Second Virial Coefficient at the Critical Point in a Fluid of Colloidal

Letter pubs.acs.org/Langmuir

Second Virial Coefficient at the Critical Point in a Fluid of Colloidal Spheres Plus Depletants Remco Tuinier*,†,‡ and Maartje S. Feenstra† †

Van’t Hoff Laboratory for Physical and Colloid Chemistry, Department of Chemistry, Debye Institute, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands ‡ DSM ChemTech Center, P.O. Box 18, 6160 MD Geleen, The Netherlands ABSTRACT: Vliegenthart−Lekkerkerker (VL) criterion B2 = −6vc for second virial coefficient B2 at the critical (colloidal) gas−liquid point is considered for a mixture of spheres with volume vc plus depletants. For the onset of fluid-phase instability, the VL criterion holds for a wide range of shapes of direct attractive forces between hard-core spheres (Vliegenthart, G. A.; Lekkerkerker, H. N. W. J. Chem. Phys. 2000, 112, 5364). In the case of long-ranged attractions imposed indirectly via depletants, it is found that the VL relation fails. Instead, B2/vc at the critical point depends strongly on the sphere/depletant size ratio. By making the hard spheres sticky, we find that B2 moves gradually toward the VL criterion upon increasing the stickiness.

■

INTRODUCTION

Colloidal dispersions show analogues of the phase states that can be found in atomic systems: gas, liquid, and crystal.1 Perrin’s proof of earlier theoretical work by Gibbs, Von Smoluchowski, and Einstein demonstrated that colloidal particles in a background solvent behave just like atoms.2 The main difference between colloidal and atomic/molecular systems is that colloidal interactions can be tuned.3,4 When polymers are grafted at the surface of the colloidal particles1 or via a double-layer interaction,5,6 it is possible to control the range and strength of the repulsive forces in addition to the hard core repulsion. Attractive interactions can be tuned, for instance, by using a critical solvent mixture7 or via depletion forces8,9 by adding nonadsorbing polymers or rodlike colloids. Driven by the insights gained after systematic studies on the influence of the range of attraction on the phase behavior of dispersions of colloidal spheres plus nonadsorbing polymers,10−13 it turned out that a sufficient range of attraction is required to be able to induce (colloidal) gas−liquid coexistence. An example of a phase diagram for a mixture of hard spheres (radius R) plus depletants (simplified as penetrable hard spheres9 with radius δ) is plotted in Figure 1 for q = δ/R = 0.6. In short, when the range of attraction is less than one-third (q < 0.33) of the sphere diameter, only a colloidal fluid−solid equilibrium is found, whereas colloidal gas−liquid coexistence equilibria appear only for longer-ranged attractions. This holds quite generally for dispersions of spheres with a repulsive core plus an attraction.14 For the second (osmotic virial) coefficient (B2) of a collection of spherical particles at the critical (colloidal) gas−liquid point, a universal value has been identified by the Vliegenthart−Lekkerkerker criterion,15 © 2014 American Chemical Society

Figure 1. Phase diagram of a mixture of hard spheres plus depletants described as penetrable hard spheres for q = δ/R = 0.6. Free volume theory:10 solid curves and open circle (critical point); simulation data redrawn from Dijkstra et al.13 The dashed and dotted curves are the gas−liquid coexistence results for sticky spheres (SS) plus penetrable hard spheres. The SS are spheres interacting via a hard core plus Yukawa attraction (see eq 8) with κσ = 2 and βϵ = −0.3 (SS1) or βϵ = −0.6 (SS2).

B2cp = −6vc

(1)

implying that a fluid phase separates into a gas and a liquid for B*2 ≲ −6, where B*2 = B2/vc and vc is the sphere volume. For Received: July 4, 2014 Revised: October 12, 2014 Published: October 20, 2014 13121

dx.doi.org/10.1021/la5023856 | Langmuir 2014, 30, 13121−13124

Langmuir

Letter

hard core particles supplemented with a direct attractive interaction, this relation can be understood by the fact that B3 and B4 are close to their hard sphere values,15 which indicates that the deviation from the thermodynamic hard sphere behavior is dominated by B2. The VL criterion turned out to be a rather good predictor of the second virial coefficient at the onset of gas−liquid phase transitions for spherical particles with a repulsive core plus an attraction that can have different shapes. Some examples of interactions that obey the VL criterion up to at least a range of attraction that corresponds to q = 2 include the square well interaction,16 the Lennard-Jones (LJ) interaction, and the hardcore Yukawa attraction.15 The VL criterion has been applied to several systems,17−19 and together with the related extended corresponding equation of state of Noro and Frenkel,20 it gained enormous attention because of the implication that the phase behavior of a collection of attractive (colloidal) spheres is rather universal. An unresolved issue is whether the VL criterion also holds for a mixture of spherical colloids plus depletants. We find that the criterion of eq 1 fails for a mixture of hard spheres plus penetrable hard spheres for q ≳ 0.5. We have re-examined the phase behavior of a mixture of hard spheres plus penetrable hard spheres (PHS) using free volume theory,10 which is extremely accurate. See, for instance, the comparison with simulations in Figure 1 and refs 12 and 21. PHS are spheres that do not feel each other (penetrable) but that are hard for the hard spheres.9 These PHS can be used to mimic ideal polymer chains accurately.22

Figure 2. Second virial coefficient at the critical point plotted as −B*2 cp versus q. Symbols are redrawn Monte Carlo computer simulations data from simulated mixtures of hard spheres and depletants (closed circles, Bolhuis et al.;25 closed diamonds, Dijkstra et al.;13 closed triangles: Moncho-Jordá et al.;21 closed squares, Lo Verso et al.26).

allows us to approximate the dimensionless semigrand potential (ω = Ωvc/(kTV)) for a mixture of HS + PHS in a background solvent:23 ω = fc − αϕRd q−3, where fc is the dimensionless canonical free energy of the dispersion of spherical colloidal particles, ϕRd is the relative concentration (ϕRd = 1 defines the PHS reservoir overlap concentration) of the depletants in the reservoir, and the free volume fraction α reads10,23 α = (1 − η) exp[−ay − by2 − cy3], where y = η/(1 − η), a = (1 + q)3 − 1, b = 9q2/2 + 3q3, and c = 3q3. Note that ω can be computed for a fluid phase of HS + PHS by using for fc the Carnahan−Starling equation of state.24 We calculated B2 at the gas−liquid critical point (cp) for a wide range of q values as follows. First we compute the critical gas−liquid cp from free volume theory. This yields ϕR,cp (some d examples given in Figure 1). This value is then inserted into the Asakura−Oosawa−Vrij depletion potential Wdep between two hard spheres mediated by the penetrable hard spheres:8,9

■

THEORY Within free volume theory (FVT), one calculates the semigrand potential Ω for a system of interest with volume V that contains colloidal spheres plus depletants and a background solvent.10 The system is held in osmotic equilibrium with a hypothetical reservoir that contains depletants and background solvent but no colloids. Hence, the fugacity of the depletants is imposed in the reservoir, which is the same in the system. This approach

⎧∞ for x < 1 ⎪ 3⎤ ⎡ Wdep(x) ⎪ ⎪ R 3⎛ x ⎞ 1⎛ x ⎞ ⎥ −1 3 ⎟+ ⎜ ⎟ 1≤x≤1+q = ⎨−ϕd (1 + q ) ⎢1 − ⎜ ⎢⎣ 2 ⎝1 + q ⎠ 2 ⎝ 1 + q ⎠ ⎥⎦ kT ⎪ ⎪ ⎪0 x>1+q ⎩

where x = r/2R, with r being the distance between the centers of two interacting spheres.

(2)

Equation 2 allows us to compute B2, defined as

⎧ ∞⎛ ⎡ W (x ) ⎤⎞ 2 ⎪12 (stat mech definition) ⎜1 − exp⎢ − ⎥ ⎟x d x ⎣ kT ⎦⎠ 0 ⎝ ⎪ ⎪ B2* = ⎨ ⎡ W (x ) ⎤⎞ 1+q ⎛ ⎪ ⎜⎜1 − exp⎢ − dep ⎥⎟⎟x 2 dx (depletion interaction) ⎪ 4 + 12 ⎪ ⎢⎣ kT ⎥⎦⎠ 1 ⎝ ⎩

∫

∫

at the gas−liquid critical point.

■

(3)

VL criterion above q ≈ 0.5, beyond which B2*cp clearly decreases with increasing q. It is observed that FVT predicts negative B*2 cp values with enormous magnitude: B*2 cp drops to values smaller than −103 for q close to 10. We have also plotted computer simulation results (see the legend for refs) for HS + PHS

RESULTS AND DISCUSSION

Results for B2* at the critical point, B2*cp, are plotted in Figure 2. The values for B2*cp from FVT (dashed curve) deviate from the 13122

dx.doi.org/10.1021/la5023856 | Langmuir 2014, 30, 13121−13124

Langmuir

Letter

⎛ 333 B4* = 18 − ϕdR ⎜54 + q − 72q2 − 147q3 − 63q 4 ⎝ 4 117 5 43 6 33 7 3 1 ⎞ q + q + q + q 8 + q 9⎟ + 8 2 4 2 8 ⎠

mixtures (closed symbols). The simulation results for HS + PHS are in line with the FVT trend in which B*2 cp decreases with q. The simulation data for HS + PHS for large q by Moncho-Jordá et al.21 corroborate the FVT trend. We note that FVT is a mean-field theory that is expected to deviate slightly near the critical point, especially at low q.26 We must conclude that the VL criterion fails for the case of a mixture of HS + PHS for large q. For small q, FVT is close to the VL criterion. In that regime, the interactions are still pairwise additive so that the VL criterion still holds. The overlap of three or more depletion layers (the effect of indirect interactions beyond pairwise additivity) becomes possible above q ≈ 0.15.12 Such indirect contributions are not accounted for in the pairwise additive approach taken when a collection of hard spheres is considered that interact through a (hard) core repulsion plus a direct attraction. However, multiple overlap of depletion zones does not add to the attraction between hard spheres. The driving force for the depletion-induced attraction is the volume that becomes available for the depletants as a result of the overlap of two depletion zones. The overlap of three or more depletion zones does not further increase the volume available for the depletants and hence reduces the net attraction as compared to a pure pairwise additive approach. This explains why so much more depletant is required (a large negative value of B2) to reach the critical point for large q. It was shown by Meijer and Frenkel,27 Bolhuis et al.,25 and Moncho-Jordá et al.21 that three-body interactions between hard spheres in a sea of depletants are repulsive and increase with increasing q. Hence, for longer-ranged attractions the indirect nature of the depletion effect is responsible for the deviation from the VL criterion. Note that Monte Carlo simulations performed with hard spheres interacting through eq 2 without explicit PHS by Dijkstra et al. do obey the VL criterion. To rationalize and semiquantify the strong increase in −B2*cp with increasing q, we computed the virial coefficients from FVT. The osmotic pressure of the hard spheres in the HS + PHS mixture follows from FVT using standard thermodynamics as Πc = Π 0+Πd[h(η) − 1]

Both B3 and B4 are strongly q-dependent: in the high-q limit, B3 > 0 but B4 < 0. The complex q dependence contrasts with the values for B3 and B4 at the critical point in a particle suspension where the spheres interact through a Lennard-Jones-like interaction.15 For the LJ(-like) systems, B3 and B4 are close to their hard sphere values, independent of the range of attraction. This supports the idea that the colligative properties of a mixture of hard spheres plus depletants fundamentally differ from those of a mixture of hard-core particles plus a direct attraction.12 From a virial expansion of the osmotic pressure of the colloidal spheres at the critical point using B2, B3, and B4, it follows that the contributions of B3 and B4 become comparable to B2 for q ≳ 0.6. Finally, we have studied the influence of making the hard spheres sticky and study the critical behavior of these sticky spheres plus penetrable hard spheres. The spherical particles were made sticky by adding a Yukawa attraction to the hard sphere interaction. An analytical expression for the Helmholtz energy fc is available for the Yukawa interaction from Tang et al.,28 which is rather accurate.29 The hard core Yukawa interaction reads for x < 1 WY(x) ⎧∞ =⎨ kT ⎩ β ϵ exp[κσ(1 − x)]/x otherwise

(8)

where σ = 2R is the sphere diameter, κ sets the range of the interaction (2/qY = κσ), and βϵ is the strength of the interaction when the spheres touch per kT. We used a (long-)ranged attraction of κσ = 2 (qY = 1) and various strengths of the attractions of βϵ = −0.3 (SS1), −0.6 (SS2), and −0.8 (SS3) (Figure 1), which includes the gas−liquid binodals of HS, SS1, and SS2 plus depletants. In Figure 3, we plotted results for B*2 cp using free volume theory for hard spheres plus depletants (as before) and for the sticky spheres plus depletants. It follows from the results in Figure 3 that adding a direct attraction between the spheres turns the q dependence

(4)

where Π0 is the osmotic pressure of a fluid of pure hard spheres, Πd is the osmotic pressure of PHS and where h = α − η∂α/∂η.23 A virial expansion of the osmotic pressure in η yields for B2 ⎛ q3 ⎞ 15 B2* = 4 − ϕdR ⎜6 + q + 3q2 + ⎟ 2 2⎠ ⎝

(7)

(5)

Already at the B2 level FVT predicts a strong q dependence which contrasts with the VL criterion. For high q, FVT predicts B*2 ≈ ϕRd q3 scaling. Because ϕRd at the cp becomes independent of q for high q,21 this corroborates B*2 ≈ −q3 observed in Figure 2. Third and fourth osmotic virial coefficients B3 and B4, normalized as B3* = B3/υc2 B4* = B4/υc3 from the FVT osmotic pressure virial expansion, are ⎛ B3* = 10 − ϕdR ⎜24 + 33q − 3q2 − 20q3 − 12q 4 − 3q5 ⎝ −

1 6⎞ q⎟ 3 ⎠

Figure 3. Results for B*2 at the critical point for spheres mixed with penetrable hard spheres. The direct interactions between the spheres are varied from the pure hard sphere interaction (HS) to hard cores plus weak (SS1), stronger (SS2), and very strong (SS3) Yukawa attractions. HS, SS1, and SS2 correspond to the binodal curves in Figure 1

(6)

and 13123

dx.doi.org/10.1021/la5023856 | Langmuir 2014, 30, 13121−13124

Langmuir

Letter

(14) Fleer, G. J.; Tuinier, R. The critical endpoint in phase diagrams of attractive hard spheres. Physica A 2007, 379, 52−58. (15) Vliegenthart, G. A.; Lekkerkerker, H. N. W. Predicting the gasliquid critical point from the second virial coefficient. J. Chem. Phys. 2000, 112, 5364−5369. (16) Elliott, J.; Hu, L. Vapor-liquid equilibria of square-well spheres. J. Chem. Phys. 1999, 3043, 3043−3048. (17) Tuinier, R.; de Kruif, C. G. Stability of casein micelles in milk. J. Chem. Phys. 2002, 117, 1290−1295. (18) Foffi, G.; Sciortino, F. On the possibility of extending the NoroFrenkel generalized law of correspondent states to nonisotropic patchy interactions. J. Phys. Chem. B 2007, 111, 9702−9705. (19) Kulinskii, V. L. The Vliegenthart-Lekkerkerker relation: the case of the Mie-fluids. J. Chem. Phys. 2011, 134, 144111. (20) Noro, M.; Frenkel, D. Extended corresponding-states behavior for particles with variable range attractions. J. Chem. Phys. 2000, 113, 2941−2944. (21) Moncho-Jordá, A.; Louis, A. A.; Bolhuis, P. G.; Roth, R. The Asakura-Oosawa model in the protein limit: the role of many-body interactions. J. Phys.: Condens. Matter 2003, 15, S3429−S3442. (22) Tuinier, R.; Vliegenthart, G. A.; Lekkerkerker, H. N. W. Depletion interaction between spheres immersed in a solution of ideal polymer chains. J. Chem. Phys. 2000, 113, 10768−10775. (23) Fleer, G. J.; Tuinier, R. Analytical phase diagrams for colloids and non-adsorbing polymer. Adv. Colloid Interface Sci. 2008, 143, 1− 47. (24) Carnahan, N. F.; Starling, K. E. Equation of state for nonattracting rigid spheres. J. Chem. Phys. 1969, 51, 635−636. (25) Bolhuis, P. G.; Louis, A. A.; Hansen, J. P. Influence of polymerexcluded volume on the phase-behavior of colloid-polymer mixtures. Phys. Rev. Lett. 2002, 89, 128302. (26) Lo Verso, F.; Vink, R.; Pini, D.; Reatto, L. Critical behavior in colloid-polymer mixtures: Theory and simulation. Phys. Rev. E 2006, 73, 061407. (27) Meijer, E. J.; Frenkel, D. Colloids dispersed in polymer-solutions - a computer-simulation study. J. Chem. Phys. 1994, 100, 6873−6887. (28) Tang, Y.; Lin, Y.-Z.; Li, Y.-G. First-order mean spherical approximation for attractive, repulsive, and multi-Yukawa potentials. J. Chem. Phys. 2005, 122, 184505. (29) Tuinier, R.; Fleer, G. J. Critical endpoint and analytical phase diagram of attractive hard-core Yukawa spheres. J. Phys. Chem. B 2006, 110, 20540−20545.

of B2 at the critical point gradually toward the Vliegenthart− Lekkerkerker criterion. Hence by mixing the direct and indirect attractions, it is possible to tune the B2 value at the critical point. In conclusion, we demonstrated that the VL criterion fails for a mixture of hard spheres plus (large) depletants. This is due to the fact that for a mixture of hard spheres plus depletants the phase behavior is determined by the colligative properties of the mixture: multiple overlap of depletion zones should be accounted for, which is incorporated into free volume theory.

■

OUTLOOK It is hoped that this work will lead to a more careful use of the depletion interaction as an effective attraction because for longranged attractions the depletion effect leads to colligative properties that are nonpairwise additive. It will be of great interest to investigate further the combination of colloidal interactions beyond hard core repulsion plus depletion effects, as anticipated in this Letter.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS R.T. is indebted to DSM for support. REFERENCES

(1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge Universtity Press: New York, 1999. (2) Perrin, J. Mouvement brownien et réalité moléculaire [Brownian movement and molecular reality]. Ann. Chim. Phys. 1909, 18, 5−114. (3) Yethiraj, A.; van Blaaderen, A. A colloidal model system with an interaction tunable from hard sphere to soft and dipolar. Nature 2003, 421, 513−517. (4) Stradner, A.; Sedgwick, H.; Cardinaux, F.; Poon, W. C. K.; Egelhaaf, S. U. Equilibrium cluster formation in concentrated protein solutions and colloids. Nature 2004, 432, 492−495. (5) Verwey, E.; Overbeek, J. T. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (6) Royall, C. P.; Leunissen, M. E.; Blaaderen, A. A new colloidal model system to study long-range interactions quantitatively in real space. J. Phys.: Condens. Matter 2003, 15, S3581−S3596. (7) Dang, M. T.; Verde, A. V.; Nguyen, V. D.; Bolhuis, P. G.; Schall, P. Temperature-sensitive colloidal phase behavior induced by critical Casimir forces. J. Chem. Phys. 2013, 139, 094903. (8) Asakura, S.; Oosawa, F. On interaction bewteen two bodies immersed in a solution of macromolecules. J. Chem. Phys. 1954, 22, 1255−1256. (9) Vrij, A. Polymers at interfaces and interactions in colloidal dispersions. Pure Appl. Chem. 1976, 48, 471−483. (10) Lekkerkerker, H. N. W.; Poon, W. C. K.; Pusey, P. N.; Stroobants, A.; Warren, P. B. Phase behavior of colloid plus polymer mixtures. Europhys. Lett. 1992, 20, 559−564. (11) Ilett, S. M.; Orrock, A.; Poon, W. C. K.; Pusey, P. N. Phasebehavior of a model colloid-polymer mixture. Phys. Rev. E 1995, 51, 1344−1352. (12) Dijkstra, M.; Brader, J. M.; Evans, R. Phase behaviour and structure of model colloid-polymer mixtures. J. Phys: Condens. Matter 1999, 11, 10079−10106. (13) Dijkstra, M.; van Roij, R.; Roth, R.; Fortini, A. Effect of manybody interactions on the bulk and interfacial phase behavior of a model colloid-polymer mixture. Phys. Rev. E 2006, 73, 041404. 13124

dx.doi.org/10.1021/la5023856 | Langmuir 2014, 30, 13121−13124