calculated using our previous data [25]:   Ssurface = −∂γ/∂ T ≈ −1 mJ/m2 · K−1 ≈ −1 × 10−3 N · K−1 m−1

(1.4.7)

Figure (1.7) shows that the surface layer could be subcooled to below T m of noctadecane (T m ≈ 28.1 °C ≈ 301 K) and the small negative slope of ∂γ /∂T continues (the positive surface excess entropy remains constant) until the nucleation for the surface freezing transition eventually takes place at the points indicated by the vertical arrows [25]. The positive surface excess entropy remains constant below T sf before the eventual transition is consistent with the subcooled and disordered liquid surface below T sf . Meanwhile, the bulk heat of fusion of n-heptadecane (C17 ), n-octadecane (C18 ), n-nonadecane (C19 ), and n + eicosane (C20 ) are 45 kJ/mol, 60 kJ/mol, 43.3 kJ/mol, and 45.6 kJ/mol, respectively [22, 31]. Many normal alkanes of intermediate chain lengths transition from liquid to a bulk rotator phase [16] at T m , as we noted above. A separate phase transition between the bulk rotator phase and the bulk crystalline phase takes place a few Kelvins below T m of each normal alkane. However, n-octadecane is a special case and happens to undergo the liquid–rotator and the rotator–crystalline transitions at the same temperature of T m [19], and consequently its bulk latent heat of fusion is larger than that of the other normal alkanes. In other words, the larger latent heat of fusion observed for n-octadecane can be attributed to the latent heat of rotator–crystalline transition, which is required to provide thermal energy to the molecules to rotate around their lattice positions. The bulk heat of fusion of liquid–rotator phase transition of n-octadecane is expected to be similar to that of n-heptadecane (C17 ) and n-nonadecane (C19 ), or about 44 kJ/mol. L = Tm dS ≈ 44 kJ/mol

(1.4.8)

If we assume that the thickness of the frozen layer to be monomolecular [16, 29], then we can calculate the head group area (α) of n-octadecane molecule from dividing the molecular volume (vm ) by the molecular length (l ≈ 2.6 nm): α = vm /l = Mw /ρNA l ≈ 2.1 × 10−19 m2

(1.4.9)

where M w is the molar weight (254.5 g/mol) [32], ρ is the density (0.78 g/cm3 ) [32], and N A is the Avogadro number. From Eq. (1.4.9), we can calculate the number of n-octadecane molecules in a unit surface area (1 m2 ) as α−1 = 4.76 × 1018 m−2

(1.4.10)

28

1 Nucleation Theory

Or, equivalently, the amount of n-octadecane in mole in a unit surface area (1 m2 ) is (αNA )−1 = 7.9 × 10−6 mol · m−2

(1.4.11)

Then, the latent heat of fusion of 1 m2 of a surface monolayer of n-octadecane can be calculated as   L surface = (44 kJ/mol) × 7.9 × 10−6 mol · m−2 = 0.348 J · m−2

(1.4.12)

Then, dS, the change of entropy of a surface monolayer of n-octadecane from solid to liquid at T m (≈301 K) for unit area (1 m2 ) can be calculated using Eq. (1.4–1.12) as dS = L surface /Tm = 0.348 J · m−2 /301K = 1.16 × 10−3 J · K−1 · m−2

(1.4.13)

This value is very similar to the absolute value of the surface excess entropy 1 × 10−3 J K−1 m−2 found in Eq. (1.4.7). The surface of n-octadecane is thus indeed frozen below the surface freezing point and its thickness is monomolecular. Such truly two-dimensional (2D) solid layers are rare in nature, and other than the normal alkanes, only limited number of straight-chain alcohols are known to form such twodimensional (2D) solid layers. These analyses further augment the idea that the use of macroscopic specific interfacial free energy terms and surface entropy values are surprisingly accurate approximations.

1.4.4 Nucleation of Two-Dimensional (2D) Solid Layers Now we are ready to discuss the nucleation of two-dimensional (2D) solid layers. That surface freezing has been observed on both liquid n-alkane beds [17] and solid substrates of silica [26] and mica [33] suggests that the effect of a solid substrate or the liquid bed of n-alkanes underneath the frozen surface monolayer may not be as important as the lateral cohesion within the monolayer. An exception can occur when the van der Waals adhesion of the n-alkane monolayer to the underlying substrate is stronger than the lateral cohesion within the monolayer. The strong adhesion between the alkyl chains of n-alkanes and a graphite substrate was found to render n-alkane chains lie parallel to a graphite substrate [34], in contrast to the surface freezing in which each n-alkane chain stands perpendicular to the surface of the monolayer and to the surface of the underlying foreign substrate. Thus, we conclude that, as pointed out by Frederick Fowkes [35], the strength of lateral cohesion between the methylene chains (CH2 –CH2 ) stabilizes the frozen monolayer against thermal motion. In bulk crystals, cohesion within each layer (CH2 –CH2 ) is stronger than the cohesion between the successive layers (CH3 –CH3 ) because of the higher energy for the methylene group, which follows from Zisman’s

1.4 Nucleation of Two-Dimensional Solid Layers

29

critical surface free energy values [23]. On an ordinary foreign substrate, cohesion within each layer (CH2 –CH2 ) is stronger than the adhesion between the layer and the underlying foreign substrate. In these cases, before the entire surface monolayer of the liquid solidifies, a two-dimensional monolayer-nucleus (disk) inevitably exposes around itself a higher energy component of methylene groups, leading to a finite (non-zero) surface subcooling which is comparable to the bulk subcooling [25]. This is equivalent to suggest that the line tension around the two-dimensional monolayer nucleus is responsible for the activation process involved in surface freezing. We will further postulate that there will be another energy barrier against subsequent bulk freezing even after the entire top monolayer has frozen, and as such, a second activation process (subcooling) between the frozen surface monolayer and the bulk freezing may be required. For non-polar compounds such as n-alkanes, the intermolecular interactions are dominated by the van der Waals forces, as described by the Lifshitz theory. The solid phase of n-alkanes has a higher density [22] and hence, according to the Lorentz–Lorenz relationship, a higher refractive index than the liquid phase. Thus, the Hamaker constant is expected to be positive for the (liquid– solidified film–vapor) system [36]. This leads to an important conclusion that the van der Waals interaction across the solid film is attractive. The thickness of such a film would be limited to a small value until a sufficient subcooling takes place. This is an example of a negative disjoining pressure (positive chemical potential) which we will detail in Chap. 4. The first-order continuum approximation we used above, though correctly capture the essence, neglects potentially important thickness dependence of the specific interfacial free energy of the film. To quantitatively estimate the propagation of freezing from the top frozen surface monolayer into the underlying bulk liquid bed, a proper account for the long-range surface forces will be essential, which is beyond the scope of this book. Here, we only qualitatively show what is expected once the thickness dependence of the surface free energy is taken into account. Again, an analogy of a reversed system may be helpful here. It is well established that two solid walls that are interacting across a thin liquid film experience the so-called oscillatory solvation forces, the period of which reflects the size of the molecules of the thin films confined in between [37]. A schematic illustration is shown in Fig. (1.9) in which the wall separation is expressed in the number of diameters of the confined molecules. Here, the height and the depth of each neighboring repulsive and attractive peaks correspond to the energy barrier that needs to be overcome in order to increase or decrease the wall separation (e.g., when one starts with an initial surface separation of 2 molecular diameters, then the activation barrier to decreasing the surface separation to 1 molecular diameter is much greater than the activation barrier required to increasing the surface separation of 3). Generally, the amplitude of such an oscillation is large at a small surface separation and rapidly diminishes as the surface separation increases, and such oscillation is superimposed on top of the continuum dependence [37]. The underlying physics is that the disjoining pressure of the thin liquid film exhibits oscillatory behavior. Simply put, the system free energy is low when the surface separation is a multiple

Fig. 1.9 A schematic illustration which shows that the wall separation is expressed in the number of diameters of the confined molecules. Here, the height and the depth of each neighboring repulsive and attractive peaks correspond to the energy barrier that needs to be overcome in order to increase or decrease the wall separation

1 Nucleation Theory

System free energy

30

1.5 2.5

3.5 3

4

2 1

Surface separation in molecular diameters of the size of the confined molecules and high when it is halfway in between. Such difference rapidly diminishes as the surface separation increases because the confined molecules can pack in a greater number of ways at large surface separations (more degrees of freedom). The readers may refer to Chap. 4 for the basic concepts of the disjoining pressure [38, 39]. Then a similar effect may be expected for the disjoining pressure of a solidified surface film: the disjoining pressure is an oscillatory function of the film thickness, and each peak height of the oscillatory curve provides effective energy barrier to be overcome to increase or decrease the thickness of the film. These considerations lead to an important conclusion that, while the freezing of successive layers (into bulk) may proceed cooperatively, freezing of the second layer from the first may require a larger subcooling than that estimated from the continuum model alone. If the amplitude of the oscillatory function is much larger than the continuum contribution, the film might grow beyond the monolayer thickness only far below the melting point, where homogeneous nucleation of a crystal in the bulk liquid may be equally probable. Importantly, the effects of the surface forces discussed above are beyond the realm of classical nucleation theory, and hence not accounted for in classical nucleation theory. The readers will see more examples in Chap. 5.

1.5 Tea Time Break: Nucleation in Capillary Phase Transitions No nucleation is required for capillary condensation in a wedge-like pore. A condensable vapor readily condenses in a wedge because there is no activation barrier. This is consistent with the absence of 1D nucleation [5].

1.5 Tea Time Break: Nucleation in Capillary Phase Transitions

31

In a slit-like pore, in contrast, capillary condensation can be an activation process. The system can remain in a metastable state in which the vapor separates two wetting films on the pore walls even though the thermodynamically stable state is the gap filled with liquid. Strange things happen in a slit-like pore. The most acute example is condensation and evaporation behavior of a condensable vapor in a slit-like pore of adjustable slit width [40–42]. There the average density of the confined fluid in the slit (over a given cross section across the slit width) can continually vary from that of the vapor to that of the liquid under right circumstances [40–42]. How can this happen? The first instinct would be to assume that the system in fact consisted of two droplets on each slit wall that were separated by a thin layer of vapor. However, as detailed in [42], such a simplistic explanation would encounter another problem of some sort that cannot be accounted for (for example, what had happened to the van der Waals forces if two droplets could truly be separated by a narrow gap of only a few nanometers?). In the end, it appears that the most logical explanation is to assume a diffusive interface that is akin to a supercritical state [42]. Elliot and Voitcu showed that such strange behavior could occur in a slit-like pore if the slit walls are curved, even slightly so [43]. The implications of this fascinating phenomenon to nucleation are not clear at this stage. It might or might not be related to nucleation in confined spaces of fixed dimensions.

References 1. C.H. Whitson, M.R. Brule, Phase Behavior, Richardson (Society of Petroleum Engineers, Texas, 2000) 2. R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Massachusetts, 1963) 3. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill Book Co., Singapore, 1965) 4. H.B. Callen, Thermodynamics and an Introduction to Thermostatistics (Wiley, New York, 1985) 5. D. Kashchiev, Nucleation (Oxford, UK, Elsevier Science & Technology, 2000) 6. J.N. Israelachvili, Intermolecular and Surface Forces, 2nd edn. (Academic Press, San Diego, 1991) 7. B.J. Murray, D. O’Sullivan, J.D. Atkinson, M.E. Webb, Ice nucleation by particles immersed in supercooled cloud droplets. Chem. Soc. Rev. 41, 6519–6554 (2012) 8. N. Maeda, Nucleation curves of model natural gas hydrates on a quasi-free water droplet. AIChE J. 61, 2611–2617 (2015) 9. N. Maeda, Nucleation curves of methane hydrate from constant cooling ramp methods. Fuel 223, 286–293 (2018) 10. A.S. Stoporev, A.Y. Manakov, L.K. Altunina, L.A. Strelets, V.I. Kosyakov, Nucleation rates of methane hydrate from water in oil emulsions. Can. J. Chem. 93, 882–887 (2015) 11. A.S. Stoporev, A.P. Semenov, V.I. Medvedev, B.I. Kidyarov, A.Y. Manakov, V.A. Vinokurov, Nucleation of gas hydrates in multiphase systems with several types of interfaces. J. Therm. Anal. Calorim. 134, 783–795 (2018)

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12. A.Y. Manakov, N.V. Penkov, T.V. Rodionova, A.N. Nesterov, E.E. Fesenko Jr., Kinetics of formation and dissociation of gas hydrates. Russ. Chem. Rev. 86, 845–869 (2017) 13. S.A. Vitale, J.L. Katz, Liquid droplet dispersions formed by homogeneous liquid-liquid nucleation: “The ouzo effect”. Langmuir 19, 4105–4110 (2003) 14. R. Pashley, Effect of degassing on the formation and stability of surfactant-free emulsions and fine teflon dispersions. J. Phys. Chem. B 107, 1714–1720 (2003) 15. B. Sowa, X.H. Zhang, K. Kozieski, P.G. Hartley, N. Maeda, Influence of dissolved atmospheric gases on the spontaneous emulsification of alkane-ethanol-water systems. J. Phys. Chem. C 115, 8768–8774 (2011) 16. B.M. Ocko, X.Z. Wu, E.B. Sirota, S.K. Sinha, O. Gang, M. Deutsch, Surface freezing in chain molecules; normal alkanes. Phys. Rev. E 55, 3164 (1997) 17. B.M. Ocko, X.Z. Wu, E.B. Sirota, S.K. Sinha, O. Gang, M. Deutsch, Surface freezing in chain molecules: normal alkanes. Phys. Rev. E 55, 3164–3182 (1997) 18. Y. Yamamoto, H. Ohara, K. Kajikawa, H. Ishii, N. Ueno, K. Seki, Y. Ouchi, A differential thermal analysis and ultraviolet photoemission study on surface freezing of n-alkanes. Chem. Phys. Lett. 304, 231–235 (1999) 19. E.B. Sirota, Supercooling, nucleation, rotator phases and surface crystallization of n-alkane melts. Langmuir 14, 3133–3136 (1998) 20. E.B. Sirota, A.B. Herhold, Transient phase-induced nucleation. Science 283, 529–531 (1999) 21. N. Maeda, V. Yaminsky, Experimental observations of surface freezing. Int. J. Mod. Phys. B 15, 3055–3077 (2001) 22. R.S. Bradley, A.D. Shellard, The rate of evaporation of droplets III. Vapour pressures and rates of evaporation of straight-chain paraffin hydrocarbons. Proc R Soc (London) A 198, 239–251 (1949) 23. A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, 6th edn. (Wiley, New York, 1997) 24. A.W. Adamson, An adsorption model for contact angle and spreading. J. Colloid Interface Sci. 27, 180–187 (1968) 25. N. Maeda, V. Yaminsky, Surface supercooling and stability of n-alkane films. Phys. Rev. Lett. 84, 698–700 (2000) 26. C. Merkl, T. Pfohl, H. Riegler, Influence of the molecular ordering on the wetting of SiO2 /air interfaces by alkanes. Phys. Rev. Lett. 79, 4625–4628 (1997) 27. N. Maeda, M.M. Kohonen, H.K. Christenson, Phase behavior of long-chain n-alkanes at one and between two mica surfaces. J. Phys. Chem. B 105, 5906–5913 (2001) 28. J.C. Earnshaw, C.J. Hughes, Surface-induced phase transition in normal alkane fluids. Phys. Rev. A 46, R4494–R4496 (1992) 29. X.Z. Wu, B.M. Ocko, E.B. Sirota, S.K. Sinha, M. Deutsch, B.H. Cao, M.W. Kim, Surface tension measurements of surface freezing in liquid normal alkanes. Science 261, 1018–1021 (1993) 30. Y. Hayami, G.H. Findenegg, Surface crystallization and phase transitions of the adsorbed film of F(CF2)12(CH2)16H at the surface of liquid hexadecane. Langmuir 13, 4865 (1997) 31. R.S. Bradley, M.G. Evans, R.W. Whytlaw-Gray, Evaporation and diffusion coefficients, and vapour pressures of dibutyl phthalate and butyl stearate. Proc. Roy. Soc. (London) A 186, 368–390 (1945) 32. D.R. Lide (ed.), CRC Handbook of Chemistry and Physics, 80th edn. (CRC Press, Boca Raton, 1999–2000) 33. N. Maeda, M.M. Kohonen, H.K. Christenson, Phase transition of n-alkane layers adsorbed on mica. Phys. Rev. E 61, 7239–7242 (2000) 34. F.Y. Hansen, K.W. Herwig, B. Matthies, H. Taub, Intramolecular and lattice melting in n-alkane monolayers: an analog of melting in lipid bilayers. Phys. Rev. Lett. 83, 2362–2365 (1999) 35. F.M. Fowkes, Calculation of work of adhesion by pair potential summation. J. Colloid Interface Sci. 28, 493–505 (1968) 36. J. Mahanty, B.W. Ninham, Dispersion Forces (Academic Press, London, 1976) 37. J.N. Israelachvili, Intermolecular and Surface Forces, 2nd edn. (Academic Press, San Diego, CA, 1991)

References

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38. P.G. deGennes, F. Brochard-Wyart, D. Quéré, Capillarity and Wetting Phenomena. (Springer, New York2004) 39. B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface Forces (Consultants Bureau, New York, 1987) 40. N. Maeda, J.N. Israelachvili, Nanoscale mechanisms of evaporation, condensation and nucleation in confined geometries. J. Phys. Chem. B 106, 3534–3537 (2002) 41. N. Maeda, J.N. Israelachvili, M.M. Kohonen, Evaporation and instabilities of microscopic capillary bridges. Proc. Natl. Acad. Sci. U.S.A. 100, 803–808 (2003) 42. N. Maeda, Phase transitions of capillary-held liquids in a slit-like pore. J. Phys. Chem. B 110, 25982–25993 (2006) 43. J.A.W. Elliott, O. Voitcu, On the thermodynamic stability of liquid capillary bridges. Can. J. Chem. Eng. 85, 692–700 (2007)

Chapter 2

Experimental Methods for Determination of Nucleation Rates

2.1 Constant Temperature Method Nucleation rate is central to nucleation phenomena as we saw in the previous chapter. The question is: how can one determine the nucleation rate of a given system? It is important to appreciate that direct detections of nuclei are impossible. There are several reasons for this: a reason is that one cannot pre-determine where in the system one should focus his instrument in advance. Another reason is that focusing of a detection instrument will likely bring about local heating. Such local heating may be small but, because of the exponential nature of Arrhenius’s law, any small local heating would lower the nucleation rate at the focused location compared to the other unheated locations because the subcooling (driving force) would become lower. These points aside, a primary reason is that nuclei would look like any other (slightly smaller) clusters that are populating the system which would have a size distribution of the Boltzmann form. In other words, an experimental technique would need to have a spatial resolution of a building block of a nucleus to be able to tell whether the critical size has been reached or not and each building block can be as small as a single molecule. As such, a nucleation event can only be detected after the fact, i.e., after a nucleus has surmounted the free energy barrier and completed an irreversible overgrowth to macroscopically detectable sizes. There are two basic approaches to this challenge. One is an isothermal method or a constant temperature (subcooling) method. The other is a linear cooling ramp (constant cooling rate) method. In this section, we will deal with the former. The latter will be covered in the next section. An experimental challenge with the measurements of induction times at a constant subcooling is that an induction time is often very long and could potentially stretch to infinity at a shallow subcooling (high temperature), as might be expected from the exponential distribution of the induction times under a constant driving force. An introduction of an arbitrary cut-off induction time (maximum waiting time) is thus common in this type of measurements for practical reasons. A related issue is that the scatter in induction time data is very large, especially at a shallow subcooling [1–4]. © Springer Nature Switzerland AG 2020 N. Maeda, Nucleation of Gas Hydrates, https://doi.org/10.1007/978-3-030-51874-5_2

35

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2 Experimental Methods for Determination of Nucleation Rates

A few example chronological histograms of experimental induction times are shown in Fig. (2.1). Here, the arbitrary maximum waiting (cut-off) time was set to 15000 s in each case. The scatter worsens the smaller the sample size and the shallower the subcooling. A numerical average of inductions times is often impossible to determine when a substantial number of experimental runs have reached the maximum waiting time without a nucleation event. Then, a numerical average cannot be used to obtain the average induction times or its inverse, the average nucleation rate. In some cases, the median could be used as an alternative measure of the average induction time where more than 50% of the experimental runs have resulted in nucleation events before the maximum waiting time is reached, as is the case shown in Fig. (2.1a). However, more than 50% of the experimental runs often reach the maximum waiting time without nucleation events, as is the case shown in Fig. (2.1b). In such cases, one cannot even determine the median of the induction time distribution, let alone the average. We will therefore have to rely on classical nucleation theory. Fig. 2.1 Typical chronological induction time distributions under a constant subcooling (driving force)

2.1 Constant Temperature Method

37

A constant temperature (subcooling) experiment means that the driving force does not change with time during an experiment. An important conclusion of Sect. (1.2) was that the survival probability, F, of such a system has an exponential distribution of induction times of the form e–ckt , where c is a constant, k is the nucleation rate, and t is time. We derived this conclusion in Sect. (1.2) and in [5] from ab initio considerations that the survival probability at a time (t + dt) is the survival probability at a time t multiplied by the probability that nucleation does not occur in the subsequent duration dt. Essentially, the same conclusion can be reached from the Poisson distribution once we assume that a single nucleation event is all that is required to initiate a phase transition [3, 4, 6, 7]. Thus, if one can measure the survival probability as a function of induction time, a plot of lnF versus t will yield a straight line with a slope of –ck. Since c = ln2 in our definition, the most probable nucleation rate can be readily calculated from the best fit to the slope. The first step of calculating F(t) is to rearrange a chronological histogram of induction times, such as the one shown in Fig. (2.1), to an ascending order. Three examples of chronological histograms and the corresponding rearranged histograms are shown in Fig. (2.2). In Fig. (2.2a), several experimental runs resulted in zero experimental induction times, which means that a nucleation event took place before the system reached the target subcooling temperature of interest. This event becomes increasingly more common as the subcooling of interest becomes deeper (as the set temperature becomes colder). In Fig. (2.2b), none of the experimental runs resulted in zero experimental induction times or none of the induction time measurements were maxed out by an arbitrary cut-off waiting time. This is undoubtedly an ideal case that is rare in reality. In Fig. (2.2c), a substantial number of experimental runs reached the maximum waiting time of 15000 s without encountering nucleation events. As may be expected, this situation occurs more commonly at shallow subcoolings (high temperatures). Even in this case, the chronological histogram shown on the left panel can be rearranged in an ascending order, as shown on the right panel. The survival probability as a function of induction time, F(t), can be calculated if one can assume equivalency among all experimental runs—i.e., each experimental run contributes equally to the whole induction time distribution, regardless of the chronological sequence. For example, when there are a total of 1000 experimental runs and if none of the 1000 runs has nucleated at a very short induction time of t = 0+, then F(0+) = 1, as may be expected. Likewise, at a very long induction time, all experimental runs would have experienced a nucleation event by then F(t → ∞) = 0. In between, F(t) at a given intermediate time, t, can be calculated by dividing the number of experimental runs that have not experienced a nucleation event until t by the total number of the experimental runs. For the above example of a total of 1000 experimental runs, F(t) will monotonically decrease with t from 1 to 0 in 1000 steps. Once F(t) is determined, the calculation of lnF(t) is a simple numerical operation of taking its natural logarithm. Where an artificial maximum waiting time is set, F(t) may not reach 0 at the maximum waiting time. Importantly, even though the numerical values of the induction times of these experimental runs that have not encountered nucleation events

38

2 Experimental Methods for Determination of Nucleation Rates

Fig. 2.2 Typical induction time distributions under a relatively deep subcooling (large driving force) (a) and an intermediate subcooling (intermediate driving force), (b) and a relatively shallow subcooling (small driving force) (c). Both the chronological raw data (left) and rearranged data (right) are shown

until the maximum waiting time cannot be determined (any more than to conclude that the induction times of these experimental runs must have been longer than the maximum waiting time), the survival probabilities up until the maximum waiting time remain valid. The reason is that the survival probabilities up until the maximum waiting time do not depend on the ultimate fate of the experimental run (when nucleation would have eventually taken place if the measurement time were extended) that is still in the future. An example survival curve, F(t), and a plot of corresponding lnF versus t of the data shown in Fig. (2.2c) are presented in Fig. (2.3a) and Fig. (2.3b), respectively.

2.1 Constant Temperature Method

39

Fig. 2.3 Typical induction time distribution shown in Fig. (2.2c) is converted to survival probability distribution (survival curve) as a function of induction time (a). The corresponding plot of lnF versus t is also shown (b)

Since a substantial number of experimental runs resulted in reaching the maximum waiting time of 15000 s without nucleation for this particular experiment, the cluster of data points at t = 15000 s shown in Fig. (2.2c) are useless on their own. Still the survival probability at 15000 s of F(15000 s) = 0.395 is a useful piece of information. As can be seen from Fig. (2.3), the 114 data points in this particular example are insufficient to yield a good linear fit of lnF versus t. Several hundreds of data points are typically required for a reasonable linear fit of lnF versus t. Especially concerning is that a plot of lnF versus t appears to have two distinct slopes, one below about 5000 s and the other above about 5000 s. This attribute of apparent dual nucleation rates in a single data set appears common [7]. Importantly, since these two domains of nucleation rates are chronologically mixed, it cannot be attributed to any real physical change in the sample during the measurements (like sample aging), unless a given sample undergoes a transition every time after it stays at a given constant temperature for about 5000 s. Such a regular change could arise as a newly prepared sample at a warmer temperature will initially take some time to adjust to a colder experimental temperature due to thermal lag. However, if this were the case, the nucleation rate at shorter times would have been lower because the sample would have been warmer.

40

2 Experimental Methods for Determination of Nucleation Rates

An alternative approach is to use multiple sample cells and do without any use of arbitrary cut-off time for the collection of the nucleation data. The constant subcooling method is suitable for the use of multiple sample cells because the isothermal nature of the method ensures that all samples are at the same temperature. This is especially useful for homogeneous nucleation, such as freezing of a liquid droplet suspended in another immiscible liquid, for which one does not need to worry about the heterogeneity of foreign container walls. The concern here is the uniformity of the sample size and a potential spatial temperature gradient across the multiple samples because, albeit constant, maintaining at a constant temperature generally requires a heating or a cooling device with feedback control that would induce heat flows across the samples. Another effort directed to shorten the total experimental time required for the collection of the nucleation data is by Svartaas and co-workers whose improved analysis method at a constant temperature enabled to lower the number of required repeat measurements to about 25 [8]. It is rare in reality that one is only interested in the nucleation rate of a given system at a single subcooling temperature. Nucleation rate depends on the driving force for nucleation, Gdriving_force , so a nucleation curve that relates the nucleation rate to the driving force for a given system is required. Then, one needs to repeat the above protocol for a number of different subcoolings and use some form of interpolation to determine the nucleation curve. This is a highly time-consuming and laborious endeavor.

2.2 Linear Cooling Ramp Method 2.2.1 Introduction An important recent innovation in the experimental investigations of nucleation rates was the use of linear cooling ramps that enabled simultaneous determination of an entire nucleation curve over a whole range of experimentally accessible subcoolings. An automated lag time apparatus (ALTA) has been used since the middle of the 1990 s for experimental investigations of ice and other liquids under atmospheric pressure [1, 2, 9–11]. These studies compiled survival probability distributions as functions of system subcoolings and used the median of the distribution as the representative measure of the most probable subcooling. These early studies did not advance so far as to determine the nucleation rates of the systems. Maeda established a systematic method that enabled simultaneous determination of an entire nucleation curve over the whole range of experimentally accessible subcoolings [5, 12]. The details of the systematic method are summarized in this section and in Sect. 1.2. A family of high-pressure automated lag time apparatus (HP-ALTA) [13, 14] has been developed to experimentally determine the survival probability distributions of clathrate hydrates as functions of system subcoolings, F(T ). Other than an

2.2 Linear Cooling Ramp Method

41

HP-ALTA, instruments such as high-pressure differential scanning calorimeter (HPμDSC) have been commercially available that can apply a large number of linear cooling ramps to a given sample under a constant elevated pressure. HP-μDSC can obtain the same type of raw data as an HP-ALTA does, with a greater utility that enables investigations of optically opaque samples and over a greater range of guest gas pressures. On the other hand, an HP-ALTA offers a greater flexibility in modifications/coating of the sample cells [5, 15, 16]. Both an HP-μDSC and an HPALTA can be used for the construction of a survival curve when the cooling rate is slow enough to eliminate thermal lags and temperature gradients within a sample. In each setup, a small quiescent sample is cooled at a constant rate until nucleation is effectively forcibly induced. The time and the temperature at which this event has taken place are recorded, and the sample is subsequently heated to melt a crystal or dissociate a clathrate hydrate. Then, the process is repeated for a large number of times as deemed necessary. The resulting data are chronological histograms of lag times. An example is shown in Fig. (2.4). Lag time is the time from the point of establishment of metastability to the point of nucleation event. Lag times would equal induction times if the system were held at a constant subcooling. Figure (2.5) shows an example of a survival probability distribution as a function of system subcooling (“survival curve”). These example data shown here are recorded using an HP-ALTA in 12 MPa of a mixed gas of 90 mol% methane–10 mol% propane. The cooling rate used was 0.05 K/s, and the dissociating condition after each clathrate hydrate formation event was at 310 K for 300 s. Rearranging the histogram shown in Fig. (2.4) from the shortest lag time to the longest, rotating the figure clockwise by 90°, multiplying the lag times by the experimental cooling rate, and normalizing each run number by the total run numbers in the experiment yield Fig. (2.5). As can be seen, the level of stochasticity in Fig. (2.4) is much smaller than that in Fig. (2.1) of the constant temperature mode. The reason for this compressed stochasticity of the linear cooling ramp data is that the progressively

Fig. 2.4 Typical chronological lag time distribution

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2 Experimental Methods for Determination of Nucleation Rates

Fig. 2.5 Typical survival probability distribution (survival curve) as a function of system subcooling

increasing driving force in a linear cooling ramp brings nucleation forward (to more immediate future) than in a constant subcooling method. These chronological histograms of lag times and the resulting survival probability distributions as functions of system subcoolings, or survival curves F(T ), are the raw data of an HP-ALTA or an HP-μDSC employed for this purpose. The question is how to analyze such survival probability distribution data. There are two mutually complementary methods, one in the subcooling domain and the other in the time domain. We will detail each of these modes below.

2.2.2 Analysis in the Subcooling Domain The goal of this mode of analysis is to establish a systematic method that determines the most probable subcooling of a given sample from an experimentally measured survival curve. The most probable subcooling is sometimes termed metastable zone width (MSZW) in the literature. A distinct feature of a survival curve is that the distribution of the data in a survival curve is always non-uniform: the number density of the data is the highest near the middle of a survival curve and becomes progressively scarce toward both ends. It is therefore necessary to establish a systematic, robust, and equitable method of determining the most probable subcooling in which each data point on the survival curve carries an equal weight and no particular data point has undue influence of the outcome. The basic theoretical framework of this mode of analysis is detailed in [17]. Here, we define the probability density of finding a particular data point in a certain location of a survival curve as q(T ). When a total number, N total , of data points exist on a survival curve and the number of the data points ranging from T j to T j + d(T )

2.2 Linear Cooling Ramp Method

43

is N(T j ) · d(T ), the probability density of finding a particular data point at a subcooling of T j , which we may term q(T j ), is N(T j )/N total :     q Tj ≡ N Tj /Ntotal

(2.2.1)

Experimentally, the survival probability, F(T ), is complementary to the cumulative of q(T ): T     F(T ) = 1 − ∫ q T  d T 

(2.2.2)

0

An experimental survival curve, F(T ), starts from 1 at the melting point (T = 0), monotonically decreases with increasing T, eventually falls to the minimum of 0 at the maximum achievable experimental subcooling, and then remains 0 beyond that T. Thus, F(T ) only changes its value where the nucleation data exist, or where q(T ) is non-zero. The incremental decrease in the survival probability, dF(T ), at a certain T along a survival curve is –q(T ) · d(T ) at that T: −dF(T ) = q(T )d(T )

(2.2.3)

Clearly, dF(T ) falls more when a larger numbers of data points exist in the range of d(T ). Integration of Eq. (2.2.3) yields 1

∞

0

0

∫ d F = − ∫ q(T )d(T )

(2.2.4)

It is vitally important to recognize at this stage that this data distribution density function, q(T ), is fundamentally different from the nucleation rate or nucleation probability density, p(T ), we detailed in Chap. 1. To illustrate the intrinsic difference, for example, q(T ) typically falls with increasing T beyond the T for which F(T ) = 0.5 and eventually falls to 0 beyond the maximum experimentally achievable subcooling. In contrast, p(T ) should not decrease with deepening subcooling, let alone fall to 0, at very deep subcoolings, because the driving force for nucleation is becoming greater with T. Now, the expected value (the most probable value) of a quantity, , is generally given by A ≡

all 

A j q j where

all 

j

qj = 1

(2.2.5)

j

for discrete q values or all

A ≡ ∫ Aqd x 0

(2.2.6)

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2 Experimental Methods for Determination of Nucleation Rates

for continuous q values. By definition, q(T ) satisfies the condition of

all 

q j = 1,

j

as per Eq. (2.2.1). Then, the most probable subcooling for which the distribution of the data is given by q(T ) is ∞

T  = ∫ T q(T )d(T )

(2.2.7)

0

Equation (2.2.7) can be simplified using Eq. (2.2.3): ∞

∞

0

0

T  = ∫ T q(T )d(T ) = − ∫ T

1 dF d(T ) = ∫ T d F d(T ) 0

(2.2.8)

Geometrically, Eq. (2.2.8) calculates the area enclosed by the three axes of F = 0, F = 1, T = 0 and the survival curve, F. Figure 2.6 shows three example survival curves for pure water, a dilute (0.5 wt%) kinetic hydrate inhibitor solution, and the net inhibition effect of the kinetic hydrate inhibitor. The areas in question are labeled with green for water in the panel (a) and blue for a dilute solution of a kinetic hydrate inhibitor (KHI) in the panel (b). The two survival curves are shown together in the panel (c), with the survival curve for the dilute KHI solution from the panel (b) in red. Here, the net inhibition effect of the KHI can be geometrically expressed as the

Fig. 2.6 Typical survival probability distribution (survival curve) as a function of system subcooling for water (a), dilute aqueous solution of a kinetic hydrate inhibitor (KHI) (b). The most probable subcooling for each case is given by the area enclosed by the three axes of F = 0, F = 1, T = 0 and the survival curve (labeled with color). The net inhibition effect of the KHI is given by the area between the two survival curves (c)

2.2 Linear Cooling Ramp Method

45

difference between the blue area in the panel (b) and the green area in the panel (a). The difference in the areas is equal to the area enclosed by F = 1, the red survival curve, F = 0 and the black survival curve in the panel (c). The net inhibition effect of the KHI can thus be expressed with the area labeled with blue in the panel (c). For finite numbers of linear cooling ramps, Eq. (2.2.8) becomes a summation instead of an integral. Then, the enclosed area in Fig. 2.6 is calculated by adding up the areas of horizontal stripes in which each horizontal stripe corresponds to the contribution from a single linear cooling ramp to the expected T, and the summation is from F = 0 to F = 1. The obtained is sometimes called the expected or the most probable metastable zone width (MSZW). The method described here is statistically far superior to the use of simple numerical averages or medians commonly found in the literature, because all data points contribute equally. For example, a few outlier data points out of 100 total data points could pose undue influence on simple numerical averages but will be systematically accounted for in the present method.

2.2.3 Analysis in the Time Domain The above analysis in the subcooling domain is useful in quantitative comparisons of most probable subcoolings between different samples or systems, but does not involve any element of time and hence is powerless in the determination of nucleation rates or nucleation curves. The basic theoretical framework that underpins simultaneous determination of an entire nucleation curve over the whole range of experimentally accessible subcoolings from an experimentally obtained survival curve is documented in [5, 12]. Equation (1.2.7) shows the basic logic behind the nucleation rate at a constant driving force. The situation becomes more complex in the linear cooling ramp mode than in the constant temperature mode, because the survival probability function, F(t), itself changes with subcooling, T. Still, a reasonably tractable protocol can be established when an experiment uses a constant cooling rate. The driving force linearly keeps increasing during a linear cooling ramp, so the nucleation probability density depends on the system subcooling which in turn depends on time: p = pT (t). Then, the nucleation probability between t and (t + dt) becomes pT (t) · dt. Analogous to Eq. (1.2.7), the survival probability at (t + dt) can be expressed in terms of the survival probability at t, F T (t), and the probability that nucleation does not occur in the subsequent duration dt, which is [1–pT (t) · dt]. Then FT +d(T ) (t + dt) = FT (t) · [1 − pT (t)dt]

(2.2.9)

Unlike in Eq. (1.2.7), though, T is not a constant but changes with t. During a short period, dt, T changes by αdt where α is the experimental cooling rate, which is a constant for a linear cooling ramp. At the same time, F T (t) will change to a different function, F T + αdt (t + dt). The key approximation we use here is that,

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2 Experimental Methods for Determination of Nucleation Rates

when there are hundreds of data points like in a typical HP-ALTA study, [F T + αdt (t + dt) – F T (t)]/F T (t) may be approximated by the numerical average of [F T + αdt (t + dt) – F T +αdt (t)]/F T (t) and [F T (t + dt) – F T (t)]/F T (t). For a first approximation, it may suffice to approximate [F T +αdt (t + dt) – F T (t)]/F T (t) with [F T (t + dt) – F T (t)]/F T (t). The assumption here is that the data density is high enough so that the impact of the difference in the functional forms of dF T +αdt (t) and dF T (t) between two neighboring data points can be neglected over a short duration dt. We saw in Chap. 1 that F T (t) at a given constant subcooling becomes an exponential function of the form exp(–ckt). Likewise, the functional form of another F T (t) at another constant T is another exponential function of the form exp(–ckt). As we will see in the Tea Time break at the end of this chapter, exponential functions form a set of orthogonal base vectors in a Hilbert space. Therefore, there is unique one-toone correspondence between an exponential function and its exponent, which is the nucleation rate here (i.e., its uniqueness is assured). Since we defined the constant of integration as ln2, and t is the time, the difference between the two exponential functions that belong to two different subcoolings boils down to the difference in the two nucleation rates (i.e., the values of k) that are specific and unique to each subcooling. To make clear that the nucleation rate is a function of subcooling and that each subcooling during a given cooling ramp has a unique and specific value of k, we may denote each k as k T . For two particular subcoolings of T and (T + αdt), F T (t) = exp(–ck T t) and F T +αdt (t) = exp(–ck T +αdt t). We also note, as we saw in Chap. 1, that the nucleation rate becomes independent of time at a constant subcooling, regardless of the sample history. A thought experiment might be in order at this stage. We may envision two identical systems of A and B that are exact replica of each other. We then suppose that we first start an induction time measurement of the system A at a constant subcooling of T 1 . We then suppose, sometime later, we start another induction time measurement of the system B at a slightly greater constant subcooling of T 2 . The survival probability of the system B, which is supposed to be at a colder temperature, should fall faster than that of the system A. Hence, there will come a time when the survival probabilities of the two systems coincide. We may take that particular moment as time zero. We then wish to know the error that is caused by jumping from one constant subcooling survival curve of the system A (at T 1 ) to another constant subcooling survival curve of the system B (at T 2 ) at a short time dt after time zero. We can further suppose that T 1 happens to have the same numerical value as T at a certain time during a linear cooling ramp and that T 2 happens to have the same numerical value as (T + αdt) of the same linear cooling ramp. We also selected the time zero to be the moment when F T +αdt (t) and F T (t) coincide, as schematically depicted in Fig. 2.7. Since dt is small, the difference between T 1 and T 2 is small for a slow cooling rate. This is a type of thought experiment which was once popular in the early twentieth century. Now, what should be the suitable measure of the error that most appropriately describes the difference that arises from jumping from one exponential curve to the other? If one stayed on the original curve, the change in the survival probability after the duration dt would have been dF T , as shown in Fig. 2.7. Instead, because

2.2 Linear Cooling Ramp Method

47

Fig. 2.7 A conceptual illustration of a thought experiment in which two otherwise identical systems are placed at two slightly different subcooled temperatures and allowed to nucleate (image reproduced from Reference [12] with permission from Elsevier)

one jumped to another curve at the time zero, the change in the survival probability after the duration dt is now dF T +αdt . We certainly should not adopt the numerical difference between dF T +αdt and dF T as the measure because the result would depend on their absolute values or, more to the point, on the timing of the time zero; such a measure cannot possibly be applied over a whole linear cooling ramp. Then, we may instead select the ratio between dF T +αdt and dF T as the measure. Noting that dF T +αdt /dt = –ck T +αdt F T +αdt and dF T /dt = –ck T F T , the ratio of dF T +αdt (t)/dF T (t) can be expressed as dFT +αd (t)/dFT (t) = kT +αdt FT +αdt /kT FT = kT +αdt /kT

(2.2.10)

We used in the last step of Eq. (2.2.10) that (F T +αdt = F T ) at the time zero. The error boiled down to the ratio between the nucleation rates to which each F T belongs to, as might have been expected. The error that arises from this approximation depends on the gap between the neighboring two functions of k T +αdt and k T . After all, if the change in T were infinitesimal, then the two functions would be identical. The data points are generally not uniformly distributed over the entire range of a given survival curve with respect to subcooling; the data points are least densely populated at both ends and most densely populated near the center of a survival curve. The validity of the approximation therefore varies along a given survival curve. Practically speaking, a survival curve obtained using an HP-ALTA typically contains hundreds (300 to 400) of data points. Then, the average of the ratios between each neighboring pair of nucleation rates over the 300 to 400 data pairs is of the order of 1% [12]. We now go back to Eq. (2.2.9) to continue our quest to determine the nucleation rates from a survival curve. [FT (t + dt) − FT (t)]/FT (t) = − pT (t)dt

(2.2.11)

d(ln FT (t))/dt = − pT (t)

(2.2.12)

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2 Experimental Methods for Determination of Nucleation Rates

The nucleation probability density, pT (t), at each moment, t, is given by the negative of the derivative of lnF T with respect to t, at that particular t and the corresponding temperature (and subcooling), T. Then, just like we did in Eq. (1.2.11), k = pT (t)/ ln 2 ≈ pT (t)/0.7

(2.2.13)

We define the most probable lag time as the time at which the survival probability becomes 0.5 (i.e., when it is equally likely that a system has nucleated or not) and define the nucleation rate as the inverse of the most probable lag time, k ≡ 1/ . To summarize, these analyses show that the nucleation probability density at a given moment and subcooling during a linear cooling ramp only depends on the local slope of the natural logarithm of the survival curve at that moment and subcooling.

2.2.4 Protocol of Converting an Experimentally Measured Survival Curve to a Nucleation Curve Now we are ready to put the above-described theoretical framework into practice. One can construct a survival curve using an HP-ALTA or an HP-μDSC by assigning measured fractions of samples or experimental runs that have nucleated at a given temperature. A survival curve expresses the survival probability of the sample as a function of the system subcooling, F = F(T ), which forms the starting point of the protocol. An example survival curve of 90 mol% methane–10 mol% propane (C1/C3) mixed clathrate hydrate in a glass sample cell is shown in Fig. (2.8), which we will use for illustrating the procedure in this section. The next step is a numerical conversion of F(T ) to lnF(T ). We can use the same data shown in the example survival curve of Fig. (2.8) for illustration. The resulting lnF(T ) curve is shown in Fig. (2.9). The next step is to find the local slope of lnF with respect to the lag time, t, for each data point. To do so, we first need to derive the corresponding lnF T (t) curve. Since linear cooling ramps of the same cooling rate of α were used for the collection Fig. 2.8 An example survival curve

2.2 Linear Cooling Ramp Method

49

Fig. 2.9 An example lnF(T) curve

Fig. 2.10 An example lnFT (t) curve

of the data, the lag time, t, can be found by dividing T by α. We may as well use the same example data for this mathematical operation, and the resulting lnF T (t) curve is shown in Fig. (2.10). Equation (2.2.12) shows that the nucleation probability density can be found from the negative of the local slope (the time derivative) of the lnF T (t) curve at each t. There are a few mathematical procedures to find such local slopes at each data point. The simplest method is probably to fit an appropriate curve to the data shown in Fig. (2.10) and analytically differentiate the fitted curve. There is a room for refinement in this regard, but for now, we may use a simple power law. The use of a power law can ascertain that lnF T (t) monotonically decreases with t, as it should. The result is what we are looking for: the nucleation curve that was derived from the same example data, as shown in Fig. (2.11). It can be seen that the nucleation rates over the entire experimentally accessible range of system subcoolings have been systematically determined. Although we used example data of clathrate hydrates for the description, the same protocol can be applied to any system of interest.

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2 Experimental Methods for Determination of Nucleation Rates

Fig. 2.11 An example nucleation curve

2.2.5 The Impact of Experimental Cooling Rate One potential concern that can arise from the use of linear cooling ramps is that the physical conditions of the system keep changing with time, and the system may not have enough time to “catch up” to the (constantly changing) new reality. This factor is known as a thermal lag. In addition, since a nucleation event can only be detected “after the fact” and the growth time of the nucleus to detectable sizes is unknown, there is always a possibility that the temperature when the nucleation is detected could be substantially lower than the temperature when the nucleation has occurred. These potential systematic errors are inevitable in a linear cooling ramp experiment and are the price one has to pay to gain in exchange for the benefit of the systematic and simultaneous determination of the nucleation rates over the entire experimentally accessible range of system subcoolings. The size of these potential systematic errors can be mitigated by the use of a slow cooling rate. The change in the system subcooling during the growth time of a nucleus to an experimentally detectable size can be lowered by slowing the cooling rate. After all, its impact will become negligible when the cooling rate is so slow that the change in the temperature during the growth of a nucleus is less than the temperature resolution of a thermometer. This point aside, use of a slow cooling rate offers multiple advantages. First, any thermal lag of the sample and its surroundings will be mitigated. A thermal lag can arise from the non-zero heat capacity of the sample and, for a clathrate hydrate system, potential undersaturation of a quiescent aqueous sample due to the increasing solubility of the guest gas with cooling. This point will be further examined in Chap. 5. Be it an HP-ALTA or an HP-μDSC, the temperature of a sample is not directly measured because the presence of a thermometer inside a sample would influence the heterogeneous nucleation probability of the sample under investigation.

2.2 Linear Cooling Ramp Method

51

Second, the use of a slow cooling rate will allow a long time, and hence more chance of nucleation, at a shallow subcooling (high temperature). Therefore, the experimentally accessible range of system subcoolings can be expanded at the low end of system subcoolings. In contrast, the experimental accessibility to the high end of system subcoolings does not appear to suffer by the use of a slow cooling rate, in part because rapid quenching that may enable attainment of deep system subcoolings cannot be carried out orderly, let alone linearly. Thus, the use of a slow cooling rate can expand the range of system subcoolings over which the nucleation curve of interest can be determined. Even though the advantage is clear, the use of a slow cooling rate has an obvious drawback in which the process becomes time-consuming, especially when one needs to collect hundreds of data points. Besides, the physical meaning of the impact of the use of a linear cooling rate is not clear, regardless of the cooling rate. Puzzlingly, it was experimentally observed that the nucleation rate at a given subcooling became progressively lower as the experimental cooling rates were lowered [15]. Even though the effect was rather minor, it was consistently observed and appears real. What makes the matter worse, the observed trend is difficult to explain. First, the detection delay due to the finite growth rate of a nucleus is expected to worsen the faster the cooling rate, because the sample temperature will go lower for a given growth time of a nucleus. This factor would thus shift a given nucleation curve to the left. Second, a thermal lag would cause the “true” sample temperature to be higher (i.e., the subcooling and the driving force to be smaller) for which the nucleation rate should be lower due to the smaller driving force for nucleation. This factor would also shift a given nucleation curve to the left. And this factor is also expected to worsen as the cooling rate becomes faster. Third, the undersaturation of a guest gas in a clathrate system would cause the “true” driving force to be smaller for which the nucleation rate should be lower. Again, this effect is expected to worsen the faster the cooling rate because the rapidly changing temperature would not allow sufficient time for the system to “catch up”. In short, all three factors of detection delay, thermal lag, and guest undersaturation for a clathrate system that are expected to be present during a linear cooling ramp would render the “true” sample temperature higher (“true” system subcooling smaller) than the experimentally determined face values [18]. Thus, each of these factors is expected to shift a given nucleation curve to the left and, as the cooling rate becomes faster, the nucleation curve would shift even more to the left. As Fig. (2.11) shows, a shift of a nucleation curve to the left would render the nucleation rate at a given subcooling to be higher, the opposite of the experimentally observed trend. It has not been possible to decouple these three factors or assess the relative impact of each factor. As for the first factor of detection delay due to the finite growth rate of a nucleus, a plot of lnF versus t at a constant subcooling (like the one shown in Fig. (2.3b)) would shift to the left but the slope of d(lnF)/dt could be very similar. Then, one may expect that the resulting nucleation rate, k, is unlikely to be materially affected by the detection delay. We also note that the size of such detection delays is

52

2 Experimental Methods for Determination of Nucleation Rates

expected to become less significant with subcoolings (the growth rate of a nucleus to detectable sizes after nucleation is expected to be faster at deeper subcoolings). As for the second factor of thermal lag, no thermal lag exists under a constant temperature experiment, so a plot of lnF versus t at a constant subcooling (like the one shown in Fig. (2.3b)) would remain the same. During a linear cooling ramp experiment, the temperature of the linearly cooling system is always slightly colder than the temperature of the sample, and the thermal lag of the sample (the temperature differential) likely remains similar over an entire linear cooling ramp. Then a plot of lnF T (t) versus t would shift to the left by a fixed offset, and consequently the local slope of d(lnF T (t))/dt at each point would remain largely the same. As for the third factor of undersaturation of guest gases for a clathrate hydrate system, the solubility of a guest gas in water is generally not a linear function of temperature, much less the timescales with which the guest gas diffuses into the aqueous phase to replenish the undersaturation during a linear cooling, and consequently its effect during a cooling ramp is expected to be highly complex. We cannot discard the possibility that the observed downward shift of the experimentally determined nucleation curve as a slower cooling rate is used may not be related to any of the three plausible physical factors considered above. At the time of this writing, the most plausible explanation is that each experimentally determined nucleation curve by the linear cooling ramp method should be interpreted to represent the upper bound of the “true” nucleation rate at each system subcooling. In other words, the “true” nucleation rate must be likely lower but cannot be higher than the experimentally determined nucleation curve. If we assume this, we may have a way of explaining the observed trend because the upper bound progressively shifted lower as the experimental cooling rate is reduced [18]. The essence of the idea is that the gap between the experimentally determined nucleation curve and the “true” nucleation curve, which is supposed to be lower, would progressively narrow as the former progressively shifts downward as the slower cooling rate is used. Then, an experimentally determined nucleation curve would eventually match the “true” nucleation curve if an infinitesimal cooling rate could be used. We note that the difference between the experimentally determined upper bound and the “true” nucleation rates using a typical cooling rate of an HP-ALTA is well within the scatter of the induction time distributions of the constant temperature method [5], and therefore one can expect that the location of the “true” nucleation rates should be largely similar to the experimentally determined nucleation rates. We expect that the same will hold when an HP-μDSC is used for linear cooling ramp measurements of lag time distributions in future. Given that a nucleation curve typically spans over many orders of magnitude, the above uncertainty by no means lessens the great value of the linear cooling ramp method.

2.3 Tea Time Break: The Fourier Transform and the Laplace Transform

53

2.3 Tea Time Break: The Fourier Transform and the Laplace Transform 2.3.1 Orthogonality One of the greatest inventions of modern times is electronics. There are many useful electronic devices but here we only focus on lock-in amplifiers that can detect minute signals buried in a noisy environment. Good lock-in amplifiers can pick up signals much (~10−6 ) smaller than the noise components. How is this possible? A lock-in amplifier effectively extracts a signal of a given Fourier component from a mixture of waves that include noise components. Its principle utilizes the orthogonality of the sine waves, namely, that any two Fourier components are “perpendicular” to each other. To understand what is meant by orthogonal or “perpendicular”, we follow the footsteps of J. J. Sakurai [19]. In our three-dimensional environment, we intuitively know that x, y, and z are “perpendicular” to each other (Fig. 2.12). It is worth considering its meaning. What is meant by “orthogonal” or “perpendicular” is that one component, say x, cannot be expressed in a linear combination of the other two, in this case y and z. That is, one can never obtain x no matter how one linearly combines y and z: |x > = α|y > +β|z >

(2.3.1)

Here α and β are constants while x, y, and z have directions and can be regarded as vectors. In fact, only directions of the vectors are important when one considers orthogonality (i.e., the lengths of the vectors do not matter). To express a vector, it is convenient to adopt Dirac’s notations with one important difference—unlike in quantum mechanics we only consider real numbers here. Fig. 2.12 Schematic illustration of orthogonality for the case of three dimensions

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2 Experimental Methods for Determination of Nucleation Rates

Then, an inner product or a scalar product is a scalar that effectively “samples” a component that is parallel to the two vectors in question: < a|b >=< b|a >= scalar

(2.3.2)

where |a> and |b> are any vectors. The meaning of taking a scalar product is especially clear for base vectors, such as |x>, |y>, and |z>, in our three dimensions. An inner product of two perpendicuar base vectors is zero: e.g., = 0. And an inner product of two like base vectors is one: e.g., = 1. For example, consider a vector |a> that is defined as |a >≡ (3, −1, 2) = 3|x > −1|y > +2|z >

(2.3.3)

We can take inner products of |a> with the three base vectors of the space, |x>, |y>, and |z>: < a|x >= (3, −1, 2) · (1, 0, 0) = 3 + 0 + 0 = 3 < a|y >= (3, −1, 2) · (0, 1, 0) = 0 − 1 + 0 = −1 < a|z >= (3, −1, 2) · (0, 0, 1) = 0 + 0 + 2 = 2

(2.3.4)

As can be seen, taking an inner product of a vector with one of the base vectors “samples” the component of the original vector that is parallel to the direction of the base vector.

2.3.2 The Fourier Transform It turned out that each component of trigonometric functions is orthogonal to every other. An interesting demonstration is presented by Feynman in his lecture series (50) [20]. In short, any one component, say sin(x), cannot be expressed in a linear combination of the others (sin(2x), etc.). That is, no matter how one linearly combines sin(2x), sin(3x), sin(0.1x), sin(–x), etc., one can never obtain sin(x) or a multiple of it. A few other examples are | sin(x) > = α| sin(2x) > +β| sin(3x) > | cos(−4x) > = α| cos(5x) > +β| cos(6x) > |ex p(7i x) > = α| exp(8i x) > +β| exp(9i x) >

(2.3.5)

The above example shows that, unlike our three-dimensional world that are limited to only three base vectors, infinite numbers of “directions” or base vectors are possible for trigonometric functions. In mathematical terms, it can be said that trigonometric functions span a Hilbert space of infinite dimensions. Then, what would be the meaning of an inner product or a scalar product in this trigonometric space? Can

2.3 Tea Time Break: The Fourier Transform and the Laplace Transform

55

we conceive an analogue of Eq. (2.3.3)? In Eq. (2.3.4), we took the inner product of an arbitrary vector |a> with all the base vectors in our three-dimensional space to find out what each coefficient of |a> in Eq. (2.3.3) is. Then, we may take the inner products of another arbitrary vector |b> in the trigonometric space with all the base vectors of the trigonometric space to find out each trigonometric component of |b>. For example, let us define the arbitrary vector |b> as |b >≡ 3| sin(2x) > −1| sin(3x) > +2| sin(4x) >

(2.3.6)

Then,

= 0 + 0 + 0 = 0 = 3 + 0 + 0 = 3 = 0 – 1 + 0 = –1

(2.3.7)

= 0 + 0 + 2 = 2 = 0 + 0 + 0 = 0 One can take inner products of |b> with trigonometry functions of non-integer or negative coefficients. However, all such inner products, except for the three components of sin(2x)>, |sin(3x)> and |sin(4x)> will be zero in this example. The question now is, how can one express infinite numbers of inner products? As it turned out, we can use an integral form instead of a discrete summation form used in Eq. (2.3.4) or Eq. (2.3.7): ∞

F(k) ≡ ∫ f (x)eikx d x −∞

(2.3.8)

This is what they call an overlap integral in quantum mechanics. eikx in Eq. (2.3.8) corresponds to sin(kx) in Eq. (2.3.6), and f (x) in Eq. (2.3.8) corresponds to |b> in Eq. (2.3.6). F(k) in Eq. (2.3.8) is called the Fourier coefficient and quantifies the “amplitude” or “amount” that is “parallel” to each eikx . The integrand in Eq. (2.3.8) is effectively an inner product of f (x) with a trigonometric function eikx , and such inner product is integrated over every possible value of k. In other words, Eq. (2.3.8) effectively “samples” a trigonometric component (a Fourier component) that is “parallel” to each base trigonometric function and the “amplitude” or “amount” of each such Fourier component is Fourier coefficient. In the above example of |b> in Eq. (2.3.7), almost all such inner products were zero and only a few non-zero components contributed to the overall integral. We used rather artificial examples of |a> in Eq. (2.3.3) and |b> in Eq. (2.3.6), for which the coefficient of each base vector is already known. This is obviously not the case in real situations. As a more realistic example, consider a certain country in

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2 Experimental Methods for Determination of Nucleation Rates

Fig. 2.13 Schematic illustration of electricity supply in AC in time domain

Fig. 2.14 Schematic illustration of the Fourier transform of the voltage profile in Fig. 2.13

which the electrical power is supplied with the alternate current (AC) of 100 V and 50 Hz (Fig. 2.13). What would be the Fourier coefficients of this Voltage(t)? If the electricity supply in question were perfectly rectified, then the only non-zero Fourier component after taking inner products over all the possible frequencies would be the |50 Hz> component. In this case, its only surviving Fourier coefficient is 100 V of the |50 Hz> component, and the Fourier transform of Voltage(t) would be a delta function: a value of 100 V at |50 Hz> and zero elsewhere (Fig. 2.14). In reality, however, there may be small amplitudes of higher harmonics of the main 100 V as “impurities”. For the sake of illustration, let us assume here that a small (0.1 V) second harmonic and an even smaller (0.001 V) third harmonic exist in the electrical power supply. Then, the Fourier transform of this electrical power supply would schematically look like Fig. (2.15) and have three non-zero Fourier coefficients: |electricity > = 100|50Hz > +0.1|100Hz > +0.001|150Hz >

Fig. 2.15 Schematic illustration of the Fourier transform of the voltage profile in Eq. (2.3.9)

(2.3.9)

2.3 Tea Time Break: The Fourier Transform and the Laplace Transform

57

2.3.3 The Laplace Transform The trigonometric functions are not the only functions that form a Hilbert space of infinite dimensions. As it turned out, exponential functions also form a Hilbert space of infinite dimensions. That is, any one component, say exp(–x), cannot be expressed in a linear combination of the other exponential functions such as exp(–2x). For example, | exp(−2x) > = α| exp(−3x) > +β| exp(−4x) >

(2.3.10)

Analogous to the Fourier transform in Eq. (2.3.8), the Laplace transform effectively “samples” an exponential component (decay constant) that is “parallel” to each base: ∞

F(s) ≡ ∫ f (t)e−st dt

(2.3.11)

0

Here, F(s) is the Laplace transform and quantifies the “amplitude” or “amount” that is “parallel” to each e–st contained in f (t). The integrand in Eq. (2.3.11) is effectively an inner product of f (t) with an exponential function e–st , and such inner product is integrated over every possible decay constant, s. In other words, Eq. (2.3.11) effectively “samples” an exponential component that is “parallel” to each base exponent and F(s) is the “amplitude” or “amount” of each such exponential component (often referred to as Laplace s-domain). Unlike the Fourier transform, though, the “infinite dimensions” of the Laplace transform is limited to the half-space of negative exponent coefficients (the number of the total dimensions is still infinite). We may consider an example: |c >≡ 5| exp(−2x) > +6| exp(−3x) > +7| exp(−4x) >

(2.3.12)

Then,

= 0 + 0 + 0 = 0 = 5 + 0 + 0 = 5 = 0 + 6 + 0 = 6 = 0 + 0 + 7 = 7 = 0 + 0 + 0 = 0 = 0 + 0 + 0 = 0 = 0 + 0 + 0 = 0 = 0 + 0 + 0 = 0

(2.3.13)

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2 Experimental Methods for Determination of Nucleation Rates

A more realistic example is a mixture of radioactive isotopes. For example, suppose the radioactive decay of a mixture of isotopes of Yttrium was measured with time. Suppose the Laplace transform of the time evolution profile showed that 18 mol% of the decay signal had a half-life of 3.35 days, 32 mol% had a half-life of 106.6 days, 21 mol% had a half-life of 2.67 days and 29 mol% had a half-life of 58.5 days. This information can be expressed in the following form: |mixture >= 0.18|3.35 days > + 0.32|106.6 days > + 0.21|2.67 days > + 0.29|58.5 days >

(2.3.14) Meanwhile, suppose it turned out that 87 Y has a half-life of 3.35 days, 88 Y has a half-life of 106.6 days, 90 Y has a half-life of 2.67 days, and 91 Y has a half-life of 58.5 days. Then, using the orthogonality of the Laplace transform, | mixture >= 0.18|87 Y > + 0.32|88 Y > + 0.2190 Y > + 0.29|91 Y > (2.3.15) The orthogonality of the Laplace transform assures the uniqueness of each exponential component that appears in Eq. (2.3.14), and one can conclude that each component in the unknown mixture must correspond to the known isotopes of Yttrium, as shown in Eq. (2.3.15). At first sight, the orthogonality of the Laplace transform might appear of remote relevance to the statistical mechanics or the nucleation theory. This is not so. The Laplace transform is highly relevant to both the statistical mechanics and the nucleation theory. In statistical mechanics, the partition function of a canonical ensemble can be given by the Laplace transform of the energy density of states, Ω(E): ∞

Z (β) ≡ ∫ Ω(E)e−β E d E

(2.3.16)

0

Here, the variable t in Eq. (2.3.11) corresponds to energy, E, and the variable s corresponds to β ≡ (k B T )−1 . The partition function thus “samples” the number of energy states of the system of interest that is “parallel” to the inverse of the temperature, β. As for the nucleation theory, the expected value of the nucleation probability density,

, can be given by the Laplace transform of the nucleation probability density with time, p(t): ∞

 p ≡ ∫ p(t)e−ckt dt

(2.3.15)

0

Here,

is essentially a nucleation curve. At a constant subcooling, the inner product of p(t) and e–ckt has only one surviving exponential component that corresponds to the nucleation rate at that subcooling.

References

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References 1. A.F. Heneghan, A.D.J. Haymet, Liquid-to-crystal nucleation: A new generation lag-time apparatus. J. Chem. Phy. 117, 5319–5327 (2002) 2. A.F. Heneghan, P.W. Wilson, G.M. Wang, A.D.J. Haymet, Liquid-to-crystal nucleation: Automated lag-time apparatus to study supercooled liquids. J. Chem. Phy. 115, 7599–7608 (2001) 3. S. Jiang, J.H. ter Horst, Crystal nucleation rates from probability distributions of induction times. Cryst. Growth Des. 11, 256–261 (2011) 4. S.A. Kulkarni, S.S. Kadam, H. Meekes, A.I. Stankiewicz, J.H. ter Horst, Crystal nucleation kinetics from induction times and metastable zone widths. Cryst. Growth Des. 13, 2435–2440 (2013) 5. N. Maeda, Nucleation curves of model natural gas hydrates on a quasi-free water droplet. AIChE J. 61, 2611–2617 (2015) 6. H.K. Abay, T.M. Svartaas, Multicomponent gas hydrate nucleation: The effect of the cooling rate and composition. Energy Fuels 25, 42–51 (2011) 7. T.P. Adamova, A.S. Stoporev, A.P. Semenov, B.I. Kidyarov, A.Y. Manakov, Methane hydrate nucleation on water-methane and water-decane boundaries. Thermochim. Acta 668, 178–184 (2018) 8. T.M. Svartaas, W. Ke, S. Tantciura, A.U. Bratland, Maximum likelihood estimation-a reliable statistical method for hydrate nucleation data analysis. Energy Fuels 29, 8195–8207 (2015) 9. T.W. Barlow, A.D.J. Haymet, Alta - an automated lag-time apparatus for studying the nucleation of supercooled liquids. Rev. Sci. Instrum. 66, 2996–3007 (1995) 10. A. Heneghan, A.D.J. Haymet, Nucleation of pure and AgI seeded supercooled water using an automated lag time apparatus, in Nucleation and Atmospheric Aerosols 2000, ed. by B.N. Hale, M. Kulmala (2000), pp. 439–442 11. A.F. Heneghan, P.W. Wilson, A.D.J. Haymet, Heterogeneous nucleation of supercooled water, and the effect of an added catalyst. Proc. Natl. Acad. Sci. U.S.A. 99, 9631–9634 (2002) 12. N. Maeda, Nucleation curves of methane hydrate from constant cooling ramp methods. Fuel 223, 286–293 (2018) 13. N. Maeda, D. Wells, N.C. Becker, P.G. Hartley, P.W. Wilson, A.D.J. Haymet, K.A. Kozielski, Development of a high pressure automated lag time apparatus for experimental studies and statistical analyses of nucleation and growth of gas hydrates. Rev. Sci. Instrum. 82, 065109 (2011) 14. N. Maeda, Measurements of gas hydrate formation probability distributions on a quasi-free water droplet. Rev. Sci. Instrum. 85, 065115 (2014) 15. N. Maeda, Nucleation curves of methane—propane mixed gas hydrates in hydrocarbon oil. Chem. Eng. Sci. 155, 1–9 (2016) 16. N. Maeda, Nucleation curves of methane-propane mixed gas hydrates in the presence of a stainless steel wall. Fluid Phase Equilib. 413, 142–147 (2016) 17. E.F. May, R. Wu, M.A. Kelland, Z.M. Aman, K.A. Kozielski, P.G. Hartley, N. Maeda, Quantitative kinetic inhibitor comparisons and memory effect measurements from hydrate formation probability distributions. Chem. Eng. Sci. 107, 1–12 (2014) 18. N. Maeda, Nucleation curve of carbon dioxide hydrate from a linear cooling ramp method. J. Phys. Chem. A 123, 7911–7919 (2019) 19. J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley Publishing Company, Reading, Massachusetts, 1994) 20. R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Massachusetts, 1963)

Chapter 3

Gas Hydrates

3.1 Physical Properties of Gas Hydrates Understanding of nucleation of gas hydrates requires basic knowledge about the physical properties of ice and clathrate hydrates. There are excellent textbooks on the physical properties of clathrate hydrates, such as by Sloan and Koh [1] and by Makogon [2, 3], and it is not our purpose here to reinvent their contents. Rather, we will briefly review the selected contents here. The thermodynamic aspects of clathrate hydrates will be discussed in Sect. 3.2.

3.1.1 Crystal Structures of Clathrate Hydrates Common ice (Ih) floats on top of liquid water because it is less dense than liquid water. That the crystalline form of a substance is less dense than its liquid counterpart is rather unusual. In the case of ice, the tetrahedral orientations of the hydrogen bonding render its structure hollow. The hollow structure of ice can accommodate small nonpolar molecules as “guests” in its cavities under the right temperature and pressure conditions [1]. These crystalline solids in which guest molecules are accommodated by water (“hosts”) are called clathrate hydrates, gas hydrates, or clathrates. Since water molecules are polar and can form hydrogen bonds whereas the guest molecules are non-polar, they have low affinity with each other and no chemical bonds form between the guest and the host. Like ice that has 17 known phases [4, 5], several lattice structures of clathrate hydrates form at different temperature and pressure conditions and guest compositions [1]. Both pure (single-component) and mixed (multi-component) gas hydrates can form [1]. There are three common crystal structures of clathrate hydrates called Structure I (sI), Structure II (sII), and Structure H (sH) [1, 6–12], as shown in Fig. 3.1. There are also less common structures of Structure III and Structure VI, but we will not discuss them any further. The unit cell of Structure I consists of two 512 cages (“small cages”) © Springer Nature Switzerland AG 2020 N. Maeda, Nucleation of Gas Hydrates, https://doi.org/10.1007/978-3-030-51874-5_3

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Fig. 3.1 Three major structures of clathrate hydrates are Structure I (sI), Structure II (sII), and Structure H (sH). Each of the structures contains more than one type of “cages” (image reproduced from Ref. [13] with permission from Elsevier)

and six 512 62 cages (“large cages”). The unit cell of Structure II consists of sixteen 512 cages (“small cages”) and eight 512 64 cages (“large cages”). The unit cell of Structure H consists of three 512 cages (“small cages”), two 43 56 63 cages (“medium cages”), and one 512 68 cage (“large cage”). Here, the large number in the base of notation of a cage structure denotes the number of sides of a face, and the small number in the exponent denotes the number of such faces in the cage. For example, a 512 cage consists of 12 pentagons, and a 512 62 cage consists of 12 pentagons and two hexagons. A 512 cage consists of 20 water molecules, a 512 62 cage consists of 24 water molecules, a 512 64 cage consists of 28 water molecules, a 43 56 63 cage consists of 20 water molecules, and a 512 68 cage consists of 36 water molecules [1]. In terms of lattice matching that is important to heterogeneous nucleation, the three common structures are similar to each other. The lattice constants of sI, sII, and sH unit cells can be found in Reference [1]. All three structures have common cages of 512 . In addition, the 512 62 , 512 64 , and 512 68 cages have common crystallographic facets. It is thus highly likely that one can epitaxially grow on one of the others if the growing structure is the thermodynamically stable phase. The composition of the guest molecules determines which crystal structure of the clathrate hydrate forms. Among the single-component guest gases, methane, ethane, carbon dioxide, xenon, and a few others form sI hydrates, while propane, iso-butane, cyclopentane, argon, krypton, hydrogen, nitrogen, and a few others form sII hydrates. Mixed gases can form sI, sII, or sH, depending on the compositions. Even for a binary mixture of guest gases, structural transitions from sI to sII, and vice versa, can occur within a small range of guest composition [14, 15].

3.1.2 Hydration Numbers Since the guests and the hosts in clathrate hydrates do not form any chemical bonds, no unique stoichiometric numbers exist for clathrate hydrates. We can therefore

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only talk about a range or an average of hydration numbers per guest molecule to describe the molar ratio of the guests to the hosts. What further complicates the matter is that more than one molecule of argon, hydrogen, methane, or nitrogen can occupy a large cavity of sII. What complicates the matter still further is that not all the cavities of a given clathrate hydrate structure need to be occupied by the guest molecules—that is, the occupancy ratio can be less than 1. Put it another way, the maximum number of guests that can be accommodated in a given clathrate hydrate can be uniquely determined (the hydration number is 5.75 for sI and 5.67 for sII for full occupancy [1]), but that does not mean a given clathrate hydrate will always contain the maximum numbers of guest gases. This non-stoichiometric nature of clathrate hydrates gives rise to variations in their physical properties. Even for a given occupancy ratio, its physical properties generally depend on the type of guests. For example, the density of a clathrate hydrate depends on the average molecular weight of the guests. So, no unique value exists for the density of a Structure I, a Structure II, or a Structure H hydrate. Therefore, only typical values, as opposed to a unique value, can be compiled for some physical properties. With this generic point in mind, we will review several known physical properties of clathrate hydrates.

3.1.3 Thermal Properties Density depends on the guest in that the clathrate hydrate of a high-molecular-weight guest is denser than the clathrate hydrate of a low-molecular-weight guest. The value typically ranges from ≈0.94 g/cm3 for sI hydrates to ≈1.29 g/cm3 for sII hydrates, in contrast to 0.91 g/cm3 for ice (Ih) [1]. Since ice (Ih) does not contain any guests, its smaller density than either clathrate hydrate makes sense. Specific heat of sI hydrate is about ≈2080 Jkg−1 K−1 which is similar to that of sII of about ≈2130 Jkg−1 K−1 [1, 16, 17]. These values are substantially greater than that of ice (Ih) of ≈1700 Jkg−1 K−1 . The greater specific heat of clathrate hydrates is expected because the presence of guests ought to increase the amount of heat required to raise the temperature of a clathrate hydrate than ice for a given number of water molecules. For comparison, the specific heat of liquid water is much larger than either of ice or clathrate hydrate, 4186 Jkg−1 K−1 . This rather surprising fact suggests that the great majority of the hydrogen bonding in ice or clathrate hydrate does not break when they dissociate but progressively break as the liquid water warms up. The enthalpy of fusion of clathrate hydrate depends on the guest and increases with the size of the guest [1, 18–20], probably as expected. For example, the enthalpy of fusion of methane hydrate is 54.2 kJ per mole of the guest gas, ethane hydrate is 71.8 kJ per mole of the guest gas, propane hydrate is 129.2 kJ per mole of the guest gas, and iso-butane hydrate is 133.2 kJ per mole of the guest gas [1]. The thermal conductivity of an sI hydrate and an sII hydrate are both about ≈0.5 Wm−1 K−1 [1, 16, 17, 21–28]. This value is similar to that of liquid water and about 75% to 80% smaller than that of ice (Ih) of ≈2.3 Wm−1 K−1 [1]. That the thermal conductivity of a clathrate is poorer than that of ice is rather counterintuitive. One

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would expect the structurally hollow ice to be a poorer thermal conductor than a “filled” clathrate hydrate. The thermal expansion coefficients of both sI and sII hydrates increase with temperature and range from 10−5 to 10−4 K−1 [1, 18, 29–36].

3.1.4 Mechanical Properties Shimizu et al. measured the adiabatic elastic moduli, the bulk moduli, and the elastic anisotropy of single-crystalline, sI-forming, methane hydrate as a function of pressure up to 600 MPa [37]. They found that the fully occupied methane hydrate has slightly greater moduli than the 75% occupied methane hydrate. The values of the elastic moduli depended on the crystallographic direction, due to the crystallographic anisotropy, and they increased more or less linearly with the pressure. For example, the bulk modulus increased from ≈8 GPa at 296 K and 20 MPa to ≈12 GPa at 296 K and 600 MPa [37]. Helgerud et al. measured the compressional- and shear-wave velocity on compacted polycrystalline sI-forming methane hydrate and sII-forming methane– ethane mixed gas hydrate [38]. Although they did not provide the numerical values of the moduli, they showed that the compressional- and shear-wave velocities of the sII hydrate were similar to those of the sI hydrate [38]. This is in agreement with Whaley’s finding earlier that the relative speed of the longitudinal sound wave in sI and sII hydrates to be about 6% less than that of ice [39]. A somewhat smaller result was obtained by Whiffen et al. who reported that the relative speed of the longitudinal sound wave in sI hydrate to be about 12% less than that of ice [40]. Kiefte et al. found that the acoustic velocity in sI hydrates decreased with increasing mass of the guest and 3% to 24% smaller than that of ice (Ih) [41]. Shimizu et al. found that single-crystalline methane hydrate was slightly more compressible than ice (Ih) at 296 K and in the pressure range of 20–600 MPa [37]. This is in contrast to the finding of Durham et al. who reported that polycrystalline methane hydrate could be as much as 40 times stronger (creep resistant) under stress than ice (Ih) in the range 260–287 K and 50–100 MPa [42].

3.1.5 Electromagnetic Properties The dispersion relations of dielectric functions are the central electromagnetic properties of a given material that involve interactions between photons and electrons. Unfortunately, dispersion relations have rarely been measured for clathrate hydrates. A notable exception is for an sI-forming tri-methylene oxide hydrate over a range of 10 Hz and 1 MHz [43]. Both the real part, ε , and the imaginary part, ε , of the dielectric function εcomplex ≡ ε + iε showed strong temperature dependence over the temperature range studied (between 1.8 K and 200 K) [43]. As for ε at the zero frequency limit, the dielectric constant was found to be ≈58 for both sI and sII,

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in contrast to 94 for I h at 273 K [1, 18]. Magnetic susceptibilities, µ, of clathrate hydrates have not been reported. The complex refractive index, ncomplex ≡ n + iκ, is also generally a complex function that depends on the frequency (wavelength) of an electromagnetic wave. The complex refractive index is related to the dielectric function through εcomplex /ε0 = n2complex , where ε0 is the dielectric constant of vacuum, so ε /ε0 = n2 – κ 2 and ε /ε0 = 2nκ [44]. The refractive index of vacuum is defined to be ncomplex ≡ 1 with no imaginary component. The imaginary part of a complex dielectric function or a complex refractive index represents a loss of electromagnetic fields (attenuation or absorption). The experimentally measurable absolute values of a dielectric function and a refractive index are defined using the complex conjugate in each case: |εcomplex |2 ≡ (ε + iε )(ε – iε ) for the dielectric function and |ncomplex |2 ≡ (n + iκ  )(n – iκ  ) for the refractive index. For the refractive index, only the real part, n, over limited ranges of wavelengths has been measured for clathrate hydrates. Using the wavelengths of λ = 640, 756, and 844 nm, Bylov and Rasmussen reported n ≈1.346 for an sI-forming methane hydrate and ≈1.350 for an sII-forming natural gas hydrate between –2.5 and 8.6 °C [45]. Electrical conductivity, σ, also generally depends on the AC frequency. Du Frane et al. reported σ of sI-forming methane hydrate to be about ≈5 × 10−5 Sm−1 at 273 K [46]. Small frequency dependence was observed between 100 kHz and 1 MHz [46]. The temperature dependence was regular, as a plot of log(σ ) versus (1/T ) yielded a straight line of a negative slope [46]. One might wonder why we should worry about the electromagnetic properties or the dielectric functions of clathrate hydrates which, at first glance, might not appear terribly relevant. However, this is not so: like many things in life, the first appearance of an entity can be deceiving. The van der Waals forces are ubiquitous and exist in every material and substance, including clathrate hydrates. They dominate many aspects of physical properties of the material and its interactions with its surroundings. The origin of the van der Waals forces is electromagnetic forces that arise between groups of electrons, as we will see in Chap. 4. The refractive index of a material is a property determined by its electron density and its strength of interactions with photons, which is none other than the dielectric functions. Simply put, the higher the electron density and the stronger their interactions with photons, the higher the refractive index. The adhesion, cohesion, and agglomeration properties of clathrate hydrates are determined by the surface forces for which the van der Waals forces are the major components. The thickness and the physical properties of quasiliquid layers, or their presence or absence on clathrate surfaces, are also determined by the surface forces, as we will see in Chap. 5.

3.2 Thermodynamic Aspects of Gas Hydrates The thermodynamic properties of clathrate hydrates that are central to their nucleation are their surface and interfacial free energies. However, their descriptions require the concept of disjoining pressure which we will not introduce until Sect. 4.1. As such,

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we will defer the surface and interfacial free energies of clathrate hydrates to Sect. 4.3, and here we only limit ourselves to a few other major thermodynamic properties in the bulk of clathrate hydrates. Clathrate hydrates are, by definition, multi-component systems. Thermodynamically, clathrate hydrates can be viewed as solid solutions of guest gases in the host water, like alloys [47]. An important feature that is common to all types of clathrate hydrates is that the concentration of the gas that can be contained inside clathrate hydrate as a guest is much higher than the solubility of the same gas in liquid water. For example, 1 m3 of methane hydrate can contain as much as 170 m3 of methane gas at the standard temperature and pressure (STP) [1]. In contrast, the solubility of methane in liquid water is of the order of 10−3 in mole fractions at pressures for which methane hydrate is stable [48]. This much greater gas contents in the clathrate form than the solubility of the same guest gas in liquid water at the same pressure and temperature gives rise to thermodynamic stability of clathrate hydrates. Freezing point depression of ice in electrolytes is a common phenomenon. Since the solubility of a salt in liquid water far exceeds the “solubility” of the same salt in ice, if it could be regarded a “solid solution” [49], the free energy reduction due to the entropy of mixing is far greater in liquid water than in ice. The consequence is that a salt solution can remain thermodynamically stable below 273.16 K. In other words, the melting point of ice can be lowered by the dissolution of salts. Likewise, clathrate hydrates could be viewed as a kind of “melting point elevation of ice”. The much higher guest gas contents in the clathrate form than the solubility of the same guest gas in liquid water renders the entropy of mixing in the clathrate form greater than that in the aqueous solution. The resulting free energy reduction aids its thermodynamic stability at higher temperatures than 273.16 K.

3.2.1 Phase Diagrams of Clathrate Hydrates at Relatively Low Pressures Phase diagrams are probably one of the most studied thermodynamic aspects of clathrate hydrates [1]. Since clathrate hydrate must consist of at least two components of the guest and the host, its phase diagram is more complex than that of pure water. For the simplest case of a single-component guest, the Gibbs phase rule (Eq. 1.1.1) states that the number of the degree of freedom is four minus the number of phases present in the system. The minimum number of the phases allowed is one where three parameters of temperature, pressure, and composition can be independently varied. Such phase diagrams require three-dimensional descriptions that are not convenient to express on a two-dimensional paper. Alternatively, either (1) pressure–temperature phase diagram at a fixed composition, (2) pressure–composition phase diagram at a fixed temperature, or (3) temperature–composition phase diagram at a fixed pressure can be expressed on a two-dimensional paper. Here, we only limit ourselves to qualitative descriptions of single-component guest clathrate hydrate phase diagrams.

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Two phases can coexist on a curved surface in a three-dimensional phase diagram. Two such curved surfaces may intersect at a curve on which three phases can coexist, i.e., form a triple-phase curve. Three such curved surfaces may intersect at a single point on which four phases can coexist. This is equivalent to state that a triple-phase curve may intersect a double-phase curved surface at a single point. Such a special single point is called a quadruple point. Since up to four phases can coexist in a two-component system before the degree of freedom is exhausted, a quadruple point has the maximum allowable number of four phases in the system. One or more quadruple points may exist in a clathrate hydrate phase diagram of a single guest. The four phases of liquid water, clathrate hydrate, liquid guest, and gas may coexist at an upper quadruple point. The four phases of ice, liquid water, clathrate hydrate, and gas may coexist at a lower quadruple point. For supercritical guests such as methane or nitrogen, only one quadruple point may exist. A method to simplify the above descriptions is to project the entire threedimensional phase diagram onto a plane of a fixed composition. Then, a triplephase curve will still be a curve after projection onto a two-dimensional plane, whereas a double-phase curved surface will become an area after projection onto a two-dimensional plane. Figure 3.2 schematically shows the general feature of such a projected phase diagram of clathrate hydrate when the system composition is such that there is an excess of the guest with respect to the amount of the host water. Here, the pressure is expressed in a logarithmic scale and only the low-pressure region of the phase diagram is shown (the high-pressure region of a clathrate hydrate phase diagram is not characterized very well yet, as we will see later). For simplicity, here we denote

Oil–Water–Clathrate

Fig. 3.2 Schematic general feature of phase diagram of clathrate hydrate for an excess guest composition

lnP

Qupper

Qlower

T

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the liquid guest phase as merely “oil” and the liquid water phase as merely “water” (strictly speaking, an “oil” phase is an oil-rich phase that is saturated with water and a “water” phase is a water-rich phase that is saturated with the guest). Three triple-phase curves on which three phases coexist meet at each quadruple point. The areas between the triple-phase curves are two-phase regions. The identity of the two phases in each such region is defined by the two phases that are common to the adjacent triple-phase curves. For example, the lower left region bounded by the two triple-phase curves of ice–clathrate–gas and ice–water–gas consists of ice and gas. Likewise, the large lower right region bounded by the three triple-phase curves of oil–water–gas, water–clathrate–gas, and ice–water–gas consists of water and gas. The physical meaning of Fig. 3.2 is as follows. We first consider isobaric processes. At an initial pressure lower than the lower quadruple point and at a high temperature, the system initially consists of two phases of liquid water and gas. As the system cools isobarically, the system reaches the triple-phase curve of ice–water–gas (i.e., ice newly forms). Below this temperature, the system consists of two phases of ice and gas (liquid water disappears). At an initial pressure between that of the upper quadruple point and that of the lower quadruple point, the system initially consists of two phases of liquid water and gas at a high temperature. As the system cools isobarically, the system reaches the triple-phase curve of water–clathrate–gas (clathrate newly forms). Below this temperature, the system consists of two phases of clathrate and gas (liquid water disappears). At an initial pressure higher than the upper quadruple point, the system initially consists of two phases of liquid water and gas at a very high temperature. As the system cools isobarically, the system reaches the triple-phase curve of oil–water–gas (oil newly forms). Below this temperature, the gas liquefies and the system consists of two phases of oil and water (gas disappears). As the system further cools, the system reaches another triple-phase curve of oil– water–clathrate (clathrate newly forms). Below this temperature, all the water in the system is consumed for the formation of clathrate, and the system consists of two phases of the guest oil, which is supposed to exist in excess of water, and the clathrate. Similar considerations can be applied to isothermal processes. At an initial temperature that is lower than the lower quadruple point and a low pressure, the system initially consists of two phases of ice and gas. As the system is pressurized isothermally, the system reaches the triple-phase curve of ice–clathrate–gas (clathrate newly forms). Above this pressure, all the ice in the system is consumed for the formation of clathrate, and the system consists of two phases of guest gas, which is supposed to exist in excess of water, and the clathrate. As the pressure is further increased isothermally, the system reaches another triple-phase curve of oil–clathrate–gas (oil newly forms). Above this pressure, all the excess gas is liquefied, and the system consists of two phases of an excess oil and the clathrate. At an initial temperature between that of the upper quadruple point and that of the triple-phase curve of ice–water–gas, the system initially consists of two phases of liquid water and gas at a low pressure. As the system is pressurized isothermally, the system reaches the triple-phase curve of water–clathrate–gas (clathrate newly forms). Above this pressure, all the water in the system is consumed for the formation of clathrate, and the system consists of two phases of an excess gas and the clathrate. As the pressure is further increased

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isothermally, the system reaches another triple-phase curve of oil–clathrate–gas (oil newly forms). Above this pressure, all the excess gas is liquefied, and the system consists of two phases of an excess oil and the clathrate. At an initial temperature that is higher than that of the upper quadruple point, the system initially consists of two phases of liquid water and gas at a low pressure. As the pressure is increased isothermally, the system reaches the triple-phase curve of oil–water–gas (oil newly forms). Above this pressure, all the gas in the system is liquefied, and the system consists of two phases of oil and liquid water. Figure 3.3 schematically shows the general feature of such a projected phase diagram of clathrate hydrate when the system composition is such that there is an excess water with respect to the guest. Again the pressure is expressed in a logarithmic scale and only the low-pressure region of the phase diagram is shown. For simplicity, we denote the liquid guest phase as merely “oil” and the liquid water phase as merely “water”, as we did for Fig. 3.2. It looks similar to Fig. 3.2 but there are important differences due to the different excess components. Unlike in the case of an excess oil, there would be no excess hydrocarbon gas or oil in the presence of clathrate here. Therefore, there would be no liquefaction of an excess hydrocarbon guest gas to oil at high pressures and low temperatures. Such hydrocarbon guest gas-to-oil transition with pressurization can only occur at a temperature above the upper quadruple point where the clathrate phase is absent. Instead, an excess liquid water can freeze to form ice as the system cools at a high pressure and reaches the triple-phase curve of ice–water–clathrate, a feature that is absent in Fig. 3.2.

Oil–Water–Clathrate

Ice–Water–Clathrate

lnP

Fig. 3.3 Schematic general feature of phase diagram of clathrate hydrate for an excess host composition

Qupper

Qlower

T

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Qupper

Ice–Water–Clathrate

lnP

Oil–Water–Clathrate

Fig. 3.4 The two cases that correspond to Figs. 3.2 and 3.3 may sometimes be shown schematically superimposed

Qlower

T In some literature, these two cases of an excess guest composition and an excess host composition are superimposed together and may be shown schematically in Fig. 3.4. However, the readers should keep in mind that the physical meaning of Figs. 3.2 and 3.3 are quite different, as we described in detail above. Although it is not explicitly shown in the schematic figures of Figs. 3.2, 3.3 or Fig. 3.4, the lower quadruple point temperature is around 273 K in each case, as might be expected from the coexistence of liquid water and ice (I h ) along the triple-phase curve. The I h -liquid water melting point of the P–T phase diagram of I h is nearly vertical upward from the triple point, and the solubility of a guest gas in liquid water is too low to meaningfully lower the melting point of I h .

3.2.2 Phase Diagrams of Clathrate Hydrates at Higher Pressures Ice has currently 17 known phases [50, 51], and it is possible that more phases may be discovered in future. So it is reasonable to expect that clathrate hydrates also have more phases than what are known today. First of such high pressure (>GPa) phase transitions was reported by van Hinsberg et al. who reported a phase transition of nitrogen hydrate from sII to sI around 2 GPa [52, 53]. Several years later, Hirai et al. reported that sI methane hydrate became thermodynamically unstable at very high pressures [54]. At 1.5 GPa, sI methane hydrate partly decomposed to ice IV and fluid methane, while the remaining methane hydrate

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maintained Structure I. At 2.1 GPa, coexisting ice VI transformed to ice VII and the fluid methane solidified to phase I, while Structure I of methane hydrate was still maintained. At 2.3 GPa, all of the remaining sI methane hydrate decomposed into ice VII and phase I of solid methane [54]. About a year later, Chou et al. reported that sI methane hydrate did not decompose into ice VII and phase I of solid methane, as earlier thought [54], but instead transformed into two new structures of methane hydrate at high pressures [55]. One of such structures was sH-like hexagonal and the other sII-like cubic, both different from the sI methane hydrate structure at low pressures [55]. Loveday et al. then elucidated the two-phase transitions in methane hydrate at ultrahigh pressures [56]. The first was from the commonly known sI to a new methane hydrate phase (MH-II) between 0.8 and 1.1 GPa [56]. The MH-II phase then underwent a transformation at 2.0 GPa to another new methane hydrate phase of III (MHIII) which remained stable up to 10 GPa [56]. Structurally, MH-II was found to be hexagonal and structurally similar to sH [57], whereas MH-III was structurally similar to common ice (Ih) [58]. At about the same time, Manakov et al. reported phase transformations of argon hydrate at high pressures and room temperature [59, 60]. Three high-pressure phase transitions were observed: from sII to the high-pressure hexagonal structure at 0.46 GPa, from the hexagonal to a tetragonal structure at 0.77 GPa and from the tetragonal structure to ice (II) at 0.96 GPa [59, 60]. Shimizu et al. then reported phase transformations of single-crystalline argon hydrate at high pressures and room temperature [61]. Single-crystalline argon hydrate at room temperature showed three phase transitions from sII to a high-pressure hexagonal structure at 0.43 GPa, from the hexagonal to a tetragonal structure at 0.66 GPa and from the tetragonal structure to ice at 1.05 GPa [61]. Loveday et al. later extended their earlier ultrahigh-pressure studies to other sI and sII hydrates at room temperature [62]. Methane, argon, nitrogen, and xenon hydrates all transformed into hexagonal (MH-II type) structure with increasing pressure. Methane, argon, and nitrogen hexagonal hydrate then transformed into an orthorhombic hydrate with further increase in pressure, whereas xenon hexagonal hydrate decomposed to ice and xenon around ≈2.5 GPa. This decomposition of xenon hydrate appears similar to the decomposition of methane hydrate at 2.3 GPa reported by Hirai et al. earlier [54]. So, decomposition of at least some clathrate hydrates at high pressures appears real. If the water host can no longer maintain hollow structures that are large enough to accommodate guest molecules due to the very high pressures, then the guest “solubility” in the solid water host would go down and the clathrate hydrate would lose its source of thermodynamic stability. Meanwhile, Yang et al. reported a new structure of xenon hydrate under atmospheric pressure at a cryogenic temperature [63]. The new phase was derived by initial pressurization of sI xenon hydrate to 2.0 GPa at room temperature followed by quenching to 77 K at atmospheric pressure. The new structure has a hexagonal symmetry that is similar to MH-II and remains stable up to 160 K before decomposing to the traditional sI phase [63].

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3.2.3 Phase Equilibrium Computation of Clathrate Hydrates The phase diagrams of clathrate hydrates have been determined experimentally for major guests. However, it is impossible to experimentally determine phase diagrams of all possible guests because there are literally infinite numbers of compositions of gas mixtures. So a computational method that can accurately determine a phase diagram of clathrate hydrate is highly desirable. We saw in (Sect. 1.1) that, for a pure component, the Clausius–Clapeyron equation can be used to find the relationship as to how the equilibrium pressure varies with the equilibrium temperature along a phase boundary. As the first approximation, the slope of a triple-phase line that extends from a quadruple point may be estimated if the change in the enthalpy, ΔH, and that of the volume, ΔV, across the triple-phase line are known at the quadruple point temperature. The applicability of the Clapeyron equation to the phase equilibria of clathrate hydrate systems is discussed in references [1, 20, 64, 65]. As we saw in the previous section, new clathrate hydrate phases are being discovered at very high pressures (>GPa), and the Clausius–Clapeyron equation cannot tell where the triple-phase curve that extends from a quadruple point should end. A more accurate method that can be applied to gas mixtures is a statistical thermodynamic approach [1]. Here, the physical state of a given system is determined so as to minimize the Gibbs free energy of the system. For a given set of pressure, temperature, and composition, the Gibbs free energy of the system can be calculated for each case of when the system consists of one phase, two phases, etc., and compared. To do so, some assumptions and models are required that enable calculations of the chemical potential of each component involved. As we saw in Chap. 1, a liquid–gas phase boundary can in principle be determined by solving an appropriate cubic equation of state. To solve a cubic equation of state, the coefficients in the cubic equation of state (e.g., a and b for the van der Waals equation of state) that are specific to each system need to be determined. These coefficients can be determined by fitting the computed output of the cubic equation of state to relevant available experimental data. Once the coefficients are determined, the cubic equation of state can be solved for a system of interest. In typical flash calculations, the input parameters are the overall composition, the total amount of the substance, the system pressure, and the temperature. The outputs are the number of phases and the composition and the amount of each phase. The Gibbs free energy of the whole system is minimized to find the answer. Alternatively, computationally simpler fugacity can be used in the place of chemical potential as a parameter because the fugacity is balanced at a phase boundary where two phases coexist in equilibrium [66]. For a phase boundary that involves a solid (crystalline) phase, a good model for the crystal in question is generally required. For clathrate hydrates, a good model for water is generally required; however, there is no consensus as to what the best model for water should be. A great deal of work has been carried out over the years for this effort, which has been summarized in [1].

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The solubility of a guest gas in liquid water that can coexist with its clathrate hydrate for an excess water composition is an important parameter that aids calculations of the chemical potential of the guest in the clathrate hydrate. Likewise, the vapor pressure of water in the atmosphere of the guest gas that can coexist with its clathrate hydrate for an excess guest composition is an important parameter that aids calculations of the chemical potential of the host in the clathrate hydrate. Yet, very few relevant data have been reported [1]: (1) methane and ethane solubility in water as a function of temperature ramping rate [67], (2) carbon dioxide solubility in water [68], (3) methane in water in the (liquid water–clathrate hydrate) two-phase region [69], and (4) the aqueous methane solubility [70]. Qualitatively, the solubility of a given guest in liquid water should decrease still further from the already low value when the clathrate hydrate phase coexists because the activity of water should decrease to match the lower chemical potential of water in the clathrate phase. Likewise, the vapor pressure of the water vapor over a clathrate hydrate in the atmosphere of a guest gas should decrease further from that over liquid water.

3.3 Gas Hydrate Inhibitors An industrially important issue that concerns clathrate hydrates is flow assurance of oil and natural gas pipelines and flow lines. Production fluids are rarely dry, and the water content generally increases as the field ages. The presence of water in a high-pressure and low-temperature environment is prone to clathrate hydrate formation. Thus, a pipeline that travels low-temperature sections poses risks of undesired clathrate hydrate formation that could cause blockage. A clathrate hydrate plug will induce a dangerous pressure differential between the upstream and the downstream sides of the plug which, if not remediated in a timely manner, could cause a major accident. Oil and gas operators have been heavily using chemicals (gas hydrate inhibitors) to prevent such adverse incident from occurring. There are three classes of gas hydrate inhibitors: Thermodynamic hydrate inhibitors (THI), kinetic hydrate inhibitors (KHI), and anti-agglomerates (AA). Since studies on the nucleation of clathrate hydrates often involve gas hydrate inhibitors, we briefly describe each of these groups of chemicals and anti-freeze proteins.

3.3.1 Thermodynamic Hydrate Inhibitors (THI) Thermodynamic inhibition is colligative in nature. A THI is miscible with water at all proportions and lowers the activity of water. Among the possible solutes for thermodynamic inhibition, salts are usually not preferred due to their adverse corrosive effects on production facilities. Rather, alcohols and glycols are the preferred THIs. In either case, a greater free energy reduction is required for clathrate hydrate

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to form in the presence of a THI than in the absence of one. As a first approximation, the temperature depression for clathrate hydrate formation may be considered to be similar to the temperature depression of the freezing point of ice by a solute. However, Nielsen and Bucklin derived an equation which indicated that the depression temperature of a clathrate hydrate is always 30–40% less than that of ice [1, 71], which is detrimental to the thermodynamic inhibition efficacy of THIs. The inhibition capability of alcohols diminishes with the increasing molecular weight because the hydrophobic moiety of an alcohol increases with the molecular weight. Since methanol has by far the smallest molecular weight among alcohols it will be the most effective THI at a given mass concentration (that corresponds to a much higher molar concentration than the other alcohols). A glycol is more hydrophilic than an alcohol because it has a higher fraction of –OH groups than an alcohol of the same molecular weight. Among glycols, monoethylene glycol (MEG) is more hydrophilic than the higher molecular weight counterparts of di-ethylene glycol (DEG) or tri-ethylene glycol (TEG). In addition, since MEG has by far the smallest molecular weight among glycols, it will be the most effective THI at a given mass concentration (that corresponds to a much higher molar concentration than the other glycols). For these reasons, methanol and MEG have been the most common THIs. The amount of thermodynamic inhibition required in a given field operation depends on the temperature and the pressure that are encountered by the production fluid. The amount of required THIs increases as the temperature becomes lower and the pressure becomes higher inside the flow lines. Thus, large quantities of THIs are required for an aging field that produces progressively more water. Logistics of supplying large quantities of THIs to remote locations is expensive and hence requires as much regeneration and recycling of THIs as possible. Then, a regeneration unit will be required on the production site. This requirement poses an additional challenge when the space is limited (e.g., an offshore platform).

3.3.2 Kinetic Hydrate Inhibitors (KHI) The above challenges and limitations of THIs led to the development of kinetic hydrate inhibitors (KHI) since the late 1980s [72]. The basic idea behind KHIs is that it may not be essential to prevent the formation of clathrate hydrates for the flow assurance of oil and natural gas pipelines as long as one can prevent the nucleation of clathrate hydrates from developing into a more serious problem of pipeline blockage by arresting the growth into a hydrate plug. Formation of a hydrate plug from an initially hydrate-free state requires several kinetic steps of development. One of such several kinetic steps that could be arrested is the crystal growth of clathrate hydrate after its nucleation. Another kinetic step that could be arrested is agglomeration or deposition of clathrate hydrate polycrystals or particles after their nucleation and crystal growth. Arresting of any such kinetic step would suffice to meet the requirement of flow assurance. The value of this approach comes from

3.3 Gas Hydrate Inhibitors

75

Fig. 3.5 Schematic drawing of polyvinylpyrrolidone (PVP) and polyvinylcaprolactan (PVCap)

reduced costs of kinetic inhibition over thermodynamic inhibition. Kelland gave detailed accounts of the history of the development of KHIs and AAs [72], and the readers are referred to the work of Kelland and the references therein for a more comprehensive coverage of the topic. The basic requirement of a KHI is that it must be soluble in water. Clathrate hydrates generally nucleate at the guest gas–aqueous interface where the guest concentration in the aqueous phase is the highest, as we will see in Chap. 5. Therefore, KHIs need to be surface-active—i.e., increasing concentration of a KHI must lower the surface free energy of water to allow positive adsorption of the KHI to the guest gas–aqueous interface, as we will see in Chap. 4 [73]. The surface-active and water-soluble KHIs can be present in the proximity of clathrate hydrates when they first start to form. It turned out that hydrophilic polymers such as polyvinylpyrrolidone (PVP), polyvinylcaprolactan (PVCap), polyisopropylmethacrylamide (pIPMA), poly-Nmethyl-N-vinylacetamide (pVIMA), and 1:1 VIMA:IPMA (isopropylmethacrylamide) copolymer are good KHIs (Fig. 3.5) [72, 74–76]. Combined with suitable synergistic chemicals and THIs, PVCap-based KHIs provided reliable kinetic hydrate inhibition up to 13–16 K in oil and up to 11–12 K in a moist gas [72]. These KHIs are generally thought to adsorb to the growth front of microscopic or submicroscopic clathrate hydrate crystals and prevent/slow their further growth to macroscopic sizes [1].

3.3.3 Anti-Agglomerates (AA) Often turbulent flow conditions in pipelines likely lead to the formation of polycrystalline clathrate hydrate particles of random crystallographic orientations as opposed to the formation of a few large single crystals. Then, another kinetic step that could be arrested to prevent formation of a dangerous hydrate plug in pipelines is agglomeration of clathrate hydrate polycrystals or particles after their nucleation and crystal growth. Anti-agglomerates (AA) refer to a group of chemicals that target arresting such agglomeration of clathrate hydrate crystals. If clathrate hydrate particles can be dispersed in a continuous oil phase by AAs, then they can be carried with the production fluid as a cold slurry. For this approach to work, the water content in the condensed phase (the water cut) of the production fluid must be low (below 50%)

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[72]. In other words, AAs do not work in a gas field that hardly produces oil (liquid hydrocarbons). The surface of clathrate hydrate particles is hydrophilic and most likely wet with a quasi-liquid layer (QLL), as we will see in the next chapter. The presence of a QLL would give rise to attractive capillary force between clathrate hydrate particles in a continuous oil phase. Thus, it is necessary to render the surface forces between clathrate hydrate particles strongly repulsive for these particles to remain dispersed in an oil phase [77]. A common way to do so is to use amphiphilic molecules like surfactants whose hydrophilic head groups adsorb to clathrate hydrate particles and whose hydrophobic tails are exposed to the continuous oil phase. This configuration provides steric repulsion between surfactant-coated clathrate hydrate particles in the oil phase [77]. It tuned out that quaternary ammonium surfactants such as tetra-butyl-ammonium bromide (TBAB) and tetra-pentyl-ammonium bromide (TPAB) are good AAs [72, 78, 79]. In particular, quaternary surfactants with two or three n-butyl, n-pentyl, or iso-pentyl groups performed especially well as AAs. Unfortunately, these AAs are toxic and have low biodegradability [72] and a search for an environmentally friendly alternative has been ongoing.

3.3.4 Anti-freeze Proteins Anti-freeze proteins prevent ice crystals from forming in fish that live in cold regions by binding to the surface of ice nuclei. Given the similarities between ice and clathrate hydrates, some proteins might bind to the surface of clathrate hydrate nuclei and prevent clathrate hydrate crystals from forming [80–84]. This line of research is ongoing; however, anti-freeze proteins have generally been found to be precious and hence expensive, and fairly poor KHIs [72]. Compatibility issues with other chemicals in pipelines such as scale inhibitors, corrosion inhibitors, or THIs pose additional challenges to the use of anti-freeze proteins as clathrate hydrate inhibitors.

3.4 Tea Time Break: Similarities of Biological Systems to the Flow Assurance Challenges of Oil and Natural Gas Pipelines Crystal growth is generally undesirable in human bodies (e.g., oxalic acid, bile stone, gallstone, gout). The mechanism with which such harmful stones form in human bodies remains unknown [85]. For example, urine is supersaturated with respect to the components that form gallstones, but gallstones do not form in the bodies of healthy people [86, 87]. This circumstance is eerily similar to the flow assurance

3.4 Tea Time Break: Similarities of Biological Systems to the Flow Assurance …

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problem in oil and natural gas pipelines: oil and gas supersaturated with water do not always form hydrate plugs. Is it possible that either nucleation or crystal growth inhibitors naturally exist in healthy people, which, for some reason, become depleted in gallstone patients?

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65. R.M. Barrer, A.V.J. Edge, Gas hydrates containing argon krypton and xenon—kinetics and energetics of formation and equilibria, in Proceedings of the Royal Society of London Series a-Mathematical and Physical Sciences, vol. 300 (1967), p. 1 66. C.H. Whitson, M.R. Brule, Phase Behavior (Society of Petroleum Engineers, Richardson, Texas) 67. K.Y. Song, G. Feneyrou, F. Fleyfel, R. Martin, J. Lievois, R. Kobayashi, Solubility measurements of methane and ethane in water at and near hydrate conditions. Fluid Phase Equilib. 128, 249–259 (1997) 68. I. Aya, K. Yamane, N. Yamada, Simulation experiment of CO2 storage in the basin pf deepocean. Energy Convers. Manag. 36, 485–488 (1995) 69. P. Servio, P. Englezos, Measurement of dissolved methane in water in equilibrium with its hydrate. J. Chem. Eng. Data 47, 87–90 (2002) 70. Y.P. Handa, Effect of hydrostatic-pressure and salinity on the stability of gas hydrates. J. Phys. Chem. 94, 2652–2657 (1990) 71. R.B. Nielsen, R.W. Bucklin, Why not use methanol for hydrate control. Hydrocarb. Process. 62, 71–78 (1983) 72. M.A. Kelland, History of the development of low dosage hydrate inhibitors. Energy Fuels 20, 825–847 (2006) 73. D.F. Evans, H. Wennerstrom, The Colloidal Domain, 2nd edn. (Wiley-Vch, New York, 1999) 74. K.S. Colle, C.A. Costello, R.H. Oelfke, L.D. Talley, J.M. Longo, E. Berluche, K. Colle, R. Oelfke, L. Talley, R.H. Oeltke, Inhibition of clathrate hydrate formation-using specific poly:(acrylamides) (Exxon Prodn Res Co) 75. K.S. Colle, R.H. Oelfke, E. Berluche, C.A. Costello, L.D. Talley, R. Oelfke, L. Talley, M.A. Kelland, Polymer Having Vinyl Amide Unit-Used for Inhibiting Formation of Gas Hydrates in e.g. Oil or Gas Pipeline (Exxon Prodn Res Co; Exxonmobil Upstream Res Co) 76. J. Long, J. Lederhos, A. Sum, R. Christiansen, E.D. Sloan, A. Gas Processors, Kinetic inhibitors of natural gas hydrates, in Seventy-Third Annual Convention—Gas Processors Association, Proceedings (1994), pp. 85–93 77. J.N. Israelachvili, Intermolecular and Surface Forces, 2nd edn. (Academic Press, San Diego, 1991) 78. U.C. Klomp, V.R. Kruka, R. Reijnhart, A.J. Weisenborn, Inhibition of conduit plugging by gas hydrates-using long chain tertiary or quaternary salts (Shell Int Res Mij Bv; Shell Canada Ltd; Shell Oil Co) 79. U.C. Klomp, R. Reijnhart, Inhibiting plugging of conduit contg. mixt. of light hydrocarbon(s) and water-by adding ammonium or phosphonium alkylated hydrate formation inhibitor, and flowing mixt. contg. inhibitor through conduit (Shell Int Res Mij Bv; Shell Canada Ltd; Shell Oil Co; Shell Int Res Mij Nv) 80. L. Jensen, H. Ramlov, K. Thomsen, N. von Solms, Inhibition of methane hydrate formation by ice-structuring proteins. Ind. Eng. Chem. Res. 49, 1486–1492 (2010) 81. C.M. Perfeldt, P.C. Chua, N. Daraboina, D. Friis, E. Kristiansen, H. Ramlov, J.M. Woodley, M.A. Kelland, N. von Solms, Inhibition of gas hydrate nucleation and growth: efficacy of an antifreeze protein from the Longhorn Beetle Rhagium mordax. Energy Fuels 28, 3666–3672 (2014) 82. J.-H. Sa, G.-H. Kwak, K. Han, D. Ahn, S.J. Cho, J.D. Lee, K.-H. Lee, Inhibition of methane and natural gas hydrate formation by altering the structure of water with amino acids. Sci. Rep. 6 (2016) 83. S.A. Bagherzadeh, S. Alavi, J.A. Ripmeestera, P. Englezos, Why ice-binding type I antifreeze protein acts as a gas hydrate crystal inhibitor. Phys. Chem. Chem. Phys. 17, 9984–9990 (2015) 84. V.K. Walker, H. Zeng, H. Ohno, N. Daraboina, H. Sharifi, S.A. Bagherzadeh, S. Alavi, P. Englezos, Antifreeze proteins as gas hydrate inhibitors. Can. J. Chem. 93, 839–849 (2015) 85. E.J. Espinosa-Ortiz, B.H. Eisner, D. Lange, R. Gerlach, Current insights into the mechanisms and management of infection stones. Nat. Rev. Urol. 16, 35–53 (2019)

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Chapter 4

Interfacial Gaseous States

4.1 Disjoining Pressure 4.1.1 The Definition We will see in the next chapter that interfacial gaseous states hold a key to the nucleation of clathrate hydrates. Nucleation in the real world is almost exclusively through heterogeneous nucleation for the reasons detailed in Chap. 1. For clathrate hydrate, since the solubility of a typical guest gas in an aqueous phase is much lower than the guest gas content in the clathrate form (see Chap. 3), the availability of the guest gas next to a solid substrate on which heterogeneous nucleation takes place dictates the heterogeneous nucleation probability of the clathrate hydrate. Thus, the physical properties of such surfaces hold the key to the heterogeneous nucleation rates of clathrate hydrates. An interfacial gaseous state refers to any gaseous phase that metastably exists next to a solid surface in a liquid phase (Fig. 4.1) [1–4]. An interfacial gaseous state on a solid substrate in liquid water can readily supply the guest gas to an emerging nucleus of a clathrate phase and hence would dramatically impact the heterogeneous nucleation rate of clathrate hydrates. As we will see later in this chapter, interfacial gaseous states are generally not thermodynamically stable. Therefore, their presence is transient in nature and they are destined to disappear sooner or later. Because of this metastability, interfacial gaseous states are small to begin with and are usually shrinking when they survive long enough to be detected. Unlike macroscopically large bubbles, the small sizes of the interfacial gaseous states render their physical states be dominated by the surface forces exerted by the solid substrate next to them. The central concept in dealing with such situations is the disjoining pressure. There are excellent textbooks on the disjoining pressure, such as by Derjaguin et al. [5] and by de Gennes et al. [6], so here we only cover the essence of the disjoining pressure that is necessary for an understanding of interfacial gaseous states. For © Springer Nature Switzerland AG 2020 N. Maeda, Nucleation of Gas Hydrates, https://doi.org/10.1007/978-3-030-51874-5_4

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Fig. 4.1 Schematic picture of an interfacial gaseous state next to a solid surface in a bulk liquid

Bulk liquid

Interfacial gas

the underlying concepts of the disjoining pressure in surface physics and interfacial chemistry, the readers are referred to standard textbooks in the relevant field [7–9]. The disjoining pressure is defined as the difference in the chemical potential of a molecule in a bulk liquid and that in a thin film and the difference is normalized by the molecular volume. It measures the relative thermodynamic stability of a thin film with respect to a bulk liquid of the same component. Π ≡ −(μfilm − μbulk )/vm

(4.1.1)

The difference in the chemical potential between a molecule in an infinitely sized bulk liquid and one in a thin film arises because the contribution of the interfacial free energy component to the chemical potential cannot be ignored in a thin film. Confusion may arise from the historical difference in the Western and the Russian conventions that define the sign of the disjoining pressure differently. In this book, we will follow the Western convention of Eq. (4.1.1). The sign is the opposite of Eq. (4.1.1) in the Russian convention. In the Western convention, the disjoining pressure of a stable wetting film (μfilm < μbulk ) is defined as positive. Materials flow from where the chemical potential is high to where the chemical potential is low, and as such any depletion of materials from a stable wetting film will be quickly replenished by a flow from the adjacent bulk. Likewise, the disjoining pressure of an unstable film (μfilm > μbulk ) is defined negative in the Western convention.

4.1.2 Phenomenological Descriptions of Disjoining Pressure in a Single-Component System The chemical potential of a liquid molecule can be measured using the vapor pressure. (μfilm − μbulk )/vm = (kT ln P − kT ln P0 )/vm = (kT /vm ) ln(P/P0 )

(4.1.2)

4.1 Disjoining Pressure

Wetting film

85

Non-wetting film Bulk liquid

Fig. 4.2 Schematic picture of three systems of a wetting film, a bulk liquid and a non-wetting film that each coexists with an undersaturated vapor, saturated vapor, and supersaturated vapor, respectively. The three systems can be connected via the vapor phase by opening a valve in between

Here P is the vapor pressure that can coexist with a thin film and P0 is the vapor pressure that can coexist with a bulk liquid. For a condensed film on a wettable solid surface in an undersaturated vapor, the chemical potential is lower the thinner the film and the more undersaturated the vapor. This setting corresponds to a large positive disjoining pressure (Π > 0) in the Western convention by definition. For a simple Van der Waals liquid, the Van der Waals forces across a thin liquid film between a solid wall and a vapor phase are most often repulsive, as we will see later, which gives rise to a positive disjoining pressure. Such a wetting film can coexist with an undersaturated vapor. If such a system is connected via the vapor phase to another system that consists of a bulk liquid and its saturated vapor above it (the left side of Fig. 4.2), materials would flow from where the chemical potential is high to where the chemical potential is low, i.e., so as to thicken the film. As the saturation mounts, the film further thickens and the disjoining pressure of the film gradually diminishes and approaches zero; i.e., ∂Π /∂h < 0 where h is the thickness of the film. For a sufficiently thick film, there is no longer any difference between the chemical potential of a molecule in the film and the chemical potential of a molecule in the bulk liquid and hence there will be no further material transfer. To summarize, Π is a function of the film thickness and for wetting films; Π = Π (h) > 0, ∂ Π/∂h < 0

(4.1.3)

In contrast, when the vapor component does not wet a surface, like water on Teflon, one would need to go beyond saturation (supersaturation) before the liquid can be forced to condense onto the non-wetting surface. This corresponds to a positive chemical potential with respect to the saturated vapor (and the bulk liquid) and, by definition, a negative disjoining pressure. Thus, such a non-wetting film can only coexist with a supersaturated vapor. If such a system is connected via the vapor phase to another system that consists of a bulk liquid and its saturated vapor above it (the right side of Fig. 4.2), materials would flow from where the chemical potential is high to where the chemical potential is low, i.e., so as to evaporate the “film”. Thus, such a non-wetting “film” that has a negative disjoining pressure is unstable and cannot exist in the presence of a bulk liquid.

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Fig. 4.3 Schematic picture of a bubble in a liquid that is pressed against a solid substrate due to buoyancy

Bubble

Bulk liquid

Alternatively, one can start with a bulk liquid (Π = 0, h = ∞) on top of a non-wetting substrate and let the liquid evaporate in an infinitely large volume of undersaturated vapor. As the evaporation proceeds and the liquid thins, the liquid film becomes more and more unstable (μfilm increases and Π becomes negative) and eventually breaks into droplets. The droplets will then keep evaporating until they are all gone. Thus Π is a function of the film thickness and Π = Π (h) < 0, ∂Π/∂h > 0

(4.1.4)

Consider a system in which a bubble is pressed against a solid wall in the bulk of a wetting liquid (Fig. 4.3). The inside of the bubble is filled with a gas and its pressure is always isotropic and uniform, regardless of the shape of the bubble. When the bubble is far away from the solid wall and has a spherical shape, the pressure of the liquid around it is also uniform and lower than that of the bubble by the Laplace pressure [10]. As the bubble approaches the solid wall from below due to buoyancy, the liquid between the solid wall and the approaching bubble resists being displaced because it wets the solid. Once the bubble starts deforming as it is pressed against the wall, the liquid pressure outside the bubble is no longer spatially uniform and depends on the shape of the bubble. A limiting case is two parallel mathematically flat interfaces between the deformed bubble, the thin trapped liquid film, and the solid wall. The pressure in the thin liquid film is the same as the pressure of the gas inside the deformed bubble from the mechanical balance across the flat interface. In contrast, the pressure of the bulk liquid around the bubble is lower than the pressure of the gas inside the bubble by the Laplace pressure because the shape of the bubble–liquid interface away from the solid wall is spherical as depicted in (Fig. 4.3). And the pressure of the gas inside the deformed bubble should still be isotropic throughout the bubble because a gaseous phase cannot have anisotropic pressures. Then, it follows that the pressure inside the trapped, flattened liquid film must be greater than the pressure in the neighboring bulk liquid.

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Here is the problem: the liquid in the wetting film does not drain out to the surrounding bulk liquid despite its greater mechanical pressure than the pressure in the adjacent bulk liquid. To the contrary, one will need to apply even more pressure to squeeze the trapped liquid film out. How can this be? The reason why one needs to apply an excess mechanical pressure to squeeze out a wetting liquid film from between the solid wall and the approaching bubble, which already has a higher pressure than the surroundings, is that the liquid film wets the solid surface and resists being removed. The thin liquid film has a positive disjoining pressure (negative chemical potential) which would keep pulling the liquid in from its surroundings. As the liquid film thins due to the compression, the negative gradient in the disjoining pressure steepens (the chemical potential differential broadens) and the film will pull even more liquid in from its surrounding bulk liquid. The external mechanical pressure applied to the thin liquid film will increase the mechanical pressure and consequently raise the chemical potential of the liquid molecules trapped inside the film. In order to squeeze the trapped liquid film out, the mechanical pressure must exceed the threshold that corresponds to the chemical potential increment that matches the negative chemical potential due to the disjoining pressure. Thus, if one can measure the tipping point when the film first starts to drain, one can deduce what the disjoining pressure inside the film has been at the tipping point. The above considerations apply equally to negative disjoining pressures. The liquid in a non-wetting film (negative disjoining pressure) would easily collapse or drain out with little or even no compression at all, because it has a higher chemical potential than the surroundings. Here the positive excess pressure inside an approaching bubble is sufficient to easily displace the liquid between the wall and the bubble.

4.1.3 The Origin of the Disjoining Pressure We showed the definition and phenomenological descriptions of the disjoining pressure to provide the readers with the conceptual ideas behind the disjoining pressure. But what is the physical origin of the disjoining pressure? When Derjaguin and Obuchov originally studied the attachment of a bubble onto a glass wall that was immersed inside a liquid, they observed that a thin liquid film separated the glass wall from the bubble. The force in the thin liquid film that held the bubble downward against the buoyancy was named disjoining pressure because it “dis-joined” (separated) the solid wall from the gas phase. Such a system can be described schematically in Fig. 4.4 in which one semi-infinite medium is denoted as 1, the other semi-infinite medium as 2 and the thin film in between as 3. Consider a common case in which the medium 1 is a gas (g), the medium 2 is a solid substrate (s), and the medium 3 is a thin liquid film (l). The Gibbs free energy per unit area of a single-component system of this setting can be written as,

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Fig. 4.4 Schematic picture of a thin film (3) trapped between two semi-infinite media (1 and 2)

G = nμ0 (T, P) + γsl + γlg + Y (h)

(4.1.5)

Here, the first term on the right-hand side is the contribution from the bulk, the second and the third terms are the contributions from the two interfaces for which γ sl and γ lg are the specific interfacial free energy of the solid–liquid and the liquid–gas interfaces, respectively, and h is the thickness of the liquid film. The volume of the thin liquid film is h because we are considering unit area here. Then, n = h/vm is the number of moles of the single component where vm is the molar volume of the single component. The fourth term on the right-hand side, Y (h), accounts for all the other effects that have not been accounted for by the first three terms. The classical treatment by Gibbs only treated cases where Y (h) = 0, that is, when there was no overlap of the interfacial zones (no overlap in the range of the surface forces exerted by each interface). Consider a hypothetical process of transferring dn moles of the single component from outside of the system to the thin film that results in an increase in the thickness of the film by dh. From n = h/vm , dh = vm dn and the change in the system Gibbs free energy per unit area is   dG = μ0 dn + (dY (h)/dh)dh = μ0 dn + (dY (h)/dh)vm dn = μ0 + (dY (h)/dh)vm dn

(4.1.6)

The chemical potential of the molecule inside the thin liquid film is a function of its thickness and μ(h) = dG/dn = μ0 + (dY (h)/dh)vm

(4.1.7)

In a limiting case of h → ∞, the chemical potential of the single component in a very thick liquid film should approach that of a bulk liquid, so Y (∞) → 0 and μ(∞) → μ0 . In the other limiting case of h → 0, the Gibbs free energy per unit area should approach that of a dry solid surface, so G(h → 0) → γ sg . Since n also approaches zero in this limit, Eq. (4.1.5) reduces to γ sg = γ sl + γ lg + Y (h). By defining Π = −dY (h)/dh, Eq. (4.1.6) becomes

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    dG = μ0 + (dY (h)/dh)vm dn = μ0 − Π vm dn

(4.1.8)

From Eq. (4.1.7) and Π = −dY (h)/dh;   Π = −dY (h)/dh = μ0 − μ(h) /vm

(4.1.9)

The right-hand side of Eq. (4.1.9) is the same as the general definition of the disjoining pressure shown in Eq. (4.1.1). A limitation in the classical treatment by Gibbs is that, in a multi-phase system that involves a thin film, like the one shown in Fig. 4.4 for which no bulk phase for the region 3 can be defined due to the overlap of the interfacial zones, Gibbs’s surface excess quantities can no longer be defined. We will deal with Gibbs’s surface excess quantities in Chap. 5. For a sufficiently thick film of the phase 3 for which bulk physical quantities can be defined, the Y (h) term becomes zero, and the system reverts back to the classical treatment by Gibbs. In other words, for thin films for which Y (h) = 0, the bulk physical quantities cannot be defined for the phase 3 and, unlike in the bulk, its chemical potential is no longer a function of only P and T but also of h. Another consequence of the overlap of the surface forces is that the interfacial tension can no longer be defined for a thin film because an interfacial tension is defined as the work required to bring a molecule from the bulk to the interface, and yet the bulk is no longer defined for a thin film like the phase 3.

4.1.4 Van der Waals Forces There are many different components of the disjoining pressure that depend on the chemical nature of the component in a given system [5]. Here, we only consider the Van der Waals component for simplicity. The origin of the Van der Waals forces is electromagnetic forces that arise between groups of electrons [10, 11]. The refractive index of a material is a property determined by its electron density and its strength of interactions with photons. Simply put, the higher the electron density and the stronger their interactions with photons, the higher the refractive index. Because of their intimate relationship with the mass density, the refractive index and the density of a material are connected through Lorentz–Lorenz relationship [12]. Both the dielectric function and the refractive index of a material are not “constant” but are functions of the frequency (wavelength), as we saw in Chap. 3. The dispersion relation describes how the dielectric constant and the refractive index of a material vary with the wavelength of light [13]. Likewise, polarizability of a molecule is determined by its electron density and its interaction with photons. It is essentially the molecular counterpart of the dielectric function of a solid material. Like the dielectric function, polarizability is not a constant but is a function of the frequency of the light.

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Of particular importance is that every molecule and every material has an intrinsic frequency called the plasma frequency above which the molecule or the material becomes transparent to light [13]. Simply put, the oscillation of the electric field of light is too fast for an electron to respond or “keep up” above this frequency, and ceases to interact with photons that have a greater frequency. We call a material “transparent” when light goes through it without interacting with the light. When a molecule is polarized by light, a dipole, quadrupole, and weaker higher order ‘poles’ are induced that can only be described in the form of tensors [14]. Since the dipole is generally by far the strongest, the higher order poles can usually be neglected for a first approximation. In an ensemble of multiple molecules, such induced dipoles generate electromagnetic forces in that the like charges repel and unlike charges attract with each other. This is the source of the Van der Waals forces. The Van der Waals forces are thus ubiquitous in that they can arise in every material and in every molecule that has an electron. The Van der Waals forces between two molecules are short-range forces whose potential energy rapidly diminishes with the inverse of the sixth power of the distance [10]. The pre-factor (proportionality constant) in the Van der Waals force law is a complex function of the polarizability and the dielectric function of the molecules in question. W (r ) = −C/r 6

(4.1.10)

We are more interested in the Van der Waals forces between condensed matters. Since both the interacting objects and the medium in between consist of atoms or molecules, the pairwise summation of the Van der Waals forces between all the constituting atoms and molecules leads to various forms of Van der Waals force laws that depend on the shape of the interacting objects [10]. For two parallel flat surfaces of two semi-infinite media interacting across a thin film, the Van der Waals potential energy per unit area diminishes with the inverse of the second power of the distance between the two surfaces (thickness of the film), h. The Van der Waals forces between macroscopic objects are thus much longer ranged than those between two molecules. The pre-factor (proportionality constant) of such a Van der Waals force law is called a Hamaker constant and customarily denoted by A instead of C [15, 16]. W (h) = −A/12π h 2

(4.1.11)

From the origin of the Van der Waals forces described above, one would expect that the Van der Waals forces, and the Hamaker constants, would be intimately related to the dielectric properties of a molecule or a material of interest. This is indeed the case and the relevant property is the refractive index. Of our particular interest in relation to the disjoining pressure is the Van der Waals force across a thin film that is sandwiched between two semi-infinite media, like the one shown in (Fig. 4.4). Useful methods of calculating a Hamaker constant from experimentally measurable dielectric functions have been developed [11, 17]. However, one is often as much interested in the sign of a Hamaker constant as its numerical size that determines

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the strength of the van der Waals forces, because the sign determines whether the Van der Waals forces in a particular setting is attractive or repulsive. The situation is somewhat analogous to the base vectors of an orthogonal series in a Hilbert space in which we are more interested in the “direction” of each vector than its length (see Chap. 2). If one is only required to know the sign of a Hamaker constant, then a very simple relationship has been established for the van der Waals forces between two semi-infinite media across a thin film: the Van der Waals forces between 1 and 2 across 3 in Fig. 4.4 is repulsive when n1 > n3 > n2 or n1 < n3 < n2 and attractive otherwise [10, 11]. A thin film is thus thermodynamically stable when the Van der Waals force is repulsive across the thin film. This is to say that the Van der Waals component of the disjoining pressure is positive; Π = Π (h) > 0. Conversely, the film is thermodynamically unstable when the Van der Waals force is attractive across a thin film. This is to say that the Van der Waals component of the disjoining pressure is negative; Π = Π (h) < 0. These considerations are important not only in the wetting properties of materials, including the pre-melting of ice and quasi-liquid layers on clathrate hydrates [18], which dictate the adhesion, cohesion, and agglomeration of clathrate hydrate particles, but also in more practical matters of sample preparations in nucleation studies [19].

4.2 Interfacial Gaseous Layers 4.2.1 Introduction The availability of the guest gases and their physical states (i.e., whether the guests are in a dissolved state or form a separate gaseous phase) next to a solid substrate have direct impacts on the heterogeneous nucleation of clathrate hydrates. Therefore, the physical properties of interfacial gaseous states on a solid surface in water are of great interest to the nucleation of clathrate hydrates. It turned out that a variety of interfacial gaseous states may exist or coexist in water under suitable physicochemical conditions. Given that a distinct attribute of a gas phase is that all types of gases are miscible with each other at all proportions, it is surprising that multiple gaseous states can coexist in an aqueous media, which presents a surprisingly rich domain [20]. It has long been known that formation of a gaseous layer can be thermodynamically favorable for sufficiently rough surfaces as shown in Fig. 4.5, left panel. The interfacial free energy of the system can decrease by the formation of an interfacial gaseous layer when (1) the water–gas interfacial area is sufficiently smaller than the gas–solid interfacial area and (2) the gas–solid specific interfacial free energy is much lower than the water–solid specific interfacial free energy. Wetting on such rough and/or heterogeneous surfaces has a long history since the days of Wenzel [21] and Cassie and Baxter [22].

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Water

Water

Water

Water

Gaseous layer Gaseous layer

Fig. 4.5 Schematic illustration of how the formation of a gaseous layer can lower the free energy of a system on a sufficiently rough surface (left), which cannot occur on a smooth surface (right)

In contrast, it has been known that the free energy per unit area in which water is in contact with a smooth solid surface always increases when a gas layer forms in between, regardless of the contact angle of water on the solid surface. The reason is (1) the large specific surface free energy of water (about 72 mJ/m2 ) compared to the specific interfacial free energy between water and a hydrophobic surface (typically 50 mJ/m2 ), and (2) the formation of a gaseous layer doubles the number of interfaces per unit area in the system and the gas–solid interface has a relatively small but nevertheless positive specific surface free energy, as shown in Fig. 4.5, the right panel. However, that formation of a gaseous layer on a smooth solid surface is always thermodynamically unfavorable does not preclude temporary formation of a thermodynamically metastable gaseous layer that may be detectable when the lifetimes of such entities exceed typical experimental timescales. In fact, temporary formations of metastable gaseous layers turned out to be ubiquitous and hence of general phenomena that have wide-ranging implications.

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Temporary formations of metastable gaseous layers are heterogeneous nucleation processes. Heterogeneous nucleation of bubbles on a solid surface is always energetically favorable to homogeneous nucleation of bubbles in the bulk of a liquid [23]. Of course, such heterogeneous nucleation or cavitation could also occur on small, sub-micrometer sized contaminant particles in the liquid which would be hard to distinguish from homogeneous nucleation experimentally. In liquid water, homogeneous nucleation of bubbles or cavitation has been known to become virtually irrelevant at normal temperatures that are far below the critical point [23]. Formation of such thermodynamically metastable interfacial gaseous states is the subject of this section.

4.2.2 Experimental Observations There are several established methods of inducing a thermodynamically metastable interfacial gaseous state and, as might be expected, each of them involves building up of supersaturation of a gas next to a foreign solid substrate [1, 2, 24–35]. One such method involves displacement of an organic solvent that has a higher solubility of a gas of interest than water (like ethanol) with water. This method is called the “solvent exchange” method. We consider a setting in which water displaces ethanol in a channel bounded by hydrophobic surfaces. A typically large advancing contact angle of water and a small receding contact angle of ethanol on a hydrophobic surface means that the displacement will initially be incomplete and would leave a thin layer of ethanol on the hydrophobic surface. Ethanol and water are fully miscible, so the ethanol initially left on the hydrophobic surface will then quickly dissolve into the water that has displaced the ethanol. A gas (like nitrogen) is more soluble in ethanol than in water so the dissolution of ethanol into the surrounding water will induce a local supersaturation of the gas on the hydrophobic surface. This phenomenon can be regarded as a special case of spontaneous emulsification we covered in Sect. 1.3 in that the “solute” here is a gas instead of an oil and the nucleation of the gas is assisted by the presence of a solid wall (unlike in Sect. 1.3, heterogeneous nucleation takes place here). Another method of inducing a thermodynamically metastable interfacial gaseous state involves displacement of cold water (≈0 °C) with warm water (≈40 °C) that has a lower solubility of a gas than the cold water [20, 30]. Yet another method involves generation of a temperature gradient by pre-heating or in situ heating of the substrate [20, 30]. For an electrically conducting surface, an application of an appropriate bias voltage can generate interfacial nanobubbles on the substrate by an electrochemical reaction [20, 27]. In each case, the generated interfacial gaseous states were flat in their shapes; their lateral dimensions were sub-micrometers and only their height were in the nanometer scale [20, 34]. The induced gaseous entities can be removed from the substrate by injection of pre-degassed water or by direct degassing using a moderate vacuum (≈0.1 atm) [20, 33].

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Ethanol is a solvent that is commonly used for cleaning of glassware and metal vessels in a laboratory. The residual ethanol is then typically rinsed with water, which is in effect an unintended “solvent exchange” process on the surface of the cleaned glassware or metal vessels. In addition, unintended and therefore uncontrolled temperature gradients are common in daily life. We thus suspect that thermodynamically metastable interfacial gaseous states are far more commonplace than people realize. Such unintended and uncontrolled supersaturation of air and the surprising ease with which interfacial gaseous states can form have broad implications. For example, even though ethanol and water are mutually miscible at all proportions, rinsing ethanol on a solid wall with water may not result in the same surface as rinsing a dry solid wall with water after completely getting rid of ethanol in a vacuum oven. Likewise, temperature gradients induced by a detection instrument could become another source of generation of interfacial gaseous states. For example, Atomic Force Microscopy (AFM) has been a common instrument used for the study of interfacial gaseous states on smooth solid substrates. The proximity of the electronics of an AFM and the heating therefrom could be a matter of significance. One of the most surprising aspects of the interfacial gaseous states is that more than one form of gaseous states can coexist on an atomically smooth (within a cleavage step of) hydrophobic Highly Ordered Pyrolytic Graphite (HOPG) [1]. Flat, quasi-twodimensional gaseous layers (so-called “micro gas pancakes”) apparently coexisted with more common spherical-cap-shaped interfacial nanobubbles at a sufficiently high supersaturation of the gas [1]. An AFM detected images of “nanobubble–micro gas pancake composites” where some of such interfacial nanobubbles apparently sat on top of micro gas pancakes [1]. The height of the micro gas pancakes was limited to the maximum of about 5 nm (and the average of less than 2 nm) whereas their lateral dimensions extended from several hundred nanometers to tens of micro meters [1]. So their aspect ratio was about three orders of magnitude. The contact angle formed by an interfacial nanobubble on a micro gas pancake (in a composite) was estimated to be about 155° (that was measured through the aqueous phase). This value is close to the contact angle of an interfacial nanobubble that was directly sitting on a bare HOPG surface [34, 36]. The formation of micro gas pancakes was influenced by several factors (type of the solvents, the flow rates of the fluids, the temperature of the system, the level of degassing, and the concentration of the ethanol used for the “solvent exchange” procedure) [1]. There was a trend that the surface coverage of the micro gas pancakes increased with the solubility of the gas in the solvents used for the solvent exchange protocol. The temperature of the solvents used in the solvent exchange protocol also significantly influenced the surface coverage of the micro gas pancakes. The surface coverage of the micro gas pancakes and the size of the interfacial nanobubbles both increased with increasing temperature of the solvent. These results are expected from the mechanism behind the solvent exchange protocol. For example, under a partial pressure of 700 mmHg, the solubility of nitrogen in ethanol is 3.611 × 10−5 in mole fraction, and the solubility of nitrogen in water is 0.1274 × 10−5 in mole fraction, both at 20 °C. The solubility of nitrogen increases to 3.639 × 10−5 in ethanol and decreases to 0.1047 × 10−5 in water at 40 °C [37, 38]. So the gap in the

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solubility of nitrogen gas in ethanol and in water increases with heating and hence the supersaturation of nitrogen after the solvent exchange protocol is expected to increase with the temperature of the solvents. Even though “nanobubble-pancake composites” can metastably exist, the two gaseous entities are not equivalent in that the driving force required to induce them appeared different. The formation of micro gas pancakes required a higher supersaturation of the gas than the formation of interfacial nanobubbles. Other notable characteristics of these systems were [1] (1) removal of the interfacial nanobubbles that had been induced required a greater reduction of supersaturation of the gas than that of induced micro gas pancakes (interfacial nanobubbles were more stable than micro gas pancakes and could exist at a lower supersaturation of the gas), (2) the two forms of the interfacial gaseous entities could coexist at a sufficiently high supersaturation of the gas where one or more of interfacial nanobubbles sat on top of a micro gas pancake, (3) a micro gas pancake could spontaneously coalesce with another nearby micro gas pancake over time, (4) after the coalescence of two neighboring micro gas pancakes which each had an interfacial nanobubble on top, the larger nanobubble grew at the expense of the smaller one, (5) the rate of the Ostwald ripening between two such interfacial nanobubbles that sat on top of the same micro gas pancake was greater than that between two interfacial nanobubbles that sat on two separate micro gas pancakes. An interesting question is whether a micro gas pancake and an interfacial nanobubble above it could have been separated by a thin film of water or not. We note that a thin water film between a micro gas pancake and an interfacial nanobubble on top of it would be very unstable because (1) its thickness must be very thin (below the spatial resolution of an AFM and must be thinner than the height of the micro gas pancake itself) and (2) the van der Waals forces across a thin water film between two gaseous phases would be attractive. The Hamaker constant for air–water–air is 3.7 × 10−20 J, which is even larger than the Hamaker constant for water–hydrocarbon–water of about 0.4 × 10−20 J [39]. In other words, an interfacial nanobubble that appeared to sit on top of a micro gas pancake could have been a “bump” of a (deformed) micro gas pancake [2]. Just like the gas inside a balloon can be deformed to a flat layer and a bump when the balloon is pressed against a flat wall by non-uniform forces, interfacial gas could be deformed by appropriate surface forces into a flat micro gas pancake and a bump that appeared to be an interfacial nanobubble. Since the aqueous phase was supersaturated with the gas and the solid wall was non-permeable to the gas, any gas trapped in between the solid wall and the supersaturated aqueous phase was somewhat similar to the gas trapped inside a deformed balloon. Then, perhaps some excess gas that could not be accommodated inside a micro gas pancake could have become a bump that looked like an interfacial nanobubble. However, interfacial nanobubbles were found to form at a lower supersaturation of the gas than the micro gas pancakes. Then, if this scenario were the case, the “bump” must have formed first and the flat layer must have spread from the “bump” afterward as the supersaturation of the gas further mounted [2].

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The flat shape of a micro gas pancake means that a micro gas pancake exposes a larger gas–aqueous interfacial area than an interfacial nanobubble for a given amount of gas. At a first glance, the free energy cost of forming a micro gas pancake appears greater than that of forming an interfacial nanobubble. Then, the chemical potential of a gas molecule inside a micro gas pancake must be higher than that inside an interfacial nanobubble, which in turn should result in mass transfer of the gas from the underlying micro gas pancake to the interfacial nanobubble above it. The end result of such a process would be a total annihilation of micro gas pancakes and growth of the interfacial nanobubbles. That this expectation has not materialized in experimental observations suggests that another factor must be in play. This point will be examined in the next section.

4.2.3 Thermodynamic Considerations That some interfacial nanobubbles coexisted with micro gas pancakes for extend time periods suggests that the chemical potential of a gas molecule inside an interfacial nanobubble cannot be very different from that inside a micro gas pancake. Then we may assume that such a system in which interfacial nanobubbles and micro gas pancakes can metastably coexist (that is not in the global free energy minimum) is in a temporary or a local equilibrium. We may then apply the principle of virtual work to predict the direction of a potential change of the system with an infinitesimal change in a parameter of interest. Once we assume a quasi-static condition, we may analyze the chemical potential of a gas molecule inside an interfacial nanobubble in terms of the Laplace pressure and the chemical potential of a gas molecule inside a micro gas pancake in terms of the disjoining pressure. Then the chemical potential of a gas molecule inside an interfacial nanobubble and that inside a micro gas pancake, at a constant temperature, become functions of the saturation of the gas in the surrounding aqueous phase and of their respective characteristic sizes (the radius, r, for the interfacial nanobubbles and the thickness, h, for the micro gas pancakes). μnanobubble = μnanobubble (P/P0 , r )

(4.2.1)

μmicropancake = μmicropancake (P/P0 , h)

(4.2.2)

Here μ is the chemical potential, P is the actual pressure of the gas and P0 is the saturation pressure of the gas in water. Laterally spreading micro gas pancakes were frequently observed experimentally but their thickness could not be measured accurately (no more accurate than to place an upper bound of about 5 nm). Now another thought experiment may be in order; what would happen if one added an infinitesimal amount of gas to the system? If the disjoining pressure had been absent, the energetically least taxing way

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to accommodate the excess amount of gas would have been to slightly increase its thickness. Given the very large aspect ratio (i.e., flat shape) of a micro gas pancake, the increase in the total interfacial area when a given amount of excess gas is added to the system would be minimized by increasing the thickness of the micro gas pancake by a little bit. Then, the experimentally observed lateral spreading of a micro gas pancake, which is contrary to the expectation, suggests that there must be a very high free energy cost to increasing the thickness. This is to say that the disjoining pressure must be a very steep function of the thickness of the gas film. The regular occurrence of the Ostwald ripening between two interfacial nanobubbles that had been sitting on a single micro gas pancake first suggests that the gas that was feeding the growing nanobubble must have come from the shrinking one. Then it follows that the thickness of the underlying micro gas pancake could not have been increasing (if the underlying micro gas pancake consumed the gas supplied by the shrinking nanobubble, then where did the gas that was feeding the growing nanobubble could have come from?). Second, the same observation suggests that the thickness of the underlying micro gas pancake could not have been decreasing during the Ostwald ripening either (if the gas that was feeding the growing nanobubble was supplied by a shrinking micro gas pancake, then why did the smaller nanobubble shrink at all?). If the thickness of the micro gas pancake can neither be increasing nor decreasing, then we may conclude that its thickness must have been constant during the Ostwald ripening between the two interfacial nanobubbles that had been sitting on top of it. The above thermodynamic considerations both suggest that a micro gas pancake must have a very well-defined thickness. In other words, the disjoining pressure of a micro gas pancake must be a very steep function of its thickness on both sides of said well-defined thickness, with a minimum confined to a very narrow range (a step function that has two steps over a very narrow range centered around the experimentally observed thickness). Unfortunately, we have no idea as to a potential mathematical format of the disjoining pressure of a micro gas pancake, let alone as a function of its thickness. Nevertheless, the qualitative implications of the above thermodynamic considerations are highly significant; they not only imply that (1) the thickness of a given micro gas pancake must remain constant with time for as long as the quasi-static condition is maintained, but also imply that (2) all micro gas pancakes in the same system (under the same supersaturation of the gas and were connected to each other through the surrounding aqueous media) must have the same thickness. We can conclude as such despite our inability to determine a mathematical expression of the disjoining pressure of the interfacial gaseous film or to accurately measure its thickness, which illustrates the power and the elegance of thermodynamics. Thermodynamics cannot, however, elucidate the underlying molecular mechanisms that limited the thickness of the micro gas pancakes to such a well-defined thickness. As we saw in the previous (Sect. 4.1), the van der Waals forces across a micro gas pancake (solid substrate–gas film–liquid) is expected to be attractive because the refractive index of a gas must be lower than that of the underlying solid substrate or that of the liquid on top [10, 11]. Such attractive van der Waals forces and the negative

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disjoining pressure are expected to suppress the growth of a gaseous film and limit its growth to a very small thickness. In fact, we saw that the same mechanism is likely in play in surface freezing where the thickness of a frozen normal alkane film on top of its liquid bed remains monomolecular over a few Kelvins above the bulk freezing point (Sect. 1.4). The experimental results in a sufficiently supersaturated system showed that interfacial nanobubbles often sat on top of the micro gas pancakes whereas interfacial nanobubbles rarely sat directly on a bare HOPG substrate. It is conceivable that interfacial nanobubbles (the “bumps”) formed on a bare HOPG substrate first while the supersaturation was still building up, which in turn then provided the nucleation sites for subsequent formations of micro gas pancakes as the supersaturation mounted still higher. This is to say that the micro gas pancakes could have nucleated from existing interfacial nanobubbles that had formed first. Unfortunately, due to the typically poor time resolution of an AFM, we could only observe the end result of the interfacial nanobubbles sitting on top of the micro gas pancakes. That interfacial gaseous states can form with ease, not only on a rough surface but also on a molecularly smooth surface under suitable conditions, and the energy barrier to such nucleation appears surprisingly low, has important implications to heterogeneous nucleation of clathrate hydrates. Given that typical surfaces involved in a clathrate hydrate study are neither molecularly smooth nor carefully controlled to avoid generation of interfacial gaseous states, their presence is almost certain. In particular, interfacial gaseous states hold the key to the understanding of the long-standing mystery of the so-called memory effect in the nucleation of clathrate hydrates, as we will see in Chap. 5.

4.3 Is the Surface of Gas Hydrates Dry? 4.3.1 Introduction The wetting properties of clathrate hydrate surfaces are not a part of interfacial gaseous states, however, the underlying physical concepts are very similar and we cover the subject at this stage. A pre-melting layer, also known as a surface melting layer, refers to a quasi-liquid layer that exists at the surface of a solid below its melting point, T m [40, 41]. An important point here is that such quasi-liquid layer is thermodynamically stable. The phenomenon itself has been known since the days of Faraday and has since been found for many solids [40–44]. Large amounts of efforts have been specifically directed to the study of pre-melting of ice because of its wideranging environmental implications. For example, the reason that ice is slippery and one can skate on ice is because a thin pre-melting layer of liquid water acts as an effective lubricant. The central question in this section is whether the same phenomenon occurs to the surface of clathrate hydrate or not. Whether the surface of a clathrate hydrate crystal

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is coated with a quasi-liquid layer or not is an important question that influences the adhesion and cohesion forces of clathrate hydrate crystals. Practically, these factors determine the agglomeration behavior of clathrate hydrate slurries in flow lines and its transportability. They also impact the efficacy of anti-agglomerates we covered in Chap. 3. We posed this question in a 2015 review article [18], and its key points, together with supplementary information, will be produced in this section. Very recently, there has been a new development in the pre-melting of ice that the quasi-liquid layer is not a continuous “layer” that suggests complete wetting of liquid water on ice, as earlier thought, but rather consists of patches of droplets of liquid water, which is suggestive of partial-wetting nature [45]. This rather surprising new finding that liquid water does not completely wet ice will shed new light to the topic, as we will see below. Pre-melting of ice and the similarities between clathrate hydrates and ice suggest that the surface of a clathrate hydrate crystal is also likely coated with a quasi-liquid layer. However, there is an important a priori difference between pre-melting of ice and pre-melting of clathrate hydrate, if it indeed occurs. For pre-melting of ice, the quasi-liquid layer is pure liquid water and thus is the same chemical compound as ice: H2 O. In contrast, dissociation of clathrate hydrate forms liquid water and a guest gas. Since the solubility of a guest gas in liquid water is much lower than that can be accommodated in the clathrate form, most of the dissociated guest gas will phase separate from the aqueous phase and form a separate gaseous phase. Then, the quasiliquid layer that coats the surface of a clathrate hydrate crystal, if existed, would be almost pure water and have a different chemical composition than the underlying clathrate hydrate crystal. Such difference might not sound significant at first, but could be of significance especially with respect to adhesion and cohesion between clathrate polycrystals and particles. Yet another thought experiment may be in order here. When two blocks of ice cubes are brought together below the melting point, the pre-melting layers on each ice cube merge when contacted and form a single continuous liquid film. Such liquid film will fill any gap between the two ice cubes that may have existed due to the surface roughness. Then, after the contact, the merged water film will no longer be exposed to a gas phase. The salient point here is that a second phase, be it a gaseous phase or a foreign solid wall, is essential for pre-melting of ice to take place. Without one, the water film will freeze and form one large merged block of ice cube. For clathrate hydrate such as methane hydrate, pre-melting, if occurred, would not “melt” the top layer to form a thin film of liquid methane aqueous solution. The solubility of methane in water is simply too low to accommodate all the methane gas that has formed by the pre-melting. The net result of pre-melting of methane hydrate would then be the formation of a thin film of almost pure water that has a different composition than the solid phase from which it originated, and methane gas will readily escape from the pre-melting layer to an open space. Then, unlike ice, two blocks of methane hydrate would not merge to form a single block of methane hydrate when brought together because the liquid water trapped between the two blocks of clathrate hydrate lacks the amount of methane that is required to re-form

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methane hydrate. Methane gas that has escaped can no longer easily diffuse back into the thin film of pure water in the narrow gap between the two blocks after they have been brought together. Consequently, the cohesion force between two methane hydrate blocks would be many orders of magnitude smaller than that between two ice blocks. It is thus not immediately clear if pre-melting of clathrate hydrate is thermodynamically favorable or, if it were, if its physical properties are similar to that on ice. In the literature, reports on pre-melting of clathrate hydrates are scarce. Aman et al. experimentally inferred the presence of a quasi-liquid layer on the surface of cyclopentane hydrate [46]. Yan et al. [47] and Ding et al. [48] reported the presence of quasi-liquid layers on the surface of clathrate hydrates from molecular dynamics simulations. In particular, Ding et al. specifically reported an occurrence of pre-melting on the surface of methane hydrate from molecular dynamics simulation [48]. Maeda et al. found that the cohesion hysteresis between two cyclopentane hydrate particles just below the thermodynamic dissociation temperature was vanishingly small [49]. Adhesion hysteresis or cohesion hysteresis, not adhesion force or cohesion force itself, is highly correlated to friction forces [50]. Thus, Maeda et al.’s finding suggest that the friction force between two cyclopentane hydrate particles is likely very small. Such small friction forces are indicative of a surface coated with a quasi-liquid layer that may act as an excellent lubricant (like what occurs to ice). In short, circumstantial evidence points to the existence of a quasi-liquid layer on the surface of clathrate hydrate, at least for temperatures just below the thermodynamic dissociation temperature of the clathrate hydrate. Below we examine the thermodynamic basis of pre-melting.

4.3.2 Thermodynamic Basis of Pre-melting An important conclusion of Sect. 1.4 is that the specific interfacial free energy of either side of a thin film may be adequately approximated by the specific interfacial free energy of two corresponding semi-infinite media. This approximation corresponds to neglecting the disjoining pressure of the thin film, as we saw in Sect. 4.1, in particular, Eq. (4.1.5). We continue to rely on this approximation for now. We start from the basics. The internal energy, U, arises from the bonding between atoms and molecules and it can be lowered by a formation of a crystalline structure. In contrast, the entropy, S, favors disorder and hence formation of an ordered crystalline structure increases the entropic component of the free energy. Since the entropy component of the Gibbs free energy, G, is −TS, the impact of the entropy component on G increases with temperature, T. Thus, there is a point in T above which the disorderly liquid phase becomes favorable and below which the ordered crystalline phase becomes favorable. This point defines the melting point, T m . On a surface of a crystalline material or a disordered liquid, the reduced numbers of the bonds of atoms or molecules result in a higher U compared to that in the bulk of the same material or the liquid bed at a given T. This reduced average bond numbers of atoms or molecules at the surface compared to that in the bulk, and the consequent

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increase in U (energy is required to break bonds), is the source of the interfacial free energy, which is the free energy required to bring atoms or molecules from inside of a bulk to a surface. Each material has a specific value for this interfacial free energy when normalized to unit area, which is the specific interfacial free energy, γ . A disordered liquid has reduced bonding density than an ordered, crystalline bulk solid of the same material. The extent of the reduction depends on the number of chemical bonds per atom or molecule that would be broken as a result of the melting. In a hexagonal close-packed structure, for example, each atom or molecule has twelve nearest neighbors. These twelve bonds will be broken when the crystal melts. An atom or molecule in the bulk of the resulting liquid has less than twelve nearest neighbors that will decrease with further heating. In contrast, an atom or molecule on a mathematically flat and smooth surface of a material has, on average, already lost half of its bonds when it has come to the surface from inside of the bulk material. On the surface of a hexagonal close-packed structure, for example, each atom or molecule has on average six nearest neighbors. These six bonds will be broken when the crystal melts. An atom or molecule on the surface of the resulting liquid will have less than six nearest neighbors that will decrease with further heating. A salient point here is that more bonds will be broken in the bulk of a material than on the surface of the same material as the material melts, unless the molten liquid retains unusually high degrees of structural order. Straight-chain hydrocarbons are one such unusual material as we saw in Sect. 1.4. It follows that (1) the change in U per mole of atoms or molecules upon melting on the surface of a material is smaller than in the bulk of the same material (i.e., U surface < U bulk ), (2) the specific surface free energy of a solid is higher than that of the liquid of the same material (i.e., γ lv < γ sv ), and (3) the crossover T above which melting becomes favorable is lower on a surface than in the bulk (i.e., T surface, m < T bulk, m ). For these reasons, pre-melting is observed for many solids [44]. These relationships are schematically illustrated in Fig. 4.6.

Fig. 4.6 Schematic illustration of changes in the internal energy, the entropy, and the interfacial free energy. Image adapted from Fig. 1 of Ref. [18]

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The left panel of Fig. 4.6 shows that, at T m , the solid phase and the liquid phase can coexist in the bulk because their two free energy components are balanced. On the surface, in contrast, the two free energy components are not balanced at T m , because the molecules would lose the same amount of entropy as in the bulk by freezing but do not lose as much bonding energy as in the bulk because not all the bonds will be formed at the surface when the bulk of the material freezes. Consequently, the liquid phase is favored at the surface, at and slightly below T m , and remains so until T becomes so small that T S, which is smaller than T m S, becomes equal to U surface . This is the thermodynamic source of pre-melting. The right panel of Fig. 4.6 schematically shows the relative positions of the four energy levels of the surface liquid, the bulk liquid, the surface solid, and the bulk solid. For a hexagonal close-packed structure, for example, an atom in the bulk solid has 12 bonds and an atom at the surface of the solid has 6 bonds. In a special case that each atom on average loses 2/3 of its bonds upon melting, an atom in the bulk liquid will have 4 bonds and an atom at the surface of the liquid will have 2 bonds. Consequently, U surface, liquid > U bulk, liquid > U surface, solid > U bulk, solid (i.e., −2 > −4 > −6 > −12, where the zero energy level corresponds to a free atom without any bond), as shown in the figure. For a material whose molten liquid retains unusually high number densities of the bonds (retains high degrees of structural order), the order of the middle two levels may be altered. For example, in a special case that each atom in a hexagonal close-packed crystal on average only loses 1/3 of its bonds upon melting, an atom in the bulk liquid will have 8 bonds and an atom at the surface of the liquid will have 4 bonds. Consequently, in U surface, liquid > U surface, solid > U bulk, liquid > U bulk, solid (i.e., −4 > −6 > −8 > −12, where the zero energy level corresponds to a free atom without any bond). From now on, we only consider the former case. At a temperature below T m , the solid phase is favored to the liquid phase for both the bulk and the surface if it were not for the entropic component. The inclusion of the entropic term would lower the free energy of both the surface liquid and the bulk liquid by T S, which is somewhat smaller than T m S, because T is supposed to be lower than T m . For small subcoolings, T S can be smaller than U bulk but still can be larger than U surface (the left panel of Fig. 4.6). Then, in the right panel of Fig. 4.6, the inclusion of the T S term will bring the free energy of the surface liquid down to below the free energy of the surface solid but still leave the free energy of the bulk liquid above that of the bulk solid. In our usual favorite scale of the rather artificial example of the hexagonal close-packed crystal, this situation corresponds to T S being larger than 4 (i.e., a shift from −2 to −6) but smaller than 8 (i.e., a shift from −4 to −12). Thus, the solid phase can be more stable in the bulk but the liquid phase can be more stable at the surface at the same temperature for small subcoolings (i.e., pre-melting is favored). For a more quantitative treatment of the subject, the thickness dependence of the pre-melting layer will need to be accounted for [40, 41]. A qualitative conclusion is that the thickness of the pre-melting layer diminishes with subcooling as the free energy cost per unit mass of retaining a liquid phase progressively increases with subcooling.

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Fig. 4.7 Schematic illustration of pre-melting on the surface of clathrate hydrate. Image adapted from Fig. 2 of Ref. [18]

Now we consider the surface of a clathrate hydrate crystal in an excess guest composition (see Chap. 3). Dissociation of a clathrate hydrate will form liquid water and the guest gas that has been trapped in the clathrate structure. The guest gas will then become indistinguishable from the surrounding bulk gas medium, and the end result will be just an almost pure liquid water film on the surface of the clathrate hydrate in an atmosphere of the guest gas. The formation of such a water film newly introduces an extra interface, as shown in Fig. 4.7. Thus the change in the interfacial free energy costs per unit area when the surface of a clathrate hydrate crystal melts is G surf = γgw + γhw − γgh

(4.3.1)

where the subscripts g, w, and h refer to gas, water, and hydrate, respectively. Among the three specific interfacial free energy terms, γ gw is well known and about 72 mJ/m2 , depending on the temperature and pressure [7]. In contrast, γ hw and γ gh are difficult to measure and consequently their estimates vary in the literature. Still, some estimated values of γ hw have been reported. Uchida et al. reported the specific clathrate hydrate–water interfacial tension values of 17 ± 3, 14 ± 3, and 25 ± 1 mJ/m2 for CH4 hydrate, CO2 hydrate, and C3 H8 hydrate, respectively [51]. Anderson et al. reported the specific clathrate hydrate–water interfacial tension values of 32 ± 3 and 30 ± 3 mJ/m2 for CH4 hydrate and CO2 hydrate, respectively [52]. Seo et al. reported the specific clathrate hydrate–water interfacial tension values of 39 ± 2 and 45 ± 1 mJ/m2 for C2 H6 hydrate and C3 H6 hydrate, respectively [53]. Using classical crystallization theory, Zhang et al. reported the specific clathrate hydrate– water interfacial tension value of 9.3 mJ/m2 for CO2 hydrate [54]. In contrast, Aman et al. estimated the specific clathrate hydrate–water interfacial tension value of 0.32 ± 0.05 mJ/m2 for cyclopentane hydrate [55]. For comparison, the specific interfacial free energy between ice and water has been reported to be in the range of 28–33 mJ/m2 [56–59]. These estimates may appear surprisingly high given that ice is supposedly hydrophilic. Nevertheless, the recent finding that the quasi-liquid layer of water on ice is not a continuous “layer” but

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rather consists of patches of droplets [45] is consistent with the larger than expected specific interfacial free energy values between ice and water. Clathrate hydrates are similar to ice in many aspects [60], however, the specific interfacial free energy between a clathrate hydrate and water is expected to be somewhat higher than that between ice and water because most guest gases are hydrophobic. In other words, the specific interfacial free energy between hydrate and water is expected to be greater than 33 mJ/m2 . The broad similarity between γ ice-w and γ hw suggests that the number of the bonds that will be broken to create a unit area of new surface is expected to be similar between ice and clathrate hydrates. This similarity may be expected given that (1) the structures of ice and clathrate hydrates are similar and the similarity includes the close lattice matching between ice and clathrate hydrate, and (2) the guest molecules trapped in a clathrate hydrate are not chemically bonded to water and only the chemical and the hydrogen bonds among the water molecules would contribute to the interfacial free energy. The specific surface free energy of clathrate hydrate, γ gh (for the clathrate hydrate– gas interface), is experimentally difficult to measure due to surface roughness and other experimental complications, and no estimate has been reported in the literature to date. To be fair, we note that the specific surface free energy of ice has also been difficult to measure. An estimate reported is around 130 mJ/m2 [61] but the value varies wildly depending on the experimental techniques used. For example, a vacuum cleavage of ice yielded a very high specific surface energy of ice that is over a thousand mJ/m2 , which has been attributed to electrostatic charging of the cleaved ice surfaces [62]. If we can assume that γ gh of a clathrate hydrate surface is similar to that of ice (≈130 mJ/m2 ), and noting that the specific interfacial free energy between ice and water is about 30 mJ/m2 and that the surface tension of water is 72 mJ/m2 , then Eq. (4.3.1) yields Gsurf ≈ −28 mJ/m2 , which is negative. Even if we assume a somewhat higher specific interfacial free energy between clathrate hydrate and water of 40 mJ/m2 , Gsurf is still negative: ≈ –18 mJ/m2 . Gsurf will become positive only when the specific interfacial free energy between clathrate hydrate and water approaches 60 mJ/m2 , which would be higher than the specific interfacial free energy between a typical oil and water. On the other hand, if the specific surface free energy of clathrate hydrate, γ gh , is much lower than that of ice of 130 mJ/m2 then Gsurf could still become positive for a realistic specific interfacial free energy between clathrate hydrate and water. Pre-melting is expected to occur when Gsurf is sufficiently negative that offsets the free energy cost of dissociation of clathrate hydrate, Gfus , below the thermodynamic dissociation temperature. These analyses suggest that pre-melting of clathrate hydrates is likely, at least for temperatures close to the thermodynamic dissociation temperature. As detailed in [18], the Gsurf gain is proportional to the surface area whereas the Gfus penalty is proportional to the mass or volume of the clathrate hydrate to be dissociated as a result of pre-melting. Gfus becomes proportional to the thickness of the film if we only consider a unit surface area. We may denote the number density of water as ρ and the thickness of the film as h. Then the volume of the pre-melting film on ice over a unit surface area is h and G per unit area is

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G(h) = ρhG fus + G surf

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(4.3.2)

The enthalpy of fusion of ice is 334 J/g or 6.01 kJ/mol [63]. In contrast, the enthalpy of dissociation of methane hydrate to water and methane gas is 56.9 kJ, per mole of methane [60]. Given that there are 5.75 water molecules per methane molecule on average in methane hydrate, this value equates to 9.9 kJ per mole of water. These calculations suggest that the Gfus cost of dissociating methane hydrates is about 40% greater than the Gfus cost of melting ice, per mole of water. Thus, it may be expected that the thickness of the pre-melting layer on clathrate hydrates is smaller than that on ice. The above estimate is for slightly below the melting point of ice or the thermodynamic dissociation temperature of clathrate hydrate. For greater subcoolings, the Gfus cost becomes proportional to the subcooling of the system, T, in addition to the mass of the pre-melted film: Gfus (T ) = (T /T m )H fus = T S fus . The specific interfacial free energy generally increases with cooling, however, the relevant interfacial free energy values for clathrate hydrates are not available in the literature. We may still conclude that Gsurf is expected to be less sensitive to a change in temperature than the specific interfacial energy itself because Gsurf is expressed in terms of the sum and the difference of the three specific interfacial free energy values (i.e., the opposite signs involved in Eq. (4.3.1) suggests that a part of the temperature dependence would cancel each other out). Thus, for a first approximation, we may ignore the temperature dependence of Gsurf . Then, G per unit area becomes G(h, T ) = (T /Tm )ρhHfus + G surf

(4.3.3)

Equation (4.3.3) shows that the Gfus cost increases with subcooling and hence one may expect that the thickness of the pre-melting layer to decrease rapidly with the system subcooling. For comparison, the thickness of the pre-melting layer of ice was indeed found to decrease rapidly with subcooling [40, 41]. Add to this the about 40% greater Gfus cost for clathrate hydrate than ice we found above, and one may expect that the pre-melting layer of clathrate hydrates to thin out even more rapidly than that on ice with subcooling. The thickness of the pre-melting layers on ice grows with warming and eventually diverges at the melting point [40, 41]. Accurate measurements of the thickness of the pre-melting layers have been difficult due to the thinness of such layers and, as a result, different experimental techniques yielded different numbers [45]. Still, it appears clear that such pre-melting layers on ice vanishes around the subcooling of 30 K. Then, with the typically large subcoolings involved in clathrate hydrate systems (e.g., the subcooling for high-pressure natural gases at the sea bed temperature is around 20 K), one may conclude that the surface of clathrate hydrates at such large subcoolings could be effectively dry. The above analysis shows (1) the quasiliquid layer of clathrate hydrate most likely exists slightly below the thermodynamic dissociation temperature, (2) its thickness is likely smaller than that on ice, and (3) its thickness likely decreases more rapidly with subcooling than that on ice.

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Up until now, we relied on the approximation that the specific interfacial free energy of either side of a thin film may be adequately approximated by the specific interfacial free energy of two corresponding semi-infinite media. This approximation corresponds to neglecting the disjoining pressure of the thin film. However, there may be another factor that can come into play that may act in the opposite direction, which is the effect of surface forces on quasi-liquid layers that have not been considered in the literature. The relevant physical property here is the disjoining pressure, Π (h), we introduced in Sect. 4.1. Using Eq. (4.1.1), G(h) per unit area in Eq. (4.3.2) becomes G(h) = ρhG fus + G surf − Π (h)vm ρh

(4.3.4)

Multiple components exist in the disjoining pressure of a complex liquid such as water (which is polar and hydrogen bonding) in addition to the structural component we discussed in Sect. 1.4 [5]. A comprehensive treatment of the disjoining pressure of a complex fluid such as water is beyond the scope of this book. For simplicity, below we restrict our discussions about the implications of surface forces to the ubiquitous van der Waals component only. Liquid water is denser than ice and the refractive index of liquid water is greater than that of ice. Sloan and Koh tabulated the refractive indices, n, of ice, sI clathrate hydrate and sII clathrate hydrate as nice = 1.3082, nsI = 1.346, and nsII = 1.35, respectively [60]. In contrast, the refractive index of liquid water is nwater = 1.33 [63]. As we saw in Sect. 4.1, the van der Waals force between the three media 1 and 2 across 3 is repulsive when n1 > n3 > n2 or n1 < n3 < n2 and attractive otherwise. Then it follows that the van der Waals forces for the (ice–liquid water–gas) system must be attractive whereas the van der Waals forces for the (clathrate hydrate–liquid water–gas) system must be repulsive. In other words, the experimentally observed rapid thinning of the pre-melting layers on ice is at least in part due to the thickness suppression by the attractive van der Waals forces. No such thickness suppression by the van der Waals forces apply to a pre-melting layer on clathrate hydrate. Thus, if we can assume that the Van der Waals component dominates the disjoining pressure, then the thickness of quasi-liquid layers on clathrate hydrate surfaces could be thicker than those on ice. Unfortunately, most fundamental data that are central to the surface and interfacial properties of clathrate hydrates are lacking in the literature. For example, very few water–clathrate hydrate interfacial free energy values are available in the literature, let alone their temperature dependence, and none for the specific interfacial free energy values for gas–clathrate hydrate interfaces. It is therefore impossible to determine whether the quasi-liquid layers on clathrate hydrate surfaces are thicker or thinner than the pre-melting layers on ice at a given subcooling. There are two opposing factors that influence the thickness of the quasi-liquid layers on clathrate hydrates. One is the enthalpy of fusion below the thermodynamic dissociation temperature, which is unfavorable for the growth of the quasi-liquid layer on clathrate hydrates and the cost is greater than that for ice. The other is the repulsive van der Waals forces which favor thick quasi-liquid layers on clathrate hydrates,

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as opposed to the attractive van der Waals forces which suppress the thickness of pre-melting layers on ice. Future experimental effort to the measurements of these important fundamental quantities is essential before we can gain better insight.

4.4 Tea Time Break: Disjoining Pressure and Polywater Polywater was a scientific scandal that engulfed the field of surface and interfacial science in the late 1960s and the early 1970s. An excellent account of Polywater was given by Franks [64]. As Franks noted, in their haste to disengage themselves from the Polywater saga, the scientists in the field ignored many important scientific questions raised by the Polywater. One such important scientific question was; why did Polywater only form through condensation of water vapor on quartz but not after prolong contact of liquid water with quartz? Can water vapor possibly be more chemically reactive than the (denser) liquid water? If so, how? The answer to this question was not known at the time Felix Franks wrote his book in 1983, more than a decade after the closure of the Polywater saga. The most plausible explanation I have heard to the above question is as follows; Derjaguin used low vacuum to control the vapor pressure of water. Since quartz is hydrophilic, a thin film of water forms on a quartz surface at a low vapor pressure of water, which will then thicken with increasing saturation of the vapor. Such thin films of water are also encountered at an initial stage of the growth of a water film. The disjoining pressure of a thin water film is positive and very large. It is conceivable that the disjoining pressure was so large that it could crack the surface of the quartz and unleash water-soluble components from within, which would not have occurred with a direct contact of bulk liquid water with quartz, even after a long time. The ultimate irony is that the concept of disjoining pressure was conceived by none other than Derjaguin himself much earlier than the Polywater saga.

References 1. X.H. Zhang, X. Zhang, J. Sun, Z. Zhang, G. Li, H. Fang, X. Xiao, X. Zeng, J. Hu, Detection of novel gaseous states at the highly oriented pyrolytic graphite-water interface. Langmuir 23, 1778–1783 (2007) 2. X.H. Zhang, N. Maeda, J. Hu, Thermodynamic stability of interfacial gaseous states. J. Phys. Chem. B 112, 13671–13675 (2008) 3. X.H. Zhang, A. Khan, W.A. Ducker, A nanoscale gas state. Phys. Rev. Lett. 98 (2007) 4. X. Zhang, N. Maeda, Interfacial gaseous states on crystalline surfaces. J. Phys. Chem. C 115, 736–743 (2011) 5. B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface Forces (Consultants Bureau, New York, 1987) 6. P.G. de Gennes, F. Brochard-Wyart, D. Quéré, Capillarity and Wetting Phenomena (Springer, New York, 2004)

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7. 8. 9. 10.

A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, 6th edn. (Wiley, New York, 1997) R.J. Hunter, Foundations of Colloid Science (Clarendon Press, Oxford, 1986) D.F. Evans, H. Wennerstrom, The Colloidal Domain, 2nd edn. (Wiley-Vch, New York, 1999) J.N. Israelachvili, Intermolecular and Surface Forces, 2nd edn. (Academic Press, San Diego, 1991) J. Mahanty, B.W. Ninham, Dispersion Forces (Academic Press, London, 1976) R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Massachusetts, 1963) C. Kittel, Introduction to Solid State Physics, 6th edn. (Wiley, New York, 1986) Y.R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984) H.C. Hamaker, The London - Van Der Waals attraction between spherical particles. Physica 4, 1058–1072 (1937) E.M. Lifshitz, The theory of molecular attractive forces between solids. Sov. Phys. Jetp-Ussr 2, 73–83 (1956) D.B. Hough, L.R. White, detailed calculation of Hamaker const. Adv. Colloid Interface Sci. 14, 3 (1980) N. Maeda, Is the surface of gas hydrates dry? Energies 8, 5361–5369 (2015) N. Maeda, Measurements of gas hydrate formation probability distributions on a quasi-free water droplet. Rev. Sci. Instrum. 85, 065115 (2014) D. Lohse, X. Zhang, Surface nanobubbles and nanodroplets. Rev. Mod. Phys. 87, 981–1035 (2015) R.N. Wenzel, Resistance of solid surfaces to wetting by water. Ind. Eng. Chem. 28, 988–994 (1936) A.B.D. Cassie, S. Baxter, Wettability of porous surfaces. Trans. Faraday Soc. 40, 0546–0550 (1944) C.E. Brennen, Cavitation and Bubble Dynamics (Oxford University Press, New York, 1995) S. Lou, J. Gao, X. Xiao, X. Li, G. Li, Y. Zhang, M. Li, J. Sun, J. Hu, Nanobubbles at the liquid/solid interface studied by atomic force microscopy. Chin. Phys. 10, S108–S110 (2001) S. Lou, J. Gao, X. Xiao, X. Li, G. Li, Y. Zhang, M. Li, J. Sun, X. Li, J. Hu, Studies of nanobubbles produced at liquid/solid interfaces. Mater. Charact. 48, 211–214 (2002) S. Lou, Z. Ouyang, Y. Zhang, X. Li, J. Hu, M. Li, F. Yang, Nanobubbles on solid surface imaged by atomic force microscopy. J. Vac. Sci. Technol. B 18, 2573–2575 (2000) L.J. Zhang, Y. Zhang, X.H. Zhang, Z.X. Li, G.X. Shen, M. Ye, C.H. Fan, H.P. Fang, H. Hu, Electrochemically controlled formation and growth of hydrogen nanobubbles. Langmuir 22, 8109–8113 (2006) X. Zhang, G. Li, Z. Wu, X. Zhang, J. Hu, Effect of temperature on the morphology of nanobubbles at mica/water interface. Chin. Phys. 14, 1774–1778 (2005) X.H. Zhang, Z.H. Wu, X.D. Zhang, G. Li, J. Hu, Nanobubbles at the interface of HOPG and ethanol solution. Int. J. Nanosci. 4, 399–407 (2005) X. Zhang, X. Zhang, S. Lou, Z. Zhang, J. Sun, J. Hu, Degassing and temperature effects on the formation of nanobubbles at the mica/water interface. Langmuir 20, 3813–3815 (2004) X.H. Zhang, W. Ducker, Formation of interfacial nanodroplets through changes in solvent quality. Langmuir 23, 12478–12480 (2007) X.H. Zhang, A. Khan, W.A. Ducker, A nanoscale gas state. Phys. Rev. Lett. 98, 136101 (2007) X.H. Zhang, G. Li, N. Maeda, J. Hu, Removal of induced nanobubbles from water/graphite interfaces by partial degassing. Langmuir 22, 9238–9243 (2006) X.H. Zhang, N. Maeda, V.S.J. Craig, Physical properties of nanobubbles on hydrophobic surfaces in water and aqueous solutions. Langmuir 22, 5025–5035 (2006) X.H. Zhang, A. Quinn, W.A. Ducker, Nanobubbles at the interface between water and a hydrophobic solid. Langmuir 24, 4756–4764 (2008) X. Zhang, D.Y.C. Chan, D. Wang, N. Maeda, Stability of interfacial nanobubbles. Langmuir 29, 1017–1023 (2013) R. Battino, T.R. Rettich, T. Tominaga, The solubility of oxygen and ozone in liquids. J. Phys. Chem. Ref. Data 12, 163–178 (1983) R. Battino, T.R. Rettich, T. Tominaga, The solubility of nitrogen and air in liquids. J. Phys. Chem. Ref. Data 13, 563–600 (1984)

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Chapter 5

Nucleation of Gas Hydrates

5.1 Nucleation of Ice 5.1.1 Introduction Clathrate hydrate is, by definition, a multi-component system that has at least two components of a guest gas and water. Nucleation of clathrate hydrate is thus more complex than nucleation of a single-component crystal like ice. Given the similarities between ice and clathrate hydrates, it is therefore pertinent to cover some aspects of nucleation of ice first, which incidentally has been studied much more extensively and for a far longer period than nucleation of clathrate hydrates. Nucleation of ice is commonplace in winters of the temperature regions, in summers of the polar regions and in the upper atmosphere of the earth. Cloud seeding has a direct impact on weather patterns and climate change because clouds reflect incoming sunlight back to the outer space. Heterogeneous nucleation of ice is also directly relevant to building up of ice on aircraft wings, which has been a wellrecognized hazard to flight safety. For these reasons, nucleation of ice has been studied for many decades. We make three general points about nucleation at this stage before we go into the main topic of nucleation of ice in this section and nucleation of clathrate hydrate in the following sections. First, even in a system that does not involve any foreign solid walls or solid impurity particles, nucleation of a crystalline phase preferentially occurs at a liquid–vapor interface than deep inside the bulk of a liquid medium, and in such cases the nucleation rate does not scale with the system volume but with the interfacial area [1, 2]. We therefore prefer to use the term heterogeneous nucleation for such nucleation at a liquid–vapor interface even no foreign solid particles or walls are involved, because the system symmetry breaks at a liquid–vapor interface, and such breaking of system symmetry is spatially inhomogeneous. As such, we refer to such nucleation as heterogeneous nucleation in this book and we reserve the term homogeneous nucleation to nucleation in a system that consists of a single metastable parent phase, like spontaneous emulsification we detailed in Chap. 1. Of © Springer Nature Switzerland AG 2020 N. Maeda, Nucleation of Gas Hydrates, https://doi.org/10.1007/978-3-030-51874-5_5

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course, this usage of the terminology is how we have consistently applied in our past publications. For clathrate hydrate nucleation, this effect is further compounded by the limitation in the availability of the guest gas, as we will see in the next section. Second, experimentally, nucleation rate is determined under the assumption that only one supernucleus induces a nucleation event for the whole system. Even if multiple nucleation events were to occur simultaneously, the resulting growth of the thermodynamically stable phase from each nucleation site would merge with each other and result in only one experimentally detectable phase transition. In short, multiple simultaneous nucleation events would be counted as one event and the rest would be effectively neglected. It is impossible to independently detect each of such possible simultaneous nucleation events for the reasons we detailed in Chap. 2. The consequent potential undercounting of nucleation events would lead to an underestimation of nucleation rates. In short, the “real” nucleation rate could have been higher if each such individual nucleation event had been separately detectable. Third, any cooperative phenomena would steer the system away from the totally random Arrhenius behavior that forms the basis of classical nucleation theory [3]. Since classical nucleation theory is built on the random probabilistic nature of surmounting of an activation energy barrier, as we detailed in Chap. 1, a presence of cooperative phenomena in a system would invalidate the foundation of the applicability of classical nucleation theory. An extreme unrealistic case (yet another thought experiment) will make this point abundantly clear: if the size distribution of the clusters were not of the Boltzmann form but uniform (independent of their sizes) then no activation barrier would exist and no nucleation would be required for the phase transition. With these three general points in mind, we briefly review three aspects of ice nucleation in this section; (1) physical properties of H2 O, (2) nucleation of ice from liquid water, and (3) nucleation of ice from water vapor.

5.1.2 Physical Properties of H2 O Water has many unusual attributes that could be relevant to nucleation of ice. H2 O has a non-zero entropy at 0 K [4]. The coordination number of H2 O only changes from about 4.0 to about 4.4 when ice melts [5], which shows that liquid water is highly structured due to the hydrogen bonding. Each –OH bond is approximately 1/3 ionic and 2/3 covalent in nature. Each H2 O molecule in ice has four nearest neighbors and acts as a hydrogen donor to two of them and acts as a hydrogen acceptor to the other two. The H–O–H bond angle in the water molecule is very close to the tetrahedral angle of 109.5° [6]. Many physical properties of subcooled water exhibit a pronounced temperature dependence. Liquid water above 273 K expands and becomes more compressible as it is cooled and becomes less viscous when compressed. Water’s unusual physical properties at ambient conditions, such as the fact that it becomes less compressible when heated, can be traced to more pronounced anomalies at lower temperatures

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[6]. Density of liquid water keeps falling with cooling below 4 °C. The results indicate that the density of subcooled liquid water approaches the low density of ice with cooling, and if crystallization had not occurred, the density of ice and that of subcooled water would match around −45 to −50 °C [3]. In short, the tetrahedral coordination established by ice results in progressively open structures with cooling [3]. The open structures can be interrupted by the presence of hydrophilic solutes such as ions and other strongly hydrogen-bonding molecules [3]. For example, structuring in subcooled liquid water increases with the addition of ethanol [3]. The origin of water’s unusual physical properties is the tendency of the H2 O molecules to attract each other strongly through hydrogen bonds. This self-loving nature of water molecules is the source of so-called hydrophobic effects [7]. The resulting low density causes a loss of orientational entropy. At sufficiently low temperatures, hydrogen bonds render (1) the energy and the volume to be negatively correlated and (2) the entropy and the volume to be negatively correlated. In an ordinary liquid, these correlations should be the opposite [6]. Interactions of water with hydrophilic and hydrophobic solutes may provide some insight. A hydrophilic molecule reduces the viscosity of water, whereas ethanol that has a hydrophobic ethyl group increases the viscosity of water. The temperature of the density maximum of water increases by doping of ethanol at low concentrations [3]. The diffusivity of the non-hydrogen-bonding molecule acetonitrile in water decreases with heating, while the diffusivity of ethanol shows increases more with pressure than the self-diffusion of water [3]. In short, the hydrogen bonding of liquid water is strengthened by the presence of hydrophobic groups and weakened by the presence of hydrophilic groups [3]. The enhancement of the hydrogen bonding of liquid water by the presence of hydrophobic groups or molecules is called hydrophobic hydration. Two distinct amorphous (glassy) water phases exist [6] in addition to the polymorphism of ice (ice has 17 known phases as of 2020 [8]). It is possible to quench liquid water faster than it crystallizes, and homogeneous nucleation can be effectively bypassed. In other words, homogeneous nucleation temperature is not a unique temperature but is a function of the cooling rate and the observation time [6]. The viscosity of such glassy water shoots up around −80 °C and 200 MPa [3]. Thus, if one cools water to lower than −80 °C under 200 MPa so fast that does not allow sufficient response time, the water can be “petrified” in the middle of a motion, without attaining a thermodynamically stable crystalline structure. Such viscous slowdown is a significant factor that could invalidate an application of classical nucleation theory to the determination of nucleation rates [3]. If a system is initially placed neither in a thermodynamically equilibrium state nor in a metastable local minimum, the system is more likely to eventually transition to the global minimum than to a local minimum [6]. Glassy water formed by rapid quenching of the liquid water has a glass transition temperature of around 136 K or 165 K. When such glassy water is heated, it crystallizes spontaneously to ice I c at around 150 K [6].

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5.1.3 Nucleation of Ice from Bulk Liquid Water Given that liquid water already possesses important structural features of ice (the coordination number of H2 O only changes from about 4.0 to about 4.4 when ice melts [5]), the deep subcoolings observed for liquid water are surprising. Water droplets of 5 µm radius emulsified in an oil using non-nucleating surfactants appear to subcool to low temperatures similar to those found for similarly sized water droplets in cloud chambers [3]. Just as the freezing point of water can be lowered by an application of hydrostatic pressures, the temperature at which the nucleation probability becomes high is also lowered by an application of hydrostatic pressures [3]. The deep subcoolings of water can therefore be further increased by increasing pressures [5], from approximately −40 °C under the atmospheric pressure to −92 °C under 0.2 GPa [3]. Silver iodide (AgI) has lattice constants that closely match those of ice to within a few percent [9]. Because of this very good lattice matching with ice, AgI has long been regarded as an excellent nucleation promotor of ice. After decades of commercial cloud seeding with AgI, however, the evidence that AgI really enhances rainfall still remains inconclusive [9]. Perhaps surprisingly, ice grows as discrete hexagonal islands on an AgI substrate, as opposed to a uniform film [9] which might be expected of an epitaxial growth of a crystal on a lattice-matching substrate. This rather apparent contradiction is not unique to AgI; BaF2 is not an effective nucleating agent despite its good lattice matching to ice [9]. It appears that the orientation of the hydrogen bonds (dipoles) of ice is an important factor in the heterogeneous nucleation potency of an underlying substrate [4]. In short, any substrate that orients dipoles at the surface of ice parallel to one another appears a poor nucleation promoter, because it reduces the entropy and raises the free energy of any nuclei growing on the substrate [4]. Then the basal crystal faces of AgI or PbI would be poor ice nucleation agents and their activity is likely confined to the prism faces [4]. A molecular dynamics simulation study found that structurally identical substrates could both inhibit and promote ice formation, depending on the interaction between the substrate surface and H2 O molecules [10]. These points are all highly relevant to nucleation of clathrate hydrates. For example, is AgI truly a good nucleation agent of ice and is ice truly a good nucleation agent of clathrate hydrates? At temperatures ice can be present, the subcoolings are already deep for clathrate hydrate nucleation. Would the resulting large driving force be sufficient for instant nucleation of clathrate hydrate, with or without the presence of ice? If so, how can one tell whether ice is truly a good nucleation promotor of clathrate hydrate or not? These points will be revisited in the next section. In contrast, water-insoluble organic compounds with no structural similarities to ice, such as steroids and cholesterols, have been found to effectively nucleate ice [11]. Formation of hexagonal ice crystals on cholesterols that have no lattice matching to ice resulted in surprisingly small subcoolings of down to 1 K [12]. Testosterone (a steroid) with no lattice matching to ice was also found to be an excellent nucleating agent [9]. Bacteria Pseudomonas Syringaed were also found to greatly promote nucleation of ice [9]. These are very surprising findings, to say the least, and more

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than half a century since the initial report, the underlying mechanisms are still under active investigations. An issue that further complicates the matter is an impact of surface roughness on the heterogeneous nucleation potency of a material. Purely geometrical argument shows that surface roughness enhances heterogeneous nucleation potency of a substrate [13]. Nanoscale roughness in the form of etch pits on a substrate surface can have a profound effect on the kinetics of heterogeneous nucleation of ice [9]. Another surprising finding is that, in nano-porous alumina, liquid water subcools to −42 °C and then freezes to metastable I c , not to thermodynamically stable I h [14]. I c thus formed remains stable during hearing, up to 273 K at which point it melts to liquid water [14]. How could this happen? Nano-porous alumina had one of the highest surface-to-volume ratios of any system studied to date and heterogeneous nucleation of ice ought to be expected more than ever, and yet liquid water confined inside nano-porous alumina subcooled to −42 °C. Of course, if the pores were so small that the space left was insufficient for a long-range positional order (that defines a crystalline phase) to manifest then no freezing of water would have been expected. However, the pores in the nano-porous alumina were not that small; indeed, a metastable I c crystalline phase could still form inside the nano-porous alumina [14]. These are all important outstanding issues that have not been resolved to date. The so-called “Contact nucleation” refers to a nucleation process of ice on a solid wall at an air–liquid interface (like around the three-phase lines of a meniscus). Such contact nucleation has been found to be more efficient than “immersion nucleation” (nucleation of ice on a solid wall that is totally immersed in liquid water) [1]. In the presence of a potent heterogeneous nucleation substrate, surface nucleation preferentially occurred at the three-phase lines where the air, liquid water, and a solid wall met. It was also found that the nanoscopic (nanometer scale) surface roughness played a significant role while the microscopic (micrometer scale) surface roughness did not [1]. It has further been shown that rough substrates enhanced contact nucleation while smooth ones did not [1]. It appears that steps, pores, cracks, or other surface features of the sizes of the order of the critical nucleus may effectively promote nucleation by lowering the free energy barrier [1, 15]. These findings are most likely relevant to clathrate hydrate nucleation as we will see in the next section.

5.1.4 Nucleation of Ice from Water Vapor Nucleation of ice from water vapor is relatively less understood compared to nucleation of ice from bulk liquid water despite more direct relevance to cloud seeding. A two-step mechanism of (1) condensation of water vapor followed by (2) freezing of the condensed water [16] and a three-step mechanism of (1) condensation of water vapor which was followed by (2) freezing of the condensed water which in turn was followed by (3) growth of bulk ice from the ice nucleus that requires surmounting of an additional activation barrier [17] have been proposed.

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It appears that homogeneous nucleation of I h does not occur around −42 °C when the droplet is smaller than the critical nucleus size. For 5 nm diameter droplets, for example, water could be subcooled to −70 °C [9]. This deep subcooling is of different nature from the viscous slowdown we detailed above. For heterogeneous nucleation of ice from water vapor in the presence of a solid wall, wedges become preferred nucleation sites because condensation of water vapor at the contacts of the wedges does not require nucleation, as we saw in Chap. 1. Condensation onto a narrow wedge is effectively a one-dimensional nucleation along the wedge that does not require any surmounting of an activation barrier [18]. However, after water vapor has condensed to such a wedge, freezing of the confined liquid water in the narrow wedge requires surmounting of a large activation barrier of the order of −35 °C of subcooling [19, 20]. Condensation of water vapor into such a wedge would deplete moisture from the surrounding vapor, and consequently the activity of the water vapor would fall that prevents ice from forming elsewhere. The end result is that only ice at the mouth of the wedges can be observed, after the fact [19, 20]. At small subcoolings, in contrast, capillary melting prevents freezing of liquid water in wedges. Instead, ice preferentially forms by condensation and freezing on flat, open surfaces [19, 20]. This latter effect is relevant to icing on aircraft wings.

5.2 Nucleation Rate of Gas Hydrates The previous section showed that nucleation of ice, despite being studied for much longer than nucleation of clathrate hydrates, is still full of surprises and mysteries. Then, is there any wonder that nucleation of clathrate hydrates, that involves more components than ice, is hardly understood? Given the similarities between ice and clathrate hydrates, many, if not all, of the complications and mysteries concerning the nucleation of ice detailed in the previous section are expected to be present in nucleation of clathrate hydrate as well. However, so little is understood about nucleation of clathrate hydrate that we have not yet reached a stage to even start addressing such outstanding complications and mysteries. We saw in Chap. 1 that nucleation rate is central to nucleation phenomena of a given system and so is the case for clathrate hydrates. In fact, a primary difficulty in the investigations of nucleation of clathrate hydrate has been the inability of researchers to determine and compare nucleation rates of clathrate hydrates across various systems of different scales and complexity, which in turn has been limiting the ability of researchers to study the nucleation process itself. Our ability to reliably determine the nucleation rate of a given clathrate hydrate must thus precede, not follow, investigations of any of the outstanding complications and mysteries surrounding the nucleation of ice that are also expected to be present for clathrate hydrate. It is conceivable that insights gained from nucleation of clathrate hydrate may in turn shed new light to our understanding of nucleation of ice. The central theme of this section is how the theoretical framework detailed in Chap. 1, and the

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experimental techniques detailed in Chap. 2 can be applied to the investigations of nucleation of clathrate hydrates.

5.2.1 Unique Attributes of Gas Hydrate Nucleation We start from highlighting the unique attributes of clathrate hydrate nucleation that are different from ice nucleation. Clathrate hydrate is, by definition, a multicomponent system. The concentrations of a guest gas that can be contained inside clathrate hydrate are many orders of magnitude higher than the solubility of the same gas in liquid water at the same pressure and temperature. This much greater gas contents in the clathrate form than the solubility of the same guest gas in liquid water renders clathrate hydrate thermodynamically stable at temperatures higher than the ice point of around 273 K, as we saw in Chap. 3. However, this essential feature of clathrate hydrates inevitably leads to the fact that the concentration of the guest gas in the aqueous phase is not uniform but has a spatial gradient that becomes the highest at the guest–aqueous interface, under a static, isothermal and isobaric condition. Stirring of the aqueous phase and turbulent flows could somewhat mitigate the steepness of the spatial concentration gradient of the guest gas in water but a spatial concentration gradient nevertheless remains. A salient point here is that the driving force for nucleation is implicitly assumed to be spatially uniform throughout the metastable phase in Eq. (1.2.4), but this is clearly not the case for clathrate hydrates. Whether such an intrinsic spatial inhomogeneity within a system under thermodynamic equilibrium can be treated by assuming local equilibria in different sections of the system remains an unresolved issue [21]. This inevitable spatial concentration gradient of a guest gas in liquid water and the resulting spatial gradient in the driving force for nucleation have important implications to the nucleation rate of clathrate hydrates. The supersaturation and hence the driving force for nucleation becomes the greatest (i.e., G*activation in Eq. (1.2.6) becomes the smallest) at a guest–aqueous interface and diminishes (i.e., G*activation increases) with the distance from the interface. The exponential dependence of the nucleation rate to G*activation in Eq. (1.2.6) thus means that homogeneous nucleation in the conventional sense cannot occur for clathrate hydrates because the nucleation rate exponentially diminishes as one moves away from the interface. Even for a single-component system that has a spatially uniform driving force, it has been postulated that homogeneous nucleation of a crystal from a liquid of a finite size preferentially occurs at the liquid–vapor interface where the system symmetry breaks [1, 2]. It follows that the homogeneous nucleation rate scales not with the system volume but with the interfacial area in many systems [1, 2]. Then, strictly speaking, “truly” homogeneous nucleation (i.e., nucleation is equally likely throughout the system volume) can only occur in an infinitely large system. For ice, it has been known that the so-called “contact nucleation” (heterogeneous nucleation of ice in the presence of a solid wall at an air–liquid interface) is more efficient than “immersion nucleation” (heterogeneous nucleation of ice in the presence of a solid

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wall immersed in liquid water), as detailed in Sect. 5.1 [1]. Similar features also likely apply to clathrate hydrates. The traditional wisdom has been that a suitable measure of the system size for homogeneous nucleation is the system volume because the number of potential nucleation sites is expected to be proportional to the system volume. Likewise, it has been considered that a suitable measure of the system size for heterogeneous nucleation is the area of a surface or an interface that is responsible for the heterogeneous nucleation [18]. The activation barriers of these two cases are then related through the effective interfacial free energy between the thermodynamically stable phase and the metastable parent phase through Eq. (1.2.37) 1/3   1/3 = γheterogeneous /γhomogeneous = (1/4)(2 + cos θ )(1− cos θ )2

(5.2.1)

where θ is the contact angle the thermodynamically stable phase forms on a foreign substrate in the metastable parent phase, with the two limiting cases of  1/3 = 0 when θ = 0° and  1/3 = 1 when θ = 180° [18]. To recap what we covered in Sect. 1.2, the effective interfacial free energy for heterogeneous nucleation, γ heterogeneous , becomes zero when the thermodynamically stable phase perfectly “wets” the solid substrate whereas γ heterogeneous approaches that of homogeneous nucleation, γ homogeneous , when the foreign substrate does not at all contribute to the reduction of the activation barrier. Such perfect “wetting” case might be realized when the underlying foreign substrate and the emerging thermodynamically stable phase have perfect lattice matching, which would give rise to an epitaxial growth of the thermodynamically stable phase on the foreign substrate. However, these traditional considerations are unlikely to apply to clathrate hydrates [22]. For clathrate hydrate, θ in Eq. (5.2.1) could correspond to either (1) the contact angle the clathrate phase forms on a foreign solid substrate in a supersaturated aqueous guest gas solution or (2) the contact angle the clathrate phase forms on a solid foreign substrate in a moist guest gas. For the limiting case of  1/3 = 0, Eq. (5.2.1) predicts that there should be no activation barrier to heterogeneous nucleation. However, due to the concentration gradient of the guest gas in an aqueous phase and the concomitant spatial gradient in the driving force, such epitaxial growth can only occur when the foreign lattice-matching substrate is in the proximity of both the aqueous phase and the guest gas phase. Presence of thermodynamically metastable interfacial gaseous states would satisfy this condition and its impacts will be discussed in Sect. 5.3. In short, the limiting case of  1/3 = 0 can only be realized either (1) at or next to the three-phase lines where the three phases of the guest gas, the aqueous phase and the foreign (lattice-matching) solid substrate meet or (2) in a thin wetting film of aqueous guest gas solution on the foreign (lattice-matching) solid substrate under the moist guest gas. The basic idea is that both the guest gas phase and the aqueous phase must be within the range of the surface forces for a foreign (latticematching) substrate to have any impact on the heterogeneous nucleation of clathrate hydrate (see Sect. 1.4 and Chap. 4). As such, the system size is unlikely to scale with the interfacial area in either case [22].

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Gas

Water Water

Fig. 5.1 Schematic illustration of three different types of possible heterogeneous nucleation next to a three-phase line. The meniscus is drawn for the case of a hydrophilic wall

Now let us consider a situation where water is half filled in an open solid container under a pressurized guest gas and the system temperature is sufficiently low for the clathrate hydrate to be thermodynamically stable. As noted above, homogeneous nucleation in the conventional sense cannot occur for clathrate hydrate. Instead, three different types of heterogeneous nucleation are possible at or next to a three-phase line, as schematically shown for the case of a hydrophilic wall in Fig. 5.1. Here, the three different interfaces for heterogeneous nucleation are (1) heterogeneous nucleation at the guest gas–aqueous interface, (2) heterogeneous nucleation at the aqueous–foreign solid substrate interface, and (3) heterogeneous nucleation at the guest gas–foreign solid substrate interface. Since both the solubility of the guest gas in the aqueous phase and the vapor pressure of water in the guest gas phase are non-zero, all three types of heterogeneous nucleation are possible at or next to a given three-phase line [22]. The first case does not involve any foreign solid substrate at all but should still be considered heterogeneous nucleation because the system symmetry or the spatial homogeneity still breaks at the interface where the nucleation event takes place (as we have seen in Sect. 1.4, a gas phase is far from “nothing”). The rate of heterogeneous nucleation of clathrate hydrate exclusively at a guest gas–aqueous interface has not been measured until very recently [23]. The large density mismatch between the guest gas and water means either suspending a water droplet in the guest gas or suspending a guest bubble in water for extended periods, both of which are experimentally non-trivial. The closest analogue until very recently may be the heterogeneous nucleation at a hydrocarbon oil–aqueous interface that does not involve any foreign solid substrate [24]. It was found that the rate of heterogeneous nucleation that does not involve any foreign solid surface was significantly smaller than that involves one [24]. If the heterogeneous nucleation of clathrate hydrate were to take place at the aqueous–solid substrate interface (Case 2), then the nucleation rate would

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depend on the solubility of the guest gas in the liquid water. If the heterogeneous nucleation of clathrate hydrate were to take place in a thin wetting film of aqueous guest gas solution on the foreign solid substrate (Case 3), then the nucleation rate would be a complex function of both the solubility of the guest gas in liquid water and the thickness of the wetting film that is dominated by a complex structural component of the disjoining pressure [25] (see Sect. 4.1 for a general description of the disjoining pressure and Sect. 1.4 for a tangible example of the structural component of the disjoining pressure). It is thus not even clear what the equivalent of Eq. (5.2.1) is or how θ should be defined for clathrate hydrate nucleation [22].

5.2.2 Historical Perspective of Gas Hydrate Nucleation The early studies on the kinetics of clathrate hydrates were mainly about the rate of crystal growth that can be measured from the rate of guest gas consumption than the nucleation process itself [26–33]. In the rare occasions when the nucleation was of interest, induction times at constant subcoolings were reported as the representative measure of the nucleation of clathrate hydrate [26–33]. A large number of induction time measurements are required to reliably determine its average for each subcooling of interest, as we saw in Chap. 2, which was hard to do prior to the recent advent of automated measurements. Induction times were often very long, especially at small subcoolings, which rendered the measurements extremely time consuming to an extent they become impractical. As we saw in Chap. 2, a maximum waiting time was often set for induction time measurements at a constant temperature. A consequence of doing so was that only the lower limit of the induction time could be known, which made it impossible to determine the average induction time. Perhaps unsurprisingly, none of these early studies advanced far enough to reach a stage that could determine the nucleation rates of clathrate hydrates. Collection of a larger number of data points within a given timeframe can be facilitated by carrying out multiple measurements in parallel, but the heterogeneous nucleation rate usually depends on specific solid walls (surface roughness, surface defects, etc.) involved in a system. A notable effort from this approach is the recent investigations of clathrate hydrate formation in water-in-oil (W/O) emulsions using a high-pressure micro differential scanning calorimeter (HP–µDSC). If each water droplet in a W/O emulsion could act as a separate “reactor”, then the use of a W/O emulsion could astronomically increase the number of data points that can be measured in a limited timeframe and at the same time avoid solid wall contacts [34, 35]. However, these studies could not determine the nucleation rate because the total interfacial area involved in an emulsion sample could not be reliably determined. Manakov and coworkers recently advanced this line of study and estimated the total water–oil interface area per unit mass of an emulsion from ex situ droplet size distribution data and used this information to determine the nucleation rate of

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clathrate hydrates in the emulsion [36–40]. However, an emulsion is not thermodynamically stable and its total water–oil interface area per unit mass of the emulsion is expected to diminish with time by many orders of magnitude. In parallel with the effort to determine the nucleation rate of clathrate hydrates, attempts have been made to directly detect the nucleation of clathrate hydrates in micro- and sub-micro scales using X-ray diffraction [41, 42], neutron diffraction [41, 43], Raman spectroscopy [41–44], and nuclear magnetic resonance (NMR) [45– 47]. These efforts also probed potential pathways of clathrate hydrate nucleation: clathrate hydrates consist of several different building units (cages) per unit cell and it is highly beneficial to know which cage forms first and how its presence may impact the nucleation of the other cages. These efforts are ongoing, but these techniques are not suited for measurements of the critical nucleus size, the corresponding nucleation work, or the nucleation rate of clathrate hydrates for both the fundamental and the practical reasons we detailed in Chaps. 1 and 2, respectively.

5.2.3 Empirical Approach Given the unique characteristics of clathrate hydrates and the uncertainties around the applicability of classical nucleation theory to clathrate hydrates, a prudent approach would be a purely empirical one that does not rely on the theoretical framework of the classical nucleation theory [22, 24, 36–40, 48–50]—one can derive empirical equations first and worry about the physical reasoning later. In our view, establishing a reliable method of determining nucleation rates will allow systematic comparisons between systems of various scales and complexity, and such systematic comparisons may provide new insights or at least some clues. It is still early days and nucleation curves of only a handful of guest gases have been determined using the theoretical and experimental framework outlined in Chaps. 1 and 2. These nucleation rates are summarized in the Appendix. This scarcity of nucleation rate data should not come as a surprise given that the experimental techniques detailed in Chap. 2 are still only several years old and refinements of these techniques are still ongoing. Some of the data summarized in Appendix were obtained before the later refinements of the techniques took place and should be regarded as first-order approximations.

5.2.4 Scaling Laws for the Nucleation Rates of Gas Hydrates One of the fundamental questions in the nucleation of clathrate hydrate we saw above is what the appropriate measure of the system size should be that can be used as the normalization constant for its heterogeneous nucleation. The conventional wisdom has been that a suitable measure of the system size for homogeneous nucleation is the system volume because the number of potential nucleation sites is expected to be

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proportional to the system volume. Likewise, it has been considered that a suitable measure of the system size for heterogeneous nucleation is the area of a surface or an interface that is responsible for the heterogeneous nucleation. However, these conventional ideas are unlikely to be valid for clathrate hydrates, as we detailed above. Perhaps as expected, it has been experimentally found that the nucleation rate of 90–10 mol% (C1/C3) mixed clathrate hydrate in the presence of a foreign solid wall was substantially greater than that in the absence of a solid wall when both nucleation rates were normalized to the area of the guest–aqueous interface in the system [24]. This discrepancy between the nucleation rate of a system that involves a foreign solid wall and one that does not may be expected since the presence of a foreign solid wall was not at all accounted for when the experimentally determined nucleation rate was normalized to the guest–aqueous interfacial area [48]. Maeda and Shen recently applied the general scaling law we detailed in Chap. 1 to clathrate hydrate nucleation in an attempt to identify the suitable measure of the “system size” [22]. They used the nucleation rate of 90–10 mol% (C1/C3) mixed gas hydrate determined by an HP-ALTA to calculate the nucleation rate of methane hydrate in a flow loop (model pipeline) that had a much larger system scale. The nucleation rate of 90–10 mol% (C1/C3) mixed gas hydrate and the nucleation rate of methane hydrate had been found to be broadly similar at comparable system subcoolings [24]. They examined five possible measures of the “system size” that could be used for scaling; (1) total nominal area of water–guest gas interface in the presence of the pipe wall, (2) total area of water–guest gas interface in the absence of the pipeline wall, (3) total lengths of the three-phase lines where the three phases of the guest gas, water, and the pipeline wall met, (4) total wetted area of the pipeline walls, (5) total volume of water in the pipeline [22]. Among these five measures, only the total lengths of the three-phase lines scaled the nucleation rates between a system of millimeter scale and a system of meter scale to within the same order of magnitude [22]. Given that the effects of flows were ignored, the agreement within an order of magnitude appears fortuitously good, which supports Maeda’s original point that the appropriate measure of the system size in a quiescent clathrate hydrate system is the total lengths of the three-phase lines [48].

5.2.5 Theoretical Front and the Applicability of Classical Nucleation Theory to Gas Hydrate Systems Nucleation of any system has been difficult to investigate computationally for two reasons. First, it is impossible to predict when and where a nucleation event will eventually occur in a given system. Second, the typically long induction times are taxing in terms of computational resources. Consequently, the progress in computationally determining the nucleation rates of clathrate hydrates has been limited. The few reported computational estimates of nucleation rates of clathrate hydrates

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have been many orders of magnitude larger than any experimentally determined nucleation rates reported in the literature [51–53]. Given the unique attributes of clathrate hydrates in the context of nucleation, it is pertinent to re-examine the applicability of classical nucleation theory to nucleation of clathrate hydrates [54]. A good starting point is Eq. (14) of Ref. [54]:   J = z f C0 exp −W ∗ /kT

(5.2.2)

where J is the nucleation rate, z is the so-called Zeldovich factor that can take a value somewhere between 0.01 and 1, f is the attachment frequency of building units to an emerging nucleus, C 0 is the concentration of nucleation sites in the system, W * is the nucleation work that is required to surmount the activation barrier, k is the Boltzmann constant, and T is the absolute temperature [54]. Simply put, the nucleation rate is a product of (1) how often the building units attach to an emerging nucleus, (2) how many such potential nucleation sites exist in the system, and (3) the probability of realizing a system energy that is greater than the activation barrier in the Arrhenius form, with a Zeldovich factor z. Nucleation rates are often reported in the form of (J/C 0 ) in the literature. For three-dimensional nucleation, the Zeldovich factor is z = (W */3π kTn*2 )1/2 where n* is the number of building blocks in the smallest supernucleus [18]. Even for a clathrate hydrate for which nucleation sites are limited to next to an interface, the thickness of such interfacial regions is much larger than the size of the molecules. Whether such interfacial regions are substantially thicker than the size of a supernucleus and hence would allow a treatment using three-dimensional nucleation remains to be seen. We follow Ref. [54] for now and assume that C 0 = 1/vw where vw is the molecular volume of water for homogeneous nucleation (which provides the minimum nucleation rate) and that C 0 is proportional to the concentration of active nucleation sites in the system, N p (whose identities are unknown) for heterogeneous nucleation. The attachment frequency f itself is a function of the driving force or supersaturation, μ, and is related to the attachment–detachment frequencies at equilibrium, f e , through f = f e exp(μ/kT ) [54]. This last point follows from the principle of detailed balance we detailed in Chap. 1. We do not quite know which physical factors will dominate f or f e for the nucleation of clathrate hydrates, however, the expression shows that f approaches f e in the limit of μ → 0 (no subcooling at all) and f approaches zero at an infinite subcooling, which partially accounts for viscous slowdowns (in reality, viscous slowdowns should render f zero long before the system approaches 0 K). For analyses, it is convenient to lump all the kinetic factors into one “kinetic parameter”, A, in the form; A = zf e C 0 . Then, noting f = f e exp(μ/kT ), Eq. (5.2.2) can be rewritten as [54]   J = A exp(μ/kT ) exp −W ∗ /kT

(5.2.3)

The nucleation work, W *, corresponds to the activation barrier in the Arrhenius equation and can be found from the calculus of variations that renders ∂W /∂n = 0.

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The activation barrier W * (or G(r)activation in our expression in Sect. 1.2) needs to be known as a function of the number of the building blocks, n, for the calculus of variations to work. For three-dimensional nucleation of single-component condensed phases, W * = 4c3 v2h γ 3ef /(27 kμ2 ) where c is a shape factor of the nucleus (c = (36π )1/3 for the spherical shape), vh is the volume of a hydrate building unit, γ ef is the effective interfacial free energy between the thermodynamically stable phase and the parent metastable phase (the interfacial tension of the cluster) [54]. The supersaturation or the driving force, μ, is given by Eq. (8) of Ref. [55] μ = μguest_gas (P, T ) + nw (P, T )μwater (P, T )−μhydrate (P, T )

(5.2.4)

μguest_gas (P, T ) and μwater (P, T ) can be estimated from the metastable guest gas pressure and the water vapor pressure. μhydrate (P, T ) is the chemical potential of the hydrate building unit in the clathrate form at P, T. For an isobaric process like linear cooling ramps, μ becomes proportional to the system subcooling T in a first approximation [55]. This first approximation is justified when the differential between the heat capacity of the clathrate hydrate and that of the guest gas plus the water that form as a result of dissociating the clathrate hydrate is much smaller than heq , the latent heat of dissociation of a building unit of hydrate crystal into guest gas and liquid water at the thermodynamic dissociation temperature, T eq [55]. Then, μ is given by μ = S eq T where S eq = heq /T eq [54]. It is now convenient to introduce the “thermodynamic parameter”, B , defined as  B = 4c3 v2h γ 3ef /(27 kS 2eq ) [54]. Then, using μ = S eq T, Eq. (5.2.3) becomes     J = A exp Seq T /kT exp −B  /T T 2

(5.2.5)

Rearranging Eq. (5.2.5) yields ln J −Seq T /kT = ln A−B  /T T 2

(5.2.6)

Classical nucleation theory thus predicts that a plot of (ln J − S eq T /kT ) versus 1/T T 2 should yield a straight line. Then the slope of the fitted straight line yields the thermodynamic parameter B and the offset (ordinate intercept) yields the natural logarithm of the kinetic parameter, ln A [54]. The functional form of A depends on the type of nucleation and accounts for the frequency of attachment of hydrate building units to the nucleus and the concentration of potential nucleation sites [54]. The thermodynamic parameter B accounts for the probability that a system can surmount the activation barrier as prescribed by the Boltzmann distribution. Such a probability exponentially diminishes with the size of the activation barrier. The only physical parameter that can substantially influence the numerical value of B is γ ef , which impacts the size of the nucleation work or activation barrier [54]. Simply put, the kinetic parameter A accounts for the frequency at which the system “attempts” to realize a nucleation event and the

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Fig. 5.2 A plot of (ln J − S eq T /kT ) versus 1/T T 2 for the nucleation of 90 mol% methane 10 mol% propane mixed clathrate hydrate in the absence of a solid wall. The original data are from the 0.002 K/s data set of Ref. [24]. The linear least squared fit to the data is also shown in a dashed straight line

thermodynamic parameter B accounts for the probability that each such “attempt” results in surmounting of the activation barrier. The numerical values of both A and B can be calculated from the physical properties of the clathrate hydrate in question [54]. For the nucleation of methane hydrate, the kinetic parameter, A, has a numerical value of 1035 m−3 s−1 and the thermodynamic parameter, B , has a numerical value of 4.8 × 106 K3 [54]. Figure 5.2 shows a plot of (ln J − S eq T /kT ) versus 1/T T 2 for the nucleation of 90 mol% methane 10 mol% propane mixed gas hydrate in the absence of a solid wall (on a quasi-free water droplet suspended in squalene). The original data are from the 0.002 K/s data set of Ref. [24]. For a first approximation, the physical properties of methane hydrate are used here for the physical properties of 90 mol% methane 10 mol% propane mixed gas hydrate here. The linear least squared fit to the data is shown together as a dashed line. The data (open symbols) appear to exhibit a small curvature that is convex to the bottom. A similar small curvature in the same direction (convex to the bottom) was also observed in Makogon’s data shown in Fig. 8 of Ref. [54]. Kashchiev and Firoozabadi excluded the last data point of Makogon (that corresponded to the lowest subcooling) from Fig. 8 of [54] during their linear fitting that deduced the kinetic parameter A and the thermodynamic parameter B . The linear fitting could have yielded very different A and B values if the last data point had not been excluded. If we assume that the linear fitting is warranted, then the nucleation curve shown in Fig. 5.2 yields the kinetic parameter A of 59 m−2 s−1 and the thermodynamic parameter B of 1.2 × 106 K3 . This numerical value of B falls within the same order of magnitude of the theoretically expected B value for the nucleation of methane hydrate in the absence of a solid wall of 4.8 × 106 K3 , which suggests that the size of the activation barrier may be in the expected range. It is reassuring that the activation barrier to nucleation in a quasi-free water droplet suspended in squalene is in line with the theoretical expectations. In contrast, the experimentally derived kinetic factor, A, of 59 m−2 s−1 is more than 30 orders of magnitude smaller than the theoretically expected value of the nucleation of methane hydrate in the absence of a solid wall of 1035 m−2 s−1 . Even if we use a heterogeneous case of θ = 90°, A is expected to be 4 × 1026 m−2 s−1 [54]

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or more than 25 orders of magnitude larger than the experimentally derived kinetic factor. In short, the frequency with which the system “attempts” to surmount the activation barrier is 25–30 orders of magnitude lower in reality. One way to account for this very large discrepancy between the theoretically calculated and the experimentally determined kinetic parameters is to assume a much smaller “real” system size in which a small number of active nucleation sites are concentrated than is expected from a uniform distribution of nucleation sites over the nominal size that characterizes the system [54]. The basic idea is that dividing the experimentally determined nucleation rate by a much smaller “real” system size would lead to a much greater normalized nucleation rate. Then, dividing the experimentally determined nucleation rate by a much smaller “real” system size would yield a much greater normalized nucleation rate. If the locations of the potential nucleation sites were limited to the three-phase lines, as we assumed, it would substantially lower the total number of potential nucleation sites in the system and hence would bring the agreement between the experimentally determined and theoretically expected kinetic factors closer. However, this factor alone is unlikely to be sufficiently large to bridge the gap of more than 20 orders of magnitude [56]. If we assume that the relevant interfacial area in a quiescent sample is the length of the three-phase lines multiplied by the diffusive length of the guest gas, that would lower the kinetic parameter by 4–5 orders of magnitude [56]. Of course, we could further assume that the width of such a band in a quiescent sample that can contribute to the nucleation of clathrate hydrate is narrower than the diffusive length of the guest gas and bring the agreement still closer. This assumption could be warranted because nucleation rate exponentially falls with diminishing supersaturation of the guest gas and the concentration gradient exists in a direction perpendicular to the three-phase lines. However, there is a problem in this approach. First of all, bridging of the discrepancy of more than 20 orders of magnitude would require an unphysical assumption that the width of the band to be smaller than the size of an atom. Second, we used nominal lengths of the three-phase lines for the normalization of the nucleation rate that do not account for any surface roughness that would render the “real” lengths of the three-phase lines much longer [48]. Such longer “real” lengths of the three-phase lines would effectively render the “real” system size larger, which would worsen the discrepancy. Then there is the issue of silver iodide (AgI). Silver iodide is a salt of particular interest in the context of nucleation although it is sparingly soluble in water. Silver iodide is also not very hydrophilic (water still wets silver iodide: the contact angle of water on AgI is about 15°). Sowa et al. reported that an addition of silver iodide did not have appreciable impact on the nucleation probability of natural clathrate hydrates [57]. Silver iodide is perceived to be an excellent nucleation agent of ice due to its lattice matching and has long been used as a cloud seeding agent, and ice has been considered to be an excellent nucleation agent of clathrate hydrates [58, 59]. The negative results of Sowa et al. thus suggest that either silver iodide is a poor nucleation promotor of clathrate hydrates despite its lattice matching with ice or a sufficient number of ubiquitous active nucleation sites already exist in a typical clathrate hydrate system that would mask any additional effect of silver iodide. If a

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small number of active nucleation sites were indeed concentrated in a much smaller “real” system size, such active nucleation sites could not be ubiquitous. What kind of active nucleation sites could have more nucleation potency than silver iodide? As we saw in Sect. 5.1, silver iodide’s lattice constants match those of ice to within a few percent [9]. However, after decades of commercial cloud seeding with silver iodide, the detailed mechanism by which silver iodide and many other inorganic substrates facilitate ice freezing still remains unclear [9]. Thus, silver iodide could in fact be a poor nucleation promotor of ice and of clathrate hydrates remains a real possibility that could explain the negative results of Sowa et al.

5.2.6 Nucleation Pathways Another possibility that could account for the very large discrepancy in the kinetic parameter is to assume a much “longer” nucleation pathway that involves many more kinetic steps than what is accounted for in classical nucleation theory. As detailed in Chap. 1, regardless of the nature of a particular nucleation pathway or the complexity of a system, a nucleation event can only be realized when a continuous path forms from the beginning to the end or all the pieces of the puzzle are in place. And the nucleation rate represents the rate at which such occurrence is realized, over a whole nucleation pathway, in a unit system size. It means that presence of a bottleneck anywhere along such nucleation pathway would lower the overall nucleation rate. Such a bottleneck that is not accounted for in classical nucleation theory is all it takes to cause a massive discrepancy between the experimental and the theoretical nucleation rates. A primary candidate for such a bottleneck in clathrate hydrate nucleation is the typically very low solubility of a hydrocarbon guest gas in liquid water. The much lower solubility of the guest gas compared to the guest gas content in the clathrate form would pose a mass transfer barrier and hence require a larger number of kinetic steps. In other words, nucleation of clathrate hydrates has a “longer” nucleation pathway than nucleation of, say, a salt from a supersaturated solution. Any factors that “lengthen” the nucleation pathways would lower the system-wide nucleation rates, because the probability for the system to “clear” an extra hurdle within a given timeframe cannot be greater than 1 and the overall nucleation rate is expected to be a product of nucleation rates of each sub-step that constitutes the nucleation pathway. That the nucleation rate of the clathrate hydrate of more soluble CO2 was significantly greater than that of methane hydrate or 90–10 mol% (C1/C3) mixed gas hydrate for a given subcooling suggests that the mass transfer barrier to the nucleation of clathrate hydrates is indeed a significant factor [56]. Another candidate that could “lengthen” the nucleation pathway is viscous slowdowns we detailed above while we covered ice nucleation. Subcoolings of more than 30 K were routinely encountered in the nucleation of clathrate hydrate in a quiescent system of a sparingly soluble hydrocarbon guest gas [24]. Such large subcoolings of more than 30 K in pure water would experience significant viscous slowdowns

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before nucleation of ice can take place. Pure water has been reported to become glassy around the subcoolings of 40 K [6], so it is reasonable to expect that comparable viscous slowdowns of water may occur at deep subcoolings routinely encountered in a quiescent system under a pressurized sparingly soluble hydrocarbon guest gas [56]. A surprising and potentially relevant phenomenon is that a small volume of clathrate hydrate can form during heating from a deeply subcooled temperature of about 240 K [60]. Such a formation of clathrate hydrate during heating was often observed at a temperature far above the ice point of about 273 K but still below the thermodynamic dissociation temperature, T eq , of the clathrate hydrate [60]. The solubility of a guest gas in water typically falls with heating between the ice point and T eq and hence is expected to contribute to the build-up of the supersaturation of the guest gas. However, it is unlikely that this effect alone is sufficient to offset the concomitant reduction in the driving force (system subcooling). It was thus very surprising that clathrate hydrates formed more frequently at a higher temperature (just below T eq ) than at a lower temperature (just above the ice point) [60]. The mystery does not end there and goes even deeper; given that (1) ice has been considered an excellent nucleation promotor of clathrate hydrates and (2) melting of ice is endothermic and would cool the sample, at least locally, one would expect that it would have been easier for clathrate hydrate to form just above the melting point of ice (and while the ice was still melting) than just below the T eq of the clathrate hydrate long after the last trace of ice has disappeared. That clathrate hydrate did not form while the ice was still melting but did form at a higher temperature long after the last trace of ice has disappeared suggests that formation somehow became easier at a higher temperature despite its smaller driving force [60]. However, we note that ice does serve as a good nucleation agent for Structure II cyclopentane hydrate for which T eq is only 7.7 K above the ice point under atmospheric pressure [61]. It thus appears likely that ice indeed facilitates the heterogeneous nucleation of Structure II clathrate hydrate to some extent. Both of the two factors of the solubility limitation and the associated mass transfer limitation of guests and the viscous slowdown of the aqueous phase would lower the effective attachment frequency during clathrate hydrate nucleation. Still, the effective attachment frequency would need to be lowered by more than 20 orders of magnitude for the theoretically determined nucleation rate to agree with the experimentally determined nucleation rate. It thus appears clear that a new and comprehensive theory is required that accounts for all the clathrate hydrate-specific issues to bring theoretically and computationally predicted nucleation rates closer to experimental observations. It is not clear at this stage how said new and comprehensive theory might look like. Heterogeneous nucleation of gas hydrates requires not only a solid substrate that could lower the nucleation work and the activation barrier but also ample supplies of guest gases in its vicinity. This unique attribute of gas hydrate systems renders interfacial gaseous states the key concept in the heterogeneous nucleation of gas hydrates. It thus appears likely that such a theory must at least incorporate the disjoining pressure of interfacial gaseous states.

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5.3 The Memory Effect A long-standing mystery in nucleation of clathrate hydrate is the so-called memory effect [27]. The memory effect refers to observations that clathrate hydrates nucleate more easily (within a shorter timeframe at a given constant subcooling or at a smaller subcooling during a linear cooling ramp) in a dissociated water than in a fresh water that has no history of clathrate hydrate formation. The memory effect has important practical implications in a recovery of methane gas from methane-bearing sediments. Since methane hydrate is 85 mol% water, and water is heavy, it is not economical to transport the mined methane in the clathrate form. It is thus preferable to decompose the methane hydrate into methane gas and water by depressurization and only transport the recovered methane gas in a pipeline. Transportation of freed methane gas in the presence of dissociated water is susceptible to reformation of methane hydrate if any section of the pipeline is inside the thermodynamically stable zone of the phase diagram of methane hydrate, and the memory effect would increase the risk by rendering nucleation of methane hydrate easier.

5.3.1 Historical Perspective The early studies on the kinetics of clathrate hydrate formation already encountered the memory effect although the term had not yet been coined back then [31]. It was found that the sample history had some influence on the induction times of clathrate hydrates but not on the growth after the nucleation [31]. Takeya et al. then reported an interesting study that the memory effect was observed for the nucleation of carbon dioxide hydrate in the water produced by melting of ice that had no history of carbon dioxide hydrate formation [62]. The memory effect was destroyed at 298 K that corresponded to 25 K of superheating [62]. Rodger reported a molecular dynamic simulation study which showed that mostly ice-like clusters, not clathrate-like structures, had formed after dissociation of methane hydrate [63]. He suggested that the memory effect was due to the suppressed diffusion of methane gas in the dissociated water, which would leave the dissociated water supersaturated for longer with the methane gas. Such persisting supersaturation of methane gas would render the nucleation of methane hydrate easier when the dissociated water is brought back inside the thermodynamically stable zone of the phase diagram of methane hydrate [63]. A distinct attribute of the memory effect is that it varied among an ensemble of samples that had the same thermal history. Ohmura et al. studied the memory effect of HCFC-141b hydrate from the dissociated water at the superheating of 0.5, 1.0, and 1.5 K [64]. Notably, they reported that the level of memory preservation differed from sample to sample despite the same thermal history of the samples [64]. Zeng et al. suggested that the memory may be imprinted on a solid surface that is present

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in the system [47]. Sowa and Maeda reported statistically significant variations in the level of memory preservation that depended on the sample cells used [65]. Buchanan et al. suggested that the memory effect reported in the literature was in systems which did not attain full equilibrium after dissociation [66]. The later neutron diffraction studies by the same authors suggested that the memory effect could be due to a greater local water density that was influenced by the presence of clathrate hydrates [67]. More recently, He et al. reported accelerated formation of methane hydrate and CO2 hydrate in “used water” when the water was pressurized with the respective guest gases from 1 atm and suggested that the residual structure present in water as the reason for the memory effect [68]. Wu and Zhang proposed that the residual structure in liquid water after clathrate hydrate has dissociated had provided a mass transfer barrier between water and the guest molecules that led to the memory effect [69]. Sefidroodi et al. examined the formation of cyclopentane hydrates from dissociated water over a range of superheating temperatures of up to 5 K and the heating time of up to 24 h [70]. They reported that a transfer of a small amount of dissociated water to fresh water induced a memory effect that was comparable to one in a 100% dissociated water [70]. A residual clathrate hydrate structure was proposed as a possible mechanism of the memory effect from this observation [70]. The intrinsic stochasticity that is inherent to nucleation was always superimposed to the systematic difference in the induction times that had arisen from the memory effect. This limitation has posed considerable difficulty in affirming the existence of the memory effect or quantifying its magnitude. Bylov and Rasmussen reported that the induction times of methane hydrate and natural gas hydrate did not become shorter when formed from the dissociated water, while noting that the limited amount of their data made it impossible to draw a definitive conclusion [71]. Fandino and Ruffine studied the formation of methane hydrate at 10 and 19.5 MPa and suggested that the stochastic nature of clathrate hydrate nucleation overcame the memory effect [72]. They found no evidence of any statistically significant decrease in the induction times at 7 K of superheating for 5 h [72]. Sefidroodi et al. suggested that the smallest of the superheating temperatures of 6.5 K used by Fandino and Ruffine might already have been high enough to destroy the memory effect [70]. Based on unprecedented numbers at the time of several thousand nucleation events, Sowa and Maeda concluded that the presence of the memory effect was undeniable (statistically beyond doubt) [65]. Wilson and Haymet studied the nucleation of tetrahydrofuran (THF) hydrate at superheating temperatures of up to 15 K using an Automated Lag Time Apparatus (ALTA) and round no evidence for a presence of statistically significant memory effect [73]. However, unlike typical guest gases of clathrate hydrates, THF is miscible with water at all proportions, so any mass-transfer limited issues that are typical for clathrate hydrates would not arise. For example, supersaturation of THF in a dissociated water cannot arise.

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5.3.2 Major Attributes of the Memory Effect and Proposed Hypotheses Several attributes of the memory effect stand out from the historical findings: (1) the memory effect is about nucleation of clathrate hydrates and does not influence the growth rate of clathrate hydrates that comes after nucleation [31], (2) the reproducibility of the memory effect is poor and the variation shows stochastic nature of the memory effect [71, 72], much like the stochastic nature of the nucleation process itself, (3) the level of memory preservation depends on specific samples even when they have the same thermal history [47, 64, 65], (4) no molecular structures that differ from the ordinary clathrate hydrates have been detected in the dissociated water, (5) melted ice that has no history of clathrate hydrate formation can also induce the memory effect that is comparable to dissociated water made from clathrate hydrate [62], (6) the sample memory can be erased after the dissociated water is heated far above the thermodynamic phase boundary and/or for prolong period (after a sample has attained its thermodynamic equilibrium) [27], and (7) transfer of a small amount of dissociated water to fresh water still can induce the memory effect that is comparable to that in 100% dissociated water [70]. Clearly, some aspect(s) of dissociated water has not attained thermodynamic equilibrium while the memory effect is present but it has been hard to pin down what that factor may be. As for the possible mechanism of the memory effect, five major hypotheses have been put forward over the years in the literature (1) structural memory hypothesis [62, 68–70, 74, 75], (2) guest supersaturation hypothesis [63], (3) impurity imprinting hypothesis [47], (4) nanobubbles hypothesis [76, 77], and most recently, (5) interfacial gaseous states we detailed in Chap. 4 [78, 79], which we may refer to as “interfacial gaseous states hypothesis” in this book. The structural memory hypothesis invokes molecular level structuring of H2 O molecules in the dissociated liquid water that is different from the structure of the ordinary liquid water [62, 68–70, 74, 75]. As we saw in Sect. 5.1, ordinary liquid water is already highly structured due to the hydrogen bonding compared to other ordinary liquid. The structural memory hypothesis assumes an even greater level of molecular level structuring in the dissociated liquid water. The guest supersaturation hypothesis postulates that the memory effect is due to the retarded diffusion of the guest gas in the dissociated water, which would leave the dissociated water supersaturated with the guest gas. Such persisting supersaturation of the guest gas would render the nucleation of clathrate hydrate easier when the dissociated water is brought back inside the thermodynamically stable zone because the real driving force for the nucleation of the clathrate hydrate is greater. The impurity imprinting hypothesis postulates that the memory is somehow imprinted to solid impurities in a sample [47]. A solid wall of the container that contains the sample can also serve as an impurity in that the solid is made of a foreign material. The hypothesis is consistent with the general observations that the level of memory preservation differed from sample to sample (that were usually contained in different sample cells or after non-identical cleaning procedures of the

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sample cells) despite the same thermal history of the samples. The mechanism of the memory imprinting is not known, nor is the mechanism as to how the imprinted memory can be erased by moderate heating of the sample. The nanobubbles hypothesis postulates that the very high level of supersaturation of the guest gas after the dissociation of clathrate hydrate would lead to formation of nanobubbles in the bulk of the supersaturated solution [76, 77]. Since nanobubbles are small, they would be invisible to naked eye and persist for a long time because they would not float quickly due to buoyancy. The interfacial gaseous states hypothesis postulates that the high level of supersaturation of the guest gas after the dissociation of clathrate hydrate would lead to formation of interfacial gaseous states on solid walls of a container [78, 79] because heterogeneous nucleation of bubbles on a solid surface is always energetically favorable to homogeneous nucleation of bubbles in the bulk of a solution [80]. Of course, such heterogeneous nucleation or cavitation could also occur on small, sub-micrometer sized contaminant particles in the liquid which would be hard to distinguish from homogeneous nucleation experimentally. Homogeneous nucleation of nanobubbles or cavitation has been known to become virtually irrelevant in water at normal temperatures that are far below the critical point [80], however, given the very high levels of supersaturation of the guest gases after the dissociation of clathrate hydrates, we cannot totally discount the nanobubble hypothesis. Among these hypotheses put forward over the years, the structural memory hypothesis is the least likely hypothesis of all for two reasons. First, clathrate hydrate formation is a first-order phase transition that is accompanied by the latent heat of fusion and the memory effect was observed after the dissociated water had been raised to a temperature that was several Kelvins higher than the thermodynamic equilibrium dissociation temperature (superheating temperature). If the memory effect had indeed been caused by some sort of residual structure in the dissociated water, it would be difficult to comprehend how it was possible to raise the temperature of the dissociated water to above the thermodynamic equilibrium dissociation temperature while preserving the residual structure. Second, evidence is mounting that there are significant variations in the level of memory preservation among an ensemble of samples that had the same thermal history. Here, either the surface history of container walls was different or different sample cells were used. One would expect the level of memory preservation to be independent of the solid container walls if the memory effect had indeed originated from some form of residual structure in the aqueous phase. We note at this stage that the finding that transfer of a small amount of dissociated water to a sample of fresh water still induced the memory effect that was comparable to the memory effect found in a 100% dissociated water [70] does not necessarily support the structural memory hypothesis. Other entities like imprinted solid impurities, supersaturated domains of the dissociated solution or nanobubbles could be transferred together with the structure of the dissociated water, regardless of whether the structure of the dissociated water differs from that of fresh water or not.

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The hypotheses (2), (3), and (4) are all plausible to some extent but still have some trouble in explaining some of the known attributes of the memory effect by themselves. On the one hand, the guest supersaturation hypothesis and the nanobubble hypothesis do not account for a major attribute of the memory effect that it varied among an ensemble of samples that had the same thermal history. Since both of these hypotheses place the origin of the memory effect to the bulk of the dissociated water, there would be little reason to expect that the walls of the container that contain the dissociated water would make a difference. On the other hand, the impurity imprinting hypothesis does not account for the primary attribute of the memory effect that the memory effect vanishes (the memory is erased) after the dissociated water is heated 10–15 K above the thermodynamic phase boundary and/or for prolong period. It is not at all clear how the perceived memory, once imprinted to a solid wall, could be erased by simply heating the sample by modest 10 or 15 K above the superheating temperatures that have failed to erase the memory effect. Maeda recently found that the memory effect was absent in a quasi-free water droplet suspended in squalene and concluded that a solid wall was required for the manifestation of the memory effect [78]. This result alone would support the impurity imprinting hypothesis. However, given the difficulty of the impurity imprinting hypothesis detailed above, Maeda proposed that interfacial gaseous states we detailed in Chap. 4 were the source of the memory effect (the interfacial gaseous states hypothesis) [78]. Guo et al. then directly detected interfacial gaseous states on the solid walls in dissociated water [79], which further supported the interfacial gaseous states hypothesis. Importantly, the interfacial gaseous states hypothesis could explain why the memory effect has been so elusive (hard to positively affirm its existence) in the past studies, let alone quantify its magnitude. Formation of a bubble or cavitation is a heterogeneous nucleation process that generally depends on the surface chemistry and the surface roughness of the solid wall present [80]. Then, the formation of interfacial gaseous states, after the dissociation of clathrate hydrates, is also itself a heterogeneous nucleation process, as we detailed in Sect. 4.2. Thus the probability of formation of interfacial gaseous states after the dissociation of clathrate hydrates likely varies for different sample cells or vessels. In a static system, clathrate hydrate forms at the surface of water and its dissociation will render the top layer of water supersaturated with the guest gas. Then, a section of the solid wall from the top of the meniscus of the dissociated water down to the thickness of the clathrate hydrate layer before it has dissociated would become susceptible to the formation of interfacial gaseous states. It is not difficult to envision that the water level would change for different measurements that use the same sample cell or container, and consequently expose different stripe sections of the solid wall of the container to the “interfacial nanobubble-susceptible regions” (Fig. 5.3). Even for a given measurement in a given sample cell or container, the water level might vary with time due to evaporation or other causes (e.g., stirring action in some experimental systems). The possibility that at least some sections of the solid walls in the system become susceptible to the formation of interfacial gaseous states

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Interfacial nanobubble –susceptible region

Water

Interfacial nanobubbles & other interfacial gaseous states

Fig. 5.3 Schematic illustration of interfacial nanobubble-susceptible regions just below the threephase lines. The original figure is from Ref. [78], reproduced with permission from the American Chemical Society

increases with the system size, which would increase the likelihood of encountering the memory effect. The presence of a hydrophobic solid wall in a system (e.g., some stirrers have Teflon coating) would render the formation of interfacial gaseous states after the dissociation of clathrate hydrates still more likely.

5.3.3 Remaining Loose Ends A primary remaining loose end of the interfacial gaseous states hypothesis is that the memory can be erased by simply heating the sample by modest 10–15 K above the thermodynamic phase boundary, which incidentally correspond to only several Kelvins warmer than the superheating temperatures that have failed to eradicate the memory effect. Can the interfacial gaseous states be eradicated by simply heating the solid walls by additional several Kelvins above the superheating temperatures that have failed to eradicate the memory effect? To answer this question, we need to take a close look at what may happen when clathrate hydrate dissociates. The dissociation of clathrate hydrate will first build-up a massive supersaturation of the guest gas in the surrounding aqueous phase because the solubility of the guest gas is much lower than the gas content in the clathrate form that is now released. Formation of interfacial gaseous states will consume a part of the released guest gas and lower the massive guest gas supersaturation in the process. The number and/or the size of these interfacial gaseous states will keep increasing and the supersaturation level of the guest gas in the surrounding aqueous phase will keep falling, until the guest gas supersaturation reaches a level that can coexist with these interfacial gaseous states. Some of the interfacial gaseous states might grow so

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large that they may detach from the substrate due to buoyancy and be removed from the system. Shinoda showed that freezing of a non-gaseous, non-polar solute in a binary solution decreased the solubility of the solute whereas freezing of the solvent (such as ordering of water around non-polar solutes) in a binary solution increased the solubility of the solute [81]. Consequently, the solubility of a non-gaseous, nonpolar solute does not monotonically increase with heating but becomes rather constant with temperature between the melting point of water and room temperature [81]. Although this mechanism also applies to gaseous, non-polar solutes in water, the solubility of a gas in a liquid usually decreases with heating because the entropy of the solute molecule is expected to be much greater in the gaseous state than in the dissolved state and that the entropic contribution to the system free energy increases with temperature. The effect of ordering of water around nonpolar solutes thus makes the solubility of nonpolar liquid solutes to be rather temperature independent, but the solubility of nonpolar gaseous solutes in water surely decreases with heating in this temperature range. Methane, whose critical pressure of about 4.6 MPa, might behave a gaseous solute or a liquid solute depending on the pressure. The solubility of inert gases such as rare gases in water typically reaches a minimum at around 300 K and then increases with heating at higher temperatures [82–84]. At elevated pressures for which clathrate hydrate formation is possible, the solubility of methane in water decreases with heating up to at least 313 K for the pressures of up to 15 MPa [85]. On the other side, the solubility of methane in water starts increasing with heating above 71 °C (344 K) for pressures greater than 20 MPa [86]. Unfortunately, the precise location of the minimum is not clear in the literature that reported the solubility of a guest gas in water at elevated pressures. Meanwhile, the thermodynamic equilibrium dissociation temperature of clathrate hydrate depends on the guest composition and the system pressure and is up to about 300 K [27]. Up until the temperature at which the solubility becomes a minimum, the solubility of the guest gas decreases with heating. The interfacial gaseous states will not go away while the solubility of the guest gas in the surrounding aqeueous phase keeps falling. Above the expected minimum solubility temperature, however, any further heating will cause the solubility of the guest gas in water to increase. The interfacial gaseous states will then be deflated as the guest gas contained therein will be dissolved into the surrounding aqueous phase. Therefore, heating to above the expected solubility minimum of the guest gas would be required to erase the memory effect. That the expected minimum solubility temperature of a typical guest gas appears to be slightly higher than the thermodynamic equilibrium dissociation temperature of clathrate hydrate renders the interfacial gaseous states hypothesis still more plausible. Another potential remaining loose end of the interfacial gaseous states hypothesis could be that the memory effect was reported to occur in melted water from ice that has no history of clathrate hydrate formation [62]. How can interfacial gaseous states influence the memory effect of melted ice with no history of clathrate hydrate formation? The answer might lie in the fact that freezing effectively degasses water (i.e., it is impossible to freeze water without degassing it) [87].

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When water freezes, dissolved gases (air) are excluded from the ice lattice. Some may be trapped between the grains of polycrystalline ice as fine bubbles which may be released from the trapped spots when the ice was subsequently melted [87]. Others may form an interfacial gaseous state because the heterogeneous nucleation of a bubble on a sloid wall is always easier than the homogenous nucleation of a bubble in water [80]. When such a sample is subsequently warmed, the melting generally takes place from the inner wall of a container because the wall should be the warmest. Importantly, the solubility of air is low in the warmer part of the water (next to the walls) than in the colder part (near the center of the container). Then the fine bubbles that have been trapped between the grains of polycrystalline ice will not immediately dissolve back into water when released, especially for those near the walls. A simple estimate shows that it will take many hours for the air to dissolve back into the degassed water [88] and the lifetime of interfacial gaseous states are surprisingly long [89]. Plus, warming of a wall has been known to induce interfacial nanobubbles [90]. In short, (1) the air in interfacial gaseous states would be the last to disappear among all the air bubbles in the container and (2) it will take many hours for them to do so. If such a sample were subsequently exposed to pressurized guest gas, the guest gas would promptly diffuse through the interfacial gaseous states made of air (the guest diffusion is expected to be faster in air than in water) and stabilize these interfacial gaseous states. That Takeya et al. indeed found that the nucleation rate of CO2 hydrate decreased (1) when their non-treated water was degassed or (2) when their melted water was heated to 298 K for 1 h, is in fact consistent with the interfacial gaseous states hypothesis [62].

5.4 Effects of Electrolytes 5.4.1 Introduction An aging oil field or gas field produces progressively more water with time as the reservoir pressure falls. Such oilfield waters are generally not fresh but mineralized. Oilfield waters can contain much higher concentrations of minerals than seawater does, as high as up to 10 times (for comparison, typical salt concentrations of seawater is 3.5 wt%). Dissolved cations commonly found in oilfield waters are Na+ , Ca2+ , Mg2+ , K+ , Ba+ , Li+ , Fe2+ , Sr2 and common anions are Cl– , SO4 2– , HCO3 – , CO3 2– , NO3 – , Br– , I– , BO3 2– , S2 – . Such oilfield water has different compositions than other brine, even those in the immediate vicinity of that field. Drilling operations, especially offshore, also encounter clathrate hydrate issues in the presence of salts. For these reasons, nucleation of clathrate hydrates in electrolyte solutions is of great practical interest in oil and gas industries. Salts are thermodynamic hydrate inhibitors (THIs) because salts are highly soluble in water. The colligative effect of lowering of water activities through the entropy of

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mixing is proportional to the number of ions and independent of the identity of ion species [91]. However, salts are not preferred as THIs due to the corrosive effects on pipes and production facilities. As a research field, nucleation of clathrate hydrates in electrolyte solutions is in its infancy and very little is understood about the impacts of electrolytes on the kinetics of clathrate hydrate formation. Nucleation of clathrate hydrates in the absence of ions is already complex and the complexities are compounded by another enormously complex issue of the impact of ions on hydrogen bonding and on the structure of liquid water. Since very little is understood about the effects of electrolytes on nucleation of clathrate hydrates beyond the ubiquitous thermodynamic inhibition, we here only briefly summarize various issues that are relevant in future research effort on the subject.

5.4.2 Locations of Ions in a Salt Solution Global electro-neutrality means that equal numbers of opposite electric charges must exist in a given closed system. How these electric charges are spatially distributed in a given system is a complex problem. Each electric charge generates an electric field around it and at the same time feels the forces by the electric fields generated by other electric charges. The electromagnetic fields generated by electric charges are governed by Maxwell’s equations and the motions of the electric charges due to the electromagnetic fields are governed by Newton’s equations of motion [92]. Maxwell’s equations are ρ ε

(5.4.1)

∇·B=0

(5.4.2)

∇·E=

∇×E =−

∂B ∂t

∇×H =i+

∂D ∂t

(5.4.3) (5.4.4)

where E is the electric field, ρ is the electric charge density, B is the magnetic field, ε is the dielectric permittivity, μ is the magnetic permeability, i is the electric current density, and B = μH and D = εE. The first two Maxwell’s equations are Gauss’s laws for electric and magnetic fields, respectively. The third is Faraday’s electromagnetic induction and the last is Ampere’s law. These laws need to be combined with the equations of motion for each ion and the resulting motion of each ion will then need to be fed back into Maxwell’s equations. These sets of equations are enormously complex multi-body problems and not

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solvable. To tackle the large numbers of ions in a salt solution and the enormously complex interplay among them, a number of simplifications and approximations have been introduced. Such first approximation is typically electrostatics; neglecting of the timedependent terms greatly simplifies Maxwell’s equations. The resulting Poisson’s equation relates the electric charge density, ρ, to the electrostatic potential, φ, and the dielectric permittivity, ε [92]; φ = −

ρ ε

(5.4.5)

Though simplified, Poisson’s equation is still not solvable in general cases. In colloid science, additional models and approximations are typically introduced; e.g., the Gouy–Chapman model that decouples the electric charge of a colloidal particle to electric double layer that consists of the Stern layer and the diffuse layer, and the Boltzmann distribution of ions in an electrostatic potential, that lead to the Poisson– Boltzmann equation [93]   all 1  −zeφ φ = − nze exp ε0 εr j kT

(5.4.6)

Here the summation is over all the electric charges, n is the number of the charges, z is the valency of each electric charge in question, e is the elementary charge (1.602 × 10−19 coulombs), k is the Boltzmann constant, and T is the absolute temperature. The Poisson–Boltzmann equation is still not generally solvable, so yet additional approximations such as the Debye–Hückel approximation (linearization approximation) for dilute electrolytes are typically introduced, which eventually leads to an exponentially decaying electrostatic potential from an electrically charged colloidal particle [93]. A consequence is that, when time averaged, more cations will be found around a given anion, and vice versa.

5.4.3 Gibbs Adsorption Isotherm Adsorption is attachment of atoms, ions, or molecules from a gas or liquid to an interface. By definition, adsorption can only occur in a multi-component system. A particular case of our interest in this section is adsorption of a solute from a solution to an interface. The concentration of the solute in a solution (that is below the solubility limit) is assumed to be uniform throughout the solution phase except for next to an interface. A real interface is not mathematically sharp. The density profile of a pure liquid at a surface is not a step function that jumps from that of the liquid to that of the vapor, which would render its spatial derivative a delta function. Instead, the density varies

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more gradually albeit over a narrow range. Then other thermodynamic properties also vary gradually over the narrow range. The question is how to treat such situations. The Gibbs convention is to define a Gibbs dividing surface so that the thermodynamic properties of one phase continue up to one side of the Gibbs dividing surface and the thermodynamic properties of the other phase continue from the other side up to the Gibbs dividing surface [94]. Then, in a two-component system that consists of a solvent and a solute, the concentration of the solute is constant in each of the two phases divided by the Gibbs dividing surface. Surface excess, , is defined as the difference between the total mole in a system and the sum of the moles in each of the two phases divided by the Gibbs dividing surface [93]. The Gibbs convention of selecting the position of the Gibbs dividing surface is such that the surface excess of the solvent becomes zero. The Gibbs adsorption isotherm relates the surface excess to the change in the surface tension as [94] dγ = −Γ dμ

(5.4.7)

where γ is the surface tension,  is the surface excess and μ is the chemical potential of a solute. Practically, it is convenient to express the chemical potential in terms of the bulk solute concentration, C, in the case of an ideal solution [91] dγ = −Γ kT d(ln C)

(5.4.8)

where k is the Boltzmann constant and T is the absolute temperature. The Gibbs adsorption isotherm shows that (1) the surface excess is positive when the surface tension decreases with increasing solute concentration in the bulk solution phase, and vice versa, (2) the greater the change in the surface tension for a given change in the solute concentration the greater the size of the surface excess. For example, the surface tension of a surfactant solution decreases with the surfactant concentration. The Gibbs adsorption isotherm shows that surfactants positively adsorb to a water surface. The greater the reduction in the surface tension for a given amount the more surface active the surfactant is. In contrast, the surface tension of a salt solution typically increases with the salt concentration. The Gibbs adsorption isotherm shows that salts negatively adsorb to a water surface, that is, less salts are present next to a water surface than they are deep inside the bulk of the salt solution. In microscopic terms, an electric field is induced when an electric charge approaches an interface where there is a discontinuity in the dielectric constant [92]. A graphically convenient way to describe such an induced electric field by an interface is to imagine that an image charge is induced inside the medium at the opposite side of the interface to which the real electric charge is approaching [92]. The consequence of such an image charge force is that an electric charge such as an ion is repelled from a gas–aqueous interface as it approaches from the aqueous phase, and the repulsive force becomes progressively stronger the closer the electric charge approaches the interface. Consequently, the ion concentrations

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near a gas–aqueous interface are lower than those in the bulk solutions, and the topmost layer of the aqueous phase is almost totally devoid of ions.

5.4.4 Concentration Gradient of Guest Gases Near a Salt Solution–Guest Gas Interface It has been reported that some THIs like methanol [95], ethylene glycol [96] and propanol [97], could enhance the rate of clathrate hydrate formation when doped at low concentrations. Farhang et al. reported a similar promotion effect of dilute sodium halide solutions for CO2 clathrate hydrate [98]. They reported that a variation in the concentration from 10 to 100 mM of the sodium halide solutions induced a transition from promotion to inhibition of CO2 hydrate formation. A promotion effect of CO2 hydrate formation was observed most acutely for sodium iodide, while sodium chloride was found to be an inhibitor at all concentrations studied [98]. These promotion effects cannot arise from the thermodynamic effect of lowering of the activity of water by the solutes and thus must be kinetic in nature. Ions could impact the nucleation of clathrate hydrates through the local structuring of water molecules around each ion (hydration). The hydrophobic hydration of nonpolar entities like alkanes impacts the local water structure differently than the hydration shells of hydrophilic solutes like ions. Hydrophilic ions generally put stresses onto the tetrahedral network of hydrogen-bonded water molecules [99]. The higher the salt concentrations the weaker the tetrahedral structure and the lower the hydration number [100]. In contrast, hydrophobic (non-polar) solutes like alkanes enhance the tetrahedral network with larger numbers of hydrogen bonds [101–103]. It is thus reasonable to expect that such different tetrahedral networks of hydrogen-bonded water would influence the nucleation rate of clathrate hydrates. As we saw in the previous sections, clathrate hydrates nucleate at an aqueous– guest gas interface. For ions to influence the nucleation of clathrate hydrates, they need to be present in the vicinity of an aqueous–guest gas interface. Even though the guest gas diffusion penetrates many micrometers from the topmost surface, the guest concentration is the highest at the topmost surface and gradually diminishes deeper into the bulk water. The presence of turbulence or rigorous stirring may facilitate mixing of guest gases into the aqueous phase and reduce the steepness of the concentration gradient of the guest gases. Nevertheless, the concentration gradient cannot be inverted. The driving force for nucleation is related to the nucleation rate by the Arrhenius law for a given supersaturation of the guest gas, as we saw in Chap. 1. The exponential dependence of the Arrhenius law means that the nucleation probability will diminish rapidly as the supersaturation falls. And yet the topmost layer of the aqueous phase where the guest gas supersaturation is the greatest could almost be totally devoid of ions due to the negative adsorption, as described by the Gibbs adsorption isotherm. If this were the whole story, then the influence of salts on the nucleation rate of clathrate

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hydrate would be limited to the thermodynamic inhibition effect of lowering the activity of water. However, this is not the end of the story. The surface tension of a salt solution typically goes down at low concentrations (1 mM and below) from that of pure water whereas it goes back up at higher concentrations [104]. This rather peculiar phenomenon is known as the Jones–Ray effect [104, 105]. The Jones–Ray effect appears ubiquitous and has been known for more than 80 years, and yet the underlying physical mechanism remains unclear to this day. According to the Gibbs adsorption isotherm, the Jones–Ray effect suggests that ions positively adsorb at low concentrations and negatively adsorb at high concentrations. In other words, topmost layer of the aqueous phase where the guest gas supersaturation is the greatest is rich in ions when the salt concentrations are low. A related matter is that some large anions have greater affinity to an aqueous surface than other ions in spite of their negative surface excesses. This individuality or uniqueness of each ion species is called surface propensity or ion specificity [106– 114]. For example, iodide ions have a greater surface propensity than chloride ions. Such ion specificity is expected to influence the nucleation of clathrate hydrates.

5.4.5 Salting-Out Effect of Ions Another factor that impacts the nucleation of clathrate hydrates in electrolytes is the salting-out effect of ions. The solubility of a hydrophobic, non-polar solute decreases when salts are added to water and causes the non-polar solute to precipitate out of the aqueous phase. This effect is called the salting-out effect and proportional to the ionic strength. The salting-out effect has a broad range of implications, from denaturing of proteins to the effectiveness of detergency. Since most guest gases are non-polar and hydrophobic, the salting-out effect is expected to further reduce the (already low) solubility of guest gases. Lu et al. showed that a salt of a stronger ionic strength (and hence stronger saltingout effect), 1M MgSO4 , resulted in much less thermodynamic inhibition of methane hydrate than a salt of a weaker ionic strength (and hence salting-out effect), 2M NaCl [115]. The chemical potential of the guest gas, and hence the driving force for nucleation, does not change with the addition of a salt because it is identical to the chemical potential in the gas phase (the chemical potential of water does change). However, the kinetics is expected to be proportional to the absolute amount of the guest gas molecules present in the salt solution especially in the topmost layer of the aqueous phase, as expected from the principle of detailed balance we detailed in Chap. 1. Since there are twice as many numbers of ions in 2M NaCl than in 1M MgSO4 , Lu et al.’s results suggest that the numbers of ions are a much more important factor than the ionic strength of the salt solution in the inhibition of clathrate hydrates (i.e., the colligative effect dominates).

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5.4.6 Experimental Observations Given that the nucleation of clathrate hydrates in the absence of salts is still full of unresolved issues as detailed in Sect. 5.2, one may conclude that it would be premature to investigate the impact of salts on clathrate hydrate nucleation. Still, some limited progress has been made. The basic idea behind such endeavor was that any insight obtained from the investigations of clathrate hydrate nucleation in the presence of salts might shed new light to the general issues of clathrate nucleation. Sowa et al. found that (1) concentrated (>1 M) strong monovalent salt solutions acted as thermodynamic inhibitors as expected, (2) some strong monovalent salts might kinetically promote natural clathrate hydrate formation at low concentrations (