J . Phys. Chem. 1990, 94, 7239-7243 curvature effects can be analyzed by using an extended YoungLaplace equation, where the surface excess pressure is related to the surface tension u and radius r by
& = - 2a - - 2ko (1 6 ) r r2 While the results presented here do not in themselves allow application of a model incorporating effects of the surfactant interface, they do demonstrate that such effects in all probability are important to understand the magnitude and dependencies of the exchange rate constant. Further studies over a wide range of micellar radii and with varying surface properties, such as the nature of the surfactant and the ionic strength, are needed to assess the validity of these concepts. The similarity between the exchange rate constants k,, in the DTAC and other investigated systems is evident: all are on the order 106-1 O7 (M-s)-I, and the fraction of collisions leading to coalescence k e x / k d iisf fof the order 104-10-2, a ratio that seems to be common to many reversed micellar systems far from critical points and phase boundaries.6 This ratio is expected to approach unity at or near the critical point where the concepts of discrete droplets and well-defined interfaces break down. Conclusions The intermicelle solubilizate exchange rates in the DTAC/ hexanol/heptane system depend on the location of the probe molecules within the micellar aggregates. For pool-solubilized solutes the exchange is governed by micelle-micelle coalescence processes, which are 2-3 orders of magnitude slower than would be inferred from simple diffusional collision rates. This supports the findings of earlier studies which indicated that the surfactant layer opening was the rate-determining step in the coalescence process. On the other hand, for solutes which are associated with (39) Overbeek, J. Th. G.; Verhoeckx, G. J.; de Bruyn, P. L.; Lekkerkerker, H. N. W. J. Colloid Interface Sci. 1987, 119, 422.
7239
the surfactant interface the exchange rates appear to be diffusion-limited, indicating that solubilizate exchange m u r s through the surfactant layers on micelle-micelle contact, and actual merging of the micelles is not required. This observation reinforces the notion that care must be exercised in the selection of indicator reactions in Drobing intermicellar coalescence behavior. There is a'need account explicitly for the statistical distribution of solubilizates over the micelle population when interpreting dynamic exchange experimental results. The population balance model is a convenient, internally consistent and unambiguous way to account for these processes. In this work, it was assumed that the distribution was random, Le., that there were no solubilizate-solubilizate interactions within the micelles that could influence the equilibrium distribution over the micelle population, and this was confirmed by experiment. However, other cases may arise, and it may be important to incorporate attractive or repulsive solubilizate interactions in the analysis of experimental data. This is readily a c c ~ m p l i s h e d . ~ ~ . The interfacial curvature is an important determinant of micellar coalescence behavior, which is probably a reflection of changes in film rigidity with changing micellar radius. Although the surfactant interfacial composition does not appear to have a large effect on the overall exchange rates at compositions far from the phase boundaries, the enthalpic and entropic components of the activation energy for the coalescence process differ between different ionic surfactant systems. However, the data base currently available does not permit general conclusions to be drawn at this time, and more detailed, systematic studies are required to put these different effects in perspective. Acknowledgment. We acknowledge the Biotechnology Process Engineering Center funded under the NSF-ERC Cooperative Agreement CDR 85 0003, the Dow Chemical Co., and an NSF Presidential Young Investigator Award to T.A.H., Contract No. 845 1593, for financial support of this work. We also thank Paul D. I . Fletcher for his invaluable critique of this work, and suggestions for improving and clarifying certain sections of this paper.
Temperature Dependence of the Growth of Diheptanoylphosphatidylcholine Micelles Studied by Small-Angle Neutron Scattering Tsang-Lang Lin,* Ming-Yu Tseng, Sow-Hsin Chen,+and Mary F. Roberts* Department of Nuclear Engineering, National Tsing- Hua University, Hsin- Chu. Taiwan 30043, ROC (Received: December 14. 1989; I n Final Form: April 18, 1990)
The growth of diheptanoylphosphatidylcholine(diheptanoyl-PC) micelles with increasing concentration was studied by small-angle neutron scattering techniques in the temperature range of 25-45 "C. Diheptanoyl-PC forms polydisperse spherocylindrical micelles in aqueous solutions. The mean micellar aggregation number increases with increasing concentration. At a constant concentration, the mean micellar aggregation number is found to decrease with increasing temperature. A thermodynamic model called the ladder model is employed in the analyses of the measured neutron scattering data. Both the thermodynamic parameters and the structural parameters that characterize the structure and size distribution of the micellar system are determined at each temperature. The growth of the rodlike micelles with increasing concentration is driven by the total free energy difference of the monomers in the two end caps of one micelle and the same number of monomers in the cylinder section of the micelle. The magnitude of this free energy difference in units of kT is found to decrease with increasing temperature, and it is responsible for the decrease of the mean aggregation number with increasing temperature.
1. Introduction In previous studies, we have employed small-angle neutron scattering (SANS) techniques to study the structure and size distribution of short chain lecithin micelles at constant temper-
'
Present address: Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA 021 39. 'Present address: Department of Chemistry, Boston College, Chestnut Hill, MA 02167.
0022-3654/90/2094-7239$02.50/0
a t ~ r e . ' - ~A "step-and-ladder" thermodynamic model, or simply the ladder ~ n o d e l , ~has . ~ -been ~ used successfully in the analysis (1) Lin, T.-L.; Chen, S.-H.; Gabriel, N. E.; Roberts, M. F. J . Am. Chem. SOC.1986, 108, 3499. (2) Lin, T.-L.; Chen, S.-H.; Gabriel, N. E.; Roberts, M. F. J . Phys. Chem. 1987, 91, 406. (3) Lin, T.-L.; Chen, S.-H.;Roberts, M. F. J . Am. Chem. SOC.1987, 109, 2321.
0 1990 American Chemical Society
7240
The Journal of Physical Chemistry, Vol. 94, No. 18. 1990
of the formation and growth of these lecithin micellar Both thermodynamic and structural parameters that characterize the micellar system can be determined from such analysis. Short-chain lecithins are useful model phospholipids for studying the activity of water-soluble pho~pholipase.'-~ To understand the phospholipase kinetics, it is important to know the micelle structure at different concentrations and at different temperatures. Diheptanoylphosphatidylcholine (diheptanoyl-PC'), which is zwitterionic, forms polydisperse spherocylindrical micelles in aqueous solutions. It has been shown by SANS studies that diheptanoyl-PC micelles grow in the longitudinal direction with increasing concentration.2 The growth is due to the free energy of insertion of a monomer in the straight section of the spherocylindrical micelle being lower than that in the end caps of the micelle. The key factor that determines how long the micelles will grow with increasing concentration is the total difference in free energy associated with the monomers in the two end caps compared with the same number of monomers in the cylinder section of the micelle. Micellar growth, measured in terms of the increase of the weight-averaged aggregation number N,is roughly proportional to the square root of the micellar concentration (X where X is the total lecithin concentration (in mole fraction) and XIis the concentration (in mole fraction) of lecithins in free monomer ~ t a t e . This ~ , ~ prediction of the thermodynamic model has been verified by SANS studies of the growth of short chain lecithin micellar systems and also by light scattering and SANS studies of the growth of sodium dodecyl sulfate micelle^.^.^ For ionic micelles, the mean micellar aggregation number is usually found to decrease with increasing temperature,I0 whereas an increase is usually found for nonionic micelles.",12 With the help of the thermodynamic model, it is possible to gain insights into the temperature effects on the growth of micellar systems. In this paper, we present the results of S A N S studies of the temperature dependence of the growth of diheptanoyl-PC micelles with increasing lecithin concentration in the temperature range of 25-45 'C. We have obtained the temperature dependence of the thermodynamic parameters and structural parameters of diheptanoyl-PC micelles. These results are helpful in understanding the nature of the micelle formation and growth.
2. Experimental Procedures 2. I . Sample Preparation. Diheptanoyl-PC was obtained from Avanti Biochemicals (Birmingham, AL) in powder form and used without further purification. Samples for SANS were prepared The lecithin concentrations were in D 2 0 as described previo~sly.~-~ 5.6, 1 1.25, 16.9, and 22.5 mM, respectively. The concentration range spans from above the critical micelle concentration (cmc) to a sufficient concentration to observe the growth of the rodlike micelles. At these concentrations, the interparticle interaction can be neglected.',* 2.2. Small-Angle Neutron Scattering Measurements. Neutron scattering measurements were done at Brookhaven National Laboratory using the Biology Small-Angle Neutron Scattering Spectrometer located at H9B beam line of the High Flux Beam Reactor.I3 The monoenergetic neutrons selected by Bragg reflection from a multilayered monochromator had a wavelength of X = 4.5 A with a spread AX/X of about 10%. Samples were placed at a distance of 1.40 m from the 20 cm by 20 cm two(4) Tausk, R. J. M.; Overbeek, J. Th. G. J . Colloid Interface Sci. 1976,
2 -319_ . _.
( 5 ) Missel, P. J.; Mazer, N . A.: Benedek, G .B.; Young, C. Y. J . Phys. Chem. 1980,84, 1044. (6) Sheu, E. Y.; Chen, S.-H. J . Phys. Chem. 1988, 92, 4466. (7) DeBose, C . D.; Burns. R. A,, Jr.; Donovan, J. M.; Roberts, M. F. Biochemistry 1985, 24, 1298. (8) Burns, R. A., Jr.; Roberts, M. F. Biophys. J . 1982, 37, 105. (9) El-Sayed. M. Y.; DeBose, C. D.; Coury. L. A.; Roberts, M. F. Biochim. Biophys. Acta 1985, 837. 3 2 5 . (IO) Malliaris, A.; LeMoigne, J.: Sturm, J.; Zana, R. J . fhys. Chem. 1985. 89, 2709. ( I I ) Kato, T.; Seimiya, T. J. f h y s . Chem. 1986, 90, 3159. (12) Herrington. T. M.; Sahi, S. S. J. Colloid Interface Sri. 1988, 121. 107. (13) Schneider, D. K.; Schoenborn, B. P. In Neutrons in Biology; Schoenborn. B. P., Ed.: Plenum Press: New York, 1984.
Lin et ai. dimensional position-sensitive detector. The available Q range covers 0.015-0.125 A-l. Q = [(4r/X) sin (0/2)] is the magnitude of the scattering vector, and 0 is the scattering angle. The neutron scattering spectra were measured with each sample at five different temperatures: 25, 30, 35, 40, and 45 "C, respectively. The measured neutron scattering intensities were corrected and normalized to obtain the normalized scattering intensity per unit sample volume, I ( @ , in units of cm-l as reported previously.14 These I(Q)'s were then analyzed to give the thermodynamic and structural parameters of the micelles at different temperatures.
3. SANS Analysis Here we will describe briefly the thermodynamic model of the micellar growth and the methods of analyzing the measured neutron scattering intensity distributions.2 On the basis of the ladder model, the formation and growth of micelles are characterized by three parameters: ( A - No6)/kT,6 / k T , and No. No is the aggregation number of the minimum size micelle or, equivalently, the number of monomers in the two end caps of te spherocylindrical micelle. 6 is the free energy change for a monomer when it is inserted into the cylinder section of the micelle. A is the free energy change when No monomers aggregate to form the minimum size micelle. Micellar growth depends strongly on the magnitude of ( A - N o 6 ) / k T , which represents the difference in free energy (in units of k T ) of monomers in the two end caps versus the same number of monomers in the cylindrical section. These parameters together with the total lecithin concentration in the solution enable one to calculate the concentration (in mole fraction) distribution of the N-mer aggregates, X,, in equilibrium with monomers as given by2
XN -- N@Ne-(h-NoJ)/kT
,TV
2 No
(1)
The parameter is determined by using a constraint that the total concentration X in mole fraction is equal to the sum of all XN's and XI as given by2
The weight-averaged aggregation number N is given by2
N = N o + - - 1- -
P
[I+
1
NO(1 - P ) + P
]
(4)
The mean aggregation number can be approximated by2 N = No + 2(X - Xc,,)1/2e(A-Nd/2kT (5) where X,,, is the critical micelle concentration in mole fraction. The concentration distribution X, is needed in the computation of the neutron scattering intensity distribution. For a dilute sample, the neutron scattering intensity is equal to the sum of the scattering intensity of each micelle in the solution. The scattering intensity of each micelle depends on its size and shape. The structure of the diheptanoyl-PC micelles can be modeled by spherocylinders. To simplify the computation of the scattering intensity, the shape of these rodlike micelles can be approximated by cylinders with the same radius R and different lengths in the computation. Since the micelles grow only in the longitudinal direction, the cross-sectional structure is similar. The length of the rodlike micelle, L, will be approximately proportional to the aggregation number N . Thus the aggregation number per unit rod length N I L , denoted as C Y , is a constant for every micelle in the solution. The spherocylindrical micelles have a transverse radius of gyration R, which is related to the radius R by R, = R / 2 1'. (14) Chen, S.-H.; Lin, T.-L. In Methods of Experimental PhysicsNeutron Scattering in Condensed Matter Research; Skold, K., Price, D. L., Eds.; Academic Press: New York, 1987: Vol. 23B, Chapter 16.
Growth of Diheptanoyl-PC Micelles (4
n
The Journal of Physical Chemistry, Vol. 94, No. 18, 1990 7241
(C)
-3
.
d
'5
'5
h
h
cp
2
e
.
cp
v I
250
1225my
16.9 mM
W I
1
n
0
30'C
200
45'C
I2 150
100
0.1
Q(8-l)
50
t 25
20
30
35
40
45
50
T( "C) Figure 2. Temperature dependence of the weight-averaged aggregation number of diheptanoyl-PC micelles at four different concentrations: (0) 22.5 mM e
30*C
1
45'C
5.6, (A) 11.25, (D) 16.9, and (0)22.5 mM.
25 9 :
250
30 OC
200
35 9:
409:
I2 150 0
QW) Figure 1. Small-angle neutron scattering spectra together with the fitted solid curves at 30 and 45 OC, respectively, for diheptanoyl-PC concentrations of 5.6, 11.25, 16.9, and 22.5 mM in D20solution.
TABLE I: Results of Fitting the SANS Data from Dibeptanoyl-PC Micelles
25 30 35 40 45
(A-Nd)JkT 6JkT 17.35 f 0.05 -10.35 f 0.08 17.01 f 0.05 -10.37 f 0.08 16.70 f 0.05 16.44 f 0.05 16.15 f 0.05
100
50
I 0
T,
'C
459:
0.1
-10.39 f 0.08 -10.40 f 0.08 -10.40 f 0.08
No 27 27 25 22 20
f f f f f
5 5 5
5 5
a, A-' 0.76 f 0.02 0.75 f 0.02 0.74 f 0.02 0.73 f 0.02 0.72 f 0.02
R,,8, 11.7 f 0.2 11.2 f 0.2 11.2 f 0.2 10.9 f 0.2 11.1 f 0.2
The measured neutron scattering spectra of different concentrations, but at the same temperature, were fit with this model to determine the five parameters (A - NoS)/kT, 6 / k T , No, R,, and a. These five parameters are in general temperature dependent. The nonlinear least-squares fittings will yield one set of parameters for each temperature. 4. Results Figure 1 shows the measured neutron scattering spectra together with the fitted curves for lecithin concentrations 5.6, 11.25, 16.9, and 22.5 mM, respectively. Only the S A N S data at 30 and 45 OC are shown to illustrate that S A N S data can be well fit with this model. The neutron scattering intensity is higher for higher lecithin concentration at each studied temperature. The effect of temperature is to lower the neutron scattering intensity as the temperature is raised. The higher the temperature, the lower the scattering intensity. This means the mean size or the mean aggregation number of the micelles decreases as the temperature is raised. The profiles of the neutron scattering intensity distribution are similar though the sample temperature is different. It can be noted from Figure 1 that the scattering intensities at large Q almost coincide for the same sample but at different temperatures. The large Q data depend on only the radial structure, not the length scale, of the rodlike micelles? This implies that the radial structure of the rodlike micelles does not change much within the investigated temperature range. For each temperature the five key parameters obtained by fitting the S A N S data to the model are listed in Table I . The
0.01
0.02
I/x-x,, Figure 3. Plot of weight-averaged aggregation number vs (X- X,,)'/* at five different temperatures: (0)25, (A) 30, (0) 35, (W) 40, and (+) 45 OC, where Xis the concentration of diheptanoyl-PC in mole fraction. The data points fit very well to straight lines for each temperature.
weight-averaged mean aggregation number can also be obtained at the same time. Figure 2 shows the effect of temperature on the mean aggregation number. The mean aggregation number decreases gradually with increasing temperature for each lecithin concentration. At 45 OC it drops to about 55% of the mean aggregation number at 25 O C . As predicted by the thermodynamic model, Figure 3 shows that the mean aggregation number is a linear function of (X- X,,)l/z for each investigated temperature. Here XI has been replaced by X,,, since they are almost equal when X >> X,,,. As mentioned earlier, the radial structure of the rodlike micelles does not change much within the investigated temperature range. Consistent with this are the fitted values of R, and a. As listed in Table I, both R, and a are found to vary slightly with increasing temperature. The values of R, decrease from 1 1.7 f 0.02 to about 11.0 f 0.02 A as the temperature is increased from 25 to 45 O C . The values of a decrease from 0.76 f 0.02 to 0.72 f 0.02 A-l as the temperature is increased from 25 to 45 OC. The effect of raising temperature is to decrease the mean length of the rodlike micelles while the radial structure remains roughly unchanged. The fitted values for the three thermodynamic parameters are also listed in Table I. The parameter ( A - NoG)/kT determines how rapidly the rodlike micelles will grow with increased concentration. The value of ( A - N o S ) / k Tat 25 "C is found to be 17.35 in this study, which is higher than the value of 16.46 found in the previous study at 24 0C.2 This could be due to a slight difference in the purity of the original lecithin materials. Since all the samples used in this study were prepared from the same bottle of lecithin and with similar procedures, they should have
7242
The Journal of Physical Chemistry, Vol. 94, No. 18, 1990
175
1
Lin et ai.
g
a
a
9 3
t
20
25
30
35
40
45
50
T( "C) Figure 6. Computed radius of the hydrocarbon core of the spherocylindrical diheptanoyl-PC micelles plotted vs temperature.
-9 4
6 - 1 0 4 -
0
-1s
-11
0
0
'
the same thermodynamic properties, though they might be slightly different from the previous ones. The variation of ( A - No6)/kT with temperature is plotted in Figure 4. The effect of raising the sample temperature is to decrease ( A - No6)/kT linearly from 17.35 at 25 OC to 16.15 at 45 OC. This means that the mean aggregation number will become smaller at higher temperature. On average, ( A - N o 6 ) / k T drops about 0.35% by raising 1 OC. The slopes of the straight lines for each temperature in Figure 3 also give roughly the values of 2e(A-N@)/zkT. The slope decreases with increasing temperature, which also means smaller values for (A - N o 6 ) / k T at higher temperatures. Figure 5 shows the effects of temperature on the parameter 6 l k T . In the temperature range of 25-45 OC, the values of 6 / k T vary from -1 0.35 to -10.40, with an estimated uncertainty of 0.08. Therefore, 6 / k T is almost a constant in this investigated temperature range. The fitted parameters in Table I show that the aggregation number of the minimum size micelle, or equivalently the number of monomers in the two end caps of the rodlike micelle, tends to decrease with increasing temperature. No decreases from 27 f 5 at 25 OC to 20 f 5 at 45 OC. Although these fitted values for No have large uncertainties, there is a clear tendency for No to decrease with increased temperature. 5. Discussion
The results of this study show that No, R,, and cy all tend to be slightly smaller at higher temperature. Smaller values of R, and cy both suggest that the radius of the cylinder section of the rodlike micelles will be smaller at higher temperature. The decrease of R, with increasing temperature would be expected as more gauche isomers and other segmental motions of the chains increase with increasing temperature. One may compute the radius of the hydrocarbon core, R,,,,, of the cylinder section according to2v3
Here Vtailis the volume of the hydrocarbon tail of one diheptanoyl-PC molecule. The hydrocarbon tails of the amphiphilic molecules are assumed to form a close-packed hydrophobic core.'-3 The computed values of R,,, are 9.56, 9.50,9.44,9.37, and 9.31 A, respectively, for 25, 30, 35, 40, and 45 "C. Figure 6 shows that the value of R,, decreases with increasing temperature. The uncertainty in R,, is about 0.12 A. The effect of temperature on 6 allows one to determine the separate contributions of enthalpic h6 and of entropic s6 to 6.s96,'5 6 can be written as 6 = h6 - T s ~ (7) Since 6 / k T is found to be almost a constant, the enthalpic contribution h6 should be very small. The major contribution to the negative 6 is a positive entropic s6. s6 is found to be about 20.6 cal/(OC mol). The mean size and size distribution of the rodlike micelles depend on the parameter ( A - No6)/kT. One could gain insight into the thermodynamics of the formation and growth of rodlike micelles by analyzing the thermodynamic parameter ( A - NOS). It can be expressed in terms of enthalpic and entropic contributions that are temperature-independent quantities5 and can be written as ( A - No6) = h, - Ts,. In fact, ( A - No6) is comprised of three independent parameters, No, A / N o , and 6, and can be written as N o ( A / N o- 6). Here both A / N o and 6 are on the same basis of per molecule, and they can be compared easily. When No is not temperature independent as in this study, it is no longer valid to express ( A - Nod) simply in terms of enthalpic and entropic contributions. If No is temperature dependent, meaningless values of h, and s, would be obtained by assuming (A - NOS)= h, - Ts,. Instead, one could separate enthalpic contributions and entropic contributions to A / N o and write A / N o = h - Ts. It is interesting to know how these three parameters, No, A/No, and 6, affect the temperature dependence of the compound parameter No(A/No- 6 ) / k T . 6 / k T is found to be about a constant, and No is found to decrease with increasing temperature. To obtain the values of ( A / N o )or ( A / N o ) / k T ,one would need accurate numbers of NO'Sat different temperatures. The fitted values of No have large uncertainties and cannot be used in the calculation. One way to get the best estimate of No is to assume that the end caps of the micelles at different temperatures have similar shape^.^ Thus, No is proportional to (R,re)3 and can be written as NO,T= N O , T , ( R ~ , ~ , (, T R )c~o/r e , ~ J 3
(8)
where T, is the reference temperature and is taken as 25 OC. No,T, is then equal to 27. The estimated values of No are 26.5, 26.0, 25.4, and 24.9, respectively, for 30, 35,40, and 45 "C. With the aid of the best estimated NO(s,the computed values of ( A / N o ) / k T are equal to -9.71, -9.73, -9.75, -9.75, and -9.75, respectively, for 25, 30, 35,40, and 45 "C. Thus, ( A / N o ) / k Tis found to be about constant over the investigated temperature range. This (15) Tanford, C . The Hydrophobic Effecr, 2nd ed.; Wiley: New York, 1980.
J . Phys. Chem. 1990, 94, 7243-7250 means ( A / N o ) ,like 6, is also of entropic origin. The entropy change of moving one lecithin molecule from solvent (D,O) to the cylindrical section of the spherocylindrical micelle can then be estimated from the mean value of ( A / N o ) / k Tand is found to be 19.35 cal/(OC mol). ( A / N o )is smaller than 6 because the lecithin molecules in the end caps have a larger area in contact with the solvent than the lecithin molecules in the straight section do. The decrease of No(A/No- 6 ) / k T with increasing temperature is due to the decrease of No with increasing temperature, while both ( A / N o ) / k Tand 6 / k T remain constant in the investigated temperature range. 6. Conclusions
This paper reports the study of the temperature dependence of the growth of diheptanoyl-PC rodlike micelles by using the small-angle neutron scattering techniques. By fitting the SANS data it is possible to extract the thermodynamic and structural parameters that characterize the micellar system at different temperatures. It is found that the mean aggregation number of the micelles decreases with increasing temperature. The radius of the rodlike micelles tends to be slightly smaller at higher temperature. The number of diheptanoyl-PC molecules in the end caps of the spherocylindrical micelle also decreases with increasing temperature. The growth of the rodlike micelles is driven by the free energy difference of the monomers in the two end caps and the same number of monomers in the cylinder section. The mean aggregation number depends strongly on this
7243
free energy difference (in units of k T ) , N o ( A / N o- 6 ) / k T . No( A / N o- 6 ) / k T is found to decrease with increasing temperature, which is responsible for the decrease of the mean aggregation number with increasing temperature. ( A / N o ) / k Tand 6 / k T are found to be almost constant over the investigated temperature range; thus they are of entropic origin. The decrease of No is then responsible for the decrease of the parameter No(A/No- 6 ) / k T . Since a small change in the value of No(A/No- 6 ) / k T would have a large effect on the mean aggregation number, a slight change of No with temperature would affect the whole micellar size distribution at different temperatures. In order to understand the effects of temperature on the number of monomers in the two end caps, chain packing of the hydrocarbon tails in the end caps will need to be considered. Using the results obtained in this study, one will be able to predict accurately the size and size distribution of the diheptanoyl-PC micelles at different concentrations and different temperatures. This method of analysis should be also applicable to other micellar systems. Acknowledgment. We are grateful to Dr. B. Shoenborn and Dr. D. K. Schneider of the Biology Department of Brookhaven National Laboratory for granting beam time for the experiment. We also thank Dr. C.-F. Wu for assistance in taking the SANS data. This research is supported in part by the National Science Council, ROC, Grants NSC78-0208-M007-56 and NSC790208-M007-33 (T.L.L.); N I H (Grant GM26762 (M.F.R.); and N S F Grant DMR-8719217 (S.H.C.).
Size-Exclusion Chromatography of Polyelectrolytes: Comparison with Theory Paul L. Dubin,**+Raima M. Larter,*qt Christine J. W U , and ~ Jerome I. Kaplanl Departments of Chemistry and Physics, Indiana University-Purdue University at Indianapolis, Indianapolis, Indiana 46205 (Received: December 27, 1989; I n Final Form: April 16. 1990)
A theoretical model for the free energy of partitioning of charged spheres into cylindrical pores of like charge, based on the linearized Poisson-Boltzmann equation, has been evaluated vis-&vis the elution behavior of anionic polyelectrolytes on size-exclusion chromatography columns packed with porous glass beads. Theoretical predictions of the dependence of the chromatographic partition coefficient on the relative dimensions of solute and pore were compared to experimental results obtained over a wide range of polyion molecular weights, pore sizes, and mobile-phase ionic strengths. Good agreement was found when the polyelectrolyte was modeled as a porous charged sphere, with the dimensions of its equivalent hydrodynamic sphere and a volume charge density dictated by an adjustable effective degree of dissociation. The last parameter, determined by empirical fit, was in good agreement with results for the effective ionization degree of similar polyelectrolytes determined by other methods.
Introduction
The permeation of macroions in porous media has been the subject of experimental and theoretical Such studies are relevant to a number of phenomena, including, for example, applications of polyelectrolytes in enhanced oil recovery, the characetization of charged polymers of natural and synthetic origin, and the transport of charged macromolecules through membranes. Analyses of the dynamic transport processes require highly complex mathematical procedures. However, an understanding of the presumably simpler equilibrium behavior of polyelectrolytes in charged cavities may be viewed as providing a foundation for the modeling of dynamic processes. The equilibrium exclusion of polyions from similarly charged cavities is encountered in size-exclusion chromatography (SEC). It is widely held that the size-exclusion chromatographic partition coefficient, KsEc, measures only the equilibrium distribution of I
'Department of Chemistry. 'Department of Physics. *Current address: Department of Chemistry, UCLA, Los Angeles, CA.
0022-3654/90/2094-7243$02.50/0
particles between interstitial and pore volumes; it is thus equivalent to the ratio of the solute concentration in the pore to that in the interstitial volume, C / C o = K . This conclusion-based on the measured insensitivity of KsEc to flow rate (at least for isotropic macromolecules) and the observed equivalence of the chromatographic value to the value measured by batch suggests that an equilibrium electrostatic model should account for the SEC behavior of polyelectrolytes on chromatographic packings of similar charge. Analysis of such SEC data may help (1) Malone, D. M.; Anderson, J. L. Chem. Eng. Sci. 1978, 33, 1429. (2) Anderson, J. L.; Brannon, J . H. J. Polym. Sci., Polym. Phys. Ed. 1981, 19, 405. (3) Deen, W . M.; Smith, F. G.,111. J. Membr. Sci. 1982, 12, 217. (4) Chen, J.-L.; Morawetz, H . Macromolecules 1982, IS, 1185. (5) Smisek, D. L.; Hoagland, D. A. Macromolecules 1989, 22, 2270. (6) Ackers, G. K. Biochemistry 1964, 3, 724. (7) Yau, W . W.; Malone, C. P.; Fleming, S. W. J. Polym. Sei., Parr B
--. -. (8) Edmond, E.; Farguhor, S.; Dunstone, J. R.; Ogston, A. G.Biochem.
1968. 6. 803. .. ---
J . 1968, 108, 755. (9) Ackers, G. K. A h . Protein Chem. 1970, 24, 343.
0 1990 American Chemical Society
