J. Phys. Chem. B 2001, 105, 2285-2290
2285
Latex Particle Thermophoresis in Polar Solvents M. E. Schimpf† and S. N. Semenov*,‡ Department of Chemistry, Boise State UniVersity, Boise, Idaho 83725, and Institute of Biochemical Physics RAS, 117977 Moscow, Kosygin Street 4, Russia ReceiVed: February 9, 2000; In Final Form: December 4, 2000
A theoretical model is presented of the thermophoresis of latex particles in polar solvents. Expressions for the thermophoretic mobility (TM) of the particles, which consist of polymer solutions emulsified in a poor solvent with the addition of salt, are obtained. The model is based on the thermophoresis of polymer chains partitioned between the emulsion droplet and the surrounding solvent. The partitioning is assumed to originate with the “salting in” of monomer units (mers) in the surfactant film that surrounds and stabilizes the latex particle. Correlation between mers then leads to a partitioning of mers between the good solvent inside the latex particle and the surrounding poor solvent. Assuming a mer concentration that decays exponentially into the surrounding solvent, the expression for the particle TM is obtained. It is found that, with the increase of the square of the derived dimensionless parameter, the latex particle TM increases from zero to a hypothetical value of the polymer’s TM in the poor solvent. The dimensionless parameter contains the product of mer concentration at the particle surface, the mer radius, and the square of the mean thickness of the mer distribution in the external solvent. Based on experimental data, the dimensionless parameter has a value greater than or equal to unity for polystyrene in the polar solvents water and acetonitrile. The TM of polystyrene in these systems is comparable with that of polystyrene in nonpolar solvents.
Introduction The experimental study of macromolecule and particle thermophoresis was renewed with the development of thermal field-flow fractionation (ThFFF), an analytical-scale technique that relies on thermophoresis to separate and characterize polymers and colloids. Over the past decade in particular, ThFFF has proven to be a superb technique for empirical studies of thermophoresis through measurements of thermal diffusion coefficients, or thermophoretic mobilities (TMs), of dilute polymer solutions and particle suspensions.1 To date, studies of particle thermophoresis have primarily been empirical in nature, rather than theoretical. Using ThFFF, particle thermophoresis has been studied in both aqueous and nonaqueous solvents, and with and without the addition of salt.2-7 In addition to silica, a variety of latex particles have been studied, including polystyrene (PS), polybutadiene (PB), and others. A variety of liquids have also been used in these studies, including water, methanol, acetonitrile, tetrahydrofuran (THF), and cyclohexane. In all of these studies, thermophoresis was shown to depend on both particle size and salt concentration. In general, both thermophoresis and its size dependence increase with salt concentration to a plateau value. The size dependence may be positive or negative, depending on the liquid, and the relative magnitude of thermophoresis in different solvents can change with salt concentration. For example, the thermophoresis of PB latex particles is very small in pure water, and without salt addition, thermophoresis in different solvents follows the order methanol > acetonitrile > water. With the addition of salt, thermophoresis increases dramatically in all * To whom correspondence should be addressed. E-mail:
[email protected]. FAX: 7-095-137-4101. † Boise State University. ‡ Institute of Biochemical Physics RAS.
liquids and the solvent order is reversed; that is, thermophoresis is highest in water and lowest in methanol. In acetonitrile, the thermophoresis of PS latex particles increases several times by the addition of small amounts of salt but depends very weakly on the actual concentration of the salt. The effect of salt addition is especially apparent with tetrabutylammonium perchlorate (TBAP) in all solvents that have been examined. In organic solvents containing 0.1 mM TBAP, the thermophoresis of PS latex particles follows the order cyclohexane > water > methanol > acetonitrile > THF, whereas the thermophoresis of silica, which is much more polar, follows the reverse order. In contrast to suspended particles, the thermophoresis of linear polystyrene dissolved in tetrahydrofuran is independent of salt concentration, even though in principle, the chemical composition of the bulk material is the same in dissolved polymers and suspended particles. The properties of acrylonitrile-butadienestyrene (ABS) plastic exhibit similar behavior.5 In ABS, particulate and polymeric components of the same material were shown to have similar TMs in the absence of salt. When TBAP was added, thermodiffusion of the polymeric component was not affected, whereas that of the particulate component increased significantly. Another notable difference in the thermophoretic behavior of latex particles versus dissolved polymers is that the TMs of latex particles have a size dependence, whereas those of dissolved polymers are independent of size and molecular weight. One of the authors of this work proposed a mechanism of thermophoresis based on the osmotic pressure gradient produced by the temperature dependence in the concentration of accumulation charges on the particle surface.8 The model explained the thermophoresis of metal particles, which is higher than expected with the common assumption that they cannot maintain a large temperature gradient across their surface due
10.1021/jp0005221 CCC: $20.00 © 2001 American Chemical Society Published on Web 03/02/2001
2286 J. Phys. Chem. B, Vol. 105, No. 12, 2001 to a high thermal conductivity. The model is also consistent with the thermophoretic behavior of silica particles, where the effect of salt addition appears to be associated with ion adsorption and the establishment of a double layer at the particle surface. However, the model does not account for certain thermophoretic properties of latex particles. For example, in a recent study of PS latex particles,6 the addition of salt was shown to have less of an effect on thermophoresis in water compared to acetonitrile. Furthermore, carboxylated PS latex particles have the same thermophoretic properties as unmodified particles in acetonitrile when TBAP is the only additive. In summary, one can say that the thermophoretic behavior of dissolved polymers, solid particles, and latex emulsions cannot be described by a common physicochemical mechanism. Even though the addition of salt increases the TM of latex particles by several (up to ten) times, the increase cannot be related to electrostatic effects, as is the case with solid particles such as silica. These facts prompt us to look for a mechanism of thermophoresis in particles that is not related to charging of the particle surface as a result of additives present in the solvent. In this paper we develop the theory of latex particle thermophoresis with a particular focus on the role of a salt addition, as this appears to be required for significant thermophoresis to occur. As we mentioned above, a latex particle consists of droplets of polymer dissolved in a good solvent and emulsified in a poor solvent using surfactants for stabilization. The surfactant is typically a soap-based detergent9 that decreased the surface tension at the interface between the droplet and the external solvent. These emulsions are unstable in an external solvent that is also a good solvent for the polymer contained within because the polymer tends to escape to the solvent. For this reason, we cannot yet directly compare the thermophoretic behavior of latex emulsions with that of dissolved polymers. However, some indirect comparisons will be made once the model is presented. In developing a model for emulsions, we must consider the film of surfactant that surrounds the particle and serves as a barrier to monomers during the emulsion polymerization process. When certain salts, such as NaCl, are added to the emulsion, this barrier breaks down, resulting in the escape of monomers and eventual breakdown of the emulsion.9 This effect is explained by an increase in the surface tension within the barrier film as a result of ion dissolution therein, which leads to a decrease in the electrostatic repulsion between the charged groups of the detergent. Salt also causes an increase in mer solubility within this film, which is related to a decrease in the polarity of the film. The increased hydrocarbon solubility in bulk polar solvents is referred to as salting in. The tetraalkylammonium compounds are known to be particularly effective agents for the salting-in process. For example, the adsorption of tetraalkylammonium ions to the micellar surface is assumed to be the most likely cause of the increase in critical micelle concentration in aqueous solutions.10 Thus, the addition of TBAP increases the mer solubility within the barrier film around the latex particle. Due to the correlation between mers in the polymer chain, mers cannot be partitioned between the latex particle interior and the detergent film only. Some mers will also be placed in the external solution, where the polymer itself is poorly soluble. Because TM increases as polymer solubility decreases,1,11 these “external” mers are expected to undergo strong thermophoresis, and in fact, will dominate the movement of latex particles in a temperature gradient. However, these external mers will also add to the hydrodynamic friction of the particle. In deriving the flow profile of solvent around the latex
Schimpf and Semenov particle, both of these forces of thermophoresis and friction will be taken into account. To include the force distribution in solving the hydrodynamic problem, we must first derive the distribution of mers around the particle. Then, the force acting on a single mer in a given temperature gradient, and its hydrodynamic friction coefficient, will be derived using the temperature distribution around the latex particle. The mathematical problem can be divided into calculation of the temperature distribution, derivation of the mer concentration distribution, and derivation of the general expressions for the solvent velocity profile around the particle. Theory Temperature Distribution around the Particle. Calculation of the temperature distribution around the particle was outlined in a previous work.8 Therefore, we present here only an outline with the final results for a spherical particle. The temperature distributions inside and outside the particle (Ti and Te, respectively), as obtained from the temperature conduction equation, take the following form:
Te ) T + ∇T‚R‚cosϑ
(
r n -1 R2 + R n + 2 r2
)
(1)
where T is the temperature at the center of the particle, r is the distance from its center, ϑ is the angle between the radius vector b r and the outer temperature gradient ∇T, n ) θi/θe (θi and θe are the thermal conductivities of the particle and external liquid, respectively). For further calculations, only the temperature gradient on the particle surface will be necessary, which is defined by
2n + 1 n+2
∇Tc ) -∇T‚sinϑ
(2)
Derivation of the Mer Concentration Distribution around the Particle. It is known that the conformation of a solvated polymer in a confined space, such as that in a latex emulsion, can be modeled by one large polymer chain, especially near the boundaries of this space, because the fraction of uncorrelated mers is small.12 In this case, the latex particle containing the polymer can be modeled using a potential well with steep walls as the boundaries, as illustrated in Figure 1. The detergent film, in turn, may be represented as a very thin (1-2 µm9) potential well at the external boundary of the larger well, in which the polymer is contained. In a theta-solvent or at low polymer concentrations, where interactions between segments are negligible, the segment concentration distribution can be written as c(r) ) φ(r)2, where φ(r) is the eigenfunction of the equation 2 l2 d (rφ) - [E + Φ(r)]rφ ) 0 6 dr2
(3)
with the eigenvalue -E.12 In eq 3, l is the persistent length (Kuhn segment length) of the polymer and Φ(r) is the potential well profile in kT units. Parameter l can be determined using dynamic light scattering13 or viscometry.14 Inside the droplet, the potential of the segments is assumed to be uniform and equal to -Φ0. This parameter may be expressed through K, the partition coefficient for the mers between the internal solvent, where the polymer is dissolved, and the external solvent, where latex particles are suspended:
Φ0 ) lnK
(4)
Latex Particle Thermophoresis in Polar Solvents
J. Phys. Chem. B, Vol. 105, No. 12, 2001 2287 B is the fluid velocity that arises from the force acting -bmT ‚∇T on the solvent around mers with thermophoretic velocity bmT in the temperature gradient given by eq 2. We assume that the angular dependence of the meridian flow velocity component has the same form as the surface temperature gradient (eq 2) and may therefore be written as uϑ ) uz(x)‚sinϑ. For the details of the calculations, see Appendix B. In the calculation of the TM of the latex particle, only the slip velocity us ) uz(∞) is necessary. Using eq 4B, that velocity can be written as
us ) u0(I0(2xβ) - 1)/I0(2xβ)
(8)
In our coordinate system, the latex particle thermophoretic velocity is equal to -us.16 The expression for the parameter β from Appendix B can be rewritten using the volume fraction of the spherical mers φs ) ns‚(4πr3m)/3;
β)
9φs λ2 2 r2
(9)
m
Figure 1. Profile of potential energy for mers in a latex particle.
Outside the droplet, the potential Φ(r) is equal to zero. In this approach, the detergent film may be modeled as an additional narrow potential well placed at r ≈ R. For the details of the calculations, see Appendix A. Neglecting parameter ∆ in comparison with the potential well depth in eq 7A, we obtain
x c0‚exp -2x6Φ0 l c(x) ) 12 hR 1+ 1 + x6Φ0 - 6Φ1 5 l l
[ (
)]
(5)
d2uz dx2
+ nse - x/λξ[u0 - uz(x)] ) 0
l 2rm
[ (
bT ) bmT
h R x6Φ0 - 6Φ1 l l
)]
(x ) (x ) β0
(7)
The parameter λ ) l/x6Φ0 is the characteristic thickness of the segment layer in the external solvent, ξ ) 6πηrm is the hydrodynamic friction coefficient for the mer, and ub0 )
1 + δ R/ l
I0 2
-1 (10)
β0
1 + δ R/ l
where bmT is the TM of the mer, and
β0 )
45φs l2 34 6Φ r2 0 m
(11)
is the β value in the situation where the eigenvalue position coincides with the bottom of the “main” potential well. Parameter δ in eq 10 is the deviation of the eigenvalue from the bottom of the well, defined as
δ)
c0 12 1+ 1+ 5
The expression for the TM of the latex particle may be written in the following convenient form:
(6)
where z ) R‚ϑ is the linear meridian coordinate on the droplet surface, η is the dynamic viscosity of the liquid, b u is its velocity. B) f nse-x/λξ[u b0 - b u(x)] is the volume force acting on unit volume of the solvent. Here, ns is the mer concentration on the droplet surface, which is related using eqs 2A and 5 to the surface segment concentration as follows:
ns )
Results and Discussion
I0 2
where x is the distance from the particle surface, and the small parameter 3/2x6Φ0 l/R is neglected. The denominator of eq 5 contains a function of the droplet radius that reflects the redistribution of mers between the droplet interior and the external solvent due to their accumulation in the detergent film. This means that the segment concentration on the droplet surface decreases with droplet size, provided the surface well depth does not exceed some critical value. In the opposite case, the segment concentration increases with droplet size. Derivation of the Solvent Velocity Profile around the Latex Particle. The flow velocity profile in the thin particle surface layer is defined by the Navier-Stokes equation
η
As eq 9 shows, when the thickness reaches about 10 mer radii, β will reach a value of about 5 when the volume fraction of mers is as small as 1%. For a polymer layer thickness comparable to the mer radius, β will be small at any reasonable value for the volume fraction of the mers.
h 17 6Φ - 6Φ1 5 x 0 l
(
)
(12)
Equation 10 allows the size dependence in bT(R/l) to be extrapolated to zero:
bT(0) ) bmT
I0(2xβ0) - 1 I0(2xβ0)
(13)
Equation 13 indicates that the TM of particles with δ R , l is defined only by the parameters of the main potential well, which include the partition coefficient for the mers. Therefore, extrapolations of the experimental dependencies of bT(R/l) for different systems should approach the same value as the particle radius approaches zero. This feature provides a method to check the consistency of the theory with experimental data because
2288 J. Phys. Chem. B, Vol. 105, No. 12, 2001
Schimpf and Semenov
TABLE 1: Thermophoretic Mobility (bT) Values of PS Latex Particles in Different Polar Solvents, Measured by ThFFF; Results of a Least-Squares Fit of bT to Predictor Variable R/l; Estimation of Parameters in Eq 9 R/l a
acetonitrile + 0.1 mM TBAP
5.59 9.43 12.22 13.97 15.71 18.30 20.95
7.25 7.5 6.9 6.6 6.5 6.3 6.1
H2O + 3 mM NaN3 bT × 108 (cm2/s‚K) 3.1 3.8 4.2 4.6 5.1 5.8 6.3
7.63-2.87‚10-2(R/l) - 2.30‚10-3(R/l)2 bmT ) 7.65 β0 ) 20.5 δ ) 0.37 a
H2O + 9 mM NaN3
H2O + 10 mM phosphate buffer
4.3 5.6 6.3 6.6 7.4 7.8
2.0 2.1 2.1 2.4 2.6 2.6 3.0
polynomial least-squares fit to experimental data 2.3 + 0.122(R/l) 2.3 + 0.385(R/l) 3.44‚10-3(R/l)2 4.51‚10-3(R/l)2
parameters obtained by least-squares fit of bT to R/l using eq 9 bmT ) 3.5 bmT ) 4 β0 ) 2.30 β0 ) 2.50 δ ) 0.11 δ ) 0.14
2.0-1.90‚10-2(R/l) + 3.15‚10-3(R/l)2 bmT )2.65 β0 ) 2.10 δ ) 0.067
(l ) 7.16 nm)
the dependence of latex TMs on size is examined in several papers.2,3,6 A crude approximation of the mer radius can be made using the hard sphere model. The possible error in this approximation will be on the order of several to 10 percent. Using data for liquid styrene,17 we calculate a mer radius rm ) 0.716 nm. According to ref 18, the persistent segment of a PS chain contains about 10 mers, thus l ) 7.16 nm. This value will be used in the calculations below. In a comprehensive study of the size dependence of the thermophoretic mobility (bT) in PS latex particles,6 seven different particle sizes were examined in acetonitrile and several aqueous solvents. The experimental data is reproduced in Table 1, along with a polynomial least-squares fit of the data to parameter R/l. The results demonstrate that the dependence of bT on particle size approaches about the same limit at R ) 0 in aqueous solutions with different salt additions. Thus, a limiting value of bT ) 2.3 × 10-8 cm3/s‚K is obtained when sodium azide is added at different concentrations. In phosphate buffer, a similar value, bT ) 2.0 × 10-8 cm3/s‚K is obtained. The discrepancy in these bT values (15%) is greater than the uncertainty in measurements of bT (5-10%), and may be due to the electrostatic contribution to the TM.8 Thus, values of bmT and β0, which determine the zero-limit behavior of the size dependence, agree within 15% among the different aqueous solutions. Similar results were obtained using the size dependence of bT in polybutadiene latex particles.2 These results demonstrate that ThFFF experiments on latex particles could provide information on the penetration of additives from the external solvent into the latex particle, as well as information on interactions between the internal solvent and polymer. For example, the presence of a common zero limit in the size dependence of the TM in experiments with different additives at different concentrations would indicate the lack of penetration of such additives into the latex particle. By contrast, a shift in the zero-limit value would indicate penetration of the additive. Of course, factors involved in the electrostatic effect on thermophoresis8 should be taken into account in such interpretations. A fit of the data in ref 6 to eq 10 was also made, using adjustable parameters δ, β0, and bmT . The results, which are summarized in Table 1, are also self-consistent. We note that it is not possible to carry out a direct nonlinear least-squares fit calculation between the parameters in eq 10, i.e., the parameters δ, β0, and bmT in Table 1 were adjusted
somewhat arbitrarily by aesthetic criteria, and the approximate expression I0(x) ≈ 1 + (x/2)2 is used in the interpretation of the data for the aqueous solutions. Nevertheless, the agreement between theoretical and experimental dependencies is satisfactory (less than 10%). Values of δ, β0, and bmT used in the theoretical calculations are the same as those in Table 1. Because the TMs of dissolved polymers are equal to those of the individual mers, the theory allows for the determination of polymer TM values in poor solvents, or even nonsolvents, using data on the thermophoresis of emulsion droplets containing the polymer. The TMs of the mers (bmT ) listed in Table 1 merit further discussion. The values indicate that the TM of dissolved PS in either acetonitrile or water should be in the range of 2-8 × 10-8 cm3/s‚K, which is comparable to that of PS in nonpolar organic solvents. In the literature, we found out only one reference (ref 11) to the TM of dissolved nonionic polymers in polar solvents. In this paper, it was noted that “...Limited data suggest that ThFFF retention in aqueous mobile phases is greater when the macromolecule has limited or poor solubility in water...”. The TMs of the polymers examined in that work, which included polystyrene sulfonates, poly(ethylene oxides), and poly(ethylene glycols), were comparable to those of the same polymers in nonpolar organic solvents. By analogy, one can expect that the TM of polystyrene in water, which is a very poor solvent (the solubility of styrene in water is 10-2 wt %19), will be comparable (but not identical to) to that of polystyrene in nonpolar solvents. Of course, this extrapolation may be used for water only, which is a chemically unique medium, and cannot be applied to organic solvents, where most of the experiments on polymer thermophoresis have been carried out. Parameter β0 provides information on the properties of the latex particle (see eq 11), as it contains the segment-to-mer size ratio (l/rm), the partition coefficient for mers in the external versus internal solvent (inherent in Φ0; see eq 4), and the mer volume fraction (φs). The segment-to-mer size ratio can be obtained with independent experiments. The partition coefficient can be calculated using solubility parameters,17,20 provided the internal and external solvents are known. Assuming the internal solvent is chemically similar to the dissolved polymer (for example, styrene is often used in the formation of PS latex particles21), the partition coefficient will equal the limiting molar concentration of mer in the external solvent. In that case, Φ0 ≈ 11-12 for PS in water, based on the solubility of styrene in
Latex Particle Thermophoresis in Polar Solvents water.19 For acetonitrile, where the solubility of styrene should be on the order of 10 wt % (K ) 0.04), Φ0 ≈ 3. Therefore, the ratio of parameter β0 for acetonitrile versus water, which is governed by the inverse ratio of Φ0 for the two solvents when φs is assumed to be constant, should be about 3.6-4. From the data in Table 1, this ratio is actually 8.2-9.8. The discrepancy could be due to partial penetration of acetonitrile inside the droplet, which results in a mixed internal solvent and subsequent decrease in the solubility of the styrene mers, i.e., a decrease in Φ0. In water, of course, solvent penetration is impossible due to the total immiscibility of the internal and external solvents. Using the data for water, the volume fraction of mers may be evaluated from the value of β0. Substituting the values discussed above into eq 11, we obtain φs ≈ 0.8. Usually, the monomer concentration is established in the emulsion droplets before polymerization is initiated, and it is practically constant for a given type of latex,21 provided the latex suspension is stable. In a typical emulsion polymerization, the monomer volume fraction is 0.5-0.6.21 This high mer concentration (φs > 0.5) may be another reason for some disagreement between theory and experiment, since this situation requires that excluded volume effects are taken into account in the calculation of the segment distribution.12 Thus, values of φs calculated from β0 in water are too high for the estimations based on an approximation of dilute polymer solution. Nevertheless, TM values obtained by ThFFF can in principle be used to estimate either mer or internal solvent concentrations in latex particles. Analysis of parameter δ, whose values are contained in Table 1, allows for an evaluation of mer solubility in the detergent film around the latex particle, provided its solubility in the internal solvent is known. From Table 1, one can see that the parameter δ is always small; therefore, the deviation of eigenvalues from the bottom of the main potential well is also small. In such a situation, we can use the following equation to establish the required solubility of mers in the detergent film: Φ1 ≈ xΦ0/6 l/h. The characteristic thickness of the detergent film is about 1 nm.8 This condition is important for obtaining measurable thermophoresis in latex particles. Using the values discussed above, Φ1 is ∼3 for acetonitrile and ∼9 for water. At lower values of Φ1, the concentration of mers on the droplet surface decreases due to their redistribution inside the droplet. At higher values of Φ1, the thickness of the layer in the external solvent, where a part of the polymer chain segment is placed, decreases. In both cases, the latex particle TM should decrease. The question may be asked as to why this condition for optimal mer solubility in the surface detergent film is fulfilled in published experiments. This question cannot be answered without an additional systematic set of experiments, but it is known that many salts and surfactants fail to yield measurable thermophoresis, and these results go unpublished. It is also likely that the optimal salt concentration in the surface detergent film coincides with an adsorption limit. Indeed, parameter δ in Table 1 changes by only 30% when the NaN3 concentration is increased from 3 to 9 mM. In ref 2, the TM of PS latex particles in acetonitrile increases several times when 0.25 mM TBAP is added to pure acetonitrile, but any further increase in TBAP concentration up to 1 mM changes the TM by less than 20%. These facts certainly can be interpreted as adsorption saturation. Further evaluation of this aspect requires more experiments with smaller concentrations of salt, in which the concentration of the adsorbed salt depends directly on the bulk concentration. In principle, however, the dependence of latex particle thermophoresis on salt concentration could be used
J. Phys. Chem. B, Vol. 105, No. 12, 2001 2289 to study the effect of salt on latex suspension stability, which is related to the polymer solubility in the surface detergent film. Conclusions Theoretical studies of latex particle thermophoresis in emulsions formed in polar solvents with the addition of salt, which have been used as model systems for ThFFF, are carried out. An expression for the TM of latex particles, which represent droplets of the polymer solution in a polar solvent emulsion, is obtained by assuming that the thermophoresis of such droplets is due to that of polymer chains partitioned between the droplet and the surrounding polar solvent. It is assumed that such a partial dissolution of the polymer chains in the solvent, where the polymer itself is poorly soluble, is due to the established mechanism of salting in, that is, the increase in hydrocarbons solubility in polar solvents with the addition of salt. In the case considered here, it causes an increase in monomer solubility in the surfactant film used to stabilize the latex particle. Due to the correlation between mers in the polymer chain, the increased solubility leads to a partitioning of mers between the good solvent inside the latex particle and the poor solvent surrounding the particle. Assuming an exponentially decaying distribution of mers in the solvent around the particle, the expression for the particle TM is obtained. It is shown that the TM of the latex particle increases from zero to its hypothetical value for polymer in the polar solvent, as the square root of the derived dimensionless parameter increases. This parameter represents the product of the mer concentration at the particle surface, the mer radius, and the square of the mean thickness of the mer distribution in the external solvent. In comparing the expressions obtained with experimental data for polystyrene latex particles in water and acetonitrile, it was shown that this dimensionless parameter has a value of 1-10 or more and that the TM of polystyrene in water and acetonitrile is comparable to that in nonpolar solvents. Acknowledgment. This work was supported by the Russian Foundation for Base Research (Project # 98-03-32728) and grant 0602236 from the National Research Council. Appendix A The eigenfunction φ(r) of eq 3 is normalized by the condition
∫0∞[φ(r)]2 4πr2dr ) 34c0πR3
(1A)
where c0 is the mean concentration of segments in the emulsion droplet when all of the segments are placed inside the droplet. We can relate c0 to the mer concentration n0 as follows:
c0 ) n0
2rm l
(2A)
where rm is the mer radius. According to ref 9, the width of the detergent film is 1-2 nm, which is smaller than the segment length for many polymers. Rather than modeling the details for the surface well, we will use the “short-range force approach” accepted in quantum mechanics.15 We assume that the eigenfunction φ(r) is not changed to a great extent in the narrow well and therefore has the same value φ(R) on both sides of the surface potential well. Then, by integrating eq 3 over the narrow interval containing the surface well, we have the following
2290 J. Phys. Chem. B, Vol. 105, No. 12, 2001
Schimpf and Semenov defining the deviation of the eigenvalue from the bottom of the potential well. Appendix B The boundary conditions to eq 6 have the form that is standard for problems dealing with surface kinetic phenomena in thin layers (see, for example, refs 8 and 16)
uz ) 0, at x ) 0;
Figure 2. Plots of the left-hand parts of eq 6A (broken lines) and the exact dependencies based on the boundary condition given by eq 3A (solid lines) as functions of the parameter x ) x6|Φ0-E| R/l. The upper family of four curves corresponds to Φ0 ) 10; the lower four curves correspond to Φ0 ) 3.
duz ) 0, at x f ∞ dx
(1B)
By introducing a new variable τ ) 2xβ‚e-x/2λ, a new function υ ) (uz - u0)/u0, and the dimensionless parameter β ) 6π‚ns‚ rmλ2, eq 6 may be cast in the following dimensionless form (see ref 15):
d2υ 1 dυ -υ)0 + dτ2 τ dτ
(2B)
with the boundary conditions: boundary condition for the eigenfunction:
[
]
(3A)
where the notations φi and φe are used for the eigenfunction inside and outside the droplet, respectively, and parameters Φ1 and h are the mean effective depth and thickness of the surface well, respectively. These parameters are directly related to the real parameters of the surface well by the expression -φlh ) ∫∞R Φ(r)dr. With the above assumptions, the eigenfunctions may be written as
R r-R φe(r) ) A exp -x6E r l
(4A)
r l R sinx6(Φ0 - E) l
(5A)
R φi(r) ) A r
sinx6(Φ0 - E)
E < Φ0
where A is the normalization coefficient calculated by eq 1A; it is equal to the eigenfunction value on the droplet surface. At E g Φ0, eq 5A is transformed by using the known relationships between the trigonometric and hyperbolic functions. Substituting eqs 4A and 5A into eq 3A, we obtain the equations for the eigenvalues near the bottom of the main potential wall:
(
h l l2 6E ) 6Φ0 + 3 6Φ1 - x6Φ0 + 2 R R 2R
)
(6A)
Equation 6A yields a relative error of less than 10% when 6|E - Φ0|(R/l)2 e 2, as illustrated by the graphical evaluation in Figure 2. Using eqs 3, 1A, and 4A, one can write the expression for the segment distribution in the external solvent as
c(x) )
x l
2x6Φ0 + 3∆‚c0‚exp -2x6Φ0 + 3∆
{
2x6Φ0 + 3∆ 1 -
where ∆ ) 6Φ1(h/R) -
x6Φ0
() }
dυ ) 0, at τ f 0 dτ
υ ) -1, at τ ) β; τ
6Φ1h dφe dφi |r ) R ) - 2 φ(R) dr dr l
12 R 2 l ∆ +3 5 l R
(7A)
l/R + l2/R2 is the parameter
(3B)
The solution of eq 2B is the modified Bessel function (the hyperbolic Bessel function) of zero order I0(τ).15 Using the first boundary condition from eq 3B, we obtain
υ(τ) ) -
I0(τ) I0(2xβ)
(4B)
References and Notes (1) Schimpf, M. E.; Giddings, J. C. J. Polym. Sci. B 1989, 27, 13171322. (2) Shiundu, P. M.; Liu, G.; Giddings, J. C. Anal. Chem. 1995, 67, 2705-2713. (3) Shiundu, P. M.; Giddings, J. C. J. Chromatogr. A 1995, 715, 117126. (4) Ratanathanawongs, S. K.; Shiundu, P. M.; Giddings, J. C. Colloids Surf. A 1995, 105, 243-250. (5) Shiundu, P. M.; Remsen, E. E.; Giddings, J. C. J. Appl. Polym. Sci. 1996, 60, 1685-1707. (6) Jeon, S. J.; Schimpf, M. E.; Nyborg, A. Anal. Chem. 1997, 69, 3442-3450. (7) Liu, G.; Giddings, J. C. Anal. Chem. 1991, 63, 296-299. (8) Giddings, J. C.; Shiundu, P. M.; Semenov, S. N. J. Colloid Interface Sci. 1995, 176, 454-458. (9) Capek, J. AdV. Coll. Interface Sci 1999, 82, 253-273. (10) Ray, A.; Nemethy, G. J. Am. Chem. Soc. 1971, 93 (25), 67876793. (11) Kirkland, J. J.; Yau, W. W. J. Chromatogr. 1995, 353, 95. (12) Grosberg, A.; Yu., A.; Khokhlov, A. R. Statistical Physics of Macromolecules; Academic International Press: Woodbury, NY, 1994, Chapters 1, 3. (13) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976. (14) Tirrell, D. A.; Pearce, E. M.; Sawamoto, M.; Amis, E. J. A HalfCentury of Journal of Polymer Science; Wiley: New York, 1997; pp 363369. (15) Morse, P. M.; Feshbach, H. Methods of Theoretical Physics, Part II; McGraw-Hill: New York, 1953, Chapters 10, 12. (16) Prieve, D. C.; Anderson, J. L.; Ebel, J. P.; Lowell, M. E. J. Fluid Mech. 1984, 148, 247-269. (17) Riddik, J. A.; Bunger, W. B.; Sakano, T. K., Eds. Organic SolVents. Physical Properties and Methods of Purification, 4th ed.; Wiley: New York, 1986. (18) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: New York, 1988. (19) Stull, D. R. In Styrene; Boundy, R. H.; Boyer, R. F., Eds.; Reinhold Publishing: New York, 1952; Chapter 3. (20) Schimpf, M. E.; Semenov, S. N. J. Phys. Chem. B 2000, 104 (42), 9935-9942. (21) Rudin, A. The Elements of Polymer Science and Engineering; Academic Press: New York, 1982; Chapter 7.
