On the Negative Surface Tension of Solutions and on Spontaneous

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On the negative surface tension of solutions and on spontaneous emulsification George Kaptay Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01968 • Publication Date (Web): 17 Sep 2017 Downloaded from http://pubs.acs.org on September 18, 2017

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On the negative surface tension of solutions and on spontaneous emulsification

George Kaptay University of Miskolc, Dep of Nanotechnology, Egyetemvaros, Miskolc-3515 Hungary MTA-ME Materials Science Research Group, Egyetemvaros, Miskolc-3515 Hungary Bay Zoltan Ltd on Applied Research, BAY-ENG, 2 Igloi, Miskolc.3519, Hungary phone: +36 30 4150002, e-mail: [email protected]

Abstract The condition of negative surface tension of a binary regular solution is discussed in this paper using the recently re-confirmed Butler equation (Langmuir, 31 (2015) 5796-5804). It is shown that surface tension becomes negative only for solutions with strong repulsion between the components. This repulsion for negative surface tension should be so strong that this phenomenon appears only within a miscibility gap, i.e. in a 2-phase region of macroscopic liquid solutions. Thus, for a macroscopic solution the negative surface tension is possible only in a non-equilibrium state. On the other hand, it is also shown that nano-emulsions and microemulsions can be thermodynamically stable against both coalescence and phase separation. Further, the thermodynamic theory of emulsion stability is developed for a 3-component (AB-C) system with A-rich droplets dispersed in a C-rich matrix, separated by the segregated Brich layer (the solubility of B is limited in both A and C, while the mutual solubility of A and C is neglected). It is shown that when a critical droplet size is achieved by forced emulsification, it is replaced by spontaneous emulsification and the droplet size is reduced further to its equilibrium value. The existence of maximum temperature of emulsion stability is shown. Using low-energy emulsification below this maximum temperature, spontaneous emulsification can appear, enhanced with further decrease of temperature. This finding can be applied to interpret experimental observations on spontaneous emulsification or for the design of stable micro-emulsions and nano-emulsions.

Keywords: negative surface tension; spontaneous emulsification; Butler equation; stable micro-emulsions; maximum temperature of emulsion stability.

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Introduction Emulsions are one of the important classes of multi-phase materials [1-33]. Usually they are considered thermodynamically unstable due to their high inner interfacial area, and can be stabilized only kinetically, for example using particles [34-39]. However, this general view is challenged by numerous observations on spontaneous emulsification [40-49]1. These observations hold a promise for thermodynamically stable emulsions. According to Gibbs [50], the thermodynamic definition of surface tension (or interfacial energy) follows as:

≡

(1)

, ,

where (J/m2) is the surface tension of a phase, (J) is the absolute (not molar) Gibbs

energy of the same phase, (m2) is the surface area of the same phase, (Pa) pressure, (K) absolute temperature, (mole) amount of component i in the same phase. Upon

emulsification the surface area increases ( > 0). Any process takes place spontaneously, if

the Gibbs energy change accompanying this process is negative ( < 0). Substituting these

sign requirements into Eq.(1) it follows that the process of emulsification will be spontaneous only, if < 0. This is not a new idea, however [51-55]. Nevertheless, the present author has no information on the existence of a comprehensive and generally valid thermodynamic theory leading to the conclusion of negative surface tension. Thus, the primary goal of this paper is to fill this gap in knowledge. As will be shown below, it is possible only for nonequilibrium macroscopic solutions or for equilibrium nano-solutions. At this point the thermodynamic stability of nano-emulsions and micro-emulsions will be also discussed, extending the present knowledge [56-65]. The temperature dependence of emulsion stability will be discussed in details.

Modeling surface tension of two-component solutions For simplicity the bulk thermodynamics of the A-B two component solution will be described by the regular solution model [66-67]:

∆ = ∆ = Ω ∙ ∙ (1 − ) ∆ , = ∆, = Ω ∙ #

= ∆,$ = Ω ∙ (1 − )# ∆ ,$

1

(2a) (2b) (2c)

There are much more papers devoted to spontaneous emulsification than cited here. According to Google Scholar, the number of papers on “spontaneous emulsification” doubles in each decade since 1900 and was around 600 in 2016.


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where ∆ , ∆ , and ∆ ,$ (all in J/mol) are excess molar Gibbs energies of the solution

and of the two components, respectively, ∆ , ∆, and ∆,$ (all in J/mol) are heats of mixing of the solution and of the two components, respectively, Ω (J/mol) is the interaction

energy in the framework of the regular solution model (the components attract each other if Ω

has negative sign and vice versa), (dimensionless) is the bulk mole fraction of component B in the solution (for macroscopic solutions it is the same as the average mole fraction of component B in the solution). In the simplest regular solution model the excess molar entropy and excess molar volume values are taken as zero for simplicity. We will consider the case of this simple solution model as it is sufficient to show the case for negative surface tension. More complex solution models will somewhat alter the findings here, but will not essentially change them. It is known since Gibbs [50] that the increased curvature (decreased size of the droplets) leads to the decrease in surface tension, so it probably helps to stabilize nanoemulsions. However, this effect becomes significant only below the droplet size of 10 nm. Thus, in the present model the size dependence of surface tension is neglected. It is done partly to keep the model simple, but also to make clear that the resulting negative surface tension is not a result of its size-dependence. Recently, the Butler equation was re-derived and re-confirmed by the present author, with its essential equations written as [68-70]: '∙

= = $

,-.

0

= % + * ∙ + ,-./ + * ∙ (1 ∙ 2# − # ) ( ( '∙

)

.

0

)

(3a) (3b)

$ = $% + (* ∙ + ./ + (* ∙ 41 ∙ (1 − 2 )# − (1 − )# 5 3

3

(3c)

where (J/m2) is the surface tension of the solution, and $ (J/m2) are the partial surface

tensions of the components defined in [69], % and $% (J/m2) are the same for pure

components, 6% and 6$% (m2/mol) are the molar surface areas of the components (in this simplified treatment they are considered concentration independent), R = 8.3145 J/molK is the universal gas constant, 2 (dimensionless) is the mole fraction of component B in the

surface region (usually being different from ), 1 (dimensionless) is the ratio of the surface bonds to the bulk bonds, being usually around 0.85 (this parameter will fall out later). Eq-s

(3b-c) are substituted into the right hand side of Eq.(3a) ( = $ ) to find the unknown value of equilibrium surface composition 2 ; then, this value is substituted back into any of Eq-s

(3b-c) to find the surface tension of the solution in agreement with Eq.(3a). This procedure 3 ACS Paragon Plus Environment

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can be performed only numerically. This is presented on one example in Fig.1 for an

arbitrarily selected set of parameters, using four different Ω parameters. The parameters are

selected such that % > $% , and thus component B is surface active, i.e. this component

segregates to the surface. As follows from Fig-s 1, in this case the Butler equation has only one solution (for other cases see [71-73]), i.e. the two lines, corresponding to the two partial surface tensions cross each other only once as function of 2 , in its whole possible interval between 0 and 1.

As follows from Fig.1a, when Ω has a negative value, the equilibrium surface

composition is close to the fixed bulk composition, so segregation is weak and surface tension is close to that of the solvent (component A). This is because in this case the components

attract each other, thus segregation is weakened. However, as parameter Ω has more and more positive values, segregation of component B becomes more and more severe, so the crossing point of the two lines gradually shifts to higher and higher 2 values, and to lower and lower values (see Fig-s 1a-c). Finally, at a high enough Ω value the surface tension becomes

negative (see Fig.1d). The evolutions of the 2 and values as function of parameter Ω are shown in Fig-s 2a-b. As follows from Fig-s 2a-b, when surface tension has negative values, the equilibrium

composition of the surface correspond to almost pure component B with 2 ≥ 0.99.

Substituting this value into Eq.(3c), this equation simplifies as follows (now it is written for the surface tension of the solution in agreement with Eq.3a): '∙

0

≅ $% − (* ∙ + − (* ∙ (1 − )# 3

3

(4)

The first two terms of Eq.(4) have positive values, only the last term of Eq.(4) can

have a negative value at Ω > 0. This is in agreement with Fig.2b, showing surface tension

decreasing almost linearly with increasing the value of Ω at its high positive values. The

critical value of parameter Ω (denoted as Ω; ) corresponding to zero surface tension can be obtained by making Eq.(4) equal zero: Ω; ≅

* * (3 ∙

(7a)

where Ω;, (J/mol) is the minimum value of Ω, which can lead to negative surface tension

at given state parameters (T, x). If this minimum interaction energy Ω;, provides such a

miscibility gap that the point (T, x) in the phase diagram appears in the two-phase region, then the case shown in Fig.3 has general validity. To prove this, let us express the interaction

energy value requested for phase separation (denoted as Ω2? ) in its dimensionless form from Eq-s (6a-b):


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0/KL '∙

0/KL

If

0H,I J '∙

>

0/KL '∙

=

'∙

=

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A BCA

(7b)

#∙.-,

.F;.G

=2

(7c)

, then the situation shown in Fig.3 has general validity. Let us first

substitute x = 0.5 into Eq.(7a):

0H,I J '∙

.F;.G

= 2.77, which is indeed larger than 2 of Eq.(7c).

For other x values the above condition can be re-written by creating the dimensionless difference of Eq-s (7a-b): NOPPQRQSQ ≡

0H,I J '∙

−

0/KL '∙

=

=. − (,-.)>

−

=

A BCA

#∙.-,

>? > 0

7(d)

To check the validity of this inequality Eq.(7d) is plotted in Fig.4 as function of x. As follows from Fig.4, inequality (7d) is obeyed as the “Difference” defined by Eq.(7d) has only

positive values at any . Thus, we can conclude that for any combination of state parameters (selected components A and B, T and x at p = 1 bar) and for any value of the interaction

energy which is high enough to provide negative surface tension, such a macroscopic solution is not thermodynamically stable, i.e. the situation is similar to Fig.3. This situation is probably not changed, if another solution model is used, or if the excess molar entropies or excess molar volumes are taken into account, or if the number of components is increased. It should be noted that the same conclusion follows from square gradient approximations [74]. In the proof above strict thermodynamics and mathematics was applied. However, the validity of this result can be demonstrated by an everyday experience of all of us: macroscopic bodies (as we see them around us) would not exist in equilibrium if surface tension was negative for them. As they do exist in reality and in equilibrium, a logical conclusion is that for equilibrium macroscopic bodies surface tension is always positive. The above conclusion means that macro-emulsions can exist only in a non-equilibrium state. However, the above analysis does not take into account the fact that the droplets of usual emulsions are very small, i.e. they have a high specific interfacial area, and thus the mutual solubility of the two liquid phases is expected to increase [75-88]. Thus, the miscibility gap shown in Fig.3 will be lower and less wide (see arrows in Fig.3) for small size droplets, and thus the characteristic point can appear in the 1-phase, stable region. This question is considered in details in the next chapter. So far, surface tension of the liquid/vapor interface was studied. A similar formalism can be extended to the interfacial energy between the liquid matrix and the dispersed nanodroplets [89]. This will lead to the understanding of spontaneous emulsification (see below). 6 ACS Paragon Plus Environment

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Modeling the thermodynamics of nano- and micro-emulsions Let us consider a 3-component system (A-B-C). The amounts of the three components

are denoted as: U > > $ (moles). All the three pure components are stable liquids at a fixed temperature T and at a fixed standard pressure of po = 1 bar. Let A and C be insoluble

liquid phases in each other with A having a smaller volume fraction compared to C, so the Arich liquid will be dispersed in the C-rich liquid upon emulsification. Let B have a limited solubility in both A and C. The repulsion between components A and B is expressed in the

positive interaction energy Ω-$ (J/mol), while the same between components B and C is expressed in the positive interaction energy ΩU-$ (J/mol) in the framework of the regular

solution model. Thus, there will be A-B droplets emulsified in a C-B matrix. Let us write the condition of bulk equilibrium for component B in the two-phase A-B / C-B system as: #

%

$,V + W ∙ ∙ +$(,X) + Ω-$ ∙ Y1 − $(,X) Z = #

% + W ∙ ∙ +$(U,X) + ΩU-$ ∙ Y1 − $(U,X) Z = $,V

(8a)

% where $,V (J/mol) is the standard Gibbs energy of pure liquid B, $(,X) (non-dimensional) is

the bulk mole fraction of component B in the A-B solution, $(U,X) (non-dimensional) is the same in the C-B solution. Generally Eq.(8a) is solved numerically for $(,X) as function of

$(U,X) at fixed value of T and given model parameters Ω-$ and ΩU-$ . If component B forms

diluted solutions in both A and C phases (i.e. both $(U,X) and $(,X) values are smaller than 0.05), then Eq.(8a) can be solved approximately as: $(,X) ≅ [ ∙ $(U,X)

(8b)

where K is the distribution coefficient of component B between the A-rich and C-rich liquid phases, defined as: [ ≡ Q

0\C3 -0)C3 '∙

(8c)

If Ω-$ > ΩU-$ , then K will have a positive value being smaller than 1, i.e. $(,X)
∙

(9f)

* gI,3

where di (m) is the thickness of the interfacial layer described by Eq.(d) of the Supporting Information. Let us substitute Eq. (8b, Ad, 9d) into Eq.(9f), taking into account Eq.(f) of the Supporting Information: $() ≅

* e.#d∙) ∙gI,) * i∙(3

∙ `1 +

l∙.3(\,a)

,-l∙.3(\,a)

b

(9g)

Now, let us substitute Eq.(9g) into Eq.(9e) and write the resulting equation in a

simplified way, taking into account 1 ≫ [ ∙ $(U,X) and 1 ≫ $(U,X) as: $ ≅ $(U,X) ∙ (U + [ ∙ ) +

Now, let us express $(U,X) from Eq.(9h): $(U,X) ≅

e.#d i

∙ ∙

gI,) * (3

(9h)

*

o j.>m 3 - n ∙) ∙ I,) p* 3

\ql∙)

(9i)

During emulsification the droplet radius (r) gradually decreases, and thus, according to Eq.(9i), xB(C,b) gradually decreases, as well. However, xB(C,b) has a physical sense only if it has


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a positive value. Therefore, the minimum droplet size ensuring at least xB(C,b) = 0 can be expressed from Eq.(9i) as: R = 3.24 ∙

* ) ∙gI,)

(9j)

* 3 ∙(3

The total Gibbs energy of the emulsified system is calculated as (see also [90-92]): .

.

? = ∙ `1 + 3(),a) b ∙ ,X,-$ + U ∙ `1 + 3(\,a) b ∙ ,X,U-$ + c ∙ 4 ∙ f ∙ R # ∙ . ,-. ,-. 3(),a)

3(\,a)

(10a)

,X,-$ = W ∙ ∙ t$(,X) ∙ +$(,X) + Y1 − $(,X) Z ∙ lnY1 − $(,X) Zw + +Ω-$ ∙ $(,X) ∙ (1 − $(,X) )

(10b)

+ΩU-$ ∙ $(U,X) ∙ (1 − $(U,X) )

(10c)

,X,U-$ = W ∙ ∙ t$(U,X) ∙ +$(U,X) + Y1 − $(U,X) Z ∙ lnY1 − $(U,X) Zw + '∙

≅ $% − (* ∙ +$(,X) − 3

'∙

= $% − (* ∙ +$(U,X) − 3

0)C3 * (3

0\C3 * (3

#

∙ Y1 − $(,X) Z =. #

∙ Y1 − $(U,X) Z

(10d)

For comparison, the total Gibbs energy of the non-emulsified system is calculated. This system is supposed to contain 3 macroscopic phases (a C-rich phase, an A-rich phase and a B-rich phase), with a negligible interfacial area between them:

%-? = ∙ x1 + .

$(,X,?y) $($-U-) ∙ `1 − bz ∙ ,X,-$,?y + 1 − $(,X,?y)

+U ∙ {1 + ,-.3(\,a,K|) ∙ 1 − 3(\,a,K|)

3(3C\C)) \

} ∙ ,X,U-$,?y + $-U- ∙ ,$

(11a)

where $(,X,?y) is the equilibrium mole fraction of component B in the saturated A solvent

calculated by Eq-s (6a-b), ,X,-$,?y (J/mol) is the molar Gibbs energy of the saturated A-B solution calculated by substituting $(,X,?y) into Eq.(10b) instead of $(,X) , $(U,X,?y) is the

equilibrium mole fraction of component B in the saturated C solvent calculated by Eq-s (6ab), ,X,U-$,?y (J/mol) is the molar Gibbs energy of the saturated C-B solution calculated by

substituting $(U,X,?y) into Eq.(10c) instead of $(U,X). The amount of component B in the Brich phase is calculated as: .

.

$($-U-) = $ − ∙ ,-.3(),a,K|) − U ∙ ,-.3(\,a,K|) 3(),a,K|)

3(\,a,K|)

(11b)

The total amount of the B-rich phase is calculated as: .

.

$-U- = $($-U-) ∙ `1 + ,-.3(),a,K|) + ,-.3(\,a,K|) b 3(),a,K|)

3(\,a,K|)

The mole fractions of components in the B-rich solution are calculated as: 9 ACS Paragon Plus Environment

(11c)

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$($) = U($) = ($) =

,q

,q

,

A3(),a,K|) A3(\,a,K|) q BCA3(),a,K|) BCA3(\,a,K|) A3(\,a,K|) BCA3(\,a,K|) A3(),a,K|) A3(\,a,K|)

q

BCA3(),a,K|) BCA3(\,a,K|) A3(),a,K|) BCA3(),a,K|) A3(),a,K|) A3(\,a,K|)

,qBCA

q

3(),a,K|) BCA3(\,a,K|)

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(11d)

(11e)

(11f)

The molar Gibbs energy of the B-rich phase is:

,$ = W ∙ ∙ t$($) ∙ +$($) + U($) ∙ +U($) + ($) ∙ +($) w + +$($) ∙ ($) ∙ Ω-$ + $($) ∙ U($) ∙ ΩU-$ ∙

(11g)

The following set of parameters is needed for the calculations: nA, nB, nC, _, , _,$ ,

T, ΩU-$ , Ω-$ . The calculations are performed only for r > rmin where rmin is calculated by

Eq.(9j). The value of $(U,X) follows from Eq.(9i) as function of r. Then, $(,X) follows from Eq.(8b). Trial calculations are shown in Fig-s.5-6. As follows from Fig.5 calculated by a

relatively low value of ΩU-$ = 10 kJ/mol, the interfacial energy remains positive, and that is why the molar Gibbs energy of the emulsified system remains more positive than that of the non-emulsified system at any droplet size. On the other hand, if a somewhat more positive

parameter ΩU-$ = 14 kJ/mol is applied, interfacial energy becomes negative above a certain droplet size and the molar Gibbs energy of the emulsified system passes through a minimum as function of droplet size (see Fig.6). Moreover, in a limited size range of 15.9 … 44.1 nm the molar Gibbs energy of the emulsified system is more negative compared to the nonemulsified mixture, thus the emulsion is stable at least in this limited size range of droplets. The stable droplet size of 22.4 nm corresponds to the minimum of the molar Gibbs energy in the bottom part of Fig.6. The emulsion stability maps as function of different state parameters (with other values fixed) are shown in Fig-s 7-10. All these stability maps show the equilibrium droplet radius corresponding to the minimum in the bottom Fig.6, and the minimum and maximum stable droplet radii, corresponding to the two cross sections of the molar Gibbs energy of the emulsion with the almost horizontal dotted line in the bottom Fig.6 (corresponding to the mixture of A-rich, B-rich and C-rich macroscopic solutions with negligible interfacial area between them). In Fig.7 an emulsion stability map is shown as function of the interaction energy between components B and C. As follows from the above, stable emulsions are obtained only if this interaction energy value is larger than a certain critical value (17.3 kJ/mol in this case). 10 ACS Paragon Plus Environment

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The minimum, equilibrium and maximum stable droplet sizes asymptotically tend to this vertical dotted line in Fig.7. In the bottom part of Fig.7 a principal possibility is shown how forced emulsification can be replaced by spontaneous emulsification when the maximum stable size is achieved by forced emulsification.

As follows from Fig-s 8, stable emulsions exist only below a certain value of $% . It is

in accordance with experimental findings, reporting usually very low interfacial energy values for spontaneous emulsification. As follows from Fig-s. 9-10, stable emulsions exist only below a certain maximum temperature. This is because at a higher temperature configurational entropy provides more and more negative molar Gibbs energy for the nonemulsified state compared to the emulsified state. The stable size ranges of droplets are approximately inversely proportional to the ratio

$ / (see Fig-s.8-10). This is because the minimum in the molar Gibbs energy is usually

only slightly higher than the minimum droplet size – see the bottom Fig.6. On the other hand, the minimum droplet size is inversely proportional to the ratio $ / , in accordance with Eq.(9j). In the bottom part of Fig. 8 a thermodynamic possibility of spontaneous emulsification is shown. If the droplet size is reduced below 4 microns by low-energy forced emulsification, then the droplet size will spontaneously reduce further in a spontaneous way, to about 2 microns. Although this effect is quite small, it can be tailored by other parameters of the above model. A considerably larger reduction in droplet size by spontaneous emulsification can be obtained if the low-energy emulsification is accompanied by slow cooling of the system below the maximum temperature of emulsion stability (see Fig-s 9-10). Let us start the experiment at a relatively high temperature, being higher than the maximum temperature of emulsion stability and provide some low-energy mixing ensuring the droplet radius in the range of 2-20 micrometers depending on composition (see Fig-s 9-10). Then, upon slow cooling, the system appears in the region of stable emulsions, and thus spontaneous emulsifications starts, and reduces the droplet size by 1-2 orders of magnitudes, while temperature slowly reaches room temperature upon further cooling. Varying the components A-B-C of the emulsion (and possibly adding to it further components D, E, etc.) and varying their molar ratios and temperature, all thermodynamic parameters of the system can be tailored, and thus the stability size range of droplets can be varied. This opens a wide technological possibility to achieve the phenomenon of spontaneous emulsification in the wide range of parameters, as already known from a large body of


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empirical findings. The model presented here hopefully provides further insight and more solid theoretical basis for further developments in the field. Let us shortly note on the mechanism of emulsification. In the present model spherical droplets of varying radii are considered for simplicity. However, the essence of reducing the spherical droplet size is in the increase of their interfacial area [77, 90-92]. The same effect is obtained if the initially large droplets roughen or form small fingers, so smaller droplets are formed in this way. This roughening and fingering effect is not modelled here in details to avoid mathematical difficulties.

Conclusions 1. The conditions of negative surface tension of a binary regular solution is found in this paper using the Butler equation. It is shown that surface tension becomes negative only for solutions with strong repulsion between the components. This repulsion should be so strong that this phenomenon appears only within a miscibility gap, i.e. in a 2-phase region of macroscopic solutions. Thus, for a macroscopic solution the negative surface tension is possible only in a non-equilibrium state. However, for nano-sized droplets negative surface tension is possible also for stable solutions. This also applies to the possible negative interfacial energy between nano-droplets and the immiscible liquid matrix in which the droplets are dispersed. 2. It is also shown that nano-emulsions and micro-emulsions can be thermodynamically stable against both coalescence and phase separation in the certain size range of the droplets. Simultaneous application of low-energy mixing and slow cooling shows the highest theoretical potential for spontaneous emulsification. This finding can be applied to interpret experimental observations on spontaneous emulsification and for the design of stable micro-emulsions and nano-emulsions.

Acknowledgement The author is grateful for the financial support from the GINOP 2.3.2 – 15 – 2016 – 00027 project. The author is also grateful to prof. Imre Dékány (University of Szeged, Hungary) and to prof. Geoffrey A. Brooks (Swinburne University, Melbourne, Australia) for motivating discussions.

References


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sigma A, sigma B, J/m2

0,3

OMEGA = -10 kJ/mol B

0,2

A

0,1 0 0

0,2

0,4

0,6

0,8

xs 1

B

-0,1

A

-0,2 -0,3

Fig.1a 0,3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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OMEGA = 0 kJ/mol

0,2

B A

0,1 0 0 -0,1

0,2

0,4

0,6

0,8

xs 1

B A

-0,2 -0,3

Fig.1b


Langmuir


0,3

OMEGA = +8 kJ/mol

0,2

A

0,1 0 0

0,2

0,4

0,6

B

0,8

1

xs

B

-0,1

A

-0,2 -0,3

Fig.1c 0,3


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OMEGA = +15 kJ/mol

0,2

A

0,1 0 0 -0,1

0,2

0,4

0,6

B

0,8

1

xs

B A

-0,2 -0,3

Fig.1d

Fig.1. The partial surface tensions of the two components calculated by Eq-s (3b-c) as function of the surface mole fraction of component B at four different interaction energy

values (see in top of the figures) and at the following fixed parameters: % = 0.072 J/m2, $% = 0.03 J/m2, 6% = 40,000 m2/mol, 6$% = 60,000 m2/mol, 1 = 0.85, T = 300 K, x = 0.1. The

vertical dotted lines correspond to the latter value of x = 0.1.


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Fig.2. Calculated equilibrium surface composition (Fig.2a) and surface tension (Fig.2b) using the cross sections of the two lines in figures similar to Fig-s 1a-d, as function of the interaction energy (bold lines). The horizontal dotted line in Fig.2a corresponds to the bulk composition, while the same in Fig.2b corresponds to the surface tension of the solvent (component A). The vertical dotted line separates the regions with positive surface tension (left from the line) and the regions with negative surface tension (right from the line).


Langmuir

Fig.3. Part of the A-B phase diagram (the solid and vapor phases are not shown), with a

miscibility gap (solid line) calculated by Eq-s (6a-b) using parameter: Ω = 9.3 kJ/mol, at which surface tension becomes negative in Fig.2b. The point corresponds to the state parameters (T = 300 K and x = 0.1) fixed in Fig-s 1-2. The arrows show the shift of the miscibility gap when the size of the liquid droplet is decreased below 100 nm.

5

Difference

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 31

4

3

2

1

0 0

0,2

0,4

0,6

0,8

x

1

Fig.4. The “Difference” defined by Eq.(7d) calculated as function of x.


Page 25 of 31

0,1

sigma, J/m2

0,075

0,05

0,025

0 0

20

40

60

80

r, nm

100

80

r, nm 100

100 Gm, J/mol

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

0 0

20

40

60

-100 -200

unstable emulsion

-300 -400

macroscopic mixture

-500

Fig.5. Interfacial energy and molar Gibbs energy as function of droplet radius calculated by % Eq-s (3, 10), using parameters: nC = 1 mol, nA = 0.1 mol, nB = 0.01 mol, _, = 19 cm3/mol, % _,$ = 20 cm3/mol, $% = 0.001 J/m2, % ≫ $% , U% ≫ $% , ΩU-$ = 10 kJ/mol, Ω-$ = 25

kJ/mol, T = 300 K. The absolute Gibbs energy in the bottom figure is divided by (nA + nB) to obtain the molar Gibbs energy. The almost horizontal dotted line in the bottom figure is calculated by Eq-s (11) and corresponds to the macroscopic mixtures of A-rich + B-rich + Crich solutions. The vertical dotted line corresponds to rmin = 9.89 nm, calculated by Eq. (9j).


Langmuir

0,05

sigma, J/m2

0,025

0 0

20

40

60

80

r, nm

80

r, nm 100

100

-0,025

-0,05

50 Gm, J/mol

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 31

0 0

20

40

60

unstable emulsion -50

macroscopic mixture

-100

stable emulsion -150

Fig.6. Interfacial energy and molar Gibbs energy as function of droplet radius calculated by % Eq-s (3, 10), using parameters: nC = 1 mol, nA = 0.1 mol, nB = 0.01 mol, _, = 19 cm3/mol, % _,$ = 20 cm3/mol, $% = 0.001 J/m2, % ≫ $% , U% ≫ $% , ΩU-$ = 14 kJ/mol, Ω-$ = 25

kJ/mol, T = 300 K. The absolute Gibbs energy in the bottom figure is divided by (nA + nB) to obtain the molar Gibbs energy. The almost horizontal dotted line in the bottom figure is calculated by Eq-s (11) and corresponds to the macroscopic mixtures of A-rich + B-rich + Crich solutions. The vertical dotted line corresponds to rmin = 9.89 nm, calculated by Eq. (9j). The emulsion is stable when the molar Gibbs energy of the emulsion (bold line in the bottom figure) is below that of the non-emulsified mixture (dotted almost horizontal line in the bottom figure).


Page 27 of 31

1500

r, nm

1200

unstable emulsion

900

unstable emulsion stable emulsion

600 300

unstable emulsion 0 10

15

20

25

30

35

OMEGA(B-C), kJ/mol 400

FE

r, nm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

unstable emulsion

300

unstable emulsion

200

SE stable emulsion

100


15

20

25

30

35

OMEGA(B-C), kJ/mol

Fig.7. An emulsion stability map as function of interaction energy in the B-C solution in two magnifications of the y-axis, found by using Fig.6 and other parameters kept constant (nC = 1 % % mol, nA = 0.1 mol, nC = 0.001 mol, _, = 19 cm3/mol, _,$ = 20 cm3/mol, $% = 0.001 J/m2,

% ≫ $% , U% ≫ $% , Ω-$ = 25 kJ/mol, T = 300 K). The middle lines show the equilibrium

droplet sizes. Spontaneous emulsification takes place from any point within the interval of the minimum and maximum lines towards this middle line. FE = forced emulsification, SE = spontaneous emulsification.


Langmuir

n(B) = 0.01; OMEGA(C-B) = 14 kJ/mol

r, nm

50 40

unstable emulsion

30

stable emulsion

unstable emulsion

20 10

unstable emulsion

0 0

0,005

0,01

0,015

0,02

0,025

sigmaB, J/m2

n(B) = 0.0001; OMEGA(C-B) = 26 kJ/mol

5000

r, nm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 31

FE unstable emulsion

4000 3000

SE unstable emulsion

2000

stable emulsion

1000

unstable emulsion

0 0

0,005

0,01

0,015

0,02

0,025

sigmaB, J/m2

Fig.8. Emulsion stability maps as function of interfacial energy of pure component B at two different nB values and corresponding selected C-B interaction energies, found by using Fig.6 % % and other parameters kept constant (nC = 1 mol, nA = 0.1 mol, _, = 19 cm3/mol, _,$ = 20

cm3/mol, % ≫ $% , U% ≫ $% , Ω-$ = 25 kJ/mol, T = 300 K). The middle lines show the

equilibrium droplet sizes. Spontaneous emulsification takes place from any point within the interval of the minimum and maximum lines towards this middle line. FE = forced emulsification, SE = spontaneous emulsification.


Page 29 of 31

r, nm

2000

unstable emulsion

C+FE stable emulsion

C+SE

1000

C+SE unstable emulsion

C+SE

0 355

356

357

358

359

360

T, K

361

200

r, nm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

unstable emulsion 150

C+SE stable emulsion

100

unstable emulsion

C+SE

50


300

320

340

360

T, K

380

Fig.9. An emulsion stability map as function of temperature in two different magnifications of the y-axis found by using Fig.6 and other parameters kept constant: nC = 1 mol, nA = 0.1 mol, % % nB = 0.01 mol, _, = 19 cm3/mol, _,$ = 20 cm3/mol, $% = 0.001 J/m2, % ≫ $% , U% ≫ $% ,

ΩU-$ = 14 kJ/mol., Ω-$ = 25 kJ/mol. The middle lines show the equilibrium size of the droplets. The vertical dotted line in the lower figure shows the maximum temperature of stability of emulsions. Spontaneous emulsification takes place from any point within the

interval of the minimum and maximum lines towards the middle line. C+FE = cooling and forced emulsification, C+SE = cooling and spontaneous emulsification.


Langmuir

25000

r, nm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 31

unstable emulsion

20000

C+FE

unstable emulsion 15000

C+SE

10000

stable emulsion

5000

unstable emulsion

C+SE

0 280

290

300

310

320

330

T, K

340

Fig.10. An emulsion stability map as function of temperature found by using Fig.6 and other % parameters kept constant: nC = 1 mol, nA = 0.1 mol, nB = 0.0001 mol, _, = 19 cm3/mol,

% _,$ = 20 cm3/mol, $% = 0.0001 J/m2, % ≫ $% , U% ≫ $% , ΩU-$ = 26 kJ/mol., Ω-$ = 25

kJ/mol. The middle line shows the equilibrium size of the droplets. The vertical dotted line shows the maximum temperature of stability of emulsions. Spontaneous emulsification takes place from any point within the interval of the minimum and maximum lines towards the middle line. C+FE = cooling and forced emulsification, C+SE = cooling and spontaneous emulsification.


Page 31 of 31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

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On the Negative Surface Tension of Solutions and on Spontaneous

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