7134
J. Phys. Chem. B 2000, 104, 7134-7143
Segment Distribution of the Micellar Brushes of Poly(ethylene oxide) via Small-Angle Neutron Scattering You-Yeon Won, H. Ted Davis, and Frank S. Bates* Department of Chemical Engineering and Materials Science, UniVersity of Minnesota, Minneapolis, Minnesota 55455
M. Agamalian† and G. D. Wignall National Center for Small-Angle Scattering Research, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 ReceiVed: February 4, 2000
Coronal structures of the micelles spontaneously formed in water by poly(ethylene oxide)-b-polybutadiene (PEO-PB) were investigated using small-angle neutron scattering (SANS). Selective deuterium labeling of the hydrophobic PB block was employed to enhance the neutron scattering contrast between the PEO and PB segments for subsequent structural characterization of their micelles. Analyses of the core- and corona-contrast SANS data based on the intramicellar interference factors led to determination of the segment density profiles of the micellar PEO brushes. Both spherical and cylindrical brushes proved to have coronal layers that are best described in form by a concave concentration profile. The results are consistent with the view that the PEO segments, despite favorable interactions with water molecules, are significantly accumulated near the hydrophobic interface, possibly due to the effect of the strong incompatibility of PB and water.
Introduction Polymer chains attached by one end to a nonpenetrable interface, often referred to as polymer brushes, are widely exploited in many technological applications, including colloidal stabilization and anything that entails the use of polymeric surfactants. Under the geometric constraints that normally accompany a high degree of crowding of the polymers, the individual chains adopt a configuration that is different from that preferred in the unbound state; they are usually stretched away from the interface. The structural properties of the tethered polymer chains have been best discussed in terms of a balance between two opposing tendencies; to maximize favorable contact with the solvent molecules, they prefer to be sparsely distributed and stretched, whereas there is an entropic resistance to this stretching.1,2 Three distinct approaches have emerged for theoretical determinations of the optimal condition that the tethered chains would achieve through the free energy balance. The simplest is the so-called Flory approximation, which adopts unrealistic mean-field assumptions of a homogeneous concentration profile and uniform stretching.3,4 A second approach makes provision for the excluded-volume effects through the use of scaling arguments, and those extended to curved interfaces economically capture the diffusely decaying character of the radial segment density distribution with characteristic power-law exponents of -3/4 for spherical brushes and -1/2 for cylindrical brushes.5-7 More rigorous treatments of the balanced chain stretching based on the self-consistent-field (SCF) models have also been advanced with both analytical8,9 and numerical10-13 solutions and have revealed quantitative structural details that * To whom correspondence should be addressed. E-mail: bates@ cems.umn.edu. † Present address: Argonne National Laboratory, Argonne, IL 60439.
are in partial agreement with the scaling predictions. Theoretical and experimental studies of the dense polymer brushes over the last two decades have arrived at a recognition that the polymer chains terminally attached to flat surfaces assume a parabolic density profile in form,8,9,13,14 while those anchored to highly curved surfaces exhibit a hyperbolic decay of the radial density profile.5-7,10,15 One factor that may influence the structural properties of polymer brushes, and was not considered in the prior studies, is the relative strength of the interaction between the solvent and the tethering substrate. Conditions at which the substratesolvent contacts are far more unfavorable than those between the substrate and brush segments, in fact, can be encountered in many practical situations. Familiar examples include hydrated poly(ethylene oxide) (PEO) chains attached to the hydrocarbon surface by their ends. PEO is one of the most prevalent of watersoluble polymers for various biomedical, pharmaceutical, and environmental applications mainly because of its biocompatibility.16 Prior experimental and theoretical studies revealed that there are specific interactions between PEO and water molecules and, as a consequence, the local rotational isomeric states of the hydrated PEO segments are strongly restricted.17,18 These configurational interactions significantly contribute to the hydrophilicity of PEO, and they are also likely responsible for the solubility gap of the polymer at elevated temperatures, where the PEO chains thermally favor local conformations that are not compatible with the polymer-water interactions.19 The global dimension of the free PEO chains in water is, however, well described by simple self-avoiding random-walk statistics.20 For hydrated PEO chains end-grafted to the hydrocarbon interface, researchers have recognized the possibility of partial condensation of PEO for local shielding of the hydrophobic
10.1021/jp000457v CCC: $19.00 © 2000 American Chemical Society Published on Web 07/12/2000
Segment Distribution of PEO Micellar Brushes domain,21 and some indirect evidence for such an effect has been reported. Carlstro¨m and Halle have suggested that the oligo(EO) segments in the mesophases formed in water from the nonionic surfactants C12H25(OCH2CH2)nOH (C12En) are collapsed near the hydrocarbon core on the basis of the 17O relaxation dynamics measurements.22 Bjo¨rling et al. also showed that application of a Flory-Huggins-type lattice theory to the modeling of the oligo(EO) part of the spherical micelles formed by C12E8 led to a density profile which supports such experimental findings.23 In this paper, we consider the structural properties of the coronal PEO brushes of the micelles formed in water by poly(ethylene oxide)-polybutadiene (PEO-PB) diblock copolymers. SANS was used to determine the segment density distribution of the micellar PEO chains. From a sensitivity point of view, this experimental probe appears to be properly suited to our purpose of investigating the quantitative details of the brush structures. Deuterium labeling of the PB blocks was utilized to enhance the neutron scattering contrast between the PEO and PB segments for subsequent structural characterization of their micelles. On the basis of the assumption that the associated segment distribution of the hydrated PEO chains are relatively insensitive to the presence of heavy water molecules, contrast-matching techniques were employed for characterizing the core and corona dimensions separately. Two basic micellar geometries, spherical and cylindrical, were examined in this work, which addresses whether or not strong solvent-substrate incompatibility induces local accumulation of polymer brushes near the interface. Experimental Section Materials. For this study, two model PEO-PB diblocks were synthesized using the two-step anionic polymerization technique which has been described in detail elsewhere.24 Partially deuterated butadiene monomer (CD4H2) was obtained through the pyrolysis of butadiene-2,2,5,5-d4 sulfone (Aldrich)25 and used for preparation of the monohydroxy-terminated PB precursor (PB-OH). The polymerization was performed with a lithium counterion in tetrahydrofuran (THF) at -60 ( 5 °C, and the resulting PB contains ∼91% 1,2 repeat units. The PEO block was attached to the PB precursor by subsequent polymerization of hydrogenous EO monomers with potassium naphthalenide.24 A melt density of the PB-d4-OH precursor of 0.950 g/cm3 at 20 °C was determined by the density gradient methods.26 The molecular characteristics of the PEO-PB diblock copolymers are summarized in Table 1. Solutions of the block copolymers were prepared by adding the solvent directly to preweighed amounts of dry bulk polymer specimens. The samples were then sealed and equilibrated at ambient temperatures under moderate magnetic stirring at several hertz for at least 1 week. Complete dissolution of the polymers was normally observed within several days. Mixtures of water (H2O, HPLC grade, Aldrich) and heavy water (D2O, 99.9% D, Cambridge Isotopes) with varying compositions were used as received as solvents. For SANS measurements, the premade aqueous solutions of the copolymers were transferred to quartz cells with 1-mm sample thicknesses. Small-Angle Neutron Scattering (SANS). SANS experiments were carried out on the NG-7 30-m instrument of the Cold Neutron Research Facility at the National Institute of Standards and Technology (NIST). Sample-to-detector distances of 1.8, 7.0, and 15.3 m with a monochromated neutron wavelength (λ) of 6 Å and a wavelength spread (∆λ/λ) of 0.11
J. Phys. Chem. B, Vol. 104, No. 30, 2000 7135 TABLE 1: Molecular Characteristics of Block Copolymers sample ID
Mna (kg/mol)
NPEOb
NPBc
fEOd
PDIe
micellar structuref
OB3g OB10h OB13h
4.93 5.38 8.18
55.8 67.2 132.2
66.3 68.8 68.8
0.457 0.494 0.658
1.15 1.10 1.09
cylinder cylinder sphere
a Number-average molecular weight determined from the reaction stoichiometry and the 1H NMR spectrum. b Number of repeat units of the PEO block. c Number of repeat units of the PB block. Here NPEO and NPB were calculated on the basis of the same monomer size. The specific volume of amorphous PEO of VEO ) 65.0 Å3 per EO segment at 20 °C (estimated from published data29) was taken as the reference monomer volume, and the corresponding monomer molecular weights for the hydrogenated and deuterated (d4) PB’s are 35.0 and 37.2, respectively. d Volume fraction of PEO determined from 1H NMR spectrum. e Polydispersity index determined from gel permeation chromatography (GPC). f Identified with cryo-transmission electron microscopy (cryo-TEM).47 g Hydrogenous polymer. h Deuterium content per C4 repeat unit of the PB block was determined to be 3.92 by 1H NMR spectroscopy.
were used to cover scattering wave vectors in the range 0.001 Å-1 < q < 0.3 Å-1. Here the scattering wave vector (q) is defined as q ) (4π/λ) sin (θ/2), where θ is the scattering angle. All measurements were taken at 20 °C. The resulting data were corrected for background electronic noise, nonuniform detector efficiency, solvent plus empty cell scattering, and sample transmission. The scattering intensities for the dissolved polymers were then scaled to absolute values on the basis of direct beam flux measurements.27 This method was checked by alternative calibrations28 obtained through the use of a water standard for the 1.8-m configuration and a silica standard for the 7-m configuration, and the results agreed within 8 and 1% discrepancies, respectively. The statistical uncertainties of the individual data points used for the least-squares analysis are calculated on the basis of the number of averaged detector counts. The polymeric micelles were studied at concentrations of 1 and 3% by weight under eight different contrast conditions. On the basis of densities of the individual materials at 20 °C (that is, dPB-d4 ) 0.950 g/cm3, dPEO ) 1.125 g/cm3,29 dD2O ) 1.105 g/cm3,30 and dH2O ) 0.998 g/cm3,30) and coherent scattering lengths for the atomic constituents (that is, bD ) 0.667 × 10-12 cm, bH ) -0.374 × 10-12 cm, bC ) 0.665 × 10-12 cm, and bO ) 0.580 × 10-12 cm),31,32 we estimate the scattering length densities, FPB-d4 ) 4.440 × 1010 cm-2, FPEO ) 0.637 × 1010 cm-2, FD2O ) 6.355 × 1010 cm-2, and FH2O ) -0.560 × 1010 cm-2. For mixtures of D2O and H2O, the scattering length density of the solvent (Fo) is
Fo ) φD2OFD2O + (1 - φD2O)FH2O
(1)
where φD2O is the volume fraction of D2O. Thus conditions for core and corona contrasts can be obtained with φD2O being equal to 0.173 and 0.723, respectively. Ultrasmall-Angle Neutron Scattering (USANS). USANS measurements were performed on the USANS facility at Oak Ridge National Laboratory using a Bonse-Hart double-crystal diffractometer equipped with triple-bounce Si(111) channel-cut crystals with an additional indentation for a cadmium absorber.33 With a neutron wavelength of 2.59 Å, the q range 3 × 10-5 Å < q < 2 × 10-3 Å was chosen to overlap that of the SANS experiments. All measurements were taken at 20 °C. The sample transmission was measured using a monitor located behind the analyzer crystal. Upon subtraction of all instrumental artifacts, the resulting data were subsequently converted to an absolute
7136 J. Phys. Chem. B, Vol. 104, No. 30, 2000
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differential scattering cross section using the corresponding precalibrated SANS data. Analytical Formulation for the Form Factor Scattering
the core-corona boundary. Using spherical symmetry, we remove imaginary numbers from eq 4 to obtain the simpler formula
∫RR nb(r) [sin(qr)/qr]r2 dr Ab(q) ) ∫RR nb(r) r2 dr m
In the dilute limit of the micellar solutions of amphiphilic molecules where the average intermicellar separation is much greater than the size of the micelles, the scattering taken at the proper q domain, where 1/q is on the order of the micellar size, is dictated by the geometric features of the individual micelles and can be analyzed on the basis of the intramicellar interference factors (alternatively called form factors). For the mathematical description of the form factor scattering from polymeric micelles, we regard the system as a mixture of three types of scattering units of the same size that constitute the micellar core (c) and brushes (b) and the solvent (s), and the subscripts c, b, and s will be used in the rest of this paper. Let us consider a system containing N micelles per unit sample volume, with each of the micelles consisting of Nc core units of scattering length bc and Nb brush units of scattering length bb. Taking the solvent species as the component that fills the vacancies left by the others and collecting all the possible cases of the pair combinations, we obtain an expression for the coherent scattering intensity (or coherent scattering cross section) for the micelles in units of inverse length:
I(q) ) (bc - bs)2NNc2Pcc(q) + (bb - bs)2NNb2Pbb(q) + 2(bc - bs)(bb - bs)NNcNbPcb(q) (2) Here three normalized (that is, Pxy(|q b| ) 0) ) 1) form factors, Pcc(q b), Pbb(q b), Pcb(q b), can be written as the multiplication of the form factor amplitude of one species and the complex conjugate of the form factor amplitude of the other
b) ) Ax(q b) Ay(q b)* Pxy(q
(3)
where the form factor amplitude is the Fourier transform of the average density distribution function for the corresponding scattering unit nx(r b):
r 〈exp(iq b‚b)〉 r db r ∫V nx(b) b) ) Ax(q r db r ∫V nx(b)
(4)
The complexity involved with the actual evaluations of the form factor amplitudes can be greatly reduced owing to the dimensional symmetry of the micellar domains under consideration. Spherical Micelles. Integration of eq 4 for a homogeneous spherical core with radius Rc leads to
Ac(q) )
3 [sin(qRc) - qRc cos(qRc)] (qRc)3
c
m
(6)
c
where Rm is the overall radius of the micelle. The form factors for self- and cross-interferences between the core and corona and the scattering intensity can now be calculated:
Pcc(q) ) [Ac(q)]2
(7)
Pbb(q) ) [Ab(q)]2
(8)
Pcb(q) ) Ac(q) Ab(q)
(9)
Cylindrical Micelles. Just as in the case of semiflexible polymers in dilute solutions,36 the form factor scattering for elongated nonlinear cylindrical (i.e., so-called wormlike) micelles has several distinct q regimes, each of which is connected to the structural features on a length scale of the order of ∼1/q. For the study of the cross-sectional structures of polymeric wormlike micelles, we focus our attention on the high-q domain where q > 1/lp. Here lp denotes the persistence length of the wormlike micelles, which is a measure of the flexibility of the polymer-like objects. In this q region, the scattered intensity will provide information that is independent of micellar length and its polydispersity, and even the flexibility of the long micelles can be effectively ignored. For a quantitative characterization of polymeric wormlike micelles, we derive the form factors for a long, slender, rodlike polymeric micelle of length L and cross-sectional core radius Rc for which L . Rc. In the limit qRc , 1, the form factors can be factored into cross-sectional and axial parts, to a reasonable approximation:
Pcc(q) ≈ [Acs,c(q)]2[Paxial(q)]
(10)
Pbb(q) ≈ [Acs,b(q)]2[Paxial(q)]
(11)
Pcb(q) ≈ [Acs,c(q) Acs,b(q)][Paxial(q)]
(12)
Here the cross-sectional form factor amplitudes for the core and corona species are
Acs,c(q) )
2J1(qRc) qRc
∫RR nb(r) J0(qr) r dr Acs,b(q) ) ∫RR nb(r) r dr
(13)
m
(5)
c
m
(14)
c
which is the classical expression for a homogeneous sphere derived by Rayleigh in 1911.34 To a reasonable approximation, we assume a uniform concentration profile within the micellar core and a sharp interfacial boundary between the core and corona domains. These assumptions are justified by the highly amphiphilic nature of the PEO-PB material, as can be anticipated from the respective Flory-Huggins interaction parameters of χPB/water ≈ 3.5 (at 25 °C and low water content) and χPEO/water < 0.5 (at 25 °C and high water content).35 To calculate the form factor amplitude for the coronal brushes, we introduce a radial distribution function nb(r), which characterizes the falloff of the polymer segment density with distance from
where J0(x) and J1(x) are the zeroth- and first-order Bessel functions of the first kind, respectively, and nb(r) is a radial density profile of the brush species from the long axis. In calculating the form factors for the elongated micelles, we need to introduce an extra step to account for all orientations of the anisometric structure with respect to the scattering momentum transfer (q b), and the proposed simplification above is derived from preaveraging of the angular distribution of the intramicellar interferences due to the finite size along the direction perpendicular to the long axis of the threadlike objects. From a mathematical standpoint, such a preaveraging ap-
Segment Distribution of PEO Micellar Brushes
J. Phys. Chem. B, Vol. 104, No. 30, 2000 7137
Figure 1. Calculated form factors for a homogeneous cylindrical rod with radius Rc (Pcc(q)). Numerical evaluations performed with (solid line) and without (points) the preaveraging approximation produce almost indistinguishable results in the low-q domain. This approximation will be utilized throughout the theoretical modeling for the SANS data taken from the cylindrical micelles.
proximation is not rigorously justified for analysis of high-q data. To test the adequacy of using this approximation in a high-q domain, we performed sample numerical calculations for the form factor (Pcc(q)) for a homogeneous cylindrical core of length L ) 5 µm and core radius Rc ) 74 Å. The results presented in Figure 1 indicate that the error involved in this approximation is less than ∼10% for q < 0.1 Å-1 and can be much reduced at lower q, while it lessens the computational loads by a factor of 100 in numerically evaluating the form factors. At sufficiently high q values (q > 1/lp), the effect of flexibility and overall length of the wormlike micelles on the form factor scattering nearly vanishes and the axial part Paxial(q), the form factor of an infinitely thin chain, further reduces to an asymptotic expression:37
Paxial(q) ≈ π/qL
(15)
Estimates of overall and persistence lengths were performed on the wormlike micelles of OB3 using combined experiments of ultrasmall-angle and small-angle neutron scattering (USANS and SANS). Figure 2 presents the scattering results covering a q range of 3 × 10-5 to 3 × 10-1 Å in a bending rod plot representation (i.e., qI(q) vs q). Data for an intermediate-q domain that presumably correspond to 1/L < q (∼1/lp) < 1/Rc were modeled on the basis of eq 10 (solid line in Figure 2). A least-squares analysis using a value of Rc ≈ 76 Å estimated with cryo-TEM38 and an established methodology for numerical evaluation of the exact form factor Paxial(q) for the KratkyPorod chain39 yielded values for the persistence length lp of 5700 ( 100 Å and the overall length L of 8.8 ( 0.2 µm. From all these considerations, it is evident that the q range suitable for our study of the cross-sectional structures of the polymeric micelles is approximately between 10-2 and 10-1 Å, which is easily accessible using SANS. The scattering intensity can be calculated using eq 2, and further analytical evaluation of the form factors beyond what has been presented in this section is not feasible in general. Smearing Corrections. Now we add two additional corrections to our formalism to account for the nonideality in actual
Figure 2. Bending rod plot representation of the combined USANS and SANS data taken from 1% OB3 wormlike micelles in D2O at 20 °C. A corresponding portion of the scattering curve was simulated using eq 10 and a method for numerical evaluation of the exact form factor Paxial(q) for the Kratky-Porod chain39 (solid curve). The anomalously high intensity in the lowest-q region is likely due to the intermicellar interferences.
SANS data. First, the scattered intensity is greatly affected by the spread of micellar sizes. In this work, we use a zeroth-order log-normal (ZOLD) probability distribution40 for the polydispersity of the core radius Rc
PRc(Rc) )
(
)
(ln Rc - ln R h c)2 1 exp (2π)2σoR h c exp(σo2/2) 2σo2 (16)
where R h c is the modal mean core radius and σo is a measure of the width of the distribution. In terms of σo, the standard deviation σ can be expressed as
σ)R h cxexp(4σo2) - exp(3σo2)
(17)
Using PRc(Rc), the corrected scattering intensity 〈I(q)〉 is
〈I(q)〉 )
∫0∞I(q, Rc) PR (Rc) dRc c
(18)
This treatment of the polydispersity of the core radius is mathematically equivalent to smearing the interfacial boundary between the core and corona regions. For the sake of simplicity, we ignore plausible polydispersity effects arising from variations in parameters that characterize the structural properties of micellar brushes. The second correction is made for the finite experimental q resolution that causes smearing of otherwise sharper features in the scattering data. Major sources of this limitation are wavelength dispersion, the collimation effect, and the spatial detector resolution. Thus, to model the measured SANS pattern, we take a convolution of 〈I(q)〉 with an instrumental resolution function Pq(q, q′) which describes the distribution of q contributing to the scattering at a nominal value of q′
〈〈I(q′)〉〉 )
∫0∞〈I(q)〉R Pq(q, q′) dq c
(19)
7138 J. Phys. Chem. B, Vol. 104, No. 30, 2000
Won et al.
The resolution function Pq(q, q′) can be approximated by a Gaussian distribution function,41 and using a formalism proposed by Pedersen et al.,42 we can write
Pq(q, q′) )
1
x2πσ
[ 21(q -σ q′) ] 2
exp -
(20)
where
σ)
x(λq) σ + (2πλ) σ + (2πλ) σ 2
2
λ
2
2
2
c
2 d
(21)
Here we introduce three variances (i.e., σλ, σc, and σd) that represent smearing contributions from the wavelength spread, the finite collimation, and the detector resolution, respectively. The individual variances can be written
σλ ) σc )
∆λ 2x2 ln 2
x( ) (
(22)
)
R1 2 1 1 + R22 + L1 L1 L2
1 2
1 ∆R σd ) x12 L2
2
(23)
1 exp[(l - lo)/s] + 1
dependent approach also requires the application of least-squares methods for which the deviation of the model prediction from the experimental data is expressed in terms of a parameter χ2: M
χ2 )
(24)
where ∆λ is the full width at half-maximum (fwhm) of the wavelength distribution, R1 is the radius of the source aperture, R2 is the radius of the sample aperture, L1 is the source-tosample distance, L2 is the sample-to-detector distance, and ∆R is the detector resolution. The values used for the SANS measurements are ∆λ ) 0.66 Å, R1 ) 2.54 cm, R2 ) 0.475 cm, L1 ) 547 or 702 cm, L2 ) 180 or 700 cm, and ∆R ) 0.5 cm. Segment Density Profiles of the Micellar PEO Brushes. Calculation of the expected scattering outcome further requires us to assume a detailed model for the density distribution of the brush component in the coronal domain. We chose the brush concentration profile of the form known as the Fermi-Dirac (FD) distribution function43
nb(l) ≈
Figure 3. Examples of the radial distribution of polymer segment density in a micellar corona that can be modeled by the functional form given in eq 25.
(25)
where the brush height l is equal to r - Rc and lo and s are the two parameters that characterize the structural properties of the coronal domain. The values of lo and s determine the curvature and the skewness of the distribution, respectively. As illustrated in Figure 3, various shapes of the concentration profile, such as steplike, Gaussian-like, hyperbolic, and parabolic, can be approximated by this two-parameter function. However, it is impossible to derive analytical expressions for the form factors (i.e., Pbb(q) and Pcb(q)) with FD-type density profiles and the integrals in eqs 6 and 14 will have to be evaluated numerically. Numerical Methods. Direct modeling of the form factor scattering therefore involves the implementation of threedimensional integrals, one associated with the evaluation of the form factor amplitude of the corona (eq 6 or 14) and the others for the smearing corrections (eqs 18 and 19). A FORTRAN code based on an integration routine using the trapezoidal rule44 was written and used to perform the numerical integration. Since the integrands consisting of Bessel or trigonometric functions are not very smooth, this unsophisticated routine appears optimal, and a fractional accuracy of 10-6 was usually achieved within a maximum number of steps of 219. This model-
∑ i)1
(
)
Iexpt(qi) - Icalc(qi) σ
2
(26)
where Iexpt(qi) represents the scattering intensities measured at M data points (i.e., qi, i ) 1, ..., M) with the statistical uncertainties σi and Icalc(qi) represents the calculated scattering intensities at the corresponding data points. Therefore, the procedure for the determination of the best fit parameters aj (j ) 1, ..., N) consists of minimizing the objective function χ2 and the optimal solution should satisfy the condition
∂χ2/∂aj ) 0
(27)
for all aj’s. Results, Analysis, and Discussion Spherical Micelles. In the previous section, we developed the methods for the mathematical description of the form factor scattering for the polymeric micelles. For comparison with the scattering curves evaluated for the proposed scattering density distributions of the core and corona, experimental data were taken at low micelle concentrations in the q range where 1/q is on the order of the cross-sectional size of the micelles probed. Figure 4 compares the scattering intensity profiles obtained for 1.0 and 3.0 wt % solutions of the spherical micelles of OB13. When normalized with respect to the micelle concentration, the two scattering curves virtually coincide over the entire q domain. A minor discrepancy evident in the region of lowest q values likely reflects the effect of intermicellar interference. This result indicates that, at micelle concentrations e3.0%, the structure factor has a negligible effect on the intensity distribution for q > ∼0.01 Å-1 and therefore will be disregarded in the quantitative characterization of the scattering data obtained from 3.0% solutions of the spherical micelles. To determine the core dimension separately as a first step, SANS measurements were performed for the 3% spherical micelles dispersed in a mixture of D2O and H2O with a scattering length density matched to that of the PEO brushes. The data were then modeled by setting the second and third terms in eq 2 to zero and using the form factor for a homogeneous sphere (eqs 5 and 7). A four-parameter fit to the data obtained from the 1.8-m sample-to-detector distance setting
Segment Distribution of PEO Micellar Brushes
J. Phys. Chem. B, Vol. 104, No. 30, 2000 7139
Figure 4. Concentration dependence of the high-q intensity profile for the spherical micelles formed from OB13 in heavy water. The SANS intensities measured for two different micelle concentrations at 20 °C were normalized with respect to the concentration. The open circles represent the 1% data. For the sake of clarity, the individual data points for 3% are not shown in the plot but are instead indicated as an interpolating solid curve.
TABLE 2: Results from Nonlinear Least-Squares Analyses of the SANS Data for the Spherical Micelles of OB13 under Contrast-Matching Conditions core contrast
corona contrast
SDDa ) 1.8 m SDD ) 7.0 m SDD ) 1.8 m SDD ) 7.0 m R h cb (Å) σo lod (Å) sd (Å) lbe (Å) Rf (cm-1) βg (cm-1) χ2/Mh
112 ( 1 0.07c ( 0.01 N/A N/A N/A 72.87 ( 0.04 0.010 ( 0.001 177.5/81
fixed fixed N/A N/A N/A 70.85 ( 0.01 fixed 1655.2/61
fixed fixed fixed fixed fixed 120.28 ( 0.01 0.020 ( 0.001 396.8/37
fixed fixed -28 ( 4 52 ( 2 178 ( 4 164.33 ( 0.02 fixed 2249.1/61
a Sample-to-detector distance. b Core radius. c Using eq 17, the standard deviation for the core radius σ is 8 Å. d Two parameters for the PEO concentration profile in the form of the FD distribution (eq 25). e Corona layer thickness, lb ) Rm (micelle radius) - Rc (core radius). f Proportionality factor which corresponds to (bc - bs)2NNc2 for the core contrast or (bb - bs)2NNb2 for the corona contrast. g Background. In terms of R and β, the calculated scattering intensity can be written as Icalc ) R〈〈P(q)〉〉 + β, where 〈〈P(q)〉〉 denotes the form factor of the core or corona corrected for the core size spread and the instrumental smearing effect. h Number of data points used in the fit.
led to the determination of the values for the core radius R h c, the parameter for the width of the core size distribution σo, a proportionality factor R ()(bc - bs)2NNc2), and background β. The fitting procedure employed was the steepest descent method45 for nonlinear minimization of the χ2 parameter with respect to the multiple variables. The resulting best-fit parameters are summarized in Table 2. In terms of R and β, the calculated scattering intensity can be expressed as Icalc ) R〈〈Pcc(q)〉〉 + β, where 〈〈Pcc(q)〉〉 denotes the form factor of the core corrected for the core size spread and the instrumental smearing. The additional constant background correction accounts for a small amount of incoherent scattering that had not been removed from the experimental data. In fact, the incoherent scattering can be estimated on the basis of the published data for the incoherent scattering cross section σinc, that is, 79.7 × 10-24 cm2 for hydrogen and 2.01 × 10-24 cm2 for deuterium,31 and we obtained the value of 0.009 cm-1 for a 3.0 wt % solution
Figure 5. SANS data measured at 20 °C from 3.0% spherical micelles of OB13 under core-contrast conditions. The scattering curves evaluated for R h c ) 112 Å and σo ) 0.07 (dark solid lines) represent least-squares fits to the experimental data (error bars) from the two experimental settings.
of OB13, which is in good agreement with the background estimated from the fitting, β ) 0.010 ( 0.001 cm-1. Fitting the 7-m core contrast data for 0.010 Å-1 < q < 0.055 Å-1 at fixed values of R h c ) 112 Å, σo ) 0.07, and β ) 0.010 led to a value of the prefactor R ) 70.85 cm-1 that did not change significantly (Table 2). These results are consistent with an alternative determination of the value of R ) (bc - bs)2NNc2 ) 83.61 cm-1 based on the measured core radius R h c. As shown in a log-log presentation in Figure 5, the resulting least-squares fits well reproduce the quantitative features of the SANS data, including the slight discrepancy of the data from the two experimental configurations due to the difference in instrumental smearing. The discernible deviation of the fit from the data evident in the lowest-q region (q < 0.01 Å-1) indicates the influence of the intermicellar correlation and is likely reflected in the marked increase in χ2 associated with the 7-m data fitting. In a next step, the scattering data taken under corona contrast conditions were modeled by setting the first and third terms in eq 2 to zero and using the form factor derived for a diffuse spherical shell (eqs 6 and 8) with the coronal density profile of the form of the FD distribution (eq 25). On the basis of the values for R h c, σo, and β obtained from the core characterization discussed above, we used a four-parameter fit to the 7-m data to determine the parameters for a description of the brush profile (lo and s; see eq 25), the coronal layer thickness (lb), and a proportionality factor (R ) (bb - bs)2NNb2). Again, the scattering intensity for this core-contrast-matched condition can be written as Icalc ) R 〈〈Pbb(q)〉〉 + β where 〈〈Pbb(q)〉〉 is the form factor of the corona corrected for the core size spread and the instrumental smearing. We ignored the polydispersity effects arising from variations in corona thickness (lb) which numerically proved to be less significant than the core size spread. The resulting fit parameters are listed in Table 2. Application of the determined values for lo, s, and lb to a description of the 1.8-m data required some modification of both R and β. This likely reflects the limitations of the functional form adopted for the PEO density profile. In a double-log presentation as used in Figure 6, the overall fits of the experimental data for the core-matching condition appear reasonably good over the entire q range evaluated, although not as good as those observed for the corona-matching condition. On the basis of the proportionality
7140 J. Phys. Chem. B, Vol. 104, No. 30, 2000
Won et al.
Figure 6. SANS data measured at 20 °C from 3.0% spherical micelles of OB13 under corona-contrast conditions. The scattering curves evaluated for lo ) -24 Å and s ) 52 Å for the PEO concentration profile (eq 25) and a coronal thickness lb of 178 Å (dark solid lines) represent least-squares fits to the experimental data (error bars) from the two experimental settings.
factors determined separately from the least-squares analyses of the 7-m core and corona contrast data, we estimated the ratio between the numbers of the core and brush scattering units per micelle as Nc/Nb ) 0.657, which compares satisfactorily with a value of 0.515 calculated from the molecular characteristics of OB13. The self-consistency of the above results with contrast variations was further examined using 3% solutions of OB13 at six different contrasts beyond the core- and corona-matching conditions. As shown in Figure 7, there is a remarkable influence of contrast on the scattering intensities. Solely on the basis of the previous model data and values of NNc2 ) 1.19 × 1025 cm-3, NNb2 ) 2.77 × 1025 cm-3, NNcNb ) 1.82 × 1025 cm-3, and β ) 0.010 cm-1, the analytical expressions for the form factor scattering derived in the previous section are capable of quantitatively reproducing the variation of the scattering curves with contrast, confirming the adequacy of the results and the fitting procedure. Cylindrical Micelles. The basic procedure for characterizing the cross-sectional structure of the cylindrical micelles of OB10 is identical to that for the previous case of the spherical micelles. To determine whether or not the modeling procedure based on the intramicellar interference factors is applicable to the systems of cylindrical micelles at a comparable concentration and q range, we examined the influence of the micellar concentration on the SANS profile. Figure 8 displays the absolute intensities from 1.0 and 3.0 wt % solutions of the cylindrical micelles normalized by the respective concentration. It is evident that the effect of intermicellar correlations is negligible over the measurement q range, and the form factor analyses were performed on the samples at the concentration of 3.0%. Figure 9 shows the scattering curve of the cylindrical core with the corona being made indistinguishable with the solvent by the contrast-matching technique. We considered only the first term of eq 2 with the form factor for a homogeneous linear cylinder with radius Rc and length L (eqs 10, 13, and 15). For L > 103 Å, the resulting shape of Pcc(q) is insensitive to L, and we chose an arbitrary value of L ) 105 Å. Over the entire q domain studied, the axial component of the form factor for the cylindrical micelles can be calculated by Paxial(q) ) 3.144 ×
Figure 7. SANS data measured at 20 °C from 3.0 wt % spherical micelles of OB13 with varying contrasts (points). Water (H2O), heavy water (D2O), and their mixtures were used as the solvents. The volume fractions of D2O in the solvent were 0, 0.17, 0.32, 0.39, 0.48, 0.60, 0.72, and 1 from top to bottom, and for the sake of clarity, the top seven data sets were shifted upward by factors of 107, 106, 105, 104, 103, 102, and 10 from top to bottom. The solid curves were produced on the basis of the best-fit parameters (R h c, σo, lo, s, lb, R, and β) that resulted from least-squares analyses of the core- and corona-contrast data.
Figure 8. Concentration dependence of the intensity profile in the relatively high q domain for the cylindrical micelles formed from OB10 in heavy water. The SANS intensities measured for two different micelle concentrations at 20 °C were divided by the concentration. The open circles represent the data taken from a 1.0% sample, and the data for a 3.0% sample are shown as a solid curve.
10-5/q to within a 0.1% accuracy. A four-parameter fit to the 1.8-m core contrast data yielded values for R h c, σo, R, and β that are given in Table 3. The measured value of the incoherent scattering from the block copolymer β ) 0.034 cm-1 is significantly greater than the value of 0.008 cm-1 estimated for a 3.0% OB10 solution. However, the resulting best-fit values
Segment Distribution of PEO Micellar Brushes
J. Phys. Chem. B, Vol. 104, No. 30, 2000 7141
TABLE 3: Results from Nonlinear Least-Squares Analyses of the SANS Data for the Cylindrical Micelles of OB10 under Contrast-Matching Conditionsa core contrast R h c (Å) σo loe (Å) se (Å) lbf (Å) Rg (104 cm-1) βh (cm-1) χ2/Mj c
corona contrast
SDDb 1.8 m
SDD ) 7.0 m
SDD ) 1.8 m
SDD ) 7.0 m
74 ( 1 0.08d ( 0.01 N/A N/A N/A 3.227 ( 0.001 0.034 ( 0.001 116.8/81
fixed fixed N/A N/A N/A 3.250 ( 0.002 fixed 107.5/61
fixed fixed fixed fixed fixed 2.0398 ( 0.0003 fixedi 1649.3/36
fixed fixed -68 ( 4 36 ( 2 132 ( 4 2.7910 ( 0.0003 fixedi 4937.0/60
a Throughout the fitting procedure, the length of the cylindrical micelles was fixed at L ) 10 µm. The results are insensitive to this assumption. Sample-to-detector distance. c Cross-sectional radius of the core. d Therefore, the standard deviation for the core radius σ is 6 Å (eq 17). e Two parameters for the PEO concentration profile in the form of the FD distribution (eq 25). f Corona layer thickness, lb ) Rm (micelle radius) - Rc (core radius). g Proportionality factor which corresponds to (bc - bs)2NNc2 for the core contrast or (bb - bs)2NNb2 for the corona contrast. h Background. In terms of R and β, the calculated scattering intensity can be written as Icalc ) R〈〈P(q)〉〉 + β, where 〈〈P(q)〉〉 denotes the form factor of the core or corona corrected for the core size spread and the instrumental smearing effect. i An estimated value of 0.008 cm-1 for the 3 wt % OB10 solution was used. j Number of data points used in the fit. b
Figure 9. SANS data measured at 20 °C from 3.0% cylindrical micelles of OB10 under core-contrast conditions. The scattering curves evaluated for R h c ) 74 Å and σo ) 0.08 (dark solid lines) represent least-squares fits to the experimental data (error bars) from the two experimental settings.
Figure 10. SANS data measured at 20 °C from 3.0% cylindrical micelles of OB10 under corona-contrast conditions. The scattering curves evaluated for lo ) -68 Å and s ) 36 Å for the PEO concentration profile (eq 25) and a coronal thickness lb of 132 Å (dark solid lines) represent least-squares fits to the experimental data (error bars) from the two experimental settings.
for the associated core radius, polydispersity, and proportionality were almost insensitive to the variation of the background between those two quantities. The validity of these results, as in the previous case for the spherical micelles, was checked by reproducing the 7-m data in the q range 0.010 < q < 0.055 with
