7763
J. Phys. Chem. 1993,97, 7163-1768
Comparison of Diffusion Coefficients of Multicomponent Solutions from Light Scattering and Macroscopic Gradient Techniques. Sodium Dodecyl Sulfate Micelles in Aqueous Salt Solutions Derek G. Leaist’ and Ling Hao Department of Chemistry. University of Western Ontario, London, Ontario, Canada N6A 5B7 Received: March 1 , 1993; In Final Form: April 6, 1993
Taylor dispersion is used to measure mutual diffusion coefficients for sodium dodecyl sulfate (NaDS) micelles in aqueous sodium chloride solutions. The diffusion coefficient of the micelles suggested by quasi-elastic light scattering (QELS) spectroscopy is larger than the coefficient of NaDS determined by the dispersion method. To clarify the apparent disagreement, an expression is developed for the spectrum of light scattered by the concentration fluctuations in a multicomponent solution. The spectrum shows that QELS gives the lower eigenvalue of the mutual diffusion coefficient matrix for NaDS-NaC1 solutions. Because the cross-diffusion coefficients are not negligible, the measured eigenvalue is up to 15% larger than the diffusion coefficient of the micelles. Caution is recommended in the interpretation of QELS diffusion data for multicomponent solutions unless coupled diffusion is known to be unimportant.
Introduction Quasi-elastic light scattering (QELS) spectroscopy is widely used to study diffusion in liquids.’-’ Although mixtures of small molecules can be studied,” this versatile technique is more often applied to macromolecules,”ll micelles,l2-15 or microemulsions16 which scatter light more efficiently. The theory behind QELS diffusion measurements is well understood for binary solutions.17 But, in practice the technique is frequently applied to multicomponent solutions. Considerable effort has been devoted, for example, to studies of the diffusion of proteins and ionic micelles in aqueous salt solutions.&ls The measured diffusion coefficients are assumed to represent the diffusion coefficients of the proteins or the micelles. However, diffusion in multicomponent solutions is coupled.1.2 It is not uncommon for a diffusing solute to produce fluxes of other solutes which exceed its own flux. The electric field generated by the diffusion of proteins or micelles together with mobile counterions can drive especially large coupled flows of supporting electrolytes.1p-20In aqueoussodium chloride solutions, each mole of diffusing micelle or protein transports hundreds of moles of It is not immediately obvious what effects, if any, coupled diffusion would have on QELS diffusion measurements. When QELS is applied to micelle solutions with or without added salt, it is generally held that the measured diffusion coefficient represents the diffusion coefficient of the micelle ~pecies.12-l~By contrast, classical macroscopicgradient diffusion e x p e r i m e n t ~ l ~give . ~ ~ the - ~ ~mutual diffusion coefficient of the total micellar component as defined by Fick’s laws. The mutual diffusion coefficient is a weighted average of the diffusion coefficient of the rapidly equilibrating micelles and free monomers. The latter diffuse relatively rapidly, so the mutual diffusion coefficient of the total micellar component is larger than the diffusion coefficient of the micelles determined by QELS.14 In the study reported here the Taylor dispersion’-3 method is used to measure diffusion in aqueous solutions of sodium dodecyl sulfate (NaDS) micelles containing added sodium chloride. Taylor dispersion, like other macroscopic gradient techniques, gives unambiguous mutual diffusion coefficients. Surprisingly, the mutual diffusion coefficient of NaDS is found to be smaller than the diffusion coefficient indicated by QELS. This unexpected result prompted the calculation of the spectrum of light scattered by the concentration fluctuations in a multicomponent solution. The spectrum shows that the diffusion coefficient determined by QELS for NaDS-NaC1 solutions is 0022-3654/93/2097-7763%04.00/0
actually the lower eigenvalue of the mutual diffusion coefficient matrix, not the micelle diffusion coefficientor the mutual diffusion coefficient of the total NaDS component. This distinction accounts for the apparent inconsistency of the Taylor and QELs data and may help to clarify the meaning of QELs diffusion measurements on multicomponent solutions.
Experimental Section Solutionswere prepared in calibratedvolumetricflasks by using distilled, deionized water, BDH “specially pure” NaDS and reagent-grade NaCl. The coupled Fick equationsl-3 J1 = -Dl,VC1
- D12VCZ
(1)
+
describe mutual diffusion in aqueous NaDS(1) NaCl(2) solutions. Mutual diffusion coefficients Dtk give the molar flux of total solute component i caused by the gradient in the concentration of solute component k. The Dlk coefficientsreported here were measured by the Taylor dispersion (peak-broadening) method.22-28 A metering pump maintained a steady flow of carrier solution through a Teflon capillary tube (3658-cm length, 0.04335-cm inner radius). The tube was coiled in a 75-cm-diameter helix and held at 25.00 f 0.05 OC in a thermostat. At the start of each run 20 mm3 of solution of composition e1 + AC1, AC2 was injected into a carrier stream of composition el, c2. Flow rates were adjusted to give retention times of about 10000 s. A differential refractometer was used to detect the eluted samples. The refractometer’s output voltage was measured at 204 intervals with a computer-controlled digital voltmeter. The detector voltage generated by the coupled diffusion of two solutes2628
+
V ( t ) = (tR/t)”2vm[Wl exp[-120,(t-t,)2/3t]
+
w2exp[-12D2(t-t,)2/3t]]
(3) closely resembles two overlapping Gaussian peaks centered on the retention time t~ with approximate variances &/24a1 and r2t~/242&. r is the inner radius of the dispersion tube, V, is the peak height relative to the base line, and z)1 and 322 are the eigenvalues of the matrix D of Drk coefficients. The normalized weighting factors W1 and W2 are functions of AC1, AC2, the four 0 1993 American Chemical Society
Leaist and Hao
7764 The Journal of Physical Chemistry, Vol. 97,No. 29, 1993 Dlkcoefficients, and also the ratio R2/R1= (an/ac2)/(an/acl) of the refractive index increments per mole of each solute. A strictly Gaussian dispersion peak of variance u2 would vary with time as V,,, exp[-(t - tR)2/2a2]. Real dispersion peaks are slightly skewed because the factor l / t in -12Bi(t-r~)2/r2t in eq 3 causes the trailing edge of the peak ( t > t ~ to) decay more slowly than the leading edge ( t < tR). If diffusion coefficients are evaluated from the moments of the dispersion peaks,2k25a small correction for skewness may be required for broad peaks. In the present study, however, the diffusion coefficients were calculated by using nonlinear least squares to fit eq 3 to the measured refractometer voltages. Details of the fitting procedure have been reported.z* Because skewness is built in to the fitting equations, corrections for the small departures from Gaussian peak shapes were not required. Initial concentration differences were typically ACI = 0.01 m ~ l d m for - ~ NaDS and AC2 = 0.025 moldm-3 for NaCI. Smaller concentration differences gave identical results within experimental precision. These check runs indicated that the refractive index differences were linear in the concentration differences and that the measured diffusion coefficients were differential values.
Theory To account for possible differences between the diffusion coefficients from QELS with those from Taylor dispersion (or other macroscopic gradient techniques), it is helpful to have an expression for the spectrum of light scattered from a multicomponent solution. For the general case of a solution containing a solvent component plus N solute components, the Nz mutual diffusion coefficients are defined by the coupled Fick equations1-’
N
(7) N
which reduce the diffusion equations to diagonal form:
(i = 1, 2, ...)N)
di = -aivei
aei/at = aiv2ei (i = i , 2 ,
..., N )
(9) (10)
321, B2, ..., BNare the eigenvalues of the N by N matrix D of Dik coefficients. Transformation coefficients Aik are chosen so
that the columns of matrix A-l are independent eigenvectors of
D. Correlation Functions from Light Scattering. When a beam of light passes through a solution the fluctuations Be(r,t) in the local dielectric constant scatter some of the incident light. The fluctuations in the dielectric constant are in turn caused by microscopic fluctuations in the temperature, pressure, and concentrations of the solutes.17J4 In the present treatment, however, temperature and pressure fluctuations are ignored:
This approximation can be justified if the thermal diffusivity is much larger than the solute diffusivities and if the coupling between heat conduction and solute diffusion is weak. These conditions are satisfied for most solutions,5 except near critical points. Statistically, the decay of microscopic concentration fluctuations is governed by the same equations that describe the decay of macroscopic concentration gradients96 N
N
is the molar flux of solute component i in the volume-fixed reference frame, and ck is the concentration of solute k in moles per unit volume. If the concentration changes along the diffusion path are sufficiently small,the diffusion coefficients may be treated as constants in which case eq 5 simplifies to Ji
Because microscopic concentrations fluctuations are small compared to the mean concentrations, the diffusion coefficients may be treated as constants in the derivation of eq 12. The fluctuations in the concentration of solute i are coupled to the fluctuations in the concentration of solute k by crossdiffusion coefficient Dik (i # k). Mathematically, it is much easier to deal with the independent concentration fluctuations N
sej(r,t)
= Z A i k sck(r,t)
(13)
k= 1
N
which decay according to the pseudobinary equations Fick equations are useful because they accurately describe the diffusion of any number of electrolyte or nonelectrolyte components in dilute or concentrated solutions. “Component” is used in the thermodynamicsense (e.g. total NaDS or NaCl components, but not Na+, DS-, or C1- ions). Consequently, explicit terms are not required for the diffusion-induced electric field. Fick equations can be used to describe the diffusion of associating solutes if the association/dissociation reactions are rapid enough to maintain local eq~ilibrium.~~ This condition is satisfied for the vast majority of association reactions, including micellar aggregation.30~3~Conveniently, it is not necessary to know the concentrations or the diffusion coefficients of the actual diffusing species. The mathematics of multicomponent diffusion can always be simplified,without loss of generality or accuracy, by taking linear combinations of the concentrations and fluxes32.33
asei(r,t)/at = BiV28@,(r,t) (14) The corresponding fluctuations in the local dielectric constant are N
Mountain and Deutch” have calculated the spectrum of light scattered by the statistically independent modes of a binary solution. Neglecting temperature and pressure fluctuations,their exact treatment leads to
for the spectrum of light scattered by the concentration fluctu-
The Journal of Physical Chemistry, Vol. 97,No. 29, 1993 7765
Diffusion Coefficients of Multicomponent Solutions ations in a multicomponent solution. q is the change in the wave vector, w the change in the frequency of the scattered light, R the gas constant, and p'i the linear combination
A,, = 1
(28)
N F'i
=pAikkk =I
of the chemical potentials p k of the solute components. The Fourier transform of S(q,w) gives the field autocorrelation function
measured in homodyne scattering experiments. In general g")(t) is a multi-exponential decay with time constants 1/q2ZIlr
i/q2z)2, 1/q2aN. The intensity autocorrelation function ns.9
g'2'(t) = 1
+ Ig'"(t)l2
diagonalize the diffusion coefficient matrix:
Dilute Nonelectrolyte Solutions. Coupled diffusion is usually negligible for dilute solutions of non-associating nonelectrolytes.2 In the limiting case where all of the cross-diffusion coefficients are zero: a1 = Dii, Ail = 1, Aik (i # k) = 0,@ i = Ci, and p'i = pi. Furthermore, if the solution is thermodynamicallyideal, then pi = poi + R T In Ci, api/aCi = RT/Ci, and the expression for the field autocorrelation function simplifies
(19)
measured in optical heterodyne experiments is a more complicated multi-exponentialdecay with time constants of the form 1/qZ(Z)i ak).Fortunately there are many cases where the expressions for g")(t) and g@)(t)simplify to only one or two exponential decays. The expressions
+
It is impossible to extract accurate diffusion coefficients from multi-exponentialdecaysif the Diivaluesare similar in magnitude. But if solute 1 is macromolecular and the other solutes have much smaller molecular weights, the exponential in Dll willdecay slowly owing to the slow diffusion of the polymer. Also, the pre-exponential factor (ae/aC1)2 will be much larger than the correspondingfactorsfor theother solutes. Under theseconditions a reliable value for the diffusion coefficient Dll for the polymer can be extracted from the correlation functions. Aqueous Na,P( CI) NaCI( C,) Solutions. Coupled diffusion caused by long-range electrostatic forces is important for dilute electrolytes, even at submillimolar ionic strengths. To illustrate the possible effects of coupled diffusion on the spectrum of scattered light, consider the diffusion of macroion P* in aqueous NaCl solutions. There are three diffusing solute species, P*, C1-, and Na+, but only two solute components are independent in view of the electroneutralityrestriction. Consequently, ternary diffusion eqs 22-25 apply. At low ionic strengths the limiting Nernst equations3*
+
may be used to estimate the relative weights of the exponential decays from the measurable concentration derivatives of the dielectric constant and the solute chemical potentials. At optical frequencies e equals the the square of the solution's refractive index, and therefore ae/aCk = 2n(dn/aCk). Ternary Solutions (N+ I = 3). In a solution containing solvent plus solute components 1 and 2 the fluctuations in the solute concentrations decay as a6c,/at = -D,,v26c,
-D , , v ~ ~ c ~
a6c2/at = -D,,v~~c, -D , , v ~ ~ c ~
Dl 1
= Dp
+ tp(DNa - DP)
(34)
(22)
(23)
which transform to
D22
= DCI + tCl(DNa - DCI)
(37)
may be used to estimate the mutual diffusion coefficients of the Na,P(Cl) + NaCl(C2) componentsfrom thediffusioncoefficients Dp, Dcl, and ha of the ions. n is the number of moles of sodium counterions per mole of macroion. tp and tc1 are the ionic transference numbers The eigenvalues are
N
where zp = -n, Z C I =: -1, Z N =: ~ +1, Cp = cl, Ccl =: C2, and CNa = nC1 C2. At 25 OC the limiting diffusion ~oefficients3~ of the aqueous sodium and chloride ions are ha = 1.33 X 10-5and DCl = 2.03 X 10-5 cm2 s-l. The values n = 10 and Dp = 0.075 X 10-5 cm2 s-1 will be used to model the diffusion of serum albumin (a water-solubleprotein) in aqueous sodium chloride solutions at pH 7.22.23,40-42Figure 1
+
It is easy to show that the Aik coefficients
Leaist and Hao
7766 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 L
No,,P(l
1
)-NoCI(Z)
0 7
*,
i
' 2 -
I
0
D ,1
2
n
-1
0,000
0.002
0.004
C,(Na,,P)
0.006
/
0.008
i 8000
0.010
10000
12000
14000
t / s
mol dm-3
Figure 3. Dispersion peaks measured against a carrier stream with el = 0.100 mol dm-3 NaDS(1) and = 0.100 mol dm-) NaCl(2). NaDS peak ACl = 0.010 mol dm-3; AC2 = 0.0o0mol dm-3. NaCl peak ACI = 0.000 mol dm-3; AC2 = 0.025 mol dm--'. (a) Calculated contribution from the Gaussian in 331 (eq 3). (b) Calculated contribution from the Gaussian in 8 2 (eq 3). The peaks have been offset for clarity.
Figure 1. Calculated mutual diffusion coefficients of Nal$( 1)-NaCl(2) in solutions containing 0.100 mol dm-3 NaCl and 0 . ~ . 0 1 mol 0 dm-3 Nalfl.
0.25
E,
Owing to the electrostatic acceleration by the mobile Na+ counterions, the mutual diffusion coefficient of N a d ( 1) is up to three times larger than the diffusioncoefficientof the macroion, D,. The model calculationsforNal,,P( I)-NaC1(2) solutions suggest that QELS would give the eigenvalue Dt which differs significantly from the mutual diffusion coefficient Dll of the polyelectrolyte component and from the diffusion coefficient Dp of the polyion. Solvent Polymer 1+ Polymer 2. Another interesting example is diffusion in a ternary solution consisting of a solvent and two polymer component^.^^ Because the fluctuations in the concentrations of both polymer components can scatter significant amounts of light, the correlation function will have two decay 'modes:4M5 ~ ~ ~p ( 't ) ~= AI ~ exp(-q2Dlt) ~ . ~ +~A2 ~exp(-q2D2t). ~ " The eigenvalues D1and 0 2 are in general complicated functions of all four of the ternary D,k coefficients (eqs 26 and 27). But if one or both of the cross-coefficients 0 1 2 and 9 1 are zero, then 01 = Dll and 02 = D22. In this limiting case the eigenvalues D1 and 0 2 may be identified as the mutual diffusion coefficients of the polymer components.
+
0
. 0 0.000
5 1 0.002
. ' 0.004
~ ~ 0.006
' ~ 0.008
~ ~ 0.010
~
C,(No,,P) / mol d m - 3 Figure 2. Calculated values of the mutual diffusion coefficient Dll of the
Nalfl( 1) component, eigenvalue 331,and the polyion diffusion coefficient Dp for solutions containing 0.100 mol dm4 NaCl(2) and O.oo(M.010
mol dm-3 NaloP(1).
shows mutual diffusion coefficients calculated for solutions containing O.OO(M.010 mol dm-3 Nal,,P(l) and 0.100 mol dm-3 NaCl(1). When a gradient in the concentration of Nal,,P is formed,the electricfield which is generated speeds up the macroion and slows down the mobile Na+ ions to keep the solution electrically neutral. The electric field also causes cocurrent coupled diffusion of C1- ions as reflected by the large positive values for cross-coefficient Dzl. In order to estimate the light scattering correlation functions according to eqs 18 and 19, the limiting expressions b1 = pol RTln [CI(CI+ 10C2)10]and ~2 = F'Z + RTln [Cz(C1+ lOC2)l may be used to evaluate the &Li/dc&derivatives. Also, the molar refractivity of NaloP( 1) may be assumed to be several hundred times larger than that for NaCl(2). The decay in Dl then completely dominates the correlation function: g")(t)= A I exp(;q2Dlf), so QELS diffusion measurements would give the eigenvalue D l. In Figure 2 the mutual diffusion coefficient Dli of the Nal,,P(1) component in 0.100 mol dm-3 NaCl is compared with the eigenvalue As the concentration of Nalg(1) is raised, D1 becomes increasingly larger than Dll. At 0.01 mol dm4 Na& (l), the eigenvalue is 12% larger than the mutual diffusion coefficient. To have D1 values within 1% of Dll requires at least a 2500-fold molar excess of NaloP(l) relative to NaCl(2).
+
'
Results and Discussion The mutual diffusion coefficients of NaDS( Cl)-NaCl( C2) solutions were measured at eight different carrier-stream compositions. Figure 3 shows dispersed solute peaks for the injection of excess NaDS or NaCl into a 0.100 mol dm-3 NaDS 0.100 mol dm-3 NaCl carrier stream. The NaCl peak consists of a relatively narrow peak superimposed on a broader base peak contributed by the relatively slow coupled diffusion of NaDS. NaDS(1) diffuses about 10 times more slowly than NaCl. The eigenvalues D1 and 0 2 were therefore well separated, and the fitting proceduregave relatively precise diffusion coefficients. The average Dik values listed in Table I were obtained by fitting eq 3 to four or five replicate pairs of peaks. Uncertainties are quoted as plus or minus two standard deviations. Because the refractive index change per mole of NaDS is about three times larger than that for NaCl(2) (R1/R2 = 3.45 f 0.03), the coefficientsDll and D12 for NaDS( 1) could be determined more precisely than 4 1 and 4 2 for NaCl(2). The present results are in good agreement with the less precise diffusion coefficients measured previously by a conductimetric method.18 In Table I1 the mutual diffusion coefficient of the NaDS component (Oil) is compared with the lower eigenvalue of the
+
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7767
Diffusion Coefficients of Multicomponent Solutions
TABLE I: Ternary Mutual Diffusion Coefficients of Aqueous NaDS(l)-NaC1(2) Solutions at 25 OC Measured by the Taylor Dispersion Metbod’ CdNaDS) C2(NaCl) Dii 012 D21 022
4
0.194 f 0.005 0.239 f 0.006 0.108 t 0.006 0.110 f 0.004 0.129 f 0.005 0.138 f 0.005 0.099 i 0.003 0.086 i 0.005
0.030 0.030 0.100 0.100 0.100 0.100 0.200 0.300
0.100 0.150 0.020 0.050 0.100 0.120 0.050 0.050
-0.271 -0.315 -0,050 -0.099 -0.161 -0,178 -0.067 -0.043
f 0.008
0.1 18 f 0.010 0.082 f 0.013 0.194 f 0.006 0.196 f 0.020 0.165 f 0.008 0.143 f 0.006 0.233 t 0.01 1 0.256 t 0.008
f 0.013 f 0.008 f 0.010
f 0.005 f 0.005 f 0.008 f 0.005
1.667 t 0.007 1.712 t 0.011 1.505 0.008 1.545 f 0.010 1.595 f 0.008 1.647 f 0.012 1.495 f 0.007 1.483 t 0.007
*
Units: Ci in mol dm-3; Dn in lC5cm2 s-I.
TABLE II: Mutual Diffusion Coefficient, Apparent Diffusion Coefficient and Eigenvalue for NaDS(1) in Aqueous NaC1(2) Solutions at 25 OC’ ~~
lOO(B1-
CdNaDS) C2(NaCU 0.030 0.030 0.100 0.100 0.100 0.100 0.200 0.300
0,100 0.150 0.020 0.050 0,100 0.120
0.050 0.050
Dll
DI
0.194 0.239 0.108 0.110 0.129 0.138 0.009 0.086
0.216 0.257 0.115 0.124 0.147 0.155 0.110 0.094
100(4pp-
Did/D11
Daw 0.226 0.271 0.131 0.135 0.155 0.171 0.123 0.109
Dd/D11 11 7 7 12 14 12 11 9
16 13 21 23 20 24 24 27
@Units:Ci in mol dm-3; Dll, B1, and Dawin 10-5 cm2s-l.
1 I
0.16
7 Y)
‘
Noel
0.1 mol dm-’
t
0.14
I
OA
i
-
5
/’
10 I
‘ -
I
0
0.12
0
/
*>* A
0.00
+
+
v(f)= (tR/t)”2v,,,
-
n
(micellar plus rapidly diffusing free ions) from the Taylor experiments. The theory developed in this article suggests an explanation: the diffusion coefficient of NaDS( 1)-NaDS(2) solutions measured by QELS is actually the eigenvalue DI of the mutual diffusion coefficient matrix. To test this suggestion, the values of D1 calculated from the Taylor mutual diffusion coefficient data are compared with the QELS coefficients in Figure 4. The rather good agreement supports the idea that QELS gives the lower eigenvalue of the diffusion coefficient matrix. The refractivity ratio Rl/R2 = 3.45 together with estimates of the chemical potential derivativesl*were used to evaluate the relative weights of the terms in the correlation functions. The calculations confirmed that terms in 331 should completely dominate the correlation functions. Therefore, in addition to the experimental evidence, theory also suggests that QELS diffusion measurements on NaDS( 1)-NaC1(2) solutions gives 31. If coupled diffusion is neglected in the analysis of the Taylor dispersion profiles for the NaDS NaCl solutions, this too would lead to apparent diffusion coefficients which were too high. A dispersion peak obtained by injecting excess NaDS into a 0.100 mol dm-3 NaDS 0.100 mol dm-3 NaCl carrier stream is shown in Figure 3. Fitting the pseudobinary dispersion equation
0.04
C,(NaDS)
0.08
/
mol d m
0.12 -3
Figure 4. Comparisonof diffusion coefficientsfor NaDS( 1) in 0.100 mol
dm-3 aqueous NaCl(2) solutionsat 25 OC. A,QELS;l30 ,mutual diffusion
coefficient Dll from Taylor dispersion (this work). calculated from Taylor dispersion data.
0, eigenvalue B1
diffusion coefficient matrix (331). At the compositionsthat were used, the eigenvalue is 7-14% larger than the mutual diffusion coefficient. Several QELS diffusion studies on aqueous NaDS NaCl solutions1*J3J5have been reported. It is generally agreed, however, that the results obtained by Corti and Degiorgio13 are the most reliable.14 In Figure 4 their values for the diffusion coefficient of NaDS micelles in 0.100 mol dm-3 NaCl solutions are compared with the Taylor results. At low NaDS concentrations the two sets of data are in reasonably good agreement. But as the concentration of NaDS is raised (and coupled diffusion becomes more important) the diffusioncoefficientsfrom the Taylor method fall about 15% below the QELS values. The discrepancy is not large, but it does seem to contradict the interpretation that QELS gives the diffusion coefficient of the micelle species. If the diffusion of the actual micelle species alone was measured, the QELS diffusion coefficients would certainly be lower than the mutual diffusion coefficient Dll of the total NaDS component
+
exp[-12D,pp(?-tR)2/rZt]
(39)
tothepeakgavell,, = 0.155 X 10-5cm2s-1,whichis20%higher than the mutual diffusion coefficient of the NaDS component at this composition. The Gaussiansin both and 332 make significant contributions to the dispersion peaks for NaDS. When eq 39 is fitted to NaDS peaks the resulting apparent diffusion coefficient, a weighted average of 2Il and 332, is about 20% larger than Dll (see Table 11)* Conclusions The spectrum of light scattered by the concentration fluctuations in a multicomponent solution can be derived from the corresponding spectrum for a binary solution. In general the QELS time-correlation functions for a multicomponent solution are multi-exponential decays with time constants governed by the eigenvalues of the mutual diffusion coefficient matrix. In the favorable case of a polymer dissolved in a salt solution, however, a single exponential decay dominates, and the lowest eigenvalue can be determined. If thecross-diffusion coefficientscan be safely neglected, the eigenvalue will equal the mutual diffusion coefficient of the polymer component. But in cases where the crossdiffusion coefficients are not negligible the measured eigenvalue will in general differ from the mutual diffusion coefficient of the polymer. For NaDS micelles in aqueous NaCl solutions, the eigenvalue measured by QELS is up to 15% higher than the mutual diffusion coefficient of the NaDS component. Acknowledgment. The authors thank the Natural Sciences and Engineering Research Council for the financial support of this research.
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