VOLUME 77, NUMBER 23 PHYSICAL REVIEW LETTERS 2DECEMBER 1996
Phase Diagram of Colloidal Solutions
Neer Asherie, Aleksey Lomakin, and George B. Benedek Department of Physics and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 (Received 1 July 1996) The phase diagram of globular colloids is studied using a combined analytic and computational representation of the relevant chemical potentials. It is shown how the relative positions of the phase boundaries are related to the range of interaction and the number of contacts made per particle in the solid phase. The theory presented successfully describes the features of the phase diagrams observed in a wide variety of colloidal systems. [S0031-9007(96)01805-4]
PACS numbers: 82.70.Dd, 64.70.Ja
The phase diagrams of colloidal solutions have been shown in Fig. 1, a few do show phase behavior closer to studied for over a century not only because of their great that pictured in Fig. 2 . theoretical interest, but also for the many industrial ap- Recently, evidence has accumulated that the interaction plications of colloids . The most commonly observed range plays a signiﬁcant role in determining the structure phase transition in colloidal solutions is solidiﬁcation. of the phase diagram . In colloid-polymer mixtures Upon a change in temperature (or other external condi- it has been found that the shape of the phase diagram tion) the colloidal particles form a condensed phase which depends on the ratio of the radius of gyration of the may have a regular structure (crystals) or be amorphous polymer molecules to the radius of the colloidal particles (aggregates). A more infrequent transition is liquid-liquid [12,14,15]. Theoretical models [16,17] of this system phase separation (coacervation). Here the colloidal solu- agree with the experimental observation that with very tion forms two distinct liquid phases: one colloid-rich, the small polymers (i.e., very short ranges of attraction) there other colloid-poor . is no colloidal liquid phase . Colloids are not the These transitions have analogous counterparts in sim- only system for which the connection between short-range ple molecular ﬂuids. The solidiﬁcation of colloids is interactions and the structure of the phase diagram has equivalent to the ﬂuid-solid transition while colloidal been noted. Fullerenes, macromolecules of carbon, also liquid-liquid phase separation corresponds to gas-liquid do not appear to exhibit a liquid phase upon cooling . coexistence. There is, however, one striking difference: Simulations of hard spheres with an attractive Yukawa when most simple ﬂuids are cooled the order of phases potential have been carried out to investigate the phase observed is gas to liquid to solid. In colloidal solutions diagram of the fullerene C60 [19,20]. It is found that for the colloidal “gas” usually transforms into a solid without passing through a liquid phase. We illustrate this “anomalous” order of phases by presenting in Fig. 1 the phase diagram of gII-crystallin protein [3–5]. This phase diagram is typical of the g-crystallins (a family of monomeric eye lens proteins) and of other small globular proteins [6–8]. The circles are points which represent the volume fractions (f) of coexisting protein-rich and protein-poor liquid phases (liquid-liquid coexistence curve). The squares (liquidus line) and the triangle (solidus line), respectively, represent the volume fractions of protein in the liquid and solid phases in equilibrium with each other. We see that there is no triple point and the coexistence curve lies below the liquidus line. The g-crystallins are an unusual colloidal system in that liquid-liquid phase separation may be observed despite it being metastable with respect to solidiﬁcation. For comparison the phase FIG. 1. The phase diagram of gII-crystallin [3–5]. The diagram of argon is shown in Fig. 2 [9–11]. We see that circles are points on the liquid-liquid coexistence curve (CC). The squares are points on the liquidus line (L). The triangle is for argon the critical point lies above the triple point (i.e., a point on the solidus line (S). The lines are guides to the eye. Tc . Tt) indicating the presence of a stable liquid phase. The critical temperature is Tc 278.4 K. The critical volume Although most colloidal phase diagrams resemble that fraction is fc 0.21.
4832 0031-9007͞96͞77(23)͞4832(4)$10.00 © 1996 The American Physical Society VOLUME 77, NUMBER 23 PHYSICAL REVIEW LETTERS 2DECEMBER 1996
centers lie in the region s#r,lsfrom a given parti- cle. The square-well model produces a very simple form for the chemical potential of the solid ms, namely, 3 ms m0 2 ns͑e͞2͒ 2 kT ln͓͑l21͔͒. (2)
The ﬁrst term on the right-hand side of Eq. (2), m0,is the standard part of the chemical potential. The second term is the total energy associated with each particle. The last term is the entropic contribution: the volume accessi- ble to the center of mass of the particle is proportional to ͑l21͒3. The proportionality factor, essentially the vol- ume of a unit cell, has been absorbed into m0. This factor can be calculated within the framework of the cell model . The cell volume, however, is practically constant, FIG. 2. The phase diagram of argon [9–11] showing the reﬂecting the incompressibility of the solid phase. We coexistence curve (CC) and the liquidus (L) and solidus may therefore choose a cell volume at zero pressure. The (S) lines. The critical temperature is Tc 150.86 K. The corresponding value of m0 is found to be identical to that critical volume fraction is fc 0.133, assuming a hard core in the expression m m0 1 kT ln f, which is appropri- diameter s 3.162 Å . The triple point temperature is ate for dilute solutions . T 83.78 K 0.56T . t c An important parameter in the chemical potential of the solid is ns. The value of ns is not known a priori, but a sufﬁciently short-range potential the coexistence curve determined by the structure of the solid. It is common lies below the liquidus curve. practice to impose a particular crystal structure upon the In this Letter we present a general analysis which solid when calculating the chemical potential by choosing explains these individual experimental and theoretical the appropriate integral value for ns. In the subsequent ﬁndings. In our previous Monte Carlo work  we stud- analysis, however, we will treat ns as a continuous, ied the liquid-liquid phase separation of globular particles phenomenological parameter so as to subsume within the with attractive interactions and obtained numerically the cell model the actual structure and detailed interactions in chemical potential of the liquid phase. Here we present the solid phase of real colloids. an analytic expression for the chemical potential of a solid The liquidus line is obtained by equating the chemical with short-range interactions. Our model of the solid in- potentials of the solid and the liquid, namely, corporates the same essential features that were used to describe the liquid phase: the range l of the interaction, mˆ s͑eˆ; ns͒ mˆ l͑f,ˆe͒. (3) the interaction energy e, and the number of contacts ns made per particle. Knowing the chemical potentials of Here mˆ s ϵ͓ms 2m0͔͞kT and thus both phases we are able to demonstrate how the relative mˆ s 2ns͑eˆ͞2͒ 2 3 ln͓͑l21͔͒ (4) locations of the phase boundaries are related to l and ns. We use the Lennard-Jones and Devonshire cell model is the reduced chemical potential of the solid. The  to obtain an approximate analytic expression for the reduced chemical potential of the liquid mˆ l has been chemical potential of a solid with short-range interactions. previously established as a numerical function of f and As in our previous work, we will assume that the effective eˆ  (eˆ ϵ e͞kT is the reduced energy). potential energy u͑r͒ for a pair of proteins (diameter s) For any solid with a given value of ns, Eq. (3) allows whose centers are separated by a distance r, is of the form us to construct the liquidus line. In Fig. 3 we show of an attractive square well with a hard core as given by the liquidus lines for several values of ns at l 1.25: 1`, for r ,s, ns 12.0 (A); ns 11.6 (B); ns 11.5 (C). The line u͑r͒ 2e, for s#r,ls, (1) 8 (D) is the solidus line obtained by using the volume < 0, for r $ls. of a unit cell appropriate to a face-centered cubic solid Here l is the reduced: range of the potential well and (ns 12) at zero pressure . The coexistence curve e is its depth. When the interactions are short ranged (E) is taken from our previous work . As the value of (l ! 1) the thermodynamic properties become univer- ns increases liquid-liquid coexistence becomes metastable sal, independent of the shape of the potential. We work with respect to solidiﬁcation. The shape of the phase with the square-well potential rather than other potentials diagram for ns 12.0 (curves A, D, and E) has the [19,20,23] because it allows for an unambiguous deﬁni- same structure as the one found experimentally for the tion of not only the range of interaction l, but also of g-crystallins (Fig. 1). ns, the number of contacts made per particle in the solid We may consider as a measure of the metastability of phase. Explicitly, ns is the number of particles whose the liquid phase the “metastability gap” ͑TL 2 Tc͒͞Tc,
4833 VOLUME 77, NUMBER 23 PHYSICAL REVIEW LETTERS 2DECEMBER 1996
TABLE I. Metastability gap parameters at different reduced ranges l [see Eq. (7)]. The quantities presented are (i) the ء average number of contacts per particle in the solid ns at the metastability boundary (TL Tc), (ii) the change in the total number of contacts per particle in the liquid nl upon the addition of an extra particle, (iii) the critical volume fraction fc, and (iv) the reduced critical energy eˆc.
ء l ns nl fc eˆc 1.25 11.59 7.84 0.205 1.269 1.20 11.03 7.47 0.216 1.443 1.15 10.45 7.00 0.227 1.673 1.10 9.75 6.57 0.244 2.038 1.05 8.95 5.62 0.246 2.667
FIG. 3. The phase diagram for a square-well system for maximum number of contacts for hard spheres with short- l 1.25. The liquidus lines for three values of ns are shown: ns 12.0 (A); ns 11.6 (B); ns 11.5 (C). The vertical line range interactions is 12. Therefore at any given range the ء (D) is the solidus for ns 12.0. Curve E is the coexistence solid will always be stable if 12 $ ns . ns . As the range ء .[curve taken from Ref. [21 decreases so does ns and fewer contacts are necessary to form a stable solid. Thus, we expect that as the range de- where Tc is the critical temperature and TL is the creases, liquid-liquid phase separation will be less likely temperature of the point on the liquidus line at the critical to be observed. volume fraction fc. According to Eq. (3) With our approach we may understand the experiments of Ilett et al. . These authors study colloid-polymer mˆ ͑eˆ ; n ͒ mˆ ͑f ,ˆe ͒, (5) s L s l c L mixtures with different ranges of interaction and ﬁnd at with eˆL e͞kTL. As we have shown earlier , the l ഠ 1.25 a transition from the type of phase diagram chemical potential of the liquid at eˆL may be expanded shown in Fig. 2 to that in Fig. 1 (they do not observe about the reduced critical energy eˆc the metastable liquid-liquid coexistence). In our analysis ͑eˆL 2eˆc͒ this transition occurs when the metastability gap changes ء (mˆ ͑f ,ˆe ͒mˆ ͑f ,ˆe ͒2 n . (6 l c L l c c 2 l sign, i.e., when ns ns. In the solid phase the colloid- polymer system forms close-packed crystals, i.e., n 12 Here n is the change in the total number of contacts in the s l . Thus from Table I we expect the transition to liquid phase upon the addition of an extra particle at the occur at l ഠ 1.25, the value observed experimentally. In as the number of contacts ءcritical point. Let us deﬁne n s their experiments only one range of interaction shorter in the solid at which T T , i.e., the value of n for L c s than the crossover value of l ഠ 1.25 is studied: l ءwhich the liquidus line touches the critical point. Thus n s 1.08 . Taking n 12, Table I implies that at this mˆ f ,ˆe . Using this deﬁnition s ءis given by mˆ eˆ ; n s͑ c s ͒ l͑ c c͒ range the metastability gap of the system is so large and substituting Eqs. (6) and (4) into Eq. (5) we ءof n s that the coexistence curve lies outside the experimentally obtain examined region. It would be interesting to search ء TL 2 Tc ns 2 ns intermediate ranges, i.e., 1.08 ,l,1.25, for metastable (7) . ء Tc ns 2 nl liquid-liquid coexistence and compare the values of the From Eq. (7) we see that the metastability gap ͑TL 2 metastability gaps with our predictions. ء Tc͒͞Tc depends on the parameters ns, ns , and nl. The We had previously suggested that the large values of ﬁrst of these is determined by the structure of the solid. fc and the broad coexistence curves observed for the g- We list the other two in Table I. In Table I we show for cystallins imply that these proteins lie in the domain l# ء each value of l the corresponding values of ns and nl. 1.25 . We now ﬁnd that this is precisely the domain ء Since ns and nl are evaluated at the critical point, these where liquid-liquid coexistence may be metastable, and, two parameters depend only on l and not on e. We note in fact, this metastability is observed. When plotted in that for all of our short-range simulations, the quantity reduced units the phase diagrams of the g-crystallins ء ns 2 nl is in the range 3.2 3.8. We also list the criti- have the same shape with almost identical values of cal volume fraction fc and the reduced critical energy eˆc fc and with approximately the same size metastability at each range. Columns 3–5 are from our previous work gaps [3,21]. The interactions between these proteins ء , while ns is obtained from a numerical solution of the may therefore be described by potentials with the same ء equation mˆ s͑eˆc; ns ͒ mˆ l͑fc,ˆec͒. range. In addition, all the protein crystals should have the For the solid phase to be more stable than any coexist- same value of ns. For ͑TL 2 Tc͒͞Tc ഠ 0.1, the relation ء ing liquid phases i.e., TL . Tc, we require ns . ns . The found experimentally for the g-crystallins , Eq. (7)
4834 VOLUME 77, NUMBER 23 PHYSICAL REVIEW LETTERS 2DECEMBER 1996
ء gives ns 2 ns ഠ 0.35. It is because ns differs little from  J. A. Thomson et al., Proc. Natl. Acad. Sci. U.S.A. 84, ء ns that we observe both liquid-liquid coexistence and 7079 (1987). solidiﬁcation for the g-crystallins.  M. L. Broide, T. Tominc, and M. Saxowsky, Phys. Rev. E Related observations are made by Broide et al.  who 53, 6325 (1996). ﬁnd that the metastability gap of lysozyme is unaffected  D. Rosenbaum, P. C. Zamora, and C. F. Zukoski, Phys. by ionic strength but that it is a larger (by 5 ±C–10±C) Rev. Lett. 76, 150 (1996).  M. Muschol et al., in Photon Correlation and Scattering, with respect to needle-shaped crystals than with prism- 1996 OSA Technical Digest Series Vol. 14 (Optical shaped crystals. In our analysis these two types of –Society of America, Washington, DC, 1996), pp. 145 ء crystals are predicted to have values of ns ns differing 147. by approximately 0.05 0.1.  V. A. Rabinovich et al., Thermophysical Properties of It has been argued that in the adhesive sphere limit Neon, Argon, Krypton, and Xenon (Hemisphere, Wash- (l ! 1,e ˆ ! `) there is no thermodynamically stable ington, 1988), pp. 380–383. liquid phase . Our analysis shows that for a close-  O. G. Peterson, D. N. Batchelder, and R. O. Simmons, packed system the absence of a stable liquid phase already Phys. Rev. 150, 703 (1966). begins when l,1.25.  P. Flubacher, A. J. Leadbetter, and J. A. Morrison, Proc. We have presented a simple analytic form for the Phys. Soc. 78, 1449 (1961). chemical potential of a short-range solid in terms of  S. M. Ilett et al., Phys. Rev. E 51, 1344 (1995).  A. Daanoun, C. F. Tejero, and M. Baus, Phys. Rev. E 50, the physically signiﬁcant parameters of the system: the 2913 (1994). interaction range l, the interaction energy e, and the  F. Leal Calderon, J. Bibette, and J. Biais, Europhys. Lett. number of contacts made per particle ns. We have shown 23, 653 (1993). how these parameters determine the order of phases  P. N. Pusey et al., J. Phys. A 6, 29 (1994). found in the phase diagram. In particular, we have  H. N. W. Lekkerkerker et al., Europhys. Lett. 20, 559 demonstrated that for given ns, a sufﬁciently short range (1992). of interaction leads to the metastability of liquid-liquid  A. P. Gast, C. K. Hall, and W. B. Russel, J. Colloid coexistence with respect to solidiﬁcation. Conversely, Interface Sci. 96, 251 (1983). experimental information on the relative order of the  P. A. Heiney et al., Phys. Rev. Lett. 66, 2911 (1991). phase boundaries of colloidal solutions gives direct insight  E. Lomba and N. G. Almarza, J. Chem. Phys. 100, 8367 into the magnitude of the physically important parameters (1994).  M. H. J. Hagen and D. Frenkel, J. Chem. Phys. 101, 4093 l and n . The theory we have presented appears capable s (1994). of explaining the phase diagrams observed in a wide  A. Lomakin, N. Asherie, and G. B. Benedek, J. Chem. variety of colloidal systems. Phys. 104, 1646 (1996). We thank Dr. George Thurston and Dr. Jayanti Pande  J. E. Lennard-Jones and A. F. Devonshire, Proc. R. Soc. for critical comments. This work was supported by the London Sect. A 163, 53 (1937). National Eye Institute of the National Institute of Health  J. P. K. Doye and D. J. Wales, Science 271, 484 (1996). under Grant No. 5-R37-EY05127.  J. A. Barker, Lattice Theories of the Liquid State (Perga- mon Press, Oxford, 1963).  E. A. Guggenheim, Thermodynamics (North-Holland, Am-  D. H. Everett, Basic Principles of Colloid Science (Royal sterdam, 1967), 5th ed., pp. 220–224. Society of Chemistry, London, 1988).  P. N. Pusey et al., Phys. Rev. Lett. 63, 2753 (1989).  Colloid Stability, edited by D. A. Young, Faraday Discus-  W. C. K. Poon et al., J. Phys. II (France) 3, 1075 sions of the Chemical Society Vol. 65 (Chemical Society, (1993). London, 1978).  G. Stell, J. Stat. Phys. 63, 1203 (1991).  C. R. Berland et al., Proc. Natl. Acad. Sci. U.S.A. 89, 1214  J. O. Hirschﬁelder, C. F. Curtiss, and R. B. Bird, Molecular (1992). Theory of Gases and Liquids (Wiley, New York, 2nd  M. L. Broide et al., Proc. Natl. Acad. Sci. U.S.A. 88, 5660 printing, 1964), p. 160. (1991).