Research: Science and Education
An Excursion into Phase Tetrahedra—Where Physical Chemistry and Geometry Meet S. Ezrahi,* A. Aserin, and N. Garti** Casali Institute of Applied Chemistry, School of Applied Science and Technology. The Hebrew University of Jerusalem, 91904 Jerusalem, Israel G. Berkovic*** Mashal Alumina Industries Ltd., Box 50406, 61500 Tel Aviv, Israel
**Corresponding author. ***Present address: Electrooptics Division, Soreq NRC, Yarne 81800, Israel.
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A
X D
P
A
XC
E
XB
*Part of the results presented in this paper were included in S. E.’s Doctoral Thesis in Applied Chemistry at The Hebrew University of Jerusalem, Israel.
B
XC
Representations of the Composition of a ThreeComponent System Although any triangle can be used to represent the composition of a ternary system (1), it is most convenient to use an equilateral triangle (2, 3). Each apex of the triangle represents a pure component (i.e., 100%); the opposite base represents 0% of this component. Each point on any side represents a unique binary mixture of the components represented by the adjacent vertices. Each point within the triangle represents a unique ternary mixture, whose composition may be determined by drawing three straight lines through the point, parallel to the three sides of the triangle (see Fig. 1a). One such straight line (e.g., DE in Fig. 1a) parallel to one of the sides (BC) represents variation in the compositional relation of two components (B and C). The fractional amount of the third component (A) remains constant and is determined from the intersection of line DE with the other two sides of the triangle, as shown in Figure 1a.
Representations of the Composition of a Four-Component System A four-component system is analogously represented by a tetrahedron where each apex represents a pure component. The composition represented by a given point inside the tetrahedron is determined as shown in Figure 1c. Making a planar cross-section through the tetrahedron parallel to face BCD gives the fractional composition of component A by its cut of any of the edges AB, AC, or AD. A similar procedure will determine the composition of the other components. It is clear that these planar cuts of the tetrahedron are analogous to the lines cut to determine the composition on a threecomponent phase triangle. Although the tetrahedron fully and formally represents all the possible compositions of a four-component system, it is difficult to visualize and to work with when plotted on (two-dimensional) paper. Thus, it is often simpler to use and
B
Background
Alternatively, as shown in Figure 1b, the fractional amount of component A is given by the length of the perpendicular to side BC, divided by the sum of the three perpendiculars. This method is based on the theorem stating that the sum of these perpendiculars is equal to the altitude of the equilateral triangle (the proof of this theorem is left as an exercise to the student).
X
The phase behavior of single-component and multicomponent systems is commonly described graphically by phase diagrams. These diagrams specify the phases that are formed under various predetermined experimental conditions and uniquely define their compositions (1, 2). Phase diagrams are frequently encountered in chemistry, chemical engineering, metallurgy, and geology (1). Phase diagrams may be represented in numerous ways. A familiar X–Y graph can represent the liquid–vapor transition of a pure material (boiling point) as a function of pressure. For a two-component system, one can plot the thermal behavior as a function of composition of the binary mixture. In this communication we discuss how triangles and tetrahedra may be used to display the phase behavior of threecomponent (ternary) and four-component (quaternary) systems. We also demonstrate how to display phase behavior of 4component systems using triangular “quasi-ternary” phase diagrams, which have the important advantage of enabling the data to be plotted as a truly planar (2-dimensional) representation. The relationship between such triangular representations and the full tetrahedron is explained by geometric considerations of the cross-sections through the tetrahedron.
XA
C
Figure 1a. A typical phase triangle for a tricomponent system A,B,C. Each vertex represents one pure component and the side opposite each vertex represents 0% of this component. The composition represented by the coordinates of a point P within the phase triangle is readily determined by drawing lines (for example DE) that connect pairs of two adjacent sides (such as AB and AC) and run parallel to the complementary side (e.g., BC). The amount of each ingredient is given by the ratio of the segment to the length of the side of the triangle. For example, X A, the proportion of A in the mixture, is given by DB/AB = EC/AC.
Journal of Chemical Education • Vol. 75 No. 12 December 1998 • JChemEd.chem.wisc.edu
Research: Science & Education
easier to visualize two-dimensional triangular slices that are passed through the phase tetrahedra (1). We will show that these triangular slices are “quasi-ternary” phase diagrams of the quaternary system, in which the ratio between two of the components is held fixed. The relationship between the triangular cross-sections and the full tetrahedron can also be understood from purely geometrical considerations. We have chosen to demonstrate this way of utilizing phase tetrahedra for the study of a quaternary system based on amphiphilic components that lead to the formation of microemulsions with high water content. In our opinion, the too-often overlooked microemulsions, which are important from both the practical and the theoretical points of view, deserve a more respected place in physical chemistry curricula. Amphiphilic Systems It is well known that water and oil can form milky, thermodynamically unstable mixtures (emulsions) by mixing these components with certain compounds called surfaceactive agents (or surfactants). Surfactants are compounds belonging to the family of amphiphilic materials. An amphiphilic molecule has a polar headgroup—such as hydroxyl, sulfate, or carboxylate ions or nonionic chains composed of CH2CH 2O (ethylene oxide) units—and a nonpolar “tail” (usually a hydrocarbon chain). Thus, it has an affinity for both water and oil: the hydrophilic headgroup interacts with water and the hydrophobic “tail” interacts with oil. In this way, water and oil may be held together. Microemulsions In some cases, by adding a suitable cosurfactant (often a medium-chain-length alcohol) to a water/surfactant/oil ternary system, a transparent, thermodynamically stable, colloidal dispersion (microemulsion) may form. From a microstructural point of view, microemulsions may sometimes be envisaged as surfactant aggregates (micelles) of water dispersed (solubilized) in oil (W/O micelles) or of oil dispersed (solubilized) in water (O/W micelles). More precisely, microemulsions may be thought of as swollen (with
water or oil, respectively) micelles (4, 5). In a micelle, the surfactant molecules are organized in monolayers with their headgroups oriented toward the water and their “tails” toward the oil. The ability to form micelles is a salient feature of surfactants, which distinguishes them within the family of amphiphiles (6 ).
Use of Microemulsions (5, 7–9) Microemulsions are utilized in a large variety of applications, such as a medium for performing organic and inorganic reactions, for enhanced oil recovery, soil remediation, liquid-liquid extraction, formulations of lubricants, cutting oils, pharmaceuticals and cosmetics, etc. Description of Surfactant Systems The phase behavior of a system consisting of water, oil, surfactant, and cosurfactant may be described on a phase tetrahedron whose apexes respectively represent the pure components. However, it is more convenient to describe the phase behavior on pseudo-ternary phase triangles. Obviously, a fixed (weight, volume, or mole) ratio must be chosen for any two of the four components, and one of the triangle vertices represents 100% of this binary mixture. Phase diagrams of such quaternary systems are generally based on constant ratios of surfactant-to-water or cosurfactant-to-surfactant (10). However, in investigations of solubilization, the use of cosurfactant-to-oil constant ratio is also quite common (11– 20). Such a presentation enables us to directly and conveniently follow the variation of the surfactant amount needed to solubilize a given amount of water. The System Studied We have studied the system water/dodecane (oil)/octaethylene glycol mono-n-dodecylether [C12(EO)8, where EO stands for ethylene oxide] (surfactant)/pentanol (cosurfactant) with the weight ratio of pentanol to dodecane fixed at 1:1. Pentanol and dodecane (purity ca. 99% or above) were purchased from Aldrich Chemical Company, Inc., USA. HPLCpurified C12(EO)8 (purity in excess of 99%) was obtained from
B
A
D' F
B'
H
P
C'
D P
A
C
G
Figure 1b. Alternative method for deriving X A, X B , X C based on perpendiculars of point P (see text).
XA =
PF ; PF + PG + PH
XB =
PG ; PF + PG + PH
XC =
PH PF + PG + PH
B
C
Figure 1c. A phase tetrahedron for a four-component system. The amount of component A in the composition represented by point P is determined by the cross section B’C’D’, parallel to the face BCD, passing through point P. The proportion XA of A in the mixture is given by XA = BB’/BA = CC’/CA = DD’/DA. XB , XC, and XD are determined analogously.
JChemEd.chem.wisc.edu • Vol. 75 No. 12 December 1998 • Journal of Chemical Education
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Research: Science and Education Dodecane / Pentanol = 1
Dodecane
B 2φ
2φ B LC
Water
C12(EO)8
Figure 2a. Pseudo-ternary phase diagram of the system water/ C 12(EO)8/pentanol/dodecane (27 °C; weight ratio of dodecane/ pentanol = 1). 2φ is a two-phase region and LC is a liquid crystalline phase. The remaining area represents the one-phase region of the phase diagram. B—water designates the water dilution line for which the weight ratio of surfactant/alcohol/oil is 2:1:1. Geometrically, it is the median to the C12(EO)8-(dodecane/pentanol = 1) side.
Nikko Chemicals Co., Japan, and stored at approximately 0 °C until used. All reagents were used without further purification. Water was double-distilled.
Construction of the Phase Diagram The construction of the phase diagram was conducted at (27 ± 0.2) °C in the following way (21–24 ). Mixtures of pentanol, dodecane, and C12(EO)8 were prepared in culture tubes sealed with Viton-lined screw-caps at predetermined weight ratios of (pentanol + dodecane) to C12(EO)8. Water was then added dropwise until its solubilization limit was reached. After this point, larger increments of water were added. All water additions were followed by vigorous stirring on a vortex mixer. Equilibration time between additions of successive aliquots was typically from a few minutes to up to 24 hours. The tubes were then inspected visually for phase separation. We then constructed a phase diagram by denoting the number and nature of phases at the appropriate point in the phase triangle representing each composition. Figure 2a shows the quasi-ternary phase diagram of the system, constructed by considering the combined amount of (pentanol + dodecane) as a single component. The phase diagram includes results from several hundred data points, and for clarity only the borderlines between the various phases are shown. All compositions that arise from diluting a starting mixture of (pentanol + dodecane + surfactant) with varying amounts of water will fall along a straight line (such as B—water in Fig. 2a) from the appropriate point along the surfactant–dodecane/pentanol base to the water apex. Such a line is consequently called a “dilution line”. It should be noted that in this system, the W/O and O/W microemulsions have been merged into one large continuous monophasic region, which may have a practical application (e.g., for the formulation of fire-resistant hydraulic fluids) (22, 23, 25). Another Presentation of the Phase Behavior The triangle depicted in Figure 2a is just one possible way of describing the data points we measured. Now, let us perform another experiment with different initial compositions, where 1650
Water
C12(EO)8 Pentanol
=2
Figure 2b. Pseudo-ternary phase diagram of the system water/ C 12(EO)8/pentanol/dodecane (27 °C; weight ratio of pentanol to C 12(EO)8 is 1:2). 2φ is a two-phase region. B—water designates the water dilution line for which the weight ratio of [C12(EO)8 + pentanol] to dodecane is 3:1.
[C 12 (EO) 8 + pentanol] is considered as a quasi-single component (see Fig. 2b). The phase diagram was constructed following the procedure previously outlined. Comparison between the Two Presentations Consider a composition comprising 25% pentanol + 25% dodecane + 50% C12(EO) 8 (all by weight), which is represented by point B on the phase diagrams of Figures 2a and 2b. Dilution of this mixture produces the data points along the “dilution line” from B to the water vertex in both figures. Looking at Figures 2a and 2b, we see that although the two phase diagrams are entirely different (note the relative sizes of the 2φ region and the absence of LC region in Fig. 2b), they both contain the same B—water dilution line and the same water solubilization limit along this line. The reason why two pseudo-ternary diagrams look very different, yet contain the same dilution line BW (as an angle bisector in phase diagram 2a, but not in phase diagram 2b), will become evident as a result of the geometrical considerations in the following section.
Relative Positions of the Phase Triangles within the Phase Tetrahedron In Figure 3 we present the full phase tetrahedron in which the vertices represent 100% water (W), dodecane (D), pentanol (P), and surfactant (S). Within the tetrahedron we have drawn the two triangular cross-sections WSC and WDE, representing the pseudo-ternary phase diagrams of Figures 2a and 2b, respectively. (Point C represents pure D/P = 1 and point E represents pure S/P = 2.) It is seen that: 1. The intersection of the planes of the two triangles is along the B—water line. 2. The quasi-ternary phase triangles WSC and WDE are not equilateral triangles, but rather isosceles.1
We now understand that we may draw an infinite number of quasi-ternary cross-sectional phase diagrams for this system, and also that a given dilution line may be represented on a multitude of such diagrams.
Journal of Chemical Education • Vol. 75 No. 12 December 1998 • JChemEd.chem.wisc.edu
Research: Science & Education D
D
C (D/P = 1) a 2
W
C B
P
B
E (S/P = 2)
S S Figure 3. Two-phase triangles W(D/P = 1)S (shaded area) and W(S/ P = 2)D (dotted edges) for the system water (W)/C12(EO) 8 (S)/ pentanol (P)/dodecane (D) within the phase tetrahedron DPSW.
Note that it is possible to prove by purely geometrical means that the two triangles intersect at the same point B as was derived from the compositional analysis. We consider only the equilateral triangle DSP of the tetrahedron and the two lines SC and DE and their intersection point B (see Fig. 4). The bisector of an angle of a triangle divides the opposite side into segments, which are proportional to the adjacent sides. Since SB bisects the angle DSE the following law holds:
EB = SE DB SD and since SD = SP we get
EB = SE = 1 DB SP 3 Thus, point B divides the line ED into two segments which are proportional to 1:3. A somewhat more complicated derivation (see Appendix 1) leads to SB = BC. Therefore, point B is also the midpoint of SC. The triangle WCS (Fig. 2a) meets the triangle WED at the point B and since they also have a common vertex, W, they intersect each other along the dilution line BW (Fig. 3). Conclusions We have drawn a parallel between the phase behavior of a real physical system composed of four definite components and the predictions of abstract mathematical laws. These laws can be applied to this system when the compositional relationships between its components are translated into geometrical terms. We have shown how triangles and tetrahedra can be used to represent the composition and phase behavior of threeand four-component systems. In the latter case, the difficulty of visualizing a tetrahedral representation can be reduced to a more convenient 2-D triangular representation as a “quasi-
a 3
E
2a 3
P
Figure 4. Intersection of the triangles W(P/D = 1)S and W(S/P = 2)D as reflected on the face SDP of the tetrahedron DPSW. For the sake of brevity, the vertices D/P = 1 and S/P = 2 are designated by C and E, respectively. DC = CP and SE/SP = 1/3.
ternary” system in which the ratio of two of the components is kept constant. The triangular representations are merely cross-sections through the tetrahedron, and the relationship between the two representations can also be derived from geometrical considerations and laws of mathematics. This method of representation has been applied to a four-component amphiphilic system. Naturally, other types of quaternary systems may be used for the demonstration. We would, however, recommend that the teacher choose a real system and give a detailed description of how to construct its phase diagram, including the reasons for passing a specific 2-D triangular slice through the phase tetrahedron. Acknowledgments We would like to thank A. Shaket for helpful discussions and Yu. Talanker for technical assistance. Note 1. It is readily shown (as the students may convince themselves) that triangle WSC is isosceles with the length WS = a and WC = SC = (a √3)/2; in triangle WDE, WD = a and WE = WD = (a √7)/3. Also, the length of the line BW equals (a √11)/4. This last result can be derived geometrically as in Appendix 2. Students may try alternative proofs, including use of the cosine rule. In Figures 2a and 2b the two representative phase diagrams for our quasi-ternary system were drawn, nevertheless, as equilateral triangles because this is the common way in the literature to describe phase behavior.
Literature Cited 1. MacCarthy, P. J. Chem. Educ. 1983, 60, 922–928. 2. Broze, G. In Liquid Detergents; Lai, K.-Y., Ed.; Surfactant Science Series 67; Dekker: New York, 1996; pp 35–65. 3. Mackay, R. A. In Encyclopedia of Emulsion Technology, Vol. 3; Becher, P., Ed.; Dekker: New York, 1988; pp 223–237. 4. Friberg, S. Chemtech 1976, 6, 124–127. 5. Friberg, S. E.; Bendiksen, B. J. Chem. Educ. 1979, 56, 553–555. 6. Davis, H. T.; Balet, J. F.; Scriven, L. E.; Miller, W. G. In Physics of Amphiphilic Layers; Meunier, J.; Langevin, D.; Boccara, N.,
JChemEd.chem.wisc.edu • Vol. 75 No. 12 December 1998 • Journal of Chemical Education
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D
B
C B 60
30
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E
A
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S
Figure 5. The equilateral triangle SDP of Fig. 4 is shown here with the addition of lines CA and AB. SA = AP.
Figure 6. The shaded isosceles triangle W(D/P = 1)S of Fig. 3 is shown here as the triangle WCS. BW is a median to side CS. CK and BL are perpendiculars to WS.
Eds.; Springer Proceedings in Physics 21; Springer: New York, 1987; pp 310–327. Schwuger, M.-J.; Stickdorn, K.; Schomäcker, R. Chem. Rev. 1995, 95, 849–864. Chhabra, V.; Free, M. L.; Kang, P. K.; Truesdail, S. E.; Shah, D. O. World Surfactant Congress, 4th; Asociacion Espanola de Productores de Sustancias para Aplicaciones Tensioactivas, Barcelona, Spain, 1996; Vol. 1, pp 67–99. Industrial Applications of Microemulsions; Solans, C.; Kunieda, H., Eds.; Surfactant Science Series 66; Dekker: New York, 1997. Hunter, R. J. Foundations of Colloid Science, Vol. 2; Clarendon: Oxford, 1989; pp 948–991. Li, G.; Kong, X.; Guo, R.; Wang, X. J. Surface Sci. Technol. 1989, 5, 29–40. Venable, R. L.; Viox, D. M. J. Dispersion Sci. Technol. 1984, 5, 73–80. Venable, R. L. J. Am. Oil Chem. Soc. 1985, 62, 128–133. Fang, J.; Venable, R. L. J. Colloid Interface Sci. 1987, 116, 269–277. Chew, C. H.; Gan, L. M.; Koh, L. L.; Wong, M. K. J. Dispersion Sci. Technol. 1988, 9, 17–31. Chew, C. H.; Gan, L. M. J. Dispersion Sci. Technol. 1990, 11, 49–68. Schurtenberger, P.; Peng, Q.; Lesen, M. E.; Luisi, P.-L. J. Colloid Interface Sci. 1993, 156, 43–51. Friberg, S. E.; Rong, G. Langmuir 1988, 4, 796–801. Friberg, S. E.; Yang, C. C.; Goubran, R.; Partch, R. E. Langmuir 1991, 7, 1103–1106. Friberg, S. E.; Jones, S.M.; Yang, C. C. J. Dispersion Sci. Technol. 1992, 13, 45–63. Garti, N.; Aserin, A.; Ezrahi, S.; Wachtel, E. J. Colloid Interface Sci. 1995, 169, 428–436. Regev, O.; Ezrahi, S.; Aserin, A.; Garti, N.; Wachtel, E.; Kaler, E. W.; Khan, A.; Talmon, Y. Langmuir 1996, 12, 668–674. Garti, N.; Aserin, A.; Ezrahi, S.; Tiunova, I.; Berkovic, G. J. Colloid Interface Sci. 1996, 178, 60–68. Waysbort, D.; Ezrahi, S.; Aserin, A.; Givati, R.; Garti, N. J. Colloid Interface Sci. 1997, 188, 282–295. Garti, N.; Feldenkriez, R.; Aserin, A.; Ezrahi, S.; Shapira, D. Lubrication Eng. 1993, 49, 404–411.
∆ SCP is a 30°-60°-90° triangle. So, the leg opposite the 30° angle is half the hypotenuse and we get CP = PA = AC = AS. It was shown that EB/BD = 1/3. Therefore,
7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
Appendix 1. Derivation of the Relation SB = BC To prove that SB = BC (Fig. 5, with all data as given in Fig. 4), draw a line CA, where A is the midpoint of the side SP, and join A to B.
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EB = EB = 1 EB + BD ED 4 Now, if SE ≡ X, then EP = 2X, SP = 3X, and SA = (3/2)X. So EA = SA – SE = 1 X and EA = 1/2X = 1 2 EP 2X 4
EB = EA . ED EP The line AB divides the sides SC and SP of the triangle CSP proportionately, and thus it is parallel to the third side, CP. The corresponding angles ABS and PCS are equal, so AB is perpendicular to CS and the triangles SBA and CBA are congruent, leading to SB = BC.
and we get
Appendix 2. Evaluation of the Length of Line WB Given the isosceles triangle WCS (with CB = BS, as demonstrated in Appendix 1), draw the lines CK and BL, both perpendicular to WS (Fig. 6). Thus, BL is parallel to CK and therefore:
BL = SL = SB = 1 CK SK SC 2 Now, if WS ≡ a, then
WK = KS = 1 a ; KL = LS = 1 a ; and WL = 3 a 2 4 4 SC is a median in the equilateral triangle SDP (Fig. 4). By the Pythagorean theorem, its length equals (a √3)/2; and since SB = BC, they both equal (a √3)/4. Now, 2
2
2
(BL) = (BS) – (LS) =
3a 2 a 2 a 2 – = 16 16 8
and 2 2 2 a2 (WB) = (WL) + (BL) = 3 a 2 + = 11 a 2 4 8 16
leading to WB = (a √11)/4.
Journal of Chemical Education • Vol. 75 No. 12 December 1998 • JChemEd.chem.wisc.edu
