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Langmuir 1997, 13, 4588-4594
Phase Behavior of a Colloid + Binary Polymer Mixture: Theory Patrick B. Warren Unilever Research Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral L63 3JW, U.K. Received February 24, 1997. In Final Form: May 23, 1997X The theory of depletion flocculation in colloid + nonadsorbing polymer mixtures is extended to cover size polydispersity of the polymer. The case of colloid + binary polymer mixture is studied in detail. The main conclusion is that phase separation is enhanced in a colloid + binary polymer mixture when compared with a nominally equivalent colloid + monodisperse polymer mixture, except for the case where the equivalent polymer has the mass average molecular weight of the mixture. This equivalent polymer though has the closest correspondence to the true binodal of the mixture. Alternatively, if it is possible to use total effective coil volume fraction as a measure of polymer concentration, then phase behavior becomes insensitive to polymer size polydispersity. Size fractionation is predicted to occur, with the larger species being concentrated in the polymer rich phase, although the extent is not great. The influence of an undetected small polymer species on the onset of phase separation in a colloid + monodisperse polymer mixture is also examined. It is found that a small amount of contaminant can significantly lower the phase boundary, corresponding to the introduction of an additional short range depletion well in the interparticle potential.
Introduction Depletion flocculation arises in colloidal systems which contain mixtures of particles and polymers, or large and small particles.1-4 Recently there has been a resurgence of interest in these systems, spurred by extensions to the theory5,6 and further experiments on model systems concerning both the equilibrium phase behavior7,8 and nonequilibrium phase transition kinetics.9 An approximate form of depletion theory has also been applied to the prediction of the phase behavior of binary colloidal suspensions,10,11 a subject which has also been readdressed experimentally.12,13 To date, most published theoretical treatments have assumed monodisperse components. This is an entirely reasonable assumption to make when uncovering the basic effects, and quantitative comparisons with experiments on model systems can be sensibly made. However, many practical situations, especially in industry, do not have the luxury of monodisperse ingredients, and it is useful to analyze the effect that polydispersity may have on the theoretical predictions. The principles of equilibrium calculations for polydisX
Abstract published in Advance ACS Abstracts, August 1, 1997.
(1) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (2) Vrij, A. Pure Appl. Chem. 1976, 48, 471. (3) Gast, A. P.; Hall, C. K.; Russel, W. B. J. Colloid Interface Sci. 1983, 96, 251. (4) Vincent, B. J.; Edwards, J.; Emmett, S.; Croot, R. Colloids Surf. 1988, 31, 267. (5) Lekkerkerker, H. N. W.; Poon, W. C. K.; Pusey, P. N.; Stroobants, A.; Warren, P. B. Europhys. Lett. 1992, 20, 559. (6) Warren, P. B.; Ilett, S. M.; Poon, W. C. K. Phys. Rev. E 1995, 52, 5205. (7) Poon, W. C. K.; Selfe, J. S.; Robertson, M. B.; Ilett, S. M.; Pirie, A. D.; Pusey, P. N. J. Phys. II 1993, 3, 1075. (8) Leal Calderon, F.; Bibette, J.; Biais, J. Europhys. Lett. 1993, 23, 653. (9) Poon, W. C. K.; Pirie, A. D.; Pusey, P. N. Faraday Discuss. 1995, 101, 65. (10) Lekkerkerker, H. N. W.; Stroobants, A. Physica A 1993, 195, 387. (11) Poon, W. C. K.; Warren, P. B. Europhys. Lett. 1994, 28, 513. (12) Dinsmore, A. D.; Yodh, A. G.; Pine, D. J. Phys. Rev. E 1995, 52, 4045. (13) Steiner, U.; Meller, A.; Stavans, J. Phys. Rev. Lett. 1995, 74, 4750. (14) Gualtieri, J. A.; Kincaid, J. M.; Morrison, G. J. Chem. Phys. 1982, 77, 521. (15) Salacuse, J. J.; Stell, G. J. Chem. Phys. 1982, 77, 3714.
S0743-7463(97)00198-4 CCC: $14.00
perse systems are known,14-16 but in practice they may be very difficult to implement. Simulations have also been able to address polydispersity effects.17 Just as difficult, but less widely appreciated, are questions concerning the representation of results in conventional phase diagrams. For instance the critical point always lies at the top of the cloud curve in an ideal monodisperse system (assuming a lower consolute temperature), but it may no longer do so in a polydisperse system.14 To obviate some of these difficulties, the system actually studied in detail in the present paper is a colloid + binary polymer mixture rather than a colloid + polydisperse polymer mixture. It is hoped that the analysis of this simplified system indicates the qualitative effect of polydispersity. The extension of the theory to the fully polydisperse case is indicated. Whilst this paper was in preparation, a publication by Sear and Frenkel18 appeared describing essentially the same theory for the polydisperse case. Where appropriate I will compare with their conclusions. Also recent work was published by Jenkins and Vincent19 on a colloid + binary polymer mixture. I will also compare with their experimental results where possible. The theory calculates the equilibrium phase behavior. In the monodisperse polymer case, it is known that this predicts the experimentally observed behavior reasonably quantitatively except when enough polymer is added to take the system well into the two-phase region where a kinetically stabilized gel ‘phase’ is observed instead.9 The results of Jenkins and Vincent19 also show the importance of kinetic effects too, as the observed behavior depends on the order of addition of the two components. Theory The theory underlying the prediction of equilibrium phase behavior has been discussed in a number of places.5,6 The free energy (technically the semigrand potential) of a colloid + monodisperse ideal polymer mixture is given approximately by (16) Briano, J. G.; Glant, E. D. J. Chem. Phys. 1984, 80, 3336. (17) Stapleton, M. R.; Tildesley, D. J.; Quirke, N. J. Chem. Phys. 1990, 92, 4456. (18) Sear, R. P.; Frenkel, D. Phys. Rev. E, in press. (19) Jenkins, P.; Vincent, B. Langmuir 1996, 12, 3107.
© 1997 American Chemical Society
Phase Behavior of a Colloid + Binary Polymer Mixture f ) f0(φ) - aPkBTR(φ;ξ)
(1)
where f is the free energy per unit volume, f0 is the free energy per unit volume of the colloid system in the absence of polymer, φ is the volume fraction of colloid, aP is the polymer activity, and R is the free volume fraction, i.e. the fraction of the total volume in which the center of mass of a polymer coil can move freely. The free volume fraction depends on the volume fraction of colloidal particles, which obstruct the polymer coils, and the size of the coils themselves, expressed in terms of the size ratio ξ ) d/a of the mean coil radius d to the colloidal particle radius a. Equation 1 gives the free energy in terms of the colloidal volume fraction and polymer activity. The conversion to polymer concentration nP is straightforward:
nP ) aPR(φ;ξ)
(2)
This shows that aP, in this ideal polymer case, is also the concentration of polymer in the free volume. More details of the approximations involved and the functions f0 and R are given elsewhere.5 The case of nonideal polymers has also been discussed elsewhere.6 The theory readily generalizes to a polydisperse polymer. Simply introduce a distribution of activity aP(ξ) depending on a continuously varying size ratio: aP(ξ) dξ gives the activity of polymers having a size ratio between ξ and ξ + dξ. The generalization of eq 1 is
f ) f0(φ) - kBT
∫
∞
0
dξ aP(ξ) R(φ;ξ)
(3)
The generalization of eq 2 gives the number of polymers nP(ξ) dξ having a size ratio between ξ and ξ+ dξ:
nP(ξ) ) aP(ξ) R(φ;ξ)
(4)
One of the difficulties in calculation lies in an inversion problem whereby the activity distribution aP(ξ) should be determined such that the mean polymer distribution in a sample has a distribution reflecting that of the stock solution from which the sample is made. As mentioned above, to simplify this and other difficulties, I focus on the case of a binary polymer mixture. Specializing eqs 3 and 4 to this case, one obtains
f ) f0(φ) - aLkBTR(φ;ξL) - aSkBTR(φ;ξS)
For two phases (I and II) in coexistence there are four polymer I II concentrations nIL, nII L , nS, and nS , two colloid volume fractions φI and φII, and a relative amount of phase I given by the Lever rule:
x)
φII - φ φII - φI
(6)
where the mean colloid volume fraction is φ. The mean polymer composition is then given by
n j L ) nILx + nII L (1 - x),
n j S ) nISx + nII S (1 - x)
(7)
Following the above discussion, the activities should be adjusted so that the system has a given mean polymer composition:
n j L ) kn jS
(8)
where k is the given ratio. Since eq 8 represents a nonlinear constraint, the phase behavior in the (φ, nL, nS) space can be projected onto the (φ, nL + nS) plane (or some other linear combination of polymer concentrations) and compared with the monodisperse case. This projection requires that a unique value of φ be chosen for each tieline. Moreover a different projection is obtained for each scheme for choosing φ. In the calculations below two schemes are used: either φ has a fixed value, the same for all tielines or φ ) φI, which amounts to forcing x ) 1. The former scheme (φ fixed) corresponds to the case where increasing amounts of polymer are added to a given colloidal solution.20 The latter scheme (φ ) φI) corresponds to the onset of phase separation, since x ) 1 implies a negligible amount of second phase. It represents the determination of a cloud curve14 as polymer is added to a range of colloidal suspensions of varying volume fractions. I now indicate briefly the inversion problem for the fully polydisperse case. I first introduce a normalized distribution function for activity:
aP(ξ) ) aPf(ξ),
where
∫
∞
0
dξ ) 1
(9)
An equation analogous to eq 7 gives the mean distribution function in the phase-separated sample:
n j (ξ) ) aPf(ξ)[xR(φI;ξ) + (1 - x)R(φII,ξ)]
nL ) aLR(φ;ξL) nS ) aSR(φ;ξS)
Langmuir, Vol. 13, No. 17, 1997 4589
(5)
Here the subscripts L and S on the activity, size ratio, and concentration refer to large and small polymer species. Without the inversion problem (to be discussed further below) the calculation of phase behavior is very straightforward. The first of eqs 5 gives the free energy for fixed values of aL, aS, ξL, and ξS, and the phase behavior is determined as for the monodisperse case. Phase behavior in (φ, nL, nS) space is obtained from that in (φ, aL, aS) space by the application of the last pair of equations in eqs 5. These calculations have the added interest of shedding further light on the conditions under which fluid-fluid phase separation and a critical point-triple point structure emerge in the phase diagram. The inversion problem alluded to above may be formulated in more detail as follows. Consider what happens in a typical experiment or product formulation situation. Samples taken from stock solutions of colloid and polymer are mixed. If phase separation occurs, the compositions and amounts of the coexisting phases will be such that the mean composition of the system is the same as that of the stock solutions. To be specific, suppose that the polymer stock solution is bidisperse with concentrations of large and small polymers in a given ratio. Any particular sample made up from this stock solution must also contain large and small polymers in the same ratio, reflecting the stock solution from which it was made. Experiment or product formulation is performed at constant mean composition, and this practical consideration should be reflected in the calculations.
) ng(ξ),
where
∫
∞
0
g(ξ) dξ ) 1
(10)
In eq 10, φI, φII, and x also depend implicitly on the distribution function f(ξ) through its appearance in the free energy. Given f(ξ), g(ξ) is readily determined from the solution to the coexistence problem and eq 10. In the practical situation g(ξ) is known, since it is the distribution function in the stock polymer solution and f(ξ) must be found. The representation problem concerns the collapse of the phase behavior in the infinitely dimensional (φ, n(ξ)) space to something that can be plotted on a sheet of paper. The projection to (φ, n) where n is the total poymer concentration (as in eq 10) may be done in the same way as for the bidisperse case, and the same remarks concerning the selection of a mean colloid composition on each projected tieline still hold. Note that tielines remain straight after projection onto any linear combination of polymer concentrations, which includes the projection onto the total polymer concentration. Nevertheless such tielines only have meaning when the value of φ is specified. Sear and Frenkel18 derive identical equations to eqs 3 and 4 for the polydisperse polymer. However they work with a Schultz distribution for the activity f(ξ) and do not attempt to solve the inversion problem. (20) The volume change due to the added polymer is assumed to be negligible: this is reasonable, since each polymer chain expands to a Gaussian coil and depletion effects are seen at rather low physical volume fractions.
4590 Langmuir, Vol. 13, No. 17, 1997
Warren
paragraphs 97-99 in ref 21 has been adopted. The dotted line is the projection of the critical line. The dash-dot line is the projection of a line of triple points. The notation ‘v-l-s’ indicates that one encounters colloidal vapor, vapor + liquid, liquid, liquid + crystal, and crystal phases as φ increases. On moving down past the line of critical points, the notation ‘vl-s’ indicates that the colloidal vapor and liquid phases have merged identity into a single colloidal fluid phase. On moving up past the line of triple points, the notation ‘v-s’ indicates that the colloidal liquid phase has been lost. There is no physical difference between ‘v-s’ and ‘vl-s’. Both represent the case where one encounters a disordered colloidal fluid phase, and an ordered colloidal crystal phase; the notation serves to indicate what happens as the lines are crossed. The lines meet at the point Figure 1. Phase behavior as a function of colloid volume fraction and large polymer activity ((φ, aL) plane), at various values of small polymer activity, aS, for the size ratio 1:ξL:ξS ) 1:0.57:0.24. The solid lines give the phase behavior in the absence of small polymer. The dotted lines, from top to bottom, give the binodal when the small polymer activity is 4/3πd3SaS ) 0.075, 0.15, 0.225, and 0.3.
4/3π d3LaL ) 0.194, 4/3π d3SaS ) 0.242
Figure 2 is interesting, as it shows how fluid-fluid phase separation appears in the equilibrium phase diagram as the activities of the two species are varied. For monodisperse polymer the condition is that ξ J 0.32.5 In the bidisperse case, Figure 2 indicates how it happens as the ratio of large (ξL > 0.32) to small (ξS < 0.32) polymer is increased. One can ask if there is an equivalent monodisperse system to the two component polymer mixture. One way to formulate this is to calculate the mean properties of the polymer reservoir and determine a mean size ratio ξj. The length, and hence molecular weight, of a species of polymer is proportional to ξ2, since the chains are supposed ideal and Gaussian. One can conceive of at least four different definitions of an equivalent size ratio:
ξjS )
aLξL + aSξS aLξ2L + aSξ2S , ξj2N ) aL + aS aL + aS ξj2M )
Figure 2. Structural map of phase behavior projected from (φ, aL, aS) space onto the (aL, aS) plane. The line of critical points is shown dotted; the line of triple points is shown dash-dotted. The notations ‘vl-s’, ‘v-l-s’, and ‘v-s’ indicate the type of phases and transitions occurring (see text for a more detailed explanation).
Results General Features of Phase Behavior. Without the inversion problem, calculations are readily made by adapting existing methods which calculate the phase behavior for the monodisperse polymer case. In Figure 1, phase behavior is shown in the (φ, aL) plane for various values of aS, where 1:ξL:ξS ) 1:0.57:0.24 (these values are chosen to correspond to possible experiments). As the activity of the small polymer increases in the absence of large polymer, the width of the hard sphere freezing transition increases (cf. Figure 1a in ref 5). As the activity of the large polymer is then increased, at a given small polymer activity 4/3πd3SaS < 0.242, fluid-fluid phase separation appears and a critical point-triple point structure emerges (cf. Figure 1b in ref 5). For 4/3πd3SaS > 0.242, the freezing transition simply widens further. Figure 2 shows the structural features of the phase behavior projected from (φ, aL, aS) space onto the (aL, aS) plane. In such a projection a convenient notation from
(11)
(1 + ξjB)3 )
aLξ4L + aSξ4S aLξ2L + aSξ2S
aL(1 + ξL)3 + aS(1 + ξS)3 aL + aS
(12)
The first of these, ξjS, is a straightforward average weighted by the number concentrations of polymers in the reservoir. The second and third, ξjN and ξjM, are what would be calculated from the number average and mass average molecular weight, respectively. The fourth, ξjB, introduced by Sear and Frenkel,18 gives a polymer with the same mean second virial coefficient (or excluded volume) against the colloid as the mixture. Using any of these, one may calculate a critical ratio aL/aS below which the equivalent monodisperse system shows no fluid-fluid phase separation and compare it to the bidisperse calculation. The former is calculated by inserting the known values of ξL and ξS and ξj ) 0.32 in any of eqs 12. The latter is calculated from eq 11. Table 1 shows the values obtained. Clearly the true extent of the critical point-triple point behavior is significantly greater than predicted from any of the averaging methods, except that using mass average molecular weight gives a particularly high weighting for (21) Landau, L. D.; Lifshitz, E. M. Statistical Physics, Part 1, 3rd ed.; Pergamon: New York, 1980. (22) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, New York, 1979.
Phase Behavior of a Colloid + Binary Polymer Mixture
Langmuir, Vol. 13, No. 17, 1997 4591
Table 1. Critical Activity Ratio for Fluid-Fluid Phase Separation method straight average virial average number average true value mass average
(aL/aS) ξjS ξjB ξjN ξjM
0.320 0.251 0.201 0.0598 0.0357
Figure 4. Phase behavior on addition of equal weights of large and small polymers to a colloidal suspension to the point where j1 ) phase separation just starts to occur (cloud curve; d21n d22n j 2on the left hand binodal). Remaining parameters as in Figure 3.
Figure 3. Phase behavior on addition of equal weights of large and small polymers to a colloidal suspension at volume fraction j 2 ) d22n j 2 at φ h ) 0.52). Solid lines are for the large 52% (i.e. d21n polymer, and dotted lines are for the small polymer. Size ratios are 1:ξL:ξS ) 1:0.1:0.05. The vertical axis is the (dimensionless) polymer mass of each individual species.
the large polymers (eqs 12). This is in agreement with the conclusion of Sear and Frenkel, who find similarly that fluid-fluid phase coexistence is possible for polydisperse mixtures whose monodisperse equivalents show only fluid-solid coexistence. This is attributed to the possibility of differential partitioning of polymer species between coexisting phases, an effect which, “on going from a pure system to a mixture, is known to increase the density jump at a first-order phase transition.”18 The phase behavior in (φ, nL, nS) space may be found using the last two equations of eqs 5. The triple point lines expand to three-phase triangles, which stack to form a wedge shaped volume where colloidal vapor, liquid, and solid are all expected to coexist. Polydispersity and Fractionation. I now turn to the case where the inversion problem is solved. As already discussed, this is done by fixing the mean polymer composition at a given value, for a given mean colloid volume fraction φ on each tieline. Throughout this section I focus on the case where the size ratios are 1:ξL:ξS ) 1:0.10:0.05. In this case there are no complicating features from fluid-fluid phase separation in the phase diagram. Figure 3 shows the phase diagram when equal masses of large and small polymers are added to a colloidal suspension at a volume fraction of 52%. This corresponds j L) d2Sn j S at φ ) 0.52 (recall the mass to the case where d2Ln of a Gaussian coil is proportional to d2 ). The vertical axis in this plot is a dimensionless measure of the added mass of polymer. Tielines are shown connecting (φI, nIL) to (φII, I II I II nII L ) (solid lines) and (φ , nS) to (φ , nS ) (dotted lines); each pair intersects at φ ) 0.52, reflecting the requirement that d2Ln j L ) d2Sn j S From the diagram it is clear that there is polymer fractionation: the small polymers are more evenly distributed between the phases, as might be expected. This is also found by Sear and Frenkel.18 The scope for polymer purification by this method may be
Figure 5. Phase behavior of the mixed system compared with various pure systems. The solid line is the left hand binodal from Figure 4 (the cloud curve); the dotted lines are the left hand binodals of the two pure components; the dashed lines are the left hand binodals of pure systems with various definitions of an equivalent size ratio (see text). The vertical axis is the (dimensionless) total added polymer mass.
limited though with only about 10-20% separation by weight achieved in the polymer rich phase for this size ratio. Figure 4 shows the predicted behavior when equal masses of large and small polymers are added to a colloidal suspension at various volume fractions, to just at the point where phase separation occurs. This corresponds to a cloud curve.14 In Figure 4, the tielines shown are similar to those in Figure 3, except that the added polymer masses are now forced to be equal on the left hand binodal curve, corresponding to φ ) φI. The right hand curves show the composition of the phase that is just starting to phase separate. Figures 5 and 6 show this cloud curve (solid line) plotted using a total polymer concentration on the vertical axis rather than individual polymer concentrations as in Figure 4. The same data are shown in both plots, the difference between them being the method of plotting total polymer concentration. In Figure 5 the vertical axis is proportional to the total added mass concentration d2LnL + d2SnS. In Figure 6 the vertical axis is instead the total effective coil volume fraction 4/3(d3LnL + d3SnS), calculated using the effective volume of a polymer coil (note that the actual
4592 Langmuir, Vol. 13, No. 17, 1997
Figure 6. Same data as in Figure 5, except that the vertical axis is now the total effective polymer volume fraction. Dashed lines have been omitted for clarity.
polymer volume fraction is much lower and is proportional to the mass concentration). In addition to the cloud curve in Figures 5 and 6, binodal curves for the two pure polymers are shown as dotted lines. The lower dotted curve corresponds to ξL ) 0.10, and the upper, to ξS) 0.05. Binodal curves for monodisperse systems with a mean size ratio calculated from eqs j L ) (dS/ 12 are also shown in Figure 5. Inserting aL ) n j S ) (dS/dL)2aS in eqs 12 obtains ξjS ) 0.06, ξjN ) 0.0632, dL)2n ξjM ) 0.0791, and ξjB ) 0.0604. The dashed lines shown in Figure 5, from bottom to top, correspond to ξjM, ξjN, and ξjS, respectively. The curve corresponding to ξjB is not shown, as it is indistinguishable from ξjS. These curves are omitted in Figure 6, as they are indistinguishable from the mixture and pure components. Figure 5 has the most immediate physical significance, since the polymer axis is closest to what would be used experimentally. From Figure 5 it is clear that the cloud curve for the bidisperse polymer system lies approximately midway between those of the two pure components. Also the curves predicted by monodisperse calculations using the average size ratios ξjS, ξjB, or ξjN (see eqs 12) all lie above the true one, whereas the curve predicted using ξjM lies slightly below. This last curve is the closest to the true binodal, and this indicates that mass average molecular weight is the best estimate of the size of a coil, as far as prediction of the position of the binodal line is concerned. This is in agreement with the findings in the previous section on the onset of fluid-fluid phase separation. In Figure 6 by contrast, there is virtually no difference between the various curves. This leads to an important conclusion: the effects of polydispersity have virtually no effect on the predicted phase boundary, provided it is possible to plot the results using total effective volume fraction, i.e. the concentration of each species, scaled by the effective volume of a solvated coil and summed over all species. Whilst this may be difficult for polymers, it is easily done for colloidal mixtures. One last point should be mentioned. In general, the phase boundary obtained depends on the precise experimental protocol, because the phase behavior is projected from a higher dimensional space onto a plane. For instance, if the data shown in Figure 3 (where phase separation occurs at constant mean colloid volume fraction φ ) 0.52) are replotted with total polymer concentration on the vertical axis, slight differences from the cloud curve shown in Figures 5 and 6 are seen. The difference is of the order of the thickness of the lines used to plot the
Warren
Figure 7. Phase behavior of a pure system (ξ ) 0.1, solid lines) and the effect on the left hand binodal line when exposed to a second small species (ξ ) 0.01) at fixed activity 4/3d3SaS ) 0.05 (upper dotted line) and 0.1 (lower dotted line).
curves though. Consequently, although the above discussion is couched in terms of the cloud curve, the conclusions should also hold for the phase behavior for other experimental protocols too. Contamination by Undetected Small Polymers or Particles. The theory described here makes it possible to say something about what happens if a monodisperse mixture is ‘contaminated’ by an unsuspected species of very small polymers, particles, or indeed micelles. Figure 7 shows the predicted binodal curves and tielines for a monodisperse colloid + polymer mixture at a size ratio ξL ) 0.1. Also shown are lines giving the binodal curves when the system is exposed to a contaminant of size ratio ξS ) 0.01, at activities 4/3d3SaS ) 0.05 and 0.10. The binodal curves are shifted down by ∼20% and ∼40%, respectively, although there is little change in the width of the freezing transition at zero added polymer (this only starts to change when the activity of the contaminant is increased still further). This indicates that contamination by a small third component can have a severe effect on the observed binodal curves and may provide a possible explanation for discrepancies between different experiments on supposed binary hard sphere mixtures: Dinsmore et al.12 find a small change in the phase boundary as the size ratio ξL is varied from 0.08 to 0.15, whereas Steiner et al.13 find a marked shift in the phase boundary for approximately the same size ratios. The theory11 predicts only a small change. Contamination by a third component at fixed activity is of course equivalent to adding in a very short range depletion attraction of constant strength between colloidal particles. The same conclusion concerning a marked shift in phase boundaries would be drawn whatever the origin of the short range attraction. Comparison with Some Experimental Results. Jenkins and Vincent19 have reported experimental results on various colloid + binary polymer mixtures. They also provide enough information to be able to compare the results with the present theory, for the experimental system which they term Si 8 + PDMS 8 + PDMS 9 (see Figure 7 and Table 3 of ref 19). The relevant parameters for this system are collected here in Table 2. In this section, polymer concentration (usually in gm/ cc) is denoted by φP. In order to convert between theory and experiment, the quantity φ*P is defined as the concentration at which 4/3πd3n ) 1. It is readily calculated from the values of d, MW, and the atomic mass unit (see Table 2). Since 4/3πd3n ) 1 implies a concentration of
Phase Behavior of a Colloid + Binary Polymer Mixture
Langmuir, Vol. 13, No. 17, 1997 4593
Table 2. Parameters for the experimental system of Jenkins and Vincent.19
polymers are in a good solvent and the osmotic pressure is much increased compared to that of an ideal solvent. The trend of the theoretical curve though is in good agreement with the trend of the data, in contrast with the conclusion of Jenkins and Vincent, who suggest that the experimental data are not in qualitative agreement with equilibrium predictions. I now explore this point further. Jenkins and Vincent calculate their expected onset of phase separation by the approximate spinodal instability condition23
Si 8 a/nm d/nm ξ MW -1 φ* P/gm cc
PDMS 8
PDMS 9
8.9 0.044 53 000 3.0 × 10-2
29.5 0.146 580 000 9.0 × 10-3
202 ( 7
B2 1 + )0 4 3 2φ πa 3
(15)
Here B2 is the second virial coefficient of the colloid particles in the sea of polymers, given by
B2 )
Figure 8. Comparison of theory (solid line) with experimental results reported by Jenkins and Vincent19 (solid line with points). Parameters are given in Table 2. The dotted line is the prediction of the spinodal instability condition given in the text. The horizontal axis is the weight percentage of the large polymer. The vertical axis is the total added polymer concentration at the onset of phase separation. The inset shows the experimental data replotted using the effective coil volume fraction.
one coil per coil volume, φ* P is close to the coil overlap concentration which marks the onset of the semidilute regime.22 Unlike the coil overlap concentration though, φ* P is precisely defined here. Given φ* P, the effective polymer coil volume fraction is found from 4/3πd3n ) φP/φ* P. Note from Table 2 that the proportionality between molecular weight and ξ2 alluded to in previous sections is followed to within approximately 1%. Figure 8 (main plot) shows the amount of polymer required to first induce phase separation in a colloidal suspension at volume fraction φ ) 0.01. The data points are taken from Figure 7 of ref 19 and are converted to gm/cc assuming the polymer density is 1 gm/cc. The theoretical curve is found from cloud curves such as that shown in Figure 4, at the point where φ ) 0.01, as the ratio of large to small polymer varies. The total polymer concentration is found from
φP ) φ*L × 4/3π d3LnL + φ*S × 4/3π d3SnS
(13)
where the subscripts L and S refer to PDMS 9 and PDMS 8 polymers, respectively. Given a certain ratio of large to small polymer, the weight percent of large polymer is found from
φ*Ld3LnL
WL ) 100 φ*Ld3LnL + φ*Sd3SnS
∫2a2(a+d)(1 - e-U/k T)r2 dr
16 3 πa + 2π 3
Conversely, given a value WL, eq 14 can be used to find the ratio nL/nS of polymers, which is then used to determine the appropriate cloud curve. It is seen that the theoretical prediction in Figure 8 (solid line) lies considerably above the observed data (solid lines with points). This is to be expected because the
(16)
where U(r) is the potential of mean force between a pair of colloid particles whose centers are separated by a distance r. Jenkins and Vincent determine U(r) from selfconsistent field theory. For the present model, one can use instead the interaction potential of Asakura and Oosawa1 and Vrij,2 generalized to a binary polymer mixture:
U(r) ) -nLkBTVov(r;ξL) - nSkBTVov(r; ξS) Vov(r;ξ) ) 4/3πa3[(1 + ξ)3 - 4/3(1 + ξ)2r + 1/16r3] (17) The dotted line in Figure 8 is the prediction of eqs 15-17. The trend of behavior is different from that of the cloud curve, and furthermore, the spinodal curve falls below the binodal curve. An expansion of the free energy eq 5 in φ indicates the reason why. It is found that the spinodal condition eq 15 is recovered, with a second virial coefficient given by
B2 ) 16/3π a3 - 4/3π d3LnLf(ξL) - 4/3πd3SnSf(ξS) f(ξ) ) 12 + 15ξ + 6ξ2 + ξ3
(18)
It is readily checked that this second virial coefficient corresponds to the replacement of (1 - e-U/kBT) with (U/ kBT), in eq 16. This is clearly an approximation appropriate for small values of U/kBT, and therefore the small φ limit of the present theory may be inaccurate in its prediction of phase behavior. Such a conclusion was also reached by Meijer and Frenkel24 in their Monte Carlo studies. As an independent check though, there is enough information to be able to calculate the effective polymer coil volume fraction experimentally, as discussed in a previous section:
) 4/3π d3SnS + 4/3π d3LnL ) φ(eff) P (14)
B
φS φL + φ*S φ*L
(19)
where φL ) WLφP/100 and φS )φP - φL are the concentrations of large and small polymer, respectively. The inset in Figure 8 shows the experimental data plotted in this way (note that no theoretical input has been made, it is simply a replot of the data). According to the analysis in (23) de Hek, H.; Vrij, A. J. Colloid Interface Sci. 1981, 84, 409. (24) Meijer, E. J.; Frenkel, D. J. Chem. Phys. 1994, 100, 6873.
4594 Langmuir, Vol. 13, No. 17, 1997
the previous section the onset of phase separation is more or less independent of the relative amounts of large and small polymer in such a plot. It is seen that the experimental data have been straightened out very considerably, showing a dip of about 10-20% in the center of the data, compared to a variation of a factor 2-3 in the original data. This analysis lends support to the idea that these experimental data do follow equilibrium phase behavior. Jenkins and Vincent also report experiments in which the order of addition of the polymer is varied, and they find systematic differences in the total amount of polymer required to induce phase separation. The arguments presented above suggest that, contrary to the conclusion reached by Jenkins and Vincent, the experiments which follow equilibrium behavior are those where premixed polymer is added, or where the small polymer is added first. However there are many caveats to this altered conclusion: the theory is probably inaccurate at low colloid volume fractions and is in any case for ideal polymers; there is no indication of the coexisting colloidal crystal phase in the experiments; there is no longer an explanation of the observed kinetic effects. Nevertheless it is intriguing that the experimental data can be collapsed as shown in the inset to Figure 8, by φS/φ* S + φL/φ* L ≈0.07. Conclusions Detailed calculations have been carried out for colloid + binary polymer mixtures. One may attempt to draw some useful conclusions concerning the effect of polymer polydispersity on equilibrium phase behavior colloid + nonadsorbing polymer mixtures from these. Firstly, with one exception, it was found that such systems show an enhanced tendency to phase separate when compared with a colloid + nominally equivalent monodisperse polymer mixture. This is true both for the onset of phase separation (cloud curves: Figures 4-6) and for the appearance of fluid-fluid phase separation (Figure 2 and Table 1) and is in agreement with the conclusion of Sear and Frenkel.18 The exception is when monodisperse polymer with molecular weight equal to
Warren
the mass average molecular weight of the mixture is used. Here it is found that the binary mixture shows a slightly reduced tendency to phase separate. However, since this method most closely approximates the observed behavior of the binary mixture, it suggests that, for polydisperse polymers, mass average molecular weight is the most appropriate indicator of the phase behavior. It should be noted though that no unique single-component polymer is exactly equivalent to a mixture. Rather, the question being addressed is what single-component polymer most closely approximates a polymer mixture, with respect to its phase behavior in the presence of nonadsorbing colloidal spheres. Secondly, the calculations predict that if it is possible to plot phase diagrams using a total effective volume fraction of polymer, polydispersity should have little effect (Figure 6). The experimental results reported by Jenkins and Vincent19 support this prediction (inset to Figure 8) although experimentally there are kinetic effects which also influence the behavior. Another calculation shows that there is fractionation of the polymer by size. In the case studied (Figure 3), only 10-20% enrichment by weight of the large polymer is predicted in the polymer rich phase. Since this was for an initially equal mixture by weight of polymers one of which is twice the size (four times the molecular weight) of the other, it suggests that practical size fractionation of polymer by this method may be difficult. As the size ratio of the large to small polymer increases though, the partitioning becomes more effective. In a final calculation (Figure 7), contamination by a third small component is found to lower the onset of phase separation. It is suggested as a possible explanation for discrepancies reported in the literature on the binodal curves for binary hard sphere mixtures,11-13 although any additional, unsuspected short range attraction would have the same effect. Acknowledgment. I acknowledge useful discussions with Wilson Poon and Brian Vincent. LA970198+
