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Effect of Polydispersity on the Depletion Interaction between Colloidal Particles
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
178, 505–513 (1996)
0145
Effect of Polydispersity on the Depletion Interaction between Colloidal Particles JOHN Y. WALZ Tulane University, Department of Chemical Engineering, New Orleans, Louisiana 70118 Received April 7, 1995; accepted September 11, 1995
The effect of polydispersity on the depletion interaction in hardsphere systems is investigated theoretically. The depletion force and energy between two parallel plates and two spherical particles in a solution of nonadsorbing, hard spherical macromolecules is calculated via a general force-balance approach. The diameter of the macromolecules is assumed to be normally distributed and the polydispersity in diameter is characterized with the coefficient of variation (ratio of the standard deviation to the mean). The primary assumption is that the density of macromolecules is low enough that higher order effects arising from interactions between macromolecules are negligible. It is found that the interaction between parallel plates can be represented in terms of a complementary error function of a dimensionless separation distance. These results are then applied to spherical particles via the Derjaguin approximation. A set of general and relatively simple equations for force and energy are developed that are shown to be valid in both monodisperse and polydisperse systems. For spherical particles in a solution of macromolecules at fixed number density, polydispersity is predicted to increase the range and magnitude of the depletion force. The effect is relatively small, however, unless the polydispersity is substantial (e.g., coefficients of variation greater than 50%). q 1996 Academic Press, Inc. Key Words: colloidal forces; depletion forces; depletion interaction; polydispersity.
INTRODUCTION
The depletion interaction between colloidal particles in the presence of nonadsorbing macromolecules has been studied theoretically and experimentally by numerous researchers. The force arises whenever the concentration of macromolecules in the gap region between two particles becomes less than the bulk. At small gap widths, the macromolecules are excluded from the gap, resulting in an attractive force due to the difference in osmotic pressure. A thorough review of the work in this area is given by Seebergh and Berg (1) and Napper (2). In prior theoretical studies, the nonadsorbing species has been modeled in several different ways. In some of the earliest work, Asakura and Oosawa (3, 4) and later Vrij (5) treated the macromolecules as hard spheres of fixed radius
in which the depletion attraction resulted from a volume exclusion mechanism. Various researchers have studied the depletion interaction produced by flexible polymer chains. Frequently, the polymer molecule is treated as a hard sphere with a diameter equal to the rms end-to-end distance of the chain (6). Feigin and Napper (7) used a rotational isomeric state–Monte Carlo approach to calculate the conformation of macromolecules between two particles. Joanny et al. (8) used mean field theory to calculate the concentration profile in the gap between two parallel plates immersed in a polymer solution and then used the results to predict the magnitude and range of the attractive force between the plates. Recently, Walz and Sharma (9) studied the effect of surface charge by modeling the macromolecules as hard charged spheres of fixed radius. All surfaces were assumed to possess a constant electric surface potential and the long-range repulsive energy between macromolecules and particles was calculated using a linear superposition method. In all of these cases, the macromolecules were assumed to be perfectly uniform with no variability in size, molecular weight, or charge. In real systems, of course, some degree of polydispersity will always exist. In natural systems (e.g., interactions in the environment or biological systems), the polydispersity can be substantial. The specific effect of such polydispersity on the depletion interaction, however, has received only minor attention. Snowden et al. (10) investigated the stability behavior of silica particles in the presence of nonadsorbing sodium carboxymethyl cellulose (CMC) and polystyrene sulphonate (PSS) ionic polymers. In the case of CMC, the authors observed a substantial difference between the stability behavior predicted and that measured experimentally. By comparison, good agreement between theory and experiment was found when using nonionic hydroxyethyl cellulose polymer. The authors suggest that the discrepancy could be have been due to variations in molecular weight and molecular charge distribution for the CMC. Specifically, the shorter CMC chains were more highly charged, which could have caused a weaker depletion attraction. When the authors used the more monodisperse PSS as the nonadsorbing polymer, much better agreement between theory and experiment was obtained. Nikolov and
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0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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Wasan (11) used microinterferometric and scattering techniques to probe the structuring of colloidal particles in dynamic thin films. To test the effect of polydispersity, the authors observed the stepwise thinning of a 20% volume aqueous mixture of 1:1 silica hydrosols of 7 and 28 nm diameters. The polydispersity tended to disrupt the ordering observed in monodisperse systems, leading to less stable films. Schmitt et al. (12) modeled the thermodynamic properties of linear worm-like micelles confined between two solid repulsive plates. When the confined system was allowed to equilibrate with an external reservoir, the micelles behaved similarly to a polydisperse system of unbreakable macromolecules. The authors calculated the depletion force acting on the plates for the cases of both flexible and rigid chains. The range of the force in the flexible case was found to be substantially larger than the rigid case, due to the fact that the flexible chains were more easily excluded from the gap. However, the fundamental behavior of the micelles was essentially the same in both cases; as the gap width decreased, the larger chains were excluded first. In this paper, the effect of polydispersity in the physical size of nonadsorbing macromolecules is studied theoretically. For simplicity, the macromolecules are modeled as hard spheres, though the governing equations are general enough to allow other representations. The polydispersity in the macromolecule diameter is characterized with a coefficient of variation (CV ), defined as the standard deviation of the diameter divided by the mean. The density of macromolecules is assumed to be low enough that interactions between the macromolecules can be neglected. It is found that when the system is characterized in this manner, the depletion force and energy between parallel plates can be calculated using relatively simple expressions. Using the Derjaguin approximation, these results are then applied to spherical particles. Again, both the force and energy are given by simple expressions. For both geometries, as CV r 0, the well-known results for monodisperse systems are obtained. The equations are thus useful in that the interaction in hard-sphere systems can be represented with a single continuous equation over all separation distances.
FIG. 1. Schematic defining the variables used in the depletion force equation. Two spherical particles of radius R are interacting across gap width h in a solution of spherical macromolecules of radius a and bulk concentration r` . The vectors xW 1 and xW 2 define the position of a macromolecule relative to the centers of particles 1 and 2, respectively (from Walz and Sharma).
FW 1 Å
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F
r(xW 1 ) Å r` exp 0
1
[1]
1
E(xW 1 ) kT
G
,
[2]
where kT is the thermal energy and E(xW 1 ) represents the potential energy of mean force acting on a macromolecule at position xW 1 . Substituting this expression into Eq. [1] yields FW 1 Å
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1
where r(xW 1 ) is the number density of macromolecules at position xW 1 and Ç1 E(xW 1 ) is the gradient of the interaction energy with respect to the surface of particle 1. This equation results by performing a simple force balance over all macromolecules in solution. In the limit of low macromolecule concentration, the distribution of macromolecules around the two particles will follow a Boltzmann distribution of the form
*** r exp F 0 E(xkTW ) GÇ E(xW )dxW . 1
`
1
1
1
[3]
V
Force Balance on Colloidal Particles in Solution
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V
THEORY
The force balance used here follows the same approach presented by Walz and Sharma (9). A schematic of the system is depicted in Fig. 1, in which two spherical particles of radius R, separated by gap width h, are immersed in a solution of nonadsorbing spherical macromolecules of radius a (diameter d). The bulk number density of macromolecules (number/volume) is r` . Walz and Sharma showed that the force exerted on particle 1 by the macromolecules can be calculated as
*** r(xW )Ç E(xW )dxW ,
If interactions between macromolecules are not significant (valid for low r` ), then the depletion force produced by macromolecules of different sizes will be additive. Thus n
FW 1 Å ∑
i Å1
*** r
` ,i
V
F
exp 0
G
Ei (xW 1 ) Ç1 Ei (xW 1 )dxW 1 , [4] kT
where r` ,i is the bulk number density of macromolecules of size i and Ei is the interaction energy for macromolecules
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of size i. For a continuously variable distribution, the summation in Eq. [4] can be replaced with an integral over all macromolecule diameters, d, yielding FW 1 Å r`
*
`
f ( d)
0
***
F
E(d, xW 1 ) exp 0 kT
V
G
dimensionless diameter: dO Å
1 Ç1 E(d, xW 1 )dxW 1dd
[5]
where r` is the total number density of macromolecules, f ( d) is the fraction of macromolecules with diameter between d and d / dd, and E(d, xW 1 ) is the interaction energy of macromolecules of diameter d at position xW 1 . Equation [5] is valid for any interaction energy and system geometry. The primary assumption that has been made is that the forces produced by the various macromolecule sizes can be added to obtain the total force acting on the particle. One of the most common interactions is the hardsphere interaction, which can be written as E(d, xW 1 ) Å
H
/` for É xW 1É õ R / a
0
for É xW 1É § R / a.
[6]
The simplest geometry is that of two parallel plates. For a system of two plates separated by gap width h in a solution of hard spherical macromolecules of diameter d, Walz and Sharma showed that the volume integral in Eq. [5] will have the solution
*** V
F
exp 0
G
Rewriting Eq. [9] gives
0 AkT for h õ d
.
[12]
Substituting these expressions into Eq. [8] yields Fpp (hO ) 1 Å0q Ar`kT 2p
*
S D
`
exp 0
hO
dO 2 ddO , 2
[13]
where the identity ddˆ Å (1/ s )dd has been used. Equation [13] has the solution Fpp (hO ) 1 Å 0 erfc(hO ), Ar`kT 2
[14]
where erfc is the complementary error function (13). The depletion energy can be related to the force as
* F(h * )dh * h
for h § d,
0
[7]
*
[15]
E(hO ) Å 0 s
*
hO
`
F(hO * )dhO *,
[16]
where dhˆ * Å (1/ s )dh *. Substituting Eq. [14] into this expression and integrating gives Epp (hO ) s Å AkTr` 2
F
hO erfc(hO ) 0
1 exp( 0hO 2 ) p
q
G
.
[17]
`
f ( d)dd,
[8]
where the subscript pp denotes two parallel plates. For a normally distributed particle size, f ( d) will be given by
F S DG
1 1 exp 0 2 s 2p
d 0 dU s
2
,
[9]
where dV is the mean particle diameter and s is the standard
/
S D
1 dO 2 exp 0 2 s 2p q
f ( dO ) Å
`
h
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h 0 dU . [11] s
or in terms of the dimensionless distance
Fpp (h) Å 0 Ar`kT
q
[10]
E(h) Å 0
where A is the area of plate 1 and the negative sign indicates an attractive force. (The vector notation has been dropped for simplicity.) Thus for a given h, Eq. [5] will equal 0 AkT times the total number of particles with d ú h. This can also be written as
f ( d) Å
d 0 dU s
dimensionless separation distance: hO Å
E(d, xW 1 ) Ç1 E(d, xW 1 )dxW 1 kT Å
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deviation of the distribution. To simplify the equations, the following dimensionless variables will be used:
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Using the identity CV Å s /dV , Eq. [17] can be written in dimensionless form as Epp (hO ) CV Å AkTr` dU 2
F
hO erfc(hO ) 0
1 exp( 0hO 2 ) p
q
G
. [18]
Now the interaction between two spherical particles will be calculated. In monodisperse hard-sphere systems in which Eq. [3] is valid, the depletion force and energy are given by the expressions (4, 9, 14)
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Fss (h) r`paRkT Å
0
F
a h h2 /20 0 R a 4aR
F
1 Fss (hO ) Å CV hO erfc(hO ) 0 q exp( 0hO 2 ) r`paV RkT p
G
for 0 £ h õ d
[19]
for h § d
0
Ess (h) Å r`pa 2 RkT
0
F
4a h 2h h2 h3 /20 0 / 2/ 3R R a 2a 12a 2 R
G
Ess (hO ) Å 02CV 2 r`paV 2 RkT 1
for h § d,
[20]
where the subscript ss denotes the interaction between two spheres. To calculate the interaction in polydisperse systems, the rigorous approach would be to solve Eq. [5]. However, because both the force for a given macromolecule size (Eq. [19]) and the size distribution, f ( d), are both functions of d, the resulting integrals are cumbersome. An alternate method is to use the Derjaguin approximation. Here, the region of closest approach between the spherical particles is modeled as a series of thin concentric rings and the interaction between opposing rings is calculated using the parallel plate expression. The approximation is valid as long as the particle radius is much greater than both the gap width, h, and the characteristic length of the interaction force (equal to dV in hard sphere systems). The general expression for calculating the force between two spherical particles using the Derjaguin approach is (15) Fss (h) Å pR
*
`
h
F
G
Fpp (h * ) dh *, A
[21]
where Fpp (h * )/A is the force per unit area between two parallel plates. Rewriting in dimensionless form and substituting Eq. [14] gives Fss (hO ) s Å0 r`pRkT 2
*
`
hO
erfc(hO * )dhO *,
[22]
which has the solution Fss (hO ) s Å r`pRkT 2
F
hO erfc(hO ) 0
1 exp( 0hO 2 ) p
q
G
.
[23]
Again, inserting the identity CV Å s /dV and writing in terms of the mean radius of the macromolecules, aV , gives
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.
[24]
Finally, inserting this equation for force into Eq. [16] yields the following expression for the dimensionless depletion energy between two spheres:
0 for 0 £ h õ d
G
FS
D
1 hO 2 hO / erfc(hO ) 0 q exp( 0hO 2 ) 4 2 2 p
G
.
[25]
Equations [24] and [25] indicate that with the Derjaguin approach, the depletion force and energy between two identical spherical particles scale as aV R and aV 2 R, respectively. Equations [19] and [20] show this to be valid in the limit of a ! R and as h r 0. A summary of the depletion force and energy equations for the parallel plate and sphere–sphere geometries is given in Table 1. Also given are the equations for monodisperse systems. Comparisons between these equations are given in the Results and Discussion section. RESULTS AND DISCUSSION
Interaction between Two Parallel Plates The dimensionless depletion force predicted by Eq. [14] at various degrees of polydispersity is shown in Fig. 2. The separation distance (gap width) between the plates is presented in terms of the dimensionless variable, h/dV , to allow a more direct comparison with the monodisperse equation given in Table 1. This variable is related to the dimensionless distance, hˆ , as h Å CVhO / 1. dU
[26]
The solid line in the figure was calculated using a very low value for CV ( É0). (It was found that using CV Å 0 resulted in numerical problems when calculating the complementary error function.) As can be seen, this plot is essentially identical to that which would be predicted with the monodisperse equation (i.e., the dimensionless force equals 01 for 0 £ h õ d and zero otherwise). The polydispersity results in a longer range depletion force and also a more gradual transition between the upper and lower limits of the force. In the parallel plate geometry, the depletion force is produced by the exclusion of macromolecules too large to fit in the gap (those with d ú h). In the polydisperse system, the bigger particles are excluded at larger separations. As the gap width decreases, the number of particles excluded from the gap
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TABLE 1 A Summary of the Depletion Force and Energy Equations for Monodisperse and Polydisperse Systems Monodisperse systems
Force Energy
Force
Energy
Fpp(h)
Polydisperse systems Two parallel plates
H
Fpp(hO )
01 for 0 £ h õ d Å 0 for h § d Ar`kT h Epp(h) 0 1 for 0 £ h £ d Å d for h § d Adr`kT 0
F
Fss(h) Å r`paRkT
0
Ess(h) Å r`pa 2RkT
0
Ar`kT Epp(hO )
AdU r`kT
G
a h h2 /20 0 R a 4aR
F
Two equal spheres
G
4a h 2h h2 h3 /20 0 / 2/ 3R R a 2a 12a 2R 0
for 0 £ h õ d
FIG. 2. The dimensionless depletion force between two parallel plates versus dimensionless gap width. The dashed lines denoted a, b, and c correspond to increasing amounts of polydispersity.
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F
G
F
G
CV 1 hO erfc(hO ) 0 q exp(0hO 2) 2 p
Ess(hO ) Å 02CV 2 r`paV 2RkT
FS D
G
1 hO 2 hO / erfc(hO ) 0 q exp(0hO 2) 4 2 2 p
for h § d
increases, until at small separations, essentially all of the particles are excluded. The calculations show that a CV of 5% produces no significant deviation from the monodisperse system. In fact, it is not until the CV is of order 15–20% that the deviations become substantial. This finding is also evident in the depletion energy plots shown in Fig. 3. It is interesting to note that at any separation distance, the interaction energy in a normally distributed polydisperse system at fixed bulk num-
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1 erfc(hO ) 2
Fss(hO ) 1 Å CV hO erfc(hO ) 0 q exp(0hO 2) r`aV RkT p
for 0 £ h õ d for h § d
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ber density will always be greater than or equal to that for a monodisperse system. The energy required to bring the plates into contact, however, is the same in all systems ( 01 in these dimensionless units), since at contact all macromolecules must be excluded, regardless of the degree of polydispersity. Interaction between Two Spherical Particles Validity of Derjaguin approximation. Before presenting the effect of polydispersity on the interaction between two
FIG. 3. The dimensionless depletion energy between two parallel plates versus dimensionless gap width.
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FIG. 4. The error introduced by using the Derjaguin approximation to calculate the depletion force between two spherical particles. The percent error was calculated using Eq. [27] and also assuming a perfectly monodisperse system (CV É 0).
spherical particles, it is informative to first review the accuracy of the Derjaguin approximation for these systems. Shown in Fig. 4 are plots of the percent error between the depletion force calculated with the rigorous approach of Eqs. [19] and [24], which utilizes the Derjaguin approximation. These calculations were performed with monodisperse macromolecules (CV É 0) in order to allow direct comparisons between the two approaches. The different curves correspond to varying values of R/a (the ratio of the particle radius to that of the macromolecule.) As this ratio decreases, the assumption that the particle radius is much larger than the range of the interaction becomes less valid, thus the Derjaguin approximation should become less accurate. The percent error in Fig. 4 is defined as F rigorous 0 F Derjaguin 1 100. F rigorous
%error Å
[27]
As expected, the magnitude of the error increases with gap width, since the Derjaguin approximation also assumes that h ! R (i.e., curvature effects are insignificant). At h Å d, both solutions correctly predict the depletion effect to vanish, so the error is zero after this point. For R/a Å 10, 100, and 1000, the maximum difference between the two methods is 9.1%, 0.99%, and 0.1%, respectively. For a given R/a, the maximum fractional error is thus roughly a/R. Assuming a 1% accuracy requirement, the Derjaguin approx-
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imation provides an accurate estimate of the depletion interaction in systems in which R/a § 100. It should be emphasized that although the fractional error is greatest at h Å d, the absolute error (difference between the two approaches) is actually largest at contact. At h Å 0, the dimensionless force predicted by Eq. [19] is 0 (2 / a/R), while the Derjaguin approximation gives 02. Thus the maximum absolute difference between the two equations will be a/R. (The fractional error at contact will be a/2R, assuming a/R ! 2.) Effect of polydispersity on interaction. Shown in Figures 5 and 6 are plots of the dimensionless depletion force and energy between two spherical particles at varying degrees of polydispersity. As with the parallel plate geometry, polydispersity increases the range over which the depletion force acts. In addition, no significant deviation from the monodisperse equation is predicted for values of CV less than 15– 20%. Finally, it is interesting to note that at any h, the depletion force and energy in a normally distributed polydisperse system are always greater than or equal to the equivalent force and energy in a monodisperse system with the same average macromolecule diameter and bulk number density. This behavior differs from the parallel plate case (compare Figs. 5 and 2) but can be explained as follows. Consider first a monodisperse system in which all macromolecules have diameter d. As two particles approach, the depletion force is zero until the gap width becomes equal to d. At this point, a depletion zone is formed between the particles. As the gap width decreases, the particles are pushed
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FIG. 5. The dimensionless depletion force between two identical spherical particles versus dimensionless gap width.
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Effect on Stability
FIG. 6. The dimensionless depletion energy between two identical spherical particles versus dimensionless gap width.
further away from the line of centers, the depletion zone enlarges, and the attractive depletion force increases. The size of the depletion zone, and hence the magnitude of the depletion force, reaches a maximum at contact (h Å 0). Now consider a polydisperse system. If the diameters of the macromolecules are normally distributed about the mean, then exactly half of the macromolecules will have a diameter larger than the mean. The depletion zone thus forms at a larger gap width. Unlike the monodisperse case, smaller particles will still remain in this depletion zone and the force will not increase as rapidly. Nonetheless, the total concentration in the zone is less than the bulk and an attractive force develops. As the gap width decreases, the force increases until at h Å dV , the difference in the force between the monodisperse and polydisperse cases is a maximum. As h decreases further, this difference decreases and at contact the two solutions for force converge. Because of the longer range of the interaction in the polydisperse case, the energy at contact is greater for the polydisperse case. It should be emphasized that these results apply only to a normally distributed macromolecule diameter in which the numbers of macromolecules with diameters above and below the mean are equal. With another type of distribution function, the force profile could be quite different. For example, in a system which contains a higher fraction of smaller particles, the depletion force would be less than the monodisperse system at small separations. Calculating the force profile thus requires knowing the appropriate size distribution, f ( d).
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It is well known that a nonadsorbing polymer can induce flocculation in an otherwise stable colloidal dispersion. Using the force and energy equations presented above, the effect of polydispersity in the macromolecule size on stability can be examined. An example system consisting of 1 mm radius hard spherical particles in a solution of hard spherical macromolecules with a 2.5-nm average radius will be considered. The bulk concentration of macromolecules is 1.53 1 10 23 macromolecules/m 3 , corresponding to a volume fraction of 0.01 (1%) for the monodisperse solution. Shown in Fig. 7 are plots of the predicted depletion energy (in number of kT’s) versus particle gap width. As the degree of polydispersity increases, both the magnitude and range of the depletion energy increases. A summary of these trends is given in Table 2. An increase in the coefficient of variation from 0 to 100% increases the attractive energy at contact from 6.0 to 8.9 kT. In addition, the separation distance at which the energy equals 01 kT increases from 3.0 to 5.8 nm. Considering the magnitude of the change in polydispersity, these changes are relatively small and would not have a significant impact on system stability. A possible exception would be particles that had been stabilized sterically through the adsorption of a nonbridging polymer. If the range of this steric repulsion was less than that of the depletion attraction (e.g., 3–4 nm), then the increased polydispersity could pro-
FIG. 7. The depletion energy (in kT ) between two 1 mm radius hard spherical particles in a solution of hard spherical macromolecules with an average radius of 2.5 nm. The bulk concentration of macromolecules, r` , is 1.53 1 10 23 macromolecules/m 3 , which corresponds to 1% volume for the monodisperse solution. The curves were calculated using Eq. [25].
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produce structuring of the macromolecules and oscillations in the resulting depletion force between two macroscopic surfaces (11, 16–20). The accuracy of the results presented above depends upon the validity of this assumption of ideal behavior. The effects of macromolecule interactions can be determined with the virial expansion of the single-particle distribution function
TABLE 2 Summary of the Results of Figure 7 CV (%)
E (0)
h at E(h) Å 01 kT
0 25 50 75 100 150
06.0 06.2 06.7 07.6 08.9 012.2
3.0 3.2 3.8 4.8 5.8 8.6
Note. Shown at each value of CV is the value of the depletion energy at contact (h Å 0) plus the separation distance at which the depletion energy becomes equal to 01 kT.
mote secondary flocculation in a shallow potential energy minimum. The calculations above assumed a constant macromolecule number density. For macromolecules with a normally distributed diameter, increasing the polydispersity also increases the volume fraction. Integrating the Gaussian size distribution function yields the expression for the total volume of macromolecules per unit volume of solution £particles Å
4 r`paV 3 (1 / 3CV 2 ), 3
[28]
indicating that the volume fraction will be proportional to the factor (1 / 3CV 2 ). For the CV values listed in Fig. 7, the volume concentrations range from 1% (for CV Å 0) to 7.75% (for CV Å 150%). Instead of fixing the number density, an alternate approach is to vary the polydispersity while maintaining a constant macromolecule volume fraction. The results of these calculation are shown in Fig. 8. For each curve, the bulk number density has been adjusted using Eq. [28] to maintain a volume concentration of 1%. The resulting curves are markedly different from those in Fig. 7. For example, the magnitude of the interaction at contact is now greatest for the monodisperse system in which the number density is greatest. In addition, the range of the interaction, defined as the separation distance at which the interaction energy equals 01 kT, is actually smallest for the most polydisperse system, opposite the trend in Fig. 7. Because of these differences, formulating a general statement about the effect of polydispersity on the depletion interaction is difficult; the result depends on how each particular system is defined (e.g., constant number density, constant volume fraction).
F
r(xW ) Å r` exp 0
G
E(xW ) [1 / b2 (xW ) r` / O( r 2` )], kT
[29]
which reduces to the Boltzmann equation as r` r 0 (21). In hard-sphere systems, the value of the second virial coefficient next to a single hard wall, b2 (x Å 0), will be four times the macromolecule volume. Although this value can be altered by the presence of another wall (21), a first-order approximation of the magnitude of the initial deviation from ideal behavior is four times the macromolecule volume fraction. Higher order concentration effects should therefore be negligible for volume concentrations up to about 1%. Note that if the macromolecules experience other long-range interactions (e.g., electrostatic repulsion), the value of b2 can be greatly increased. The upper limit for the volume fraction would then be much lower.
Interactions between Macromolecules In using the Boltzmann equation to calculate the density distribution of macromolecules around the particles (Eq. [2]), interactions between the macromolecules themselves have been ignored. Such interactions have been shown to
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FIG. 8. The depletion energy (in kT) between two 1 mm radius hard spherical particles in a solution of hard spherical macromolecules with an average radius of 2.5 nm. The concentration of macromolecules is 1% volume for all cases.
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SUMMARY
A set of general equations has been developed to calculate the depletion force and energy in a solution of normally distributed polydisperse macromolecules. It is assumed that the density of macromolecules is low enough that higher order effects arising from interactions between the macromolecules are negligible. As the degree of polydispersity approaches zero, the equations match the predictions made with equations developed for monodisperse systems. The equations are thus general to any system and require knowing only the appropriate size distribution function. For two spherical particles in a solution of nonadsorbing macromolecules at fixed number density, polydispersity is predicted to increase the range and magnitude of the depletion energy. The effect is relatively small, however, even for systems with a coefficient of variation as large as 25%. Calculations on a typical system show that polydispersity can lead to increased flocculation through the formation of secondary potential energy minima. The polydispersity would have to be substantial, however, to produce any significant changes in the potential energy profile between the two particles. By comparison, calculations at constant volume fraction predict the depletion energy to actually decrease with increasing polydispersity. Finally, although the equations were derived assuming hard-sphere interactions, they could also be applied to systems in which other long-range interactions are important. For example, the effect of charged macromolecules could be simulated by replacing the hard-sphere diameter with an effective diameter which accounted for the size of the charge double layer. The size distribution function, f ( d), could then
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account for variations in both the size and charge of the macromolecules. REFERENCES 1. Seebergh, J. E., and Berg, J. C., Langmuir 10, 454 (1994). 2. Napper, D. H., ‘‘Polymeric Stabilization of Colloidal Dispersions,’’ pp. 332–413. Academic Press, London, 1983. 3. Asakura, S., and Oosawa, F., J. Chem. Phys. 22, 1255 (1954). 4. Asakura, S., and Oosawa, F., J. Polym. Sci. 33, 183 (1958). 5. Vrij, A., Pure Appl. Chem. 48, 471 (1976). 6. Sieglaff, C. L., J. Polym. Sci. 41, 319 (1959). 7. Feigin, R. I., and Napper, D. H., J. Colloid Interface Sci. 75, 525 (1980). 8. Joanny, J. F., Leibler, J. F., and de Gennes, P. G., J. Polym. Sci. Polym. Phys. Ed. 17, 1073 (1979). 9. Walz, J. Y., and Sharma, A., J. Colloid Interface Sci. 168, 485 (1994). 10. Snowden, M. J., Clegg, S. M., and Williams, P. A., J. Chem. Soc. Faraday Trans. 87, 2201 (1991). 11. Nikolov, A. D., and Wasan, D. T., Langmuir 8, 2985 (1992). 12. Schmitt, V., Lequeux, F., and Marques, C. M., J. Phys. II France 3, 891 (1993). 13. Abramowitz, M., and Stegun, I. A., Eds., ‘‘Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,’’ pp. 297– 299. Wiley, New York, 1972. 14. Gast, A. P., Hall, C. K., and Russel, W. B., J. Colloid Interface Sci. 96, 251 (1983). 15. Israelachvili, J. N., ‘‘Intermolecular and Surface Forces,’’ 2nd ed., p. 163. Academic Press, London, 1992. 16. Henderson, D., J. Colloid Interface Sci. 121, 486 (1988). 17. Richetti, P., and Ke´kicheff, P., Phys. Rev. Lett. 68, 1951 (1992). 18. Wasan, D. T., Nikolov, A. D., Kralchevski, P., and Ivanov, I., Colloids Surf. 67, 139 (1992). 19. Chu, X. L., Nikolov, A. D., and Wasan, D. T., Langmuir 10, 4403 (1994). 20. Pollard, M. L., and Radke, C. J., J. Chem. Phys. 101(8), 6979 (1994). 21. Glandt, E. D., J. Colloid Interface Sci. 77, 512 (1980).
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0145
Effect of Polydispersity on the Depletion Interaction between Colloidal Particles JOHN Y. WALZ Tulane University, Department of Chemical Engineering, New Orleans, Louisiana 70118 Received April 7, 1995; accepted September 11, 1995
The effect of polydispersity on the depletion interaction in hardsphere systems is investigated theoretically. The depletion force and energy between two parallel plates and two spherical particles in a solution of nonadsorbing, hard spherical macromolecules is calculated via a general force-balance approach. The diameter of the macromolecules is assumed to be normally distributed and the polydispersity in diameter is characterized with the coefficient of variation (ratio of the standard deviation to the mean). The primary assumption is that the density of macromolecules is low enough that higher order effects arising from interactions between macromolecules are negligible. It is found that the interaction between parallel plates can be represented in terms of a complementary error function of a dimensionless separation distance. These results are then applied to spherical particles via the Derjaguin approximation. A set of general and relatively simple equations for force and energy are developed that are shown to be valid in both monodisperse and polydisperse systems. For spherical particles in a solution of macromolecules at fixed number density, polydispersity is predicted to increase the range and magnitude of the depletion force. The effect is relatively small, however, unless the polydispersity is substantial (e.g., coefficients of variation greater than 50%). q 1996 Academic Press, Inc. Key Words: colloidal forces; depletion forces; depletion interaction; polydispersity.
INTRODUCTION
The depletion interaction between colloidal particles in the presence of nonadsorbing macromolecules has been studied theoretically and experimentally by numerous researchers. The force arises whenever the concentration of macromolecules in the gap region between two particles becomes less than the bulk. At small gap widths, the macromolecules are excluded from the gap, resulting in an attractive force due to the difference in osmotic pressure. A thorough review of the work in this area is given by Seebergh and Berg (1) and Napper (2). In prior theoretical studies, the nonadsorbing species has been modeled in several different ways. In some of the earliest work, Asakura and Oosawa (3, 4) and later Vrij (5) treated the macromolecules as hard spheres of fixed radius
in which the depletion attraction resulted from a volume exclusion mechanism. Various researchers have studied the depletion interaction produced by flexible polymer chains. Frequently, the polymer molecule is treated as a hard sphere with a diameter equal to the rms end-to-end distance of the chain (6). Feigin and Napper (7) used a rotational isomeric state–Monte Carlo approach to calculate the conformation of macromolecules between two particles. Joanny et al. (8) used mean field theory to calculate the concentration profile in the gap between two parallel plates immersed in a polymer solution and then used the results to predict the magnitude and range of the attractive force between the plates. Recently, Walz and Sharma (9) studied the effect of surface charge by modeling the macromolecules as hard charged spheres of fixed radius. All surfaces were assumed to possess a constant electric surface potential and the long-range repulsive energy between macromolecules and particles was calculated using a linear superposition method. In all of these cases, the macromolecules were assumed to be perfectly uniform with no variability in size, molecular weight, or charge. In real systems, of course, some degree of polydispersity will always exist. In natural systems (e.g., interactions in the environment or biological systems), the polydispersity can be substantial. The specific effect of such polydispersity on the depletion interaction, however, has received only minor attention. Snowden et al. (10) investigated the stability behavior of silica particles in the presence of nonadsorbing sodium carboxymethyl cellulose (CMC) and polystyrene sulphonate (PSS) ionic polymers. In the case of CMC, the authors observed a substantial difference between the stability behavior predicted and that measured experimentally. By comparison, good agreement between theory and experiment was found when using nonionic hydroxyethyl cellulose polymer. The authors suggest that the discrepancy could be have been due to variations in molecular weight and molecular charge distribution for the CMC. Specifically, the shorter CMC chains were more highly charged, which could have caused a weaker depletion attraction. When the authors used the more monodisperse PSS as the nonadsorbing polymer, much better agreement between theory and experiment was obtained. Nikolov and
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Wasan (11) used microinterferometric and scattering techniques to probe the structuring of colloidal particles in dynamic thin films. To test the effect of polydispersity, the authors observed the stepwise thinning of a 20% volume aqueous mixture of 1:1 silica hydrosols of 7 and 28 nm diameters. The polydispersity tended to disrupt the ordering observed in monodisperse systems, leading to less stable films. Schmitt et al. (12) modeled the thermodynamic properties of linear worm-like micelles confined between two solid repulsive plates. When the confined system was allowed to equilibrate with an external reservoir, the micelles behaved similarly to a polydisperse system of unbreakable macromolecules. The authors calculated the depletion force acting on the plates for the cases of both flexible and rigid chains. The range of the force in the flexible case was found to be substantially larger than the rigid case, due to the fact that the flexible chains were more easily excluded from the gap. However, the fundamental behavior of the micelles was essentially the same in both cases; as the gap width decreased, the larger chains were excluded first. In this paper, the effect of polydispersity in the physical size of nonadsorbing macromolecules is studied theoretically. For simplicity, the macromolecules are modeled as hard spheres, though the governing equations are general enough to allow other representations. The polydispersity in the macromolecule diameter is characterized with a coefficient of variation (CV ), defined as the standard deviation of the diameter divided by the mean. The density of macromolecules is assumed to be low enough that interactions between the macromolecules can be neglected. It is found that when the system is characterized in this manner, the depletion force and energy between parallel plates can be calculated using relatively simple expressions. Using the Derjaguin approximation, these results are then applied to spherical particles. Again, both the force and energy are given by simple expressions. For both geometries, as CV r 0, the well-known results for monodisperse systems are obtained. The equations are thus useful in that the interaction in hard-sphere systems can be represented with a single continuous equation over all separation distances.
FIG. 1. Schematic defining the variables used in the depletion force equation. Two spherical particles of radius R are interacting across gap width h in a solution of spherical macromolecules of radius a and bulk concentration r` . The vectors xW 1 and xW 2 define the position of a macromolecule relative to the centers of particles 1 and 2, respectively (from Walz and Sharma).
FW 1 Å
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F
r(xW 1 ) Å r` exp 0
1
[1]
1
E(xW 1 ) kT
G
,
[2]
where kT is the thermal energy and E(xW 1 ) represents the potential energy of mean force acting on a macromolecule at position xW 1 . Substituting this expression into Eq. [1] yields FW 1 Å
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1
where r(xW 1 ) is the number density of macromolecules at position xW 1 and Ç1 E(xW 1 ) is the gradient of the interaction energy with respect to the surface of particle 1. This equation results by performing a simple force balance over all macromolecules in solution. In the limit of low macromolecule concentration, the distribution of macromolecules around the two particles will follow a Boltzmann distribution of the form
*** r exp F 0 E(xkTW ) GÇ E(xW )dxW . 1
`
1
1
1
[3]
V
Force Balance on Colloidal Particles in Solution
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V
THEORY
The force balance used here follows the same approach presented by Walz and Sharma (9). A schematic of the system is depicted in Fig. 1, in which two spherical particles of radius R, separated by gap width h, are immersed in a solution of nonadsorbing spherical macromolecules of radius a (diameter d). The bulk number density of macromolecules (number/volume) is r` . Walz and Sharma showed that the force exerted on particle 1 by the macromolecules can be calculated as
*** r(xW )Ç E(xW )dxW ,
If interactions between macromolecules are not significant (valid for low r` ), then the depletion force produced by macromolecules of different sizes will be additive. Thus n
FW 1 Å ∑
i Å1
*** r
` ,i
V
F
exp 0
G
Ei (xW 1 ) Ç1 Ei (xW 1 )dxW 1 , [4] kT
where r` ,i is the bulk number density of macromolecules of size i and Ei is the interaction energy for macromolecules
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of size i. For a continuously variable distribution, the summation in Eq. [4] can be replaced with an integral over all macromolecule diameters, d, yielding FW 1 Å r`
*
`
f ( d)
0
***
F
E(d, xW 1 ) exp 0 kT
V
G
dimensionless diameter: dO Å
1 Ç1 E(d, xW 1 )dxW 1dd
[5]
where r` is the total number density of macromolecules, f ( d) is the fraction of macromolecules with diameter between d and d / dd, and E(d, xW 1 ) is the interaction energy of macromolecules of diameter d at position xW 1 . Equation [5] is valid for any interaction energy and system geometry. The primary assumption that has been made is that the forces produced by the various macromolecule sizes can be added to obtain the total force acting on the particle. One of the most common interactions is the hardsphere interaction, which can be written as E(d, xW 1 ) Å
H
/` for É xW 1É õ R / a
0
for É xW 1É § R / a.
[6]
The simplest geometry is that of two parallel plates. For a system of two plates separated by gap width h in a solution of hard spherical macromolecules of diameter d, Walz and Sharma showed that the volume integral in Eq. [5] will have the solution
*** V
F
exp 0
G
Rewriting Eq. [9] gives
0 AkT for h õ d
.
[12]
Substituting these expressions into Eq. [8] yields Fpp (hO ) 1 Å0q Ar`kT 2p
*
S D
`
exp 0
hO
dO 2 ddO , 2
[13]
where the identity ddˆ Å (1/ s )dd has been used. Equation [13] has the solution Fpp (hO ) 1 Å 0 erfc(hO ), Ar`kT 2
[14]
where erfc is the complementary error function (13). The depletion energy can be related to the force as
* F(h * )dh * h
for h § d,
0
[7]
*
[15]
E(hO ) Å 0 s
*
hO
`
F(hO * )dhO *,
[16]
where dhˆ * Å (1/ s )dh *. Substituting Eq. [14] into this expression and integrating gives Epp (hO ) s Å AkTr` 2
F
hO erfc(hO ) 0
1 exp( 0hO 2 ) p
q
G
.
[17]
`
f ( d)dd,
[8]
where the subscript pp denotes two parallel plates. For a normally distributed particle size, f ( d) will be given by
F S DG
1 1 exp 0 2 s 2p
d 0 dU s
2
,
[9]
where dV is the mean particle diameter and s is the standard
/
S D
1 dO 2 exp 0 2 s 2p q
f ( dO ) Å
`
h
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or in terms of the dimensionless distance
Fpp (h) Å 0 Ar`kT
q
[10]
E(h) Å 0
where A is the area of plate 1 and the negative sign indicates an attractive force. (The vector notation has been dropped for simplicity.) Thus for a given h, Eq. [5] will equal 0 AkT times the total number of particles with d ú h. This can also be written as
f ( d) Å
d 0 dU s
dimensionless separation distance: hO Å
E(d, xW 1 ) Ç1 E(d, xW 1 )dxW 1 kT Å
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deviation of the distribution. To simplify the equations, the following dimensionless variables will be used:
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Using the identity CV Å s /dV , Eq. [17] can be written in dimensionless form as Epp (hO ) CV Å AkTr` dU 2
F
hO erfc(hO ) 0
1 exp( 0hO 2 ) p
q
G
. [18]
Now the interaction between two spherical particles will be calculated. In monodisperse hard-sphere systems in which Eq. [3] is valid, the depletion force and energy are given by the expressions (4, 9, 14)
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Fss (h) r`paRkT Å
0
F
a h h2 /20 0 R a 4aR
F
1 Fss (hO ) Å CV hO erfc(hO ) 0 q exp( 0hO 2 ) r`paV RkT p
G
for 0 £ h õ d
[19]
for h § d
0
Ess (h) Å r`pa 2 RkT
0
F
4a h 2h h2 h3 /20 0 / 2/ 3R R a 2a 12a 2 R
G
Ess (hO ) Å 02CV 2 r`paV 2 RkT 1
for h § d,
[20]
where the subscript ss denotes the interaction between two spheres. To calculate the interaction in polydisperse systems, the rigorous approach would be to solve Eq. [5]. However, because both the force for a given macromolecule size (Eq. [19]) and the size distribution, f ( d), are both functions of d, the resulting integrals are cumbersome. An alternate method is to use the Derjaguin approximation. Here, the region of closest approach between the spherical particles is modeled as a series of thin concentric rings and the interaction between opposing rings is calculated using the parallel plate expression. The approximation is valid as long as the particle radius is much greater than both the gap width, h, and the characteristic length of the interaction force (equal to dV in hard sphere systems). The general expression for calculating the force between two spherical particles using the Derjaguin approach is (15) Fss (h) Å pR
*
`
h
F
G
Fpp (h * ) dh *, A
[21]
where Fpp (h * )/A is the force per unit area between two parallel plates. Rewriting in dimensionless form and substituting Eq. [14] gives Fss (hO ) s Å0 r`pRkT 2
*
`
hO
erfc(hO * )dhO *,
[22]
which has the solution Fss (hO ) s Å r`pRkT 2
F
hO erfc(hO ) 0
1 exp( 0hO 2 ) p
q
G
.
[23]
Again, inserting the identity CV Å s /dV and writing in terms of the mean radius of the macromolecules, aV , gives
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.
[24]
Finally, inserting this equation for force into Eq. [16] yields the following expression for the dimensionless depletion energy between two spheres:
0 for 0 £ h õ d
G
FS
D
1 hO 2 hO / erfc(hO ) 0 q exp( 0hO 2 ) 4 2 2 p
G
.
[25]
Equations [24] and [25] indicate that with the Derjaguin approach, the depletion force and energy between two identical spherical particles scale as aV R and aV 2 R, respectively. Equations [19] and [20] show this to be valid in the limit of a ! R and as h r 0. A summary of the depletion force and energy equations for the parallel plate and sphere–sphere geometries is given in Table 1. Also given are the equations for monodisperse systems. Comparisons between these equations are given in the Results and Discussion section. RESULTS AND DISCUSSION
Interaction between Two Parallel Plates The dimensionless depletion force predicted by Eq. [14] at various degrees of polydispersity is shown in Fig. 2. The separation distance (gap width) between the plates is presented in terms of the dimensionless variable, h/dV , to allow a more direct comparison with the monodisperse equation given in Table 1. This variable is related to the dimensionless distance, hˆ , as h Å CVhO / 1. dU
[26]
The solid line in the figure was calculated using a very low value for CV ( É0). (It was found that using CV Å 0 resulted in numerical problems when calculating the complementary error function.) As can be seen, this plot is essentially identical to that which would be predicted with the monodisperse equation (i.e., the dimensionless force equals 01 for 0 £ h õ d and zero otherwise). The polydispersity results in a longer range depletion force and also a more gradual transition between the upper and lower limits of the force. In the parallel plate geometry, the depletion force is produced by the exclusion of macromolecules too large to fit in the gap (those with d ú h). In the polydisperse system, the bigger particles are excluded at larger separations. As the gap width decreases, the number of particles excluded from the gap
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TABLE 1 A Summary of the Depletion Force and Energy Equations for Monodisperse and Polydisperse Systems Monodisperse systems
Force Energy
Force
Energy
Fpp(h)
Polydisperse systems Two parallel plates
H
Fpp(hO )
01 for 0 £ h õ d Å 0 for h § d Ar`kT h Epp(h) 0 1 for 0 £ h £ d Å d for h § d Adr`kT 0
F
Fss(h) Å r`paRkT
0
Ess(h) Å r`pa 2RkT
0
Ar`kT Epp(hO )
AdU r`kT
G
a h h2 /20 0 R a 4aR
F
Two equal spheres
G
4a h 2h h2 h3 /20 0 / 2/ 3R R a 2a 12a 2R 0
for 0 £ h õ d
FIG. 2. The dimensionless depletion force between two parallel plates versus dimensionless gap width. The dashed lines denoted a, b, and c correspond to increasing amounts of polydispersity.
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F
G
F
G
CV 1 hO erfc(hO ) 0 q exp(0hO 2) 2 p
Ess(hO ) Å 02CV 2 r`paV 2RkT
FS D
G
1 hO 2 hO / erfc(hO ) 0 q exp(0hO 2) 4 2 2 p
for h § d
increases, until at small separations, essentially all of the particles are excluded. The calculations show that a CV of 5% produces no significant deviation from the monodisperse system. In fact, it is not until the CV is of order 15–20% that the deviations become substantial. This finding is also evident in the depletion energy plots shown in Fig. 3. It is interesting to note that at any separation distance, the interaction energy in a normally distributed polydisperse system at fixed bulk num-
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1 erfc(hO ) 2
Fss(hO ) 1 Å CV hO erfc(hO ) 0 q exp(0hO 2) r`aV RkT p
for 0 £ h õ d for h § d
0
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ber density will always be greater than or equal to that for a monodisperse system. The energy required to bring the plates into contact, however, is the same in all systems ( 01 in these dimensionless units), since at contact all macromolecules must be excluded, regardless of the degree of polydispersity. Interaction between Two Spherical Particles Validity of Derjaguin approximation. Before presenting the effect of polydispersity on the interaction between two
FIG. 3. The dimensionless depletion energy between two parallel plates versus dimensionless gap width.
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FIG. 4. The error introduced by using the Derjaguin approximation to calculate the depletion force between two spherical particles. The percent error was calculated using Eq. [27] and also assuming a perfectly monodisperse system (CV É 0).
spherical particles, it is informative to first review the accuracy of the Derjaguin approximation for these systems. Shown in Fig. 4 are plots of the percent error between the depletion force calculated with the rigorous approach of Eqs. [19] and [24], which utilizes the Derjaguin approximation. These calculations were performed with monodisperse macromolecules (CV É 0) in order to allow direct comparisons between the two approaches. The different curves correspond to varying values of R/a (the ratio of the particle radius to that of the macromolecule.) As this ratio decreases, the assumption that the particle radius is much larger than the range of the interaction becomes less valid, thus the Derjaguin approximation should become less accurate. The percent error in Fig. 4 is defined as F rigorous 0 F Derjaguin 1 100. F rigorous
%error Å
[27]
As expected, the magnitude of the error increases with gap width, since the Derjaguin approximation also assumes that h ! R (i.e., curvature effects are insignificant). At h Å d, both solutions correctly predict the depletion effect to vanish, so the error is zero after this point. For R/a Å 10, 100, and 1000, the maximum difference between the two methods is 9.1%, 0.99%, and 0.1%, respectively. For a given R/a, the maximum fractional error is thus roughly a/R. Assuming a 1% accuracy requirement, the Derjaguin approx-
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imation provides an accurate estimate of the depletion interaction in systems in which R/a § 100. It should be emphasized that although the fractional error is greatest at h Å d, the absolute error (difference between the two approaches) is actually largest at contact. At h Å 0, the dimensionless force predicted by Eq. [19] is 0 (2 / a/R), while the Derjaguin approximation gives 02. Thus the maximum absolute difference between the two equations will be a/R. (The fractional error at contact will be a/2R, assuming a/R ! 2.) Effect of polydispersity on interaction. Shown in Figures 5 and 6 are plots of the dimensionless depletion force and energy between two spherical particles at varying degrees of polydispersity. As with the parallel plate geometry, polydispersity increases the range over which the depletion force acts. In addition, no significant deviation from the monodisperse equation is predicted for values of CV less than 15– 20%. Finally, it is interesting to note that at any h, the depletion force and energy in a normally distributed polydisperse system are always greater than or equal to the equivalent force and energy in a monodisperse system with the same average macromolecule diameter and bulk number density. This behavior differs from the parallel plate case (compare Figs. 5 and 2) but can be explained as follows. Consider first a monodisperse system in which all macromolecules have diameter d. As two particles approach, the depletion force is zero until the gap width becomes equal to d. At this point, a depletion zone is formed between the particles. As the gap width decreases, the particles are pushed
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FIG. 5. The dimensionless depletion force between two identical spherical particles versus dimensionless gap width.
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FIG. 6. The dimensionless depletion energy between two identical spherical particles versus dimensionless gap width.
further away from the line of centers, the depletion zone enlarges, and the attractive depletion force increases. The size of the depletion zone, and hence the magnitude of the depletion force, reaches a maximum at contact (h Å 0). Now consider a polydisperse system. If the diameters of the macromolecules are normally distributed about the mean, then exactly half of the macromolecules will have a diameter larger than the mean. The depletion zone thus forms at a larger gap width. Unlike the monodisperse case, smaller particles will still remain in this depletion zone and the force will not increase as rapidly. Nonetheless, the total concentration in the zone is less than the bulk and an attractive force develops. As the gap width decreases, the force increases until at h Å dV , the difference in the force between the monodisperse and polydisperse cases is a maximum. As h decreases further, this difference decreases and at contact the two solutions for force converge. Because of the longer range of the interaction in the polydisperse case, the energy at contact is greater for the polydisperse case. It should be emphasized that these results apply only to a normally distributed macromolecule diameter in which the numbers of macromolecules with diameters above and below the mean are equal. With another type of distribution function, the force profile could be quite different. For example, in a system which contains a higher fraction of smaller particles, the depletion force would be less than the monodisperse system at small separations. Calculating the force profile thus requires knowing the appropriate size distribution, f ( d).
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It is well known that a nonadsorbing polymer can induce flocculation in an otherwise stable colloidal dispersion. Using the force and energy equations presented above, the effect of polydispersity in the macromolecule size on stability can be examined. An example system consisting of 1 mm radius hard spherical particles in a solution of hard spherical macromolecules with a 2.5-nm average radius will be considered. The bulk concentration of macromolecules is 1.53 1 10 23 macromolecules/m 3 , corresponding to a volume fraction of 0.01 (1%) for the monodisperse solution. Shown in Fig. 7 are plots of the predicted depletion energy (in number of kT’s) versus particle gap width. As the degree of polydispersity increases, both the magnitude and range of the depletion energy increases. A summary of these trends is given in Table 2. An increase in the coefficient of variation from 0 to 100% increases the attractive energy at contact from 6.0 to 8.9 kT. In addition, the separation distance at which the energy equals 01 kT increases from 3.0 to 5.8 nm. Considering the magnitude of the change in polydispersity, these changes are relatively small and would not have a significant impact on system stability. A possible exception would be particles that had been stabilized sterically through the adsorption of a nonbridging polymer. If the range of this steric repulsion was less than that of the depletion attraction (e.g., 3–4 nm), then the increased polydispersity could pro-
FIG. 7. The depletion energy (in kT ) between two 1 mm radius hard spherical particles in a solution of hard spherical macromolecules with an average radius of 2.5 nm. The bulk concentration of macromolecules, r` , is 1.53 1 10 23 macromolecules/m 3 , which corresponds to 1% volume for the monodisperse solution. The curves were calculated using Eq. [25].
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produce structuring of the macromolecules and oscillations in the resulting depletion force between two macroscopic surfaces (11, 16–20). The accuracy of the results presented above depends upon the validity of this assumption of ideal behavior. The effects of macromolecule interactions can be determined with the virial expansion of the single-particle distribution function
TABLE 2 Summary of the Results of Figure 7 CV (%)
E (0)
h at E(h) Å 01 kT
0 25 50 75 100 150
06.0 06.2 06.7 07.6 08.9 012.2
3.0 3.2 3.8 4.8 5.8 8.6
Note. Shown at each value of CV is the value of the depletion energy at contact (h Å 0) plus the separation distance at which the depletion energy becomes equal to 01 kT.
mote secondary flocculation in a shallow potential energy minimum. The calculations above assumed a constant macromolecule number density. For macromolecules with a normally distributed diameter, increasing the polydispersity also increases the volume fraction. Integrating the Gaussian size distribution function yields the expression for the total volume of macromolecules per unit volume of solution £particles Å
4 r`paV 3 (1 / 3CV 2 ), 3
[28]
indicating that the volume fraction will be proportional to the factor (1 / 3CV 2 ). For the CV values listed in Fig. 7, the volume concentrations range from 1% (for CV Å 0) to 7.75% (for CV Å 150%). Instead of fixing the number density, an alternate approach is to vary the polydispersity while maintaining a constant macromolecule volume fraction. The results of these calculation are shown in Fig. 8. For each curve, the bulk number density has been adjusted using Eq. [28] to maintain a volume concentration of 1%. The resulting curves are markedly different from those in Fig. 7. For example, the magnitude of the interaction at contact is now greatest for the monodisperse system in which the number density is greatest. In addition, the range of the interaction, defined as the separation distance at which the interaction energy equals 01 kT, is actually smallest for the most polydisperse system, opposite the trend in Fig. 7. Because of these differences, formulating a general statement about the effect of polydispersity on the depletion interaction is difficult; the result depends on how each particular system is defined (e.g., constant number density, constant volume fraction).
F
r(xW ) Å r` exp 0
G
E(xW ) [1 / b2 (xW ) r` / O( r 2` )], kT
[29]
which reduces to the Boltzmann equation as r` r 0 (21). In hard-sphere systems, the value of the second virial coefficient next to a single hard wall, b2 (x Å 0), will be four times the macromolecule volume. Although this value can be altered by the presence of another wall (21), a first-order approximation of the magnitude of the initial deviation from ideal behavior is four times the macromolecule volume fraction. Higher order concentration effects should therefore be negligible for volume concentrations up to about 1%. Note that if the macromolecules experience other long-range interactions (e.g., electrostatic repulsion), the value of b2 can be greatly increased. The upper limit for the volume fraction would then be much lower.
Interactions between Macromolecules In using the Boltzmann equation to calculate the density distribution of macromolecules around the particles (Eq. [2]), interactions between the macromolecules themselves have been ignored. Such interactions have been shown to
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FIG. 8. The depletion energy (in kT) between two 1 mm radius hard spherical particles in a solution of hard spherical macromolecules with an average radius of 2.5 nm. The concentration of macromolecules is 1% volume for all cases.
coida
AP: Colloid
EFFECT OF POLYDISPERSITY ON DEPLETION INTERACTION
SUMMARY
A set of general equations has been developed to calculate the depletion force and energy in a solution of normally distributed polydisperse macromolecules. It is assumed that the density of macromolecules is low enough that higher order effects arising from interactions between the macromolecules are negligible. As the degree of polydispersity approaches zero, the equations match the predictions made with equations developed for monodisperse systems. The equations are thus general to any system and require knowing only the appropriate size distribution function. For two spherical particles in a solution of nonadsorbing macromolecules at fixed number density, polydispersity is predicted to increase the range and magnitude of the depletion energy. The effect is relatively small, however, even for systems with a coefficient of variation as large as 25%. Calculations on a typical system show that polydispersity can lead to increased flocculation through the formation of secondary potential energy minima. The polydispersity would have to be substantial, however, to produce any significant changes in the potential energy profile between the two particles. By comparison, calculations at constant volume fraction predict the depletion energy to actually decrease with increasing polydispersity. Finally, although the equations were derived assuming hard-sphere interactions, they could also be applied to systems in which other long-range interactions are important. For example, the effect of charged macromolecules could be simulated by replacing the hard-sphere diameter with an effective diameter which accounted for the size of the charge double layer. The size distribution function, f ( d), could then
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513
account for variations in both the size and charge of the macromolecules. REFERENCES 1. Seebergh, J. E., and Berg, J. C., Langmuir 10, 454 (1994). 2. Napper, D. H., ‘‘Polymeric Stabilization of Colloidal Dispersions,’’ pp. 332–413. Academic Press, London, 1983. 3. Asakura, S., and Oosawa, F., J. Chem. Phys. 22, 1255 (1954). 4. Asakura, S., and Oosawa, F., J. Polym. Sci. 33, 183 (1958). 5. Vrij, A., Pure Appl. Chem. 48, 471 (1976). 6. Sieglaff, C. L., J. Polym. Sci. 41, 319 (1959). 7. Feigin, R. I., and Napper, D. H., J. Colloid Interface Sci. 75, 525 (1980). 8. Joanny, J. F., Leibler, J. F., and de Gennes, P. G., J. Polym. Sci. Polym. Phys. Ed. 17, 1073 (1979). 9. Walz, J. Y., and Sharma, A., J. Colloid Interface Sci. 168, 485 (1994). 10. Snowden, M. J., Clegg, S. M., and Williams, P. A., J. Chem. Soc. Faraday Trans. 87, 2201 (1991). 11. Nikolov, A. D., and Wasan, D. T., Langmuir 8, 2985 (1992). 12. Schmitt, V., Lequeux, F., and Marques, C. M., J. Phys. II France 3, 891 (1993). 13. Abramowitz, M., and Stegun, I. A., Eds., ‘‘Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,’’ pp. 297– 299. Wiley, New York, 1972. 14. Gast, A. P., Hall, C. K., and Russel, W. B., J. Colloid Interface Sci. 96, 251 (1983). 15. Israelachvili, J. N., ‘‘Intermolecular and Surface Forces,’’ 2nd ed., p. 163. Academic Press, London, 1992. 16. Henderson, D., J. Colloid Interface Sci. 121, 486 (1988). 17. Richetti, P., and Ke´kicheff, P., Phys. Rev. Lett. 68, 1951 (1992). 18. Wasan, D. T., Nikolov, A. D., Kralchevski, P., and Ivanov, I., Colloids Surf. 67, 139 (1992). 19. Chu, X. L., Nikolov, A. D., and Wasan, D. T., Langmuir 10, 4403 (1994). 20. Pollard, M. L., and Radke, C. J., J. Chem. Phys. 101(8), 6979 (1994). 21. Glandt, E. D., J. Colloid Interface Sci. 77, 512 (1980).
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AP: Colloid