Substantia. An International Journal of the History of Chemistry 2(1): 7-16, 2018 Firenze University Press www.fupress.com/substantia ISSN 1827-9635 (print) | ISSN 1827-9643 (online) | DOI: 10.13128/substantia-37 Citation: S.E. Friberg (2018) Emulsion Thermodynamics – In from the Cold. Substantia 2(1): 7-16. doi: 10.13128/ substantia-37 Copyright: © 2018 S.E. Friberg. This is an open access, peer-reviewed arti- cle published by Firenze University Press (http://www.fupress.com/substan- tia) and distribuited under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All rel- evant data are within the paper and its Supporting Information files. Competing Interests: The Author(s) declare(s) no conflict of interest. Feature Article Emulsion Thermodynamics – In from the Cold Stig E. Friberg Ugelstad Laboratory, NTNU, Trondheim, Norway Email: stic30kan@gmail.com Abstract. Thermodynamics has played virtually no role in traditional emulsion research, because emulsions are inherently thermodynamically unstable. The problem with commercial emulsions needing to exist with none or only small changes during use and the industrial stability problem was resolved by formulating colloidally stable emulsions, i.e. the rate of destabilization was reduced. This approach was successful for single-oil emulsions, but encountered problems for double emulsions, for which the simultaneous stabilization of several interfaces within one drop encountered difficul- ties. Naturally, even for such an emulsion, colloid stability is the only option to sta- bilize the outer surface towards the continuous phase. In fact, the destabilization by flocculation/coalescence proceeds similarly to a single-oil emulsion. But experiments have demonstrated that complex emulsions with a thermodynamically stabilized inner interface retain the individual drop topology during the process. This result opens an avenue to significantly facilitate the formulation of a group of commercially important emulsions, because the cumbersome multiple emulsion stabilization is reduced to the more trivial single-oil emulsion case. Keywords. Emulsion stability, colloidal stability, interfacial thermodynamics, janus emulsions, emulsion coalescence. INTRODUCTION The first meal humans and many animals enjoy is an oral emulsion that replaces the nutrients through the umbilical cord. The emulsion in question is a convenient and efficient means to provide the needed mix- ture of fats, sugars and proteins, which otherwise would be solid and not useful to a toothless newborn. Since these emulsions are truly essen- tial, baby milk emulsions have been commercially available for 150 years for the cases, where the mother does not have milk, or does not want to breast feed. The initial modest introduction by Henry Nestlé has grown to a company with almost $100 billion in yearly sales, reflecting the com- mercial expansion to other food emulsions, such as cream, butter and ice cream, to mention the most obvious (Among the miniscule, but common, applications of milk is to remove the bitter taste from coffee - and in some societies from tea - an insult to the refined Oriental cuisine culture). In addition, emulsions also have a long history in personal care and cosmet- ics with Cleopatra taking baths in donkey milk about 2000 years ago. 8 Stig E. Friberg This pioneering use of emulsions has developed into a Schueller/Bettencourt company, L’Oreal (actually part- ly owned by Nestlé), which has grown from a mod- est start in 1909 to an internationally large company with sales at the level of tens of billions of dollars. These are examples of large applications of emulsions for humans, like emulsions in the paint and coatings industry, which add another huge sum. All these tremendous volumes of products, which are daily handled all over the world, are, in fact, ther- modynamically unstable compositions, doomed to final separation of the components. Nonetheless, the compo- sitions are required to remain virtually unchanged for specific times, varying from seconds to years, depend- ing on the actual purpose. Hence, an extensive volume of research1-3 is found on their properties and espe- cially their “stability”. This “stability”, is defined as an unchanged appearance reflecting the commercial needs. A more quantitative measure of emulsion stability would be its half-life, e.g. the time for the number of drops to be reduced to one half by coalescence, Figure 1. As a contrast, the so called microemulsions are not emulsions; they are equilibrium colloidal solutions4,5 and have been investigated for different aspects of their interfacial properties.6-8 Because they are thermody- namically stable, their preparation is completely differ- ent from that of emulsions. These latter are prepared, in most cases, by mechanically dividing a liquid into drops and dispersing these in a continuous liquid, while mircroemulsions are spontaneously formed,5 requir- ing only the most minute mixing for large volumes. The microemulsions are of considerable commercial interest, (reflected in about half a million Google hits) but are not the subject of the present contribution, which is focused on emulsions. Their comprehensive and advanced treatments1-3, in turn, examine emulsion stabilization from the viewpoint of colloidal stability, applying the concepts from the DLVO theory,9-11 in which a repulsive force is introduced between drops to reduce the rate of flocculation and coa- lescence. This action naturally does not make the emul- sions thermodynamically stable, but tender serviceable information to retain the properties of traditional emul- sions for commercial purposes. This colloidal stabil- ity approach was also employed for double emulsions,12 whose more complex topology caused challenging stabil- ity problems.13,14 In summary, except for in the area of microemul- sions, there was no significant need, nor even a role, for thermodynamics in emulsion research; a mindset that has recently experienced an initial conversion, when Janus emulsions were introduced.15,16 These emulsions consist of combined drops of two mutually insoluble oils, Figure 2, in which total free interfacial energy of the single drop is significantly less than that of the oils in separate drops. Hence, their “inner” topology, Figure 2, is thermodynamically stabilized,17-19 and actually has been shown to survive most of the flocculation/coales- cence process.20 A brief comment at this stage is necessary to avoid misinterpretation. Janus emulsions are not thermody- namically stable, like microemulsions. The Janus emul- sions, as a contrast, undergo a flocculation/coalescence process and, if left for sufficiently long time, will separate into three, or four liquid layers, depending on the rela- tive density, because of gravitational forces. Neverthe- less, the feature mentioned in the last part of the former paragraph is significant, because it brings to light a new approach to formulating emulsions with two-oil drops. Traditionally, commercial double emulsions are pre- pared either by first forming an oil/oil emulsion followed Figure 1. Two emulsion drops of one oil, A, are colloidally stabi- lized and, when they aggregate (Flocculation, A to B), they form an aggregated drop of transitory stability, B. The transient interface dividing them (white line, B) is destabilized and coalescence (B to C) occurs to a single drop, C. Figure 2. A. The geometry and interfacial tensions in a Janus drop. B. Black, O1 volume limited by the contact line, White, O2 volume limited by the contact line, Grey, O1 volume penetrating the O2 space from the contact line. 9Emulsion Thermodynamics – In from the Cold by its emulsification in an aqueous phase (O1/O2)/W or by emulsifying one oil in water, followed by emulsi- fication of this emulsion into the second oil plus a final emulsification into an aqueous phase (O1/W/O2/W). All the preparations necessitate independently stabiliz- ing the different interfaces by colloid stabilization; a methodology, beset with a main problem. The transfer of surfactant stabilizers between the liquid phases, which would markedly reduce their stabilizing action, was suc- cessfully overcome for individual emulsions after a sub- stantial amount of outstanding research.13,14 Conversely, in the alternative methodology using thermodynamic stabilization, the stabilizer is allowed freely to equili- brate between phases, as long as the level of surface ten- sions are in the order defined in later sections. With this approach, colloidal stabilization is limited to the outer interface towards the aqueous phase, a simpler problem, which has been extensively treated.1-3 In summary, the focus of the attention is shifted from colloidal stabilization, a kinetic phenomenon, to interfacial tensions, a quantifiable equilibrium entity. The transition from one preparation method to the other requires a change in the mind set and the present con- tribution is an attempt partially to outline the relevant thermodynamic framework. THERMODYNAMIC STABILIZATION AND INTERFACIAL TENSIONS The basis for the thermodynamic framework is the equilibrium of three tensions in one plane, Figure 3. These tensions act at the contact line, Figure 2, and, combined with the relative volume of the two oil lobes, determine the equilibrium topology of the combination drops. As a result, they are central to this contribution and a summary of their ramifications is useful for the continued analysis with appropriate limitation. The fea- tures of Figure 3 are relevant only for a case in which γO1/W < γO2/W + γO2/O1 and γO1/W > γO2/W and γO1/W > γO2/O1. An analysis of the consequences from inequalities, out- side the ones mentioned, is of algebraic interest, but offer no contribution of value to the problem at hand. Instead, the main theme of this examination is to survey the thermodynamic effect on the topology of Janus and double emulsion drops and it is convenient to divide the range of tensions into two parts. The first part of the range covers γO1/W > γO2/W + γO2/O1 (relevant for double emulsion drops), followed by γO1/W < γO2/W + γO2/ O1, defining the equilibrium in a Janus drop, Figure 2. In addition, the intermediate case of γO1/W = γO2/W + γO2/O1 has some features of interest, which will be briefly men- tioned. The stipulation γO1/W > γO2/W + γO2/O1 portends a non-equilibrium spreading of O2 or O1 on W, while γO1/W < γO2/W + gO2/O1 means a stable equilibrium with defined angles β and δ, Figure 2 and 3. For these, one has, with the ratios γO2/W/γO1/W = a and γO2/O1/γO1/W = b, equations [1] and [2]. β = acos((1 + a2 – b2)/2a) [1] δ = acos((1 - a2 + b2)/2b) [2] It should be observed that these equations are appli- cable only to the equilibria in Figure 3, and may be used to calculate the interfacial free energy as such for a selected drop topology, but the calculations fail to identify the thermodynamically preferred topology. This topology is obtained first, when the free energy quantity is contrasted with that of a counterpart. The free ener- gy number per se actually implies a counterpart with no interfacial free energy and the approach would show any selected topology to be thermodynamically disfa- vored and is of no use to judge thermodynamic stability. In addition, the free energy of two-oil emulsion drops also depends on the volume ratios of the two oils and an equitable, but injudicious, choice of a reference topology will result in erroneous conclusions. This fact will later be brought to light. Equations [1] and [2] relate the angles β and δ to tensions at equilibrium, but not in an explicitly illustra- tive manner and a few numbers from a model system are informative as a graphic. Figure 4 shows the limi- tations of the angles β and δ versus the γO2/O1/γO1/W (b) with γO2/W/γO1/W (a) as parameter. The range of the two variables in the figure is limited in order to reflect the conditions in Figure 3. So, are numbers for b > 1 exclud- ed, because they would represent a reorganization of the angles in the figure. The δ limit for a =1 varies as δ = 60 + 30(1 - b) (degrees) for the same reason. The numbers for the d angle shows the develop- ment of the equilibrium/spreading border, Figure 4 C, Figure 3. Angles β and δ for three equilibrium tensions in one plane. 10 Stig E. Friberg with varied γO2/W/γO1/W (a). The artificial δ value for a = 0 (The oils are mutually completely soluble) is a sin- gle point with δ = 0, because, since γO2/W = 0 and γO2/ O1 = γO1/W, the number for angle b becomes irrelevant. In fact, the entire area (except b = 1) denotes spreading. Increasing the γO2/W/γO1/W (a) values from zero, results in an expansion of the equilibrium part of the area, reaching the entire area for a = 1. Any number for a < 1, however close, e.g. 1 - ε (epsilon small) means a 2ε wide area of spreading along the b = 0 axis and ε broad along the axis for maximum δ. After this brief review of the ramifications for the conditions in Figure 3, the following sections analyze the interfacial free energy of a single drop, omitting the emulsion inter-drop dynamics. The analysis of the double emulsion drop is built on the inequality γO1/W > γO2/W + γO2/O1, and the spreading means that the O2/W interface does not exist. The drop instead consists of a larger drop O1/W with an O2/O1 drop inside, in accord- ance with the thermodynamic condition. Conversely, the O/W interface is thermodynamically unstable for virtu- ally all positive γO1/W and an assembly of such drops will coalesce like the simpler single-oil emulsions. Hence, the potential thermodynamic stabiliza- tion is limited to the inner interface of the drop, while the initial coalescence is exclusively concerned with its colloidally stabilized outer surface, as has been shown for Janus emulsions.20 Hence, at a first glance, the ther- modynamic stabilization of the inner interface may be considered only of minor importance, but the extensive research on double emulsions12-14 indicates otherwise. In fact, with the “inner” interface of Janus or double emul- sion drops thermodynamically stabilized, the only stabi- lization needed for a commercial double emulsion would be for the interface towards the continuous phase; i.e. a problem, that has been solved using the colloidal stabil- ity approach.1-3 The only requirement for the O1/W sta- bilizers is that the interfacial tensions obey the stated inequality; a non-specific condition. Hence, considering the future potential for a new line of formulations and the fact that virtually no infor- mation exists, a review of the thermodynamics of both a double emulsion and a Janus drop has merits. The exam- ination is initiated at a double emulsion drop, because of the extensive technical and commercial relevance for such emulsions.12-14 DOUBLE EMULSION DROP Double emulsions have wide use within a number of industries and technologies, as described in a recent and comprehensive review.12 The extensive and high- quality research within the area13,14 has applied the col- loidal stability approach, illustrating the enhanced dif- ficulties, compared to those of simple emulsions. The problem arises, because of the fact that two interfaces have to be independently stabilized.13,14 A surfactant, act- ing effectively on the O/W interface, has to be prevented from diffusing towards the O/O interface and vice versa, reducing its stabilizing action. However, the results of recent studies of the desta- bilization of Janus emulsions20 have demonstrated a sub- stantial effect of interfacial thermodynamic factors to modify the coalescence process of these emulsions. In fact, the Janus topology was retained during coalescence until the very last stages.20 No such experimental results have been reported for double emulsions, but the poten- tial for such an outcome is estimated sufficiently positive to justify a section on their interfacial thermodynamics in the present publication. Hence, the interfacial ther- modynamics of a double emulsion drop is examined to outline its prospective stabilizing effect, with a view towards the effect found for Janus emulsions. The interfacial free energy basis for a double emul- sion drop is the inequality γO1/W > γO2/W + γO2/O1, which, as mentioned, shows non-equilibrium spreading with an appealing application of this condition to estimate the thermodynamic stability of a double emulsion drop. However, this kind of interpretation has to be made with caution, because the term thermodynamic stability is defined only against a specific counterpart. A seemingly Figure 4. Angles β (A) and δ (B) and areas of equilibrium and spreading (C) versus the ratio γO2/O1/γO1/W (b). The ratio γO2/W/ γO1/W (a) is the parameter with the following numbers. , 0.99; , 0.90; , 0.75; , 0.5. The minimum b number for each a is marked with an arrow; , 0.90, dotted; , 0.75, dashed; , 0.5, full line. For , gO1/W implicitly equals unity. 11Emulsion Thermodynamics – In from the Cold attractive such system is two separate drops of the single oils, but leads to thermodynamic contradictions, show- ing this stabilization outside the mentioned inequal- ity. Nonetheless, such a choice per se has distinct alge- braic interest, combining a well delineated interfacial free energy and an easily comprehended connection to the destabilization of physical emulsions. The follow- ing evaluation focuses on the overall free energy differ- ence between a double emulsion drop and two single-oil drops. The volume fraction of oil 2 in the drop is vO2 and the interfacial free energy of the drops are offered in Table 1. Table 1. Interfacial free energy of double emulsion drops. Configuration, drops IFE, O1 IFE, O2 Double drop 4π(0.75/π)(2/3)γO1/W 4π(0.75vO2/π)(2/3)γO2/O1 Separate drops 4π(0.75(1 –vO2)/π)(2/3)γO1/W 4π(0.75vO2/π)(2/3)γO2/W Regrettably, there is no direct algebraic expression for the relationship between free energy and volumes for the calculation.18,21 Instead a realistic example was selected to illustrate the variation in free energy during a coalescence process. The particular emulsion consists of 1.09.109 internally thermodynamically stabilized double emulsion drops, each with a volume of 4.188.10-15 m3. The two oils O1 and O2, with interfacial tensions 0.004N/m (γO1/W), 0.00262 N/m (γO2/W) and 0.00116 (γO1/O2) each occupy one half. The drops are coalesced, two and two, 30 times, leaving only one final drop with a volume of 4.05.10-6 m3 and retained topology. The coalescence cov- ers the free energy change due to interface size increase of both the outer sphere and the inner one, of which the former contributes a majority of the free energy reduc- tion. As is obvious and expected, the emulsion interfa- cial free energy is exponentially reduced, Figure 5, dur- ing coalescence; a reflection of the thermodynamic over- all instability of emulsions; even when the drops contain more than one interface. The overall reduction in interfacial free energy is certainly expected, but a more essential issue is a com- parison of the free energy change, when two double emulsion drops coalesce into one double emulsion drop or to two single oil drops, O1 and O2, during the coa- lescence. Figure 6 depicts this difference between inter- facial free energy change from two Janus drops, when a double emulsion drop, , or two separate drops, , form. The results in Figures 5 and 6 are unequivocal; the coalescence to a single double emulsion drop implies an expected reduction of free energy, while the alterna- tive means an increase. These results are remarkable and conclusions would unquestionably be tempting; both about the general validity of the result and vis-à-vis the technical and commercial effects. Nevertheless, such inferences would be premature at this stage, because a more complex geometry after coalescence will, in some cases, lead to a modified result. Even so, the results encourage future experimental and numerical evalu- ations. Leaving that aspect temporarily aside, even the more fundamental aspects offer some unexpected results, illustrated by the ratio, RC/D between the interfa- cial free energy of the combination drop and that of the two separate drops, equations [3] and [4]. vO2 is fraction of O2 volume. RC/D = IFEcomb dr/IFEsep dr [3] -1,00E-04 1,00E-04 3,00E-04 5,00E-04 7,00E-04 9,00E-04 0 10 20 30 Coalescence step Figure 5. The difference in interfacial free energy (See text). Figure 6. The free energy changes, when two double emulsion drops (Example in text) coalesce to form one double emulsion drop, , or two single oil drops, , of O1 and O2, respectively. 12 Stig E. Friberg RC/D = [(1 – vO2)2/3γO1/O2 +γO2/W] / [(1 – VO2)2/3γO1/W + vO22/3γO2/W] [4] According to these conditions, there are examples, Figure 7, which indicate ranges outside the condition γO1/W > γO2/W + γO2/O1, at which the combined drop is thermodynamically favorable to two separate drops. The interfacial free energy of the double emulsion drop is unquestionably less than that of the two separate drops for all volume ratios, when γO1/W > γO2/W + γO2/O1. Then again, for γO1/W/(γO2/W + γO2/O1) = 0.85 (curve  in Figure 7) the ratio in question is still less than one; an obvious thermodynamic contradiction. If there is no spreading, it is difficult to accept that two separate drops should spontaneously unite to a double emulsion drop. Nonetheless, neglecting the fundamentals, focus- ing on the algebra per se, the trend as such of the curves in Figure 7 is anticipated from equations [3] and [4]. The initial increase of the first term of the equation denomi- nator is less than that of the numerator of the equation, giving a downward slope, while the final change is the opposite, giving the minimum of the curve. As a result of the shape of the curves, the RC/D < 1 within a limited range of relative volumes, even for γO1/W < γO2/W + γO2/O1, a purely algebraic result. However, the indisputable con- clusion is that the use of two separate spheres is not the correct counterpart to gauge the thermodynamic stabili- ty of a double emulsion drop. In fact, the choice of a cor- rect counterpart to evaluate the thermodynamic stability of the double emulsion drop needs a more comprehen- sive evaluation of the entire tension range. Another small detail in Figure 7 might be men- tioned, for which the inequality γO1/W > γO2/W + γO2/O1 actually is directly applicable to a physical emulsion. The final outcome of the coalescence of a double emul- sion, whose interfacial tensions obey the inequality in question, is in the form of three layers of the liquids. At that point, gravity decides the order of the layers in the container. If the liquid densities vary as ρW > ρO2/W > ρO1/W, the order of the three layers will be O1, O2 and W from the top and three layers are found. Con- versely, if the densities are ranged ρW > ρO1/W > ρO2/W, four layers are found, since the O1 layer is not in direct contact with the water layer, but separated from it by an (infinitely thin) O2 layer, because O2 spreads on W. Needless to say, these results are based on sin- gle-oil drops as the alternative to the double emulsion drop. In reality, a double emulsion drop will not change to two individual drops, if interfacial tension ratio is moved outside the condition γO1/W > γO2/W + γO2/O1. This will become evident in the analysis of the entire range of interfacial tensions, which shows that, for γO1/W < γO2/W + γO2/O1, the preferred topology becomes that of a Janus drop. JANUS DROP The introductory publication on Janus emulsions15 was based on microfluidics16 emulsification. This prepa- ration guarantees emulsions at virtual internal equilibri- um and led to a number of investigations into the differ- ent aspects of Janus drops,17-22 the results of which con- firmed the agreement between equilibrium predictions and experimental results. These studies formed the basis for an extensive foray into several important aspects of emulsions in biology and medicine, led by Weitz.23 The method, as such, enabled the preparation of emulsions, to all intents and purposes, of any complex topology, but was inherently limited to diminutive volumes, prevent- ing applications into commodities. This condition was changed in 2011, when Hasinovic et al prepared Janus emulsions by traditional vibrational emulsification,24 opening an avenue to large scale production. This pio- neering contribution showed microscopy photos of well- defined Janus emulsions, Figure 8. The image shows well defined drops of an O/W Janus emulsion of a vegetable oil, weight fraction 0.18 Figure 7. The IFE ratios between the interfacial free energy of a double emulsion drop and two separate single-oil drops with γO1/W = 1. Symbol γO2/W γO2/O1  0.5 0.25  0.58 0.29  2/3 1/3  0.78 0.39  0.88 0.44  0.98 0.49 13Emulsion Thermodynamics – In from the Cold and a light silicone oil, weight fraction 0.72, while the continuous aqueous phase comprises only 0.1. It is note- worthy that an O/W emulsion with such limited volume of continuous phase is formed in a standard vibrational emulsification; an early indication of the unexpected effect of interfacial thermodynamics on vibrational emulsification process. Furthermore, but equally impor- tant, the regular Janus topology was first achieved by shear after the initial emulsification. Both processes were necessary in most cases and deserve separate comments. The emulsification as such is a process, in which a large number of transitory small drops of irregular shape are formed and the freshly prepared emulsion is a result of these drops rapidly coalescing to larger enti- ties.25 This process will favor irregular Janus drops for kinetic reasons, because there is virtually no colloidal stability effect involved. Assume an equal number, n, of equally sized drops of two mutually insoluble oils, which are allowed to coalesce at a rate, which is independ- ent of specific drop topology. Subsequent coalescence of these drops leads to an overwhelming fraction of irreg- ular shape Janus drops, in addition to their larger sizes. During the ensuing shear the small attached O2 drops coalesce to a regular Janus lobe. The effect of shear was cursory illustrated 26,27 by optical microscopic images, before and after a cover glass was applied on the micro- scope slide, Figure 9. The minute shear from the cover glass resulted in fewer drops, as expected, but also in an extensive topology change to better defined Janus drops. In addition, the results of shear also – albeit indi- rectly – serve to confirm the internal thermodynamic stability of the Janus drops. Contrary to the case for single-oil emulsions, for which the effect of shear at low rates is to form larger spherical drops, low rate shear of the initial Janus drops, left micro-photo Figure 9, leads to coalescence of the attached drops and a more regular Janus drop. As such, the information in Figure 9 also complements and supports later experimental proofs of the thermodynamic stability of the structure.20 These results are concerned with the kinetic factors of the process, leaving the thermodynamics unexam- ined. The equilibrium angles and tensions of the Janus drop are given in Figure 2A and the algebra for equilib- rium has been reported16-19,27,28 with the following sum- mary. Balancing the forces in Figure 2A along and perpen- dicularly to the γO1/ W direction gives the angles β and δ, Figure 2A, which, in turn, define the angles μ and ε. μ = η + β [5] ε = η - δ [6] Furthermore, assuming rO1/W = 1, the radii rO2/W and rO2/O1 are rO2/W = sinη/sinμ [7] rO2/O1 = sinη/sinε [8] These equations control the equilibrium at the con- tact line, while the entire drop configuration, Figure 2B, also depends on the relative volumes of the two dispersed liquids. Unfortunately, the latter feature is not easily calculated from given volume fractions. The expressions become prohibitively complex and Guzowski et al18 opted to use a computer program to correlate vol- umes and topology. As an alternative, the volumes are calculated in the present contribution from the geomet- rical features in Figure 2 and the correlations between Figure 8. An optical microscopy image of a Janus emulsion, pre- pared by vibrational emulsification. Figure 9. A simple experiment illustrating the effect by shearing on a Janus emulsion, prepared by vibrational emulsification27. (From reference 27 with permission). 14 Stig E. Friberg volume ratios and topology are evaluated ex post fac- to.26,28 The volumes of O1 and O2 are calculated (equations [12] and [13]) via pre-volumes, φO1, black, Figure 2B, and φO2 white + grey, Figure 2B, separated by the plane through the visible contact line. φO1 = π(1 + cosη)2(2 + cosη)/3 [9] φO2 = π(rO2/W - cosμ)2(3 - rO2/W + cosμ)/3 [10] The volumes VO1 and VO2 are attained from φO1, φO2 and φO1/O2 (Grey, Figure 2B), φO2/O1 =πrO2/O13(1 – cosε)2(2 + cosε)/3 [11] VO1 = φO1 + φO2/O1 [12] and VO2 = φO1 - φO2/O1 [13] In the comparison of interfacial free energies, the single-oil drops as counterparts are now replaced by a direct comparison between the free energies of double emulsion and Janus drops. As will be demonstrated, this new comparison is more relevant, removing the “anoma- lous” results in Figure 7. This figure showed the double emulsion drop to have lower interfacial free energy than two separate single-oil drops in a limited range of vol- ume fractions, even for γO1/W < γO2/W + γO2/O1. The rea- son for this result is that the free interfacial free energy of the Janus drop was neglected, as has commonly been the case. Figure 10 shows the ratio of the free interfacial energies of a Janus drop to those of a double emulsion drop of identical volume. The ratio was limited to the ten- sions γO1/W < γO2/W + γO2/O1; since the drop equilibrium free energies for Janus drops with γO1/W > γO2/W + γO2/O1 is outside the equilibrium conditions and cannot be exact- ly calculated. The figure demonstrates the Janus drop to have a lower free energy than the double emulsion drop in the inequality range γO1/W < γO2/W + γO2/O1. The faulty con- clusion from using two single-oil drops as counterpart is now corrected. In addition, there are two details that are of inter- est. The topology change, when the inequalities change from γO1/W < γO2/W + γO2/O1 to γO1/W > γO2/W + γO2/O1 is of special interest, because it is highly prominent. A complete analysis involves a large number of variables and in the present contribution a simplified example is used to graphically illustrate the phenomenon. The basis for the example is the specific and well defined case γO1/W = γO2/W + γO2/O1. In the calculation the expression is divided by γO1/W, giving γO2/W = a and γO2/O1 = 1 - a. The change of inequalities mentioned is represented by γO1/W altered from 1 - ε to 1 + ε, in which ε is a small positive quantity. The angles b and d are calculated with the γO2/W and γO2/O1 equal as are the O2 and O1 volumes. The specific selection of these is not essential for the cen- tral theme, and the angles β and δ are calculated versus ε. cosβ = cosδ = 1 – (2 - ε)ε [14] The radical topology change, caused by the minute alteration of γO1/W from 0.995 to 1.005 is illustrated in Figure 11. The minute increase (1%) of the interfacial tension γO1/W causes a drastic topology change with O1 spread- ing on the large sphere of O2. The activity is the same for smaller ε, but the graphics becomes less instructive, because the visible contact line is transferred to greater η angles with reduced ε. Another, perhaps even more drastic consequence, is found of the interfacial tension variation for model sys- Figure 10. The ratio of Interfacial free energy of a Janus drop and a double emulsion drop of identical oil volumes. Figure 11. With e = 0.005, γO1/W < γO2/W + gO2/O1, a Janus drop is thermodynamically preferred. When ε changes to -0.005, O1 spreads on O2 and a double emulsion drop is favored. 15Emulsion Thermodynamics – In from the Cold tems with realistic interfacial tensions.28 As an example, the variation of rO2/W and rO2/O1 (rO1/W =1) with η is truly remarkable, Figure 12, for β = 27.8°, δ = 35.4°, γO1/W= 1 and rO1/W = 1. Each curve has a discontinuity, at which an infinite radius switches to the opposite sign with changing η. The rO2/O1 versus η switches from -∞ to +∞ at η = δ = 35.4°, while rO2 versus η discloses a disconti- nuity at η = π - β, approximately η = 152.4°, Figure 12. The discontinuity of rO2/O1 2 at η = 35.4°, has only a small effect on the volumes of the two lobes, Figure 2. Instead, the influence is felt on the Laplace pressure cor- relation. Σ2ΔPXX/rXX = 0 [15] The radius rO2/O1 is negative for η < 35.4°, Figure 2. In this η range the terms γO2/W/rO1/W < 1 and hence necessi- tates a negative rO2/O1 to satisfy the LaPlace requirements, equation [15]. For η = 35.4°, the γO2/W and rO2/W both are 0.65 and, again the Laplace pressure condition com- plies with those in equation [15]. For η > 35.4°, the sign of the radius is opposite to that at η < 35.4°. Nonetheless, as mentioned, the effect is not decisive for the size of the volumes, since φO2/O1 is usually small compared to vol- umes φO1 and φO2, equations [9] and [10]. Conversely, the change in rO2 is accompanied by an extreme change in the O2 volume. As shown by Friberg28 as well as by Ge et al26 in different examples, at η ≈ 154.2° for the Janus drop in question, the change represents a partial inversion of the Janus drop from (O1 + O2)/W, η = 140°, Figure 13, to (O1 + W)/O2 η = 164°. At η = 140°, the rO2/W has reached a value of 3.0 with rO1/W = 1 and rO2/O1 = 0.66. The O1 drop is formed by two lobes, one reaching 0.23 (fraction of rO1/W) into W, O1/W, and a second lobe entering 0.83 fraction into O2, O2/O1. There is, needless to say, no interface between the lobes. Together they form a non-spherical drop, O1, with an abrupt sign change of the radius as well as its dimension at the contact line. As expected, the added Laplace pressure over the interfaces O2/O1 and O2/W equals the pressure over the O1/W interface. When η is increased to 152.4°, the O1 drop is located at the interface between two infinite phases W and O2. Nor in this case is the O1 drop symmetrical, since the γO1/W is different from γO2/O1 and the LaPlace pressure is equal across the two interfaces of O1. Increasing γ to 164°, Figure 13, shows an (O1 + W)/O2 Janus emulsion and a further reduction of the O1 drop size. These results and those in the preceding para- graphs are correct illustrations of the drop topology, as directed by the thermodynamic requirements. However, the applications to a physical emulsion are fraught with uncertainties and a few comments on the prerequisites for the model system are useful. The γO1/W = 1 is not a cause of concern; it only implies that instead of numeri- cal values for γO2/W and γO2/O1, their ratios γO2/W/γO1/W and γO2/12/γO1/W are used to simplify the algebra. Con- versely, the second condition rO1/W = 1 causes artificial restrictions on the physical image. It indicates that for each η change, a slice of O1 between the η:s in question is removed and a modified section of O2 is added at the contact line with exactly correct b angle. As is obvious, the conditions, in spite of being thermodynamically correct, are difficult to reconcile with any physical sys- tem, especially to the close packing of drops. The sec- ond alternative, retaining the O1 volume constant, gives a result similar to the one for constant rO1/W, while the more artificial choice of keeping the entire drop vol- Figure 12. Radii of lobes for the Janus drop, dashed line in Figure 2, with rO1 equal to unity. Squares rO2/W, triangles rO2/O1 Figure 13. Schematic representation of the inversion of the Janus drop in the range η = 140° - 164°, Black areas are O2, grey are- as O1 and white W. Expanded O1 areas with correct radii ratios are shown on top with the extension of the contact line as dotted/ dashed. 16 Stig E. Friberg ume constant leads to some modification. Nevertheless, from a physical point of view, the second alternative is the most realistic with O2 added to an already formed O1/W emulsion. For this case, adding O2 brings about a greater and greater O2 lobe of the drop, but causes no emulsion inversion, until the volume of O2 is greater than that of the initial continuous phase O1. As a summary, adding the oils to an initial aqueous liquid gives an O1/W emulsion. Adding O2 to this emul- sion results in a Janus emulsion, (O1+ O2)/W, an emul- sion with increasingly larger O2 lobes. When the O2 volume exceeds that of the W, an inversion takes place to an (O1 + W)/O2 emulsion. Continued addition of O2 gives rise to a diminution of the relative size of the (O1 + W) drop. CONCLUSIONS The conclusions to include Janus emulsions as a counterpart, when considering the thermodynamic fac- tors for double emulsion drops have been proven correct for selected examples. The extension of these conclusions to Janus and double emulsion drops in general would be premature, but, so far, the indications are that the inference has more general validity. ACKNOWLEDGMENT The author is deeply grateful to his wife, Susan for her unfailing loyalty and support during the research and to the Ugelstad Laboratory, Trondheim, Norway for support. REFERENCES 1. B. P. Binks, (Ed), Modern Aspects of Emulsion Science, The Royal Society of Chemistry, Cambridge, 1998 2. J. Sjöblom, (Ed), Emulsions and Emulsion Stability. 2 ed. Taylor and Francis; 2006 3. Y. Liu, in Th. F. Tadros (Ed), Phase Inversion, Encyclo- pedia of Surface and Colloid Science, 2nd Ed., Spring- er, Amsterdam, 2013 4. D. J. Mitchell, B. W. Ninham, J. Chem. Soc. Faraday Trans. 1981, 77, 601. 5. S.I. Ahmad, S.E. 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Colloid Interface Sci. 2014, 416, 167. Substantia An International Journal of the History of Chemistry Vol. 2, n. 1 - March 2018 Firenze University Press Why Chemists Need Philosophy, History, and Ethics Emulsion Stability and Thermodynamics: In from the cold Stig E. Friberg Finding Na,K-ATPase Hans-Jürgen Apell Mechanistic Trends in Chemistry Louis Caruana SJ Cognition and Reality F. Tito Arecchi A Correspondence Principle Barry D. Hughes1,* and Barry W. Ninham2 From idea to acoustics and back again: the creation and analysis of information in music1 Joe Wolfe Snapshots of chemical practices in Ancient Egypt Jehane Ragai The “Bitul B’shishim (one part in sixty)”: is a Jewish conditional prohibition of the Talmud the oldest-known testimony of quantitative analytical chemistry? Federico Maria Rubino Michael Faraday: a virtuous life dedicated to science Franco Bagnoli and Roberto Livi