11044
J. Phys. Chem. B 2001, 105, 11044-11051
Diffusion Coefficient Matrix in Nonionic Polymer-Solvent Mixtures Alessandro Vergara, Luigi Paduano, Gaetano Mangiapia, and Roberto Sartorio* Dipartimento di Chimica dell'UniVersita` “Federico II” di Napoli, Complesso Monte S. Angelo, Via Cinthia, 80126 Napoli, Italy ReceiVed: March 19, 2001; In Final Form: June 22, 2001
Recent predictive equations for evaluating the value of diffusion coefficients, developed for two hard spheresolvent systems, have been extended to n-hard-sphere-solvent systems. This model has been used to predict the diffusion behavior of nonionic polymer-solvent mixtures as a first-order function of volume fraction from the knowledge of the distribution function. Hence, the entire diffusion matrix for polydisperse systems, which include both main- and cross-term diffusion coefficients, has been built up for the first time. The main-terms diffusion coefficients Dii are mostly affected by the variation in viscosity. The magnitude of the cross-term diffusion coefficients Dij is strictly related to the abundance and diffusivity of solute i and to the molecular volume corresponding to solute j. The predicted diffusion coefficients have been used to calculate the apparent, average diffusion coefficients, which are compared with the experimental values measured by the Gouy interferometry method in the case of the poly(ethylene glycol) homologous series. The agreement between the calculated and experimental Gouy parameters is good. The cross-term diffusion coefficients are found to be significant in the calculation of the apparent diffusion coefficient for a polydisperse solute. Some comments on the study of ternary systems containing a polydisperse solute are done.
I. Introduction The polydispersity is a characteristic of many synthetic or natural macromolecules. It can have a constitutional origin, as for synthetic polymers,1 or can depend on several chemicalphysical properties as concentration,2 pH,3 and temperature,4 as for biological macromolecules. The polydispersity affects both thermodynamic5-8 and dynamic1,9,10 properties of macromolecular solutions. Any property that can distinguish molecules for their size can be used, in principle, to study polymer polydispersity. Because the transport properties are strongly dependent on the molecular size and hence on the molecular weight of diffusing particles, diffusion experiments on polydisperse samples can provide information on their polydispersity.11 Actually, many experimental techniques for determining transport properties (dynamic light scattering,1 NMR-PGSE,12 interferometry,11,13 EPR14) are used to study polydispersity for different applications.15,16 We are interested here in the effect of polymer polydispersity on the mutual diffusion coefficients. A polymer-solvent mixture is a pseudobinary system from a massive point of view. One may choose one of the constituent components, e.g., one with chain length n, to identify the pseudobinary component and choose the solvent as the other component. If the concentration of the polymer chain length is specified, then the concentrations of the other constituents of the polymer mixture can be obtained from the molecular weight distribution function. From a diffusive point of view, such a system can be only treated as a pseudobinary one because only one diffusion coefficient can be measured, instead of the n2 Dij’s (i, j ) 1, 2, ..., n) necessary to describe the diffusion process in a system constituted by n + 1 independent components (solvent plus n polymeric components).17 In fact, the measurement of these n2 * Corresponding author. Fax +3981674090. E-mail: chemistry.unina.it.
sartorio@
Dij’s requires at least n experiments at different ratios of the solute concentration gradients. Because in a polydisperse sample the different components are present in a fixed molar ratio, it is not possible to perform these n independent experiments. Therefore, a polydisperse system must be treated as a pseudobinary one and is described only by an experimental average diffusion coefficient, DA. DA is a complicated function of all the Dij’s, of the sample composition and of other quantities related to the specific technique used for its experimental determination. From DA itself, it is impossible to obtain a complete diffusive description of the polydisperse sample in terms of the Dij’s, as discussed in section III. Some of us recently developed a procedure for analyzing the diffusive polydispersity from Gouy parameters at infinite dilution.13 At infinite dilution of the polymeric sample, all cross diffusion coefficients, D∞ij with i * j, are null, and the main terms D∞ii coincide with the diffusion coefficients of the corresponding binary systems (oligomer i-solvent) at infinite dilution, D∞i . Because the D∞i for all the components of the polydisperse sample can be evaluated, using the Flory equation18 (from the correspondent values of the first oligomers), it is possible to obtain information on the molecular weight distribution function just from D∞A. The extension of this analysis to finite concentration is impossible without knowing both the main- and cross-term diffusion coefficients. Recently we have proposed very simple predictive equations for the main- and cross-term diffusion coefficients in ternary systems containing uncharged hard spheres of different molecular volumes.19 We have tested the reliability of these equations for several experimental data, corresponding to ternary systems containing oligomers of the poly(ethylene glycol), PEG.19-22 For these systems, the prediction of the Dij is quite good in dilute solution, whereas systematic differences between estimated and experimental values are observed as the total solute concentration increases.21
10.1021/jp011034b CCC: $20.00 © 2001 American Chemical Society Published on Web 10/11/2001
Nonionic Polymer-Solvent Mixtures
J. Phys. Chem. B, Vol. 105, No. 44, 2001 11045
In the present paper, these equations are extended to ncomponent systems, and the diffusion coefficient matrix of a polymer-solvent mixture is built up for the first time. Finally, from these values the pseudobinary diffusion properties for a solution containing a polymer, composed of uncharged hardsphere-like molecules, have been evaluated. The calculated Gouy parameters are compared with the experimental ones in the case of some PEG-water systems, for which the weight distribution function is known to be a Poisson one.23 In real cases, the situation is more complicated. At increasing average molecular weight, each oligomer in the polydisperse sample cannot be described as a hard-sphere-like molecule. Therefore, systematic deviations from the results predicted by the hard-sphere model are expected at increasing the polymer molecular weight.24 In section II we have (1) extended the excluded volume analysis on main- and cross-term diffusion coefficients to n-component systems and (2) examined the polydispersity effect on the Gouy parameters at finite concentration. In section III, we have applied the theory of section II to polymer-solvent mixtures where (1) the entire diffusion matrix has been built and (2) the elements of the matrix are used to calculate Gouy parameters for the PEG-water mixtures, and we have (3) compared calculated values with experimental ones to demonstrate the validity of the predictive procedure. In section IV, we give the main conclusions of the work. II. Theory Diffusion and Gouy Interferometry in (n + 1)-Component Systems. In a solution containing n solutes (1, 2, ..., n) and a solvent (0), the mutual diffusion is described by the Fick’s laws17
∑ j)1
∂Ci
n
)
∂t
Dij
Dij ∑ j)1
n
∂2Cj
∂nλ ∂x
)
∆nλ
2 n
∑
n
Γj erf ∑ j)1
Γj
2xπt j)1 xλj
(x ) x
(5)
2 λjt
( ) x2
exp 4λjt
(6)
where h nλ and ∆nλ are, respectively, the refractive index at the mean concentration and its total difference across the diffusion boundary and Γj is a constant dependent on Kij and Rj, with the condition28 n
Γj ) 1 ∑ j)1
(7)
Equation 6 describes a Gaussian for binary systems, whereas it is the sum of n Gaussians for the (n + 1)-component system. The usual two experimental quantities obtained by the Gouy interferometric technique are DA and Q0, which represent respectively the apparent diffusion coefficient29
DA )
(2)
∂x2
Kij erf ∑ j)1
(4)
∆nλ ∂nλ 4πt ∂x
( )
2
(8)
max
and the “area under the deviation graph”, Q0, related to deviation from Gaussianity of the curve of the refractive index gradient30
where Ji is the flux of component i in the direction x, Cj is the concentration of component j, and Dij’s are the mutual diffusion coefficients. Equation 2 is valid when the coefficients can be treated as constant. All the Dij’s in this paper are expressed in the volume fixed reference frame.25 Under the hypothesis of free and differential diffusion,26 eq 2 can be integrated to give for the ith component
hi + Ci(x,t) ) C
∆nλ
nλ(x,t) ) nhλ +
(1) ∂x
Rj(Cj - C h) ∑ j j)1
where Rj ) (∂nλ/∂Cj). The refractive index and its derivative respect to x can be expressed as
∂Cj
n
Ji ) -
n
nλ ) nhλ +
(x ) x
(3)
2 λjt
where Ci is the mean concentration of component i, Kij’s are constants dependent on Dij and ∆Ci, erf is the error function operator, and λj’s are the eigenvalues of the diffusion coefficient matrix D. The Gouy and Rayleigh interferometric methods, which are the most precise methods for measuring the mutual diffusion coefficients,27 are based on the refractive index, nλ, distribution in the diffusion boundary region and its spread with time. The sensitivity of these methods permits precise experiments in which the solute concentrations are always close to the experimental mean solute concentration. Therefore, it is usually safe to assume constant diffusion coefficients and a linear concentration dependence of the refractive index, nλ. Thus
Q0 )
∫01Ω(ζ) df(ζ)
(9)
where
ζ)
x 2xπDAt
f(ζ) ) erf(ζ) -
2 exp(-ζ2) xπ
Ω(ζ) ) exp(-ζ2) -
(∂nλ/∂ζ) (∂nλ/∂ζ)max
(10)
(11)
(12)
For ideal binary systems, DA coincides with the Fick diffusion coefficient Di, while Q0 is null.13 For (n + 1)-component systems, both DA and Q0 are a function of all the diffusion coefficients and of the refractive index increments.13 At least n independent diffusion runs with different values of the refractive index ratio (Ri) are necessary for obtaining the n2 diffusion coefficients
Ri )
Ri∆Ci n
Rj∆Cj ∑ j)1
(13)
11046 J. Phys. Chem. B, Vol. 105, No. 44, 2001
Vergara et al.
The quantities DA and Q0 are related to the Dij and Ri values by the relations13,31 n
1
)
xDA n
Q0 )
Γj
∑ j)1 xλj
(14)
n
∑∑ΓiΓjgij 1 i)1 j)1 2
n
(15)
Γj
∑ j)1 xλj
where
gij )
1 xλi + xλj 1 ) gji 2x2 xλi + λj xλi + λj
RT κπriη0
Di ) MixBix
(18)
where
(
) (
)
∂ ln γi ∂ ln γ0 ) 1+ ∂ lnxi ∂ ln x0
(19)
and γi and xi are, respectively, the activity coefficient and the molar fraction. In dilute solution of hard spheres, the dependence of Di on the concentration can be expressed by the CarterPhillies equation33
Di )
RT (1 - 0.898Φi) ) D∞i (1 - 0.898Φi) ) κπriη0 D∞i (1 - 0.898CiVi) (20)
where Φi is the “effective” volume fraction of ith component and Vi its “effective” molar volume expressed in dm3 mol-1. Therefore, the experimental binary diffusion coefficients, Di, can be used to obtain the effective volume Vi by fitting eq 20 to experimental data. For a ternary system, we have proposed the following equations for the main- and cross-term diffusion coefficients:19
ηi Dii ) Di ηij CiΦj
(21) CiVj
Dij ) Dii ) Dii Cj(1 - Φj)2 (1 - CjVj)2
ηij ) 1 + 2.5(Φi + Φj) ) 1 + 2.5(CiVi + CjVj) (24) Substituting eqs 20, 23, and 24 in eqs 21 and 22, the following expressions for the diffusion coefficients can be obtained
1 + 2.5CiVi Dii ) D∞i (1 - 0.898CiVi) 1 + 2.5(CiVi + CjVj)
(25)
1 + 2.5CiVi CiVj Dij ) Di∞(1 - 0.898CiVi) 1 + 2.5(CiVi + CjVj)(1 - CjVj)2 (26) Main- and cross-diffusion coefficients in eqs 25 and 26 are expressed in terms of these quantities: the solute concentrations, the limiting diffusion coefficients of the binary systems, and the “effective” volumes of the diffusing particles in solution. The values of the cross-term diffusion coefficients depend on the choice of concentration scale for the flux and gradients. Nevertheless, eq 26 does not depend on a specific concentration scale; therefore, both the molar concentration and the weight concentration (often used in polymer chemistry) can be used. These equations have been tested for several ternary systems containing poly(ethylene glycol) oligomers.19-22 The analysis showed that the proposed equations give good prediction of the diffusion coefficients in dilute solutions, while systematic differences are observed as the total solute concentration increases.21 These predictive equations were compared elsewhere34 with other two theoretical approaches, due to Batchelor35 and Cussler.36 In the present paper, we have extended eqs 25 and 26 to (n + 1)-component systems, obtaining
Dii ) D∞i (1 - 0.898CiVi)
1 + 2.5CiVi (27)
n
1 + 2.5
∑CkVk
k)1
Dij ) D∞i (1 - 0.898CiVi)
(22)
(23)
while for the ternary system, we proposed an extension of the Einstein equation19
(17)
where κ is a parameter depending on the relative size of solute and solvent,32 R is the gas constant, and T is the absolute temperature. The dependence of the diffusion coefficient on the concentration is due to a hydrodynamic, Mix, and to a thermodynamic factor, Bix
Bix ) 1 +
ηi ) 1 + 2.5Φi ) 1 + 2.5CiVi
(16)
Predictive Equations for the Dij. For a binary solution of hard spheres at infinite dilution in a continuum, the StokesEinstein equation relates the limit diffusion coefficient Di∞ to the radius of the diffusing particle, ri, and to solvent viscosity, η026
Di∞ )
where ηi and ηij are the viscosity corresponding to the binary and ternary system, respectively. A brief comment is necessary on these last equations. The main-term diffusion coefficients are calculated using a hydrodynamic correction on binary diffusion coefficients, due to the presence of the other solute. Crossterm diffusion coefficients are corrected for a hydrodynamic contribution (because of the Dii presence) and for a thermodynamic one (elsewhere described through a jump model34). The hydro-thermodynamic corrections are estimated including only the exclusion concept and no specific interaction. Therefore, the following analysis can be applied only to nonionic polymers. According to eq 21, Dii can be obtained from Di, which can be obtained from eq 20, and the viscosity of ternary (solute j-solute i-solvent) and binary (solute i-solvent 0) systems. The Einstein predictive equation can be used to estimate the viscosity of binary systems26
1 + 2.5CiVi
CiVj
n
∑C V
1 + 2.5
k k
k)1
(28)
n
[1 -
∑C V (1 - δ )]
2
k k
k)1
ik
Nonionic Polymer-Solvent Mixtures
J. Phys. Chem. B, Vol. 105, No. 44, 2001 11047
where δik is the Kronecker’s delta. Equations 27 and 28 can be used for the evaluation of the elements of the diffusion coefficient matrix corresponding to any nonionic polydisperse solute in a solvent. III. Applications Sample Polydispersity. Equations 27 and 28 have been applied to the specific case of PEG water mixtures, for which experimental mutual diffusion data were widely collected.37 The PEG samples, HO-(CH2-CH2-O)i-H, with i polymerization degrees, exhibit a Poisson molecular weight distribution function, typical of living polymers.23 Indicating with in the average number chain length, the concentration fraction distribution is given by
pi )
exp(-ν) i-1 ν (i - 1)!
(29)
Figure 1. Main-term diffusion coefficients, Dii, for some oligomers for the PEG 300-water system at different total concentration: at infinite dilution (curve 1), at 0.1 mol dm-3 (curve 2), and at 0.5 mol dm-3 (curve 3).
where ν ) in - 1. In this paper, we will consider PEG samples at different numerical average molecular weight, ranging from 200 up to 1000 uma. In any case, the applications below presented are generally independent of the specific distribution function used. Dij Matrix for a (n + 1)-Component System. Using eqs 27 and 28, it is possible to build up the Dij matrix for a system containing n solutes (1, 2, ..., n) and the solvent (0), from the knowledge of the D∞i , the Vi, and the concentration of all the solutes in solution. For the D∞i of linear and flexible polymers as a function of i, we have used the Flory equation18
D∞i ) KFMiγ
(30)
where KF and γ are constants that have been determined previously for PEG in water13,37 (10.6 ( 0.3 × 10-5 and -0.527 ( 0.006 cm2 s-1, respectively). For the “effective” volume of the single oligomer, we have used the equation21
Vi ) a + bi
(31)
with a and b obtained by fitting a straight line to the experimental effective volumes of the first oligomers, in turn obtained from the diffusion binary data relative to PEG 2(2)H2O(0), PEG 3(3)-H2O(0), PEG 4(4)-H2O(0), PEG 5(5)H2O(0), and PEG 6(6)-H2O(0) systems38 (a ) 14.9 ( 8.9 and b ) 52.7 ( 2.1 cm3 mol-1). Once the number-average chain length in, the number-average molecular weight M h n, and the total solute concentration Ct are specified, the concentrations of all the components are computed from eq 29. For simplicity, we have truncated the distribution imax at a value of imax for which (Ct - ∑k)1 Ck) is less than 1% of the total concentration. We have calculated the elements of the diffusion coefficient matrix for the system PEG 300-H2O system at Ci ) 0.1 mol dm-3 and Ct ) 0.5 mol dm-3 (see Table 1). The PEG samples have been considered as composed of the first 17 oligomers, i ) 1-17. The analysis of these two concentrations allows discussing the general behavior of main- and cross-term diffusion coefficients. It is evident from eq 27 that the main-term diffusion coefficients Dii are systematically lower than the corresponding D∞i and Di ) D∞i (1 - 0.898CiVi) calculated at the same molar
Figure 2. Cross-term diffusion coefficients, Dij changing component j, for the PEG 300-water system, corresponding to D3j (curve 1), D5j (curve 2), D6j (curve 3), and D7j (curve 4) at total concentration 0.5 mol dm-3.
concentration of component in the polymeric solution. In Figure1, the Dii for the components of PEG 300 are reported as a function of i at two total solute concentration; for comparison, the trend of D∞i versus i is also reported. As can be seen the main diffusion coefficients, Dii’s decrease with increasing total solute concentration and decrease with increasing i. This trend is due to a combination of the Flory equation for the D∞i , of the “effective” volume Vi, and of the number distribution function (see eq 29). The analyses of the cross diffusion coefficients Dij (i * j) have been done keeping constant, one at a time, i and j. At constant i and varying j, we analyze the cross diffusion coefficients of a single polymeric component, i, with respect to all the other components, j. In this case, eq 28 can be written as
Dij ) KiVj
(32)
where Vj is a linear function of j (eq 31), which makes Dij a linear function of j. In Figure 2, Dij’s, for i ) 3,5,6,7, are reported versus j for the PEG 300-water system. It should be noted that Ki is always positive, and it is not a monotonic function of i. At constant j, and varying i, we analyze the cross diffusion coefficients of all the polymeric components, i, with respect to
11048 J. Phys. Chem. B, Vol. 105, No. 44, 2001
Vergara et al.
TABLE 1: Diffusion Coefficient Matrix for PEG 300a PEG 300. Dij × 105 cm2 s-1 Ct ) 0.1 mol dm-3 i/j
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1.101
2
0.001 0.001 0.001 0.001
0.831 0.001 0.001 0.001 0.001 0.001 0.001
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
0.693 0.001 0.002 0.002 0.001 0.001 0.001
0.001 0.607 0.002 0.002 0.002 0.001 0.001
0.001 0.001 0.002 0.546 0.003 0.002 0.002 0.001 0.001
0.001 0.002 0.003 0.003 0.501 0.003 0.002 0.001 0.001
0.001 0.002 0.003 0.004 0.004 0.464 0.002 0.001 0.001
0.001 0.002 0.003 0.004 0.004 0.003 0.434 0.002 0.001
0.001 0.002 0.004 0.005 0.004 0.004 0.003 0.408 0.001 0.001
0.001 0.003 0.004 0.005 0.005 0.004 0.003 0.002 0.386 0.001
0.001 0.003 0.005 0.006 0.005 0.005 0.003 0.002 0.001 0.367
0.001 0.003 0.005 0.006 0.006 0.005 0.004 0.002 0.001 0.001 0.351
0.002 0.003 0.005 0.006 0.006 0.005 0.004 0.002 0.001 0.001
0.002 0.004 0.006 0.007 0.007 0.006 0.004 0.003 0.002 0.001
0.002 0.004 0.006 0.007 0.007 0.006 0.004 0.003 0.002 0.001
0.002 0.004 0.007 0.008 0.008 0.007 0.005 0.003 0.002 0.001
0.002 0.004 0.007 0.008 0.008 0.007 0.005 0.003 0.002 0.001
0.337 0.324 0.313 0.303 0.293 Ct ) 0.5 mol dm-3
i/j
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0.832 0.001 0.002 0.003 0.003 0.003 0.003 0.002 0.001 0.001
a
2 0.628 0.003 0.005 0.006 0.006 0.005 0.003 0.002 0.001 0.001
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
0.002 0.527 0.007 0.008 0.008 0.007 0.005 0.003 0.002 0.001
0.001 0.003 0.006 0.466 0.011 0.010 0.009 0.006 0.004 0.002 0.001 0.001
0.001 0.003 0.007 0.011 0.424 0.013 0.011 0.008 0.005 0.003 0.001 0.001
0.001 0.004 0.008 0.013 0.016 0.391 0.013 0.009 0.006 0.003 0.002 0.001
0.001 0.004 0.010 0.015 0.018 0.018 0.363 0.011 0.007 0.004 0.002 0.001
0.001 0.005 0.011 0.017 0.021 0.020 0.017 0.338 0.008 0.005 0.002 0.001
0.001 0.005 0.012 0.019 0.023 0.023 0.019 0.014 0.316 0.005 0.003 0.001 0.001
0.001 0.006 0.014 0.021 0.026 0.025 0.021 0.015 0.010 0.296 0.003 0.001 0.001
0.002 0.007 0.015 0.023 0.028 0.028 0.023 0.017 0.011 0.006 0.280 0.002 0.001
0.002 0.007 0.016 0.025 0.031 0.030 0.025 0.018 0.012 0.007 0.003 0.267 0.001
0.002 0.008 0.018 0.027 0.033 0.033 0.027 0.020 0.013 0.007 0.004 0.002 0.255
0.002 0.008 0.019 0.030 0.036 0.035 0.029 0.021 0.014 0.008 0.004 0.002 0.001 0.245
0.002 0.009 0.020 0.032 0.038 0.037 0.031 0.023 0.015 0.008 0.004 0.002 0.001
0.002 0.010 0.022 0.034 0.041 0.040 0.033 0.024 0.015 0.009 0.005 0.002 0.001
0.002 0.010 0.023 0.036 0.043 0.042 0.035 0.026 0.016 0.009 0.005 0.002 0.001
0.236 0.229 0.222 The values lower than 0.001 × 10
-5
2
cm s
-1
are not drawn.
a single component, j. In this case, eq 28 can be written as
Dij ) KjDi∞(1 - 0.898CiVi)(1 + 2.5CiVi)Ci
(33)
where Dij’s have a complicated dependence on i. The relative magnitude of the Dij is a combination of abundance, diffusivity, and size factors of the components. In Figure 3, the Dij, for j ) 14, 16, 17, are reported versus i for the PEG 300-water system. Inspection of Figure 3 shows that the Dij, at constant j, have a maximum corresponding to i of the most abundant component in the polymeric mixture and reflecting the distribution function shape of the oligomers in the polymeric mixture. As a consequence of these trends, the largest cross-term diffusion coefficients Dij can be found for i ) 5 and 6 and for j ) 15, 16, and 17 (see also Table 1 with Ct ) 0.5 mol dm-3). Generally, the largest cross-terms are ones resulting from a combination of high values of the abundance Ci, diffusivity Dii, and volume Vj. Applications to the Gouy Interferometry. If all elements Dij of the diffusion coefficient matrix are known, it is possible to calculate DA and Q0 and to compare these quantities with the experimental ones.37 Several approaches were proposed for studying free diffusion in multicomponent systems.39-42 In what follows, we will use the Toor’s analysis,42 briefly described below. We start from eq 5: in this equation, the Γi parameters are related to the refractive index gradients by the relation
Figure 3. Cross-term diffusion coefficients, Dij, changing component i, for the PEG 300-water system, corresponding to Di14 (curve 1), Di16 (curve 2), and Di17 (curve 3) at total concentration 0.5 mol dm-3.
reported by Miller42 n
Γi )
n
Rj
∑ ∑ Rk(Zk)ij j)1 k)1R
(34)
k
where (Zk)ij is the (i,j) element of the matrix Zk that can be obtained through the following considerations. Let us start from
Nonionic Polymer-Solvent Mixtures
J. Phys. Chem. B, Vol. 105, No. 44, 2001 11049
TABLE 2: Comparison between Experimental Gouy Parameters and Calculated One (Including and Neglecting the Cross-Term Diffusion Coefficients) for PEG i-Water Systems PEG 200
experimental values
weight percent
conc. (mol dm-3)
105DA (cm2 s-1)
0.000 1.093 2.162 4.133 6.000 8.002 9.949 14.172 16.999 21.162 21.624 30.020 45.000
0.0000 0.0546 0.1081 0.2073 0.3016 0.4033 0.5027 0.7199 0.8663 1.0835 1.1078 1.5510 2.3540
0.629 0.622 0.610 0.600 0.582 0.575 0.568 0.546 0.534 0.509 0.513 0.470 0.366
PEG 300
104Q0 18 27 15 23 15 14 13 19 17 23 16 20 18
0.608 0.598 0.589 0.574 0.562 0.551 0.543 0.531 0.528 0.530 0.531 0.563 0.744
experimental values
weight percent
conc. (mol dm-3)
105DA (cm2 s-1)
0.000 1.183 2.906 3.679 4.302 5.650 8.253 15.244 23.071 32.425 39.757
0.0000 0.0394 0.0971 0.1230 0.1439 0.1894 0.2778 0.5189 0.7954 1.1350 1.4086
0.505 0.500 0.492 0.494 0.493 0.479 0.467 0.454 0.410 0.378 0.344
weight percent
conc. (mol dm-3)
105DA (cm2 s-1)
0.000 1.004 1.999 3.920 5.990 7.594 14.999 25.006 45.003
0.0000 0.0251 0.0500 0.0983 0.1507 0.1916 0.3829 0.6489 1.2075
0.438 0.440 0.433 0.429 0.423 0.419 0.409 0.372 0.293
PEG 400
weight percent
conc. (mol dm-3)
0.000 2.356 4.016 5.657 8.196 10.622 15.252 26.443 32.410 39.739
0.0000 0.0393 0.0672 0.0949 0.1380 0.1796 0.2598 0.4590 0.5684 0.7058
0.352 0.346 0.344 0.339 0.338 0.331 0.332 0.301 0.305 0.287
PEG 1000 conc. (mol dm-3)
105DA (cm2 s-1)
0.000 1.012 2.001 4.028 5.803 7.957 14.911 24.985 45.069
0.0000 0.0101 0.0200 0.0404 0.0584 0.0804 0.1524 0.2598 0.4849
0.269 0.269 0.270 0.272 0.275 0.273 0.274 0.268 0.264
104Q0
23.1 23.3 23.5 23.8 24.1 24.5 24.9 25.7 26.4 27.4 27.5 30.4 44.8
0.608 0.592 0.576 0.549 0.526 0.504 0.484 0.446 0.424 0.395 0.392 0.346 0.287
23.1 23.2 23.2 23.4 23.5 23.7 23.9 24.2 24.5 24.8 24.8 25.6 26.7
calculated values (no Di,j)
(cm2 s-1)
104Q0
10 5DA (cm2 s-1)
104Q0
14 11 15 17 14 16 15 13 19 18 15
0.500 0.491 0.479 0.474 0.470 0.463 0.451 0.436 0.443 0.492 0.577
18.6 18.8 19.0 19.0 19.1 19.3 19.6 20.5 21.9 24.4 28.3
0.500 0.485 0.464 0.455 0.448 0.434 0.409 0.355 0.309 0.268 0.242
18.6 18.7 18.7 18.8 18.8 18.8 18.9 19.2 19.6 20.1 20.5
A
calculated values
104Q0
105DA (cm2 s-1)
16 19 15 14 17 15 17 18 16
0.434 0.427 0.421 0.410 0.440 0.394 0.379 0.393 0.613
calculated values (no Di,j)
104Q0
10 5DA (cm2 s-1)
104Q0
15.4 15.5 15.6 15.7 15.9 16.0 16.8 18.1 26.2
0.434 0.423 0.412 0.392 0.373 0.359 0.308 0.258 0.194
15.4 15.4 15.4 15.5 15.5 15.6 15.8 16.1 16.8
calculated values 105D
calculated values (no Di,j)
104Q0
(cm2 s-1)
104Q0
10 5DA (cm2 s-1)
104Q0
8 9 12 7 8 8 7 9 6 8
0.355 0.342 0.334 0.328 0.320 0.315 0.310 0.328 0.360 0.433
11.3 11.4 11.5 11.6 11.7 11.9 12.2 13.1 13.9 15.5
0.355 0.333 0.319 0.306 0.289 0.274 0.249 0.203 0.185 0.167
11.3 11.3 11.3 11.3 11.4 11.4 11.5 11.7 11.8 12.0
experimental values
weight percent
104Q0
104Q0
experimental values 105DA (cm2 s-1)
calculated values (no Di,j) 10 5DA (cm2 s-1)
calculated values 105D
experimental values
PEG 600
calculated values 105DA (cm2 s-1)
A
calculated values
104Q0
105DA (cm2 s-1)
8 6 10 9 10 6 5 10 12
0.274 0.269 0.265 0.258 0.252 0.247 0.239 0.251 0.423
calculated values (no Di,j)
104Q0
10 5DA (cm2 s-1)
104Q0
7.3 7.3 7.3 7.4 7.5 7.5 7.8 8.2 10.8
0.274 0.266 0.259 0.245 0.235 0.223 0.192 0.159 0.118
7.3 7.3 7.3 7.3 7.3 7.3 7.4 7.5 7.7
11050 J. Phys. Chem. B, Vol. 105, No. 44, 2001
Vergara et al.
Figure 4. Comparison between the experimental Gouy parameter DA with calculated ones: including (curve 1) and neglecting (curve 2) the cross-term diffusion coefficients as a function of PEG 300 total concentration.
Figure 5. Comparison between the experimental Gouy parameter Q0 with calculated ones (curve 1) as a function of PEG 300 total concentration.
the following eigenvalues equations
Dvk ) λkvk D†rk ) λkrk
(35)
In these equations, vk and rk are the eigenvectors of the diffusion coefficient matrix and of its transpose respectively, both with the same eigenvalue λk. Finally the Zk matrix is defined as
Zk ) Wkrk†
(36)
Once the n matrices Zk are calculated, we can evaluate Γi parameters through eq 34; then, by substitution of the λi and Γi values in eqs 14 and 15, it is easy to calculate DA and Q0. Comparison between the Experimental and Calculated Gouy Parameters. Matrixes similar to those in Table 1 have been calculated for several PEG samples and used for the pseudobinary analysis described below. For our calculations, we have used the Ri values collected elsewhere.36 The calculated DA and Q0 values for the PEG 200-H2O, PEG 300-H2O, PEG 400-H2O, PEG 600-H2O, PEG 1000-H2O systems are reported in Table 2, and they are compared with the corresponding experimental values.37 In Figures 4 and 5, the data DA and Q0 corresponding to the PEG 300-water system are presented.
The agreement between the experimental and calculated DA and Q0 is fairly good at low PEG concentration. We want to stress here that the predictive procedure uses only diffusion data of binary systems corresponding to the first six oligomers of the PEG series,38 and therefore, the agreements can be considered surprising. At finite concentration (Figure 4), the (DA)calc’s are, in dilute PEG solutions, always smaller than the experimental values, while at increasing PEG concentratio,n they pass through a large minimum and then increase, reaching values much larger than the experimental ones. This can be due to the inability of the eqs 27 and 28 to predict correctly the diffusion coefficients at high polymer concentration.34 This issue was already discussed for the ternary system PEG4-PEG2-H2O.21 At increasing molecular weight or hydrodynamic volume of the diffusion particles, the concentration range corresponding to a good prediction of DA decreases. A fairly good prediction can be obtained in any case up to 20% in weight. We point out that the use of the pseudobinary approximation, which neglects the cross diffusion coefficients Dij with i * j, leads to calculated values much smaller than the experimental ones. This is clearly shown in Figure 4 for the system PEG 300-H2O and confirms even for this case that the pseudobinary approximation can cause very misleading analyses of multicomponent diffusion processes. In the absence of the cross diffusion coefficients, the (DA)calc’s are always a descending function of the polymer concentration. The presence of a large minimum and then an ascending trend, observed considering the entire diffusion coefficient matrix, must be ascribed to an incorrect evaluation of the cross diffusion coefficients through eq 28 at high concentrations. Calculated values of “the area under the deviation graph” at finite PEG concentration are always larger than the experimental ones (Figure 5); the differences between the two values increase at increasing PEG concentration. It seems that a better agreement between (Q0)calc and (Q0) exp is obtained for PEG samples at higher molecular weight, probably because both the molecular weight distribution function and Flory equation work better. Moreover, with increasing average molecular weight, the width of the molecular weight distribution function increases, while the difference between the diffusion coefficients of two consecutive pure oligomers decreases. As a consequence, the system, although more polydisperse from the massive point of view, is less polydisperse from the diffusive point of view. This issue can imply values of Q0 less sensitive to the parameters used in the computation. For “the area under the deviation graph” (Q0), the use of the pseudobinary approximation does not lead to different values. The (Q0)calc values seem to be very insensitive to the presence of cross diffusion coefficients in the calculation. Samples of PEG with average molecular weight greater than 1000 Da are poorly described by our model because of thermodynamic and hydrodynamic reasons.37 In fact, for samples of PEG with average molecular weight greater than 1000 Da, the DA increases with increasing PEG concentration, due to the high value of the thermodynamic factor Bxi of eq 18. The Carter-Phillies approach (eq 20) cannot account for this behavior; therefore, we have not done an analysis for a polydisperse PEG solute with an average molecular weight greater than 1000 Da. IV. Conclusions Equations for the prediction of Dij that had been given and tested for ternary systems are here generalized to n-component
Nonionic Polymer-Solvent Mixtures systems, obtaining the entire diffusion matrix for polymersolvent mixtures. These equations are based on the exclusion concept. They do not include charge interaction and in principle should only be applied to nonionic polymers. The magnitude of cross-term diffusion coefficients, Dij (i * j), is a combination of abundance and diffusivity of the component i and of the volume of the component j. In this work, the validity of our predictive procedure is tested by comparing accurate experimental data, collected by Gouy interferometry, for several PEG-water systems with our calculations. The agreement with the experimental data is good. We point out that only the experimental diffusive behavior of the first oligomers of the PEG series is necessary to predict the DA and Q0 values of polydisperse PEG samples. The predictive procedure can be extended to polymers with any distribution function. The cross-term diffusion coefficients sensitively affect the value of the apparent diffusion coefficient for a polydisperse solute. In contrast, the effect of Dij on the other Gouy parameter (Q0) can be neglected. This issue about the Gouy parameters will be useful in the future analysis of the pseudoternary systems (ternary systems with a constituent being polydisperse). Acknowledgment. This research was supported by an ASI (Italian Space Agency) contract. References and Notes (1) Harnau, L.; Winkler, R.; Reinehker, P. Macromolecules 1999, 32, 5956. (2) Wennerstro¨m, H.; Lindmann, B. Phys. Rep. 1979, 52, 1. (3) Kadima, W.; McPherson, A.; Dunn, M. F.; Jurnak, F. A. Biophys. J. 1990, 57, 125. (4) Bergstro¨mem, M.; Eriksson, J. C. Langmuir 1998, 14, 288. (5) Mennen, M. G.; Smit, J. A. M. Pol. Bull. 1990, 23, 67. (6) Gujirati, P. D. J. Chem. Phys. 1998, 108, 6952. (7) Kang, C. H.; Sandler, S. I. Macromolecules 1988, 21, 3088. (8) Yuste, S. B.; Santos, A.; de Haro, M. L. J. Chem. Phys. 1998, 108, 3683. (9) Daune, M.; Benoit, H. J. Chim. Phys. 1954, 51, 25. (10) Sundelo¨f, L.-O.; So¨dervi, I. Ark. Kemi 1963, 21(15), 143. (11) Sundelo¨f, L.-O. Ark. Kemi 1965, 25 (1), 1. (12) Ambrosone, L.; Ceglie, A.; Colafemmina, G.; Palazzo, G. J. Chem. Phys. 1997, 107, 10756. (13) Vergara, A.; Paduano, L.; Vitagliano, V.; Sartorio, R. J. Phys. Chem. B 1999, 102, 8756.
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