Interfaces in Self-Assembling Diblock Copolymer Systems

Filip Uhlík, Zuzana Limpouchová, Pavel Matějíček, and Karel Procházka , Zdeněk Tuzar , Stephen E. Webber. Macromolecules 2002 35 (25), 9497-950...
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J. Phys. Chem. B 1999, 103, 2487-2495

2487

Interfaces in Self-Assembling Diblock Copolymer Systems: Characterization of Poly(isoprene-b-methyl methacrylate) Micelles in Acetonitrile J. P. S. Farinha,† Karin Schille´ n,‡ and M. A. Winnik* Department of Chemistry, UniVersity of Toronto, 80 St. George Street, Toronto, Ontario, Canada M5S 3H6 ReceiVed: NoVember 10, 1998

The kinetics of dipolar nonradiative energy transfer (DET) between dyes confined to the core-corona interface region of poly(isoprene-b-methyl methacrylate) block copolymers (PI-PMMA) in acetonitrile was analyzed using a new distribution model for energy transfer in spherical micelles. The distribution of block junction points was described by the model of Helfand and Tagami (HT) for the strong segregation limit, adapted for the spherical geometry of the core-corona interface. We used this model to analyze experimental fluorescence decay curves for block copolymer micelles made up of polymers containing a donor dye or an acceptor dye covalently attached to the PI-PMMA junction. The analysis yielded an interface thickness between the PI core and the PMMA corona of δ ) (0.9 ( 0.1) nm. In the past, the experimental fluorescence decay curves measured for similar systems have been fitted with the Klafter and Blumen (KB) equation for energy transfer, which has a stretched exponential form. To relate these results to topological characteristics of the system, we simulated donor decay profiles for different interface thickness values using the new distribution model for energy transfer and a modified HT equation. Subsequent analysis by the stretched exponential KB equation proved that the magnitude of the fitted exponent is directly related to the interface thickness between the blocks for a given dye concentration in the core-corona interface. Within a certain range of interface thickness values, this relation can be used to determine the interface thickness from the fitting parameters of the KB equation.

Introduction Block copolymers consist of sequences of two types of monomers (A and B) linked at a common junction. If the block copolymer is dissolved in a solvent selective for polymer A, then the polymer will spontaneously form micelles with a corona of solvent-swollen A chains surrounding a core of insoluble B chains. A broad variety of applications have been suggested for these micelles, ranging from small-molecule delivery vehicles1 to surface modification agents.2 Because of these potential applications and because of the intrinsic interest in self-assembling systems, block copolymer micelles have been studied for many years.3-9 Several techniques have been used to examine the structure of block copolymer micelles. The most reliable results on micelle structure have come from small-angle scattering experiments involving light,10,11 X-rays,12,13 and neutrons.14,15 Transmission electron microscopy has also been used to characterize micelle structures, particularly for nonspherical micelles,16-18 and to follow the equilibration of the micelle morphology with time.19 One pathway to equilibrated structures involves the exchange of individual block copolymer molecules between the micelles and the solution.20-22 The rate of this process depends sensitively on the length of the chains making up the micelle core and their solubility in the selective solvent. * To whom correspondence should be addressed. E-mail: mwinnik@ chem.utoronto.ca. † On leave from Centro de Quı´mica-Fı´sica Molecular, complexo I, Instituto Superior Te´cnico, Av. Rovisco Pais, 1096 Lisboa codex, Portugal. E-mail: [email protected]. ‡ Present address: Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, S-221 00 Lund, Sweden. E-mail: [email protected]

Despite the strong interest in block copolymer micelle structure, little is known about the nature of the interface between the core and the corona. We have a better understanding of the structure of the polymer-polymer interface for block copolymers in the bulk state. In bulk systems, the interface structure results mainly from the balance between the unfavorable energy of mixing the segments of the two blocks and the entropy loss due to the chain stretching necessary to minimize the interfacial area. For example, thin films of poly(styrene-bmethyl methacrylate) block copolymers (PS-PMMA)23,24 and poly(styrene-b-butyl methacrylate) block copolymers (PSPBMA)25 have been studied by neutron reflectivity. These experiments have been interpreted in terms of an interface thickness of 5.1 nm for PS-PMMA and 7.7 nm for PS-PBMA. For poly(isoprene-b-styrene) block copolymer films (PS-PI), an interface thickness of δ ) 1.8 nm was inferred from SAXS measurements.26,27 Energy transfer (DET) experiments on films of poly(isoprene-b-methyl methacrylate) block copolymers (PIPMMA) were interpreted to indicate an interface thickness of δ ) 2.5 nm.28 For block copolymer micelles, one also has to consider the swelling of the polymer by the solvent. For example, if the solvent quality decreases with respect to the core polymer, one would expect the interface thickness to decrease, because the polymer segments at the core will try to avoid contact with the solvent-swollen corona. We recently reported energy transfer experiments on poly(isoprene-b-methyl methacrylate) block copolymer (PI-PMMA) micelles in acetonitrile solution.29 To obtain information about the core structure of these micelles, we used PI-PMMA labeled with an energy donor (phenanthrene) or an acceptor (anthracene) dye at the junction between the two blocks. When dissolved in acetonitrile (CH3CN), a solvent selective for PMMA, the PI-

10.1021/jp9843858 CCC: $18.00 © 1999 American Chemical Society Published on Web 03/16/1999

2488 J. Phys. Chem. B, Vol. 103, No. 13, 1999 PMMA copolymer formed micelles with a PI core and a PMMA corona. In these micelles, the phenanthrene and anthracene dyes attached to the junction points between the blocks become concentrated in the interface. Measurements of the rate of energy transfer from the donor to the acceptor are sensitive to the distribution of the dyes in the interface. With an appropriate model, these measurements can give information about the size of the core, and, as we show here, they can also provide information about the nature of the core-corona interface. Direct nonradiative energy transfer (DET) experiments have been used by us29-31 and others32 to obtain qualitative or semiquantitative information about the core-corona interface structure of micelles composed of donor- and acceptor-labeled block copolymers. The experimental fluorescence decay curves obtained for these systems were analyzed using the Klafter and Blumen (KB) equation for energy transfer, initially derived for fractal systems33 but applied phenomenologically to systems of restricted geometry. The fitting parameters obtained from this analysis give some information on the sharpness of the interface between the micelle core and shell.29-31 To gain deeper structural information about the nature of this interface, a new model is proposed. It represents a special case of the more general model describing direct energy transfer in systems in which the distribution of donor and acceptor dyes is characterized by spherical symmetry.34-36 This model takes into account the size and shape of the domains in which the dyes are distributed and the variation in their concentration over the domains. To proceed with the analysis, one must input a function describing the radial variation of the dye concentration. In the present case, this variation is related to the distribution of block junction points at the interface. Since acetonitrile is a very poor solvent for PI,29 we can assume that the solvent does not swell the PI core. Under these circumstances, the interface maintains its general symmetric shape and the solvent affects the core only through a change in the interface thickness, as suggested by simulation results.37 In this case, the distribution of junction points will be described by an inverse hyperbolic cosine function according to the model of Helfand and Tagami (HT) for interfaces in strongly segregated polymers.38 Because the HT model was derived for lamella structures, it has to be modified to account for the curvature of the interface in block copolymer micelles. One such function was derived by Semenov and used to examine the structure of an isolated spherical micelle.39 The function derived by Semenov, however, is only valid when the core radius is much larger than the interface region so that the micelle core-corona interface can be considered approximately planar. We suggest a new modification of the HT model that can be used even when the interface region covers a large fraction of the core volume. By using this function to describe the distribution of dyes localized at the block junction and the proposed DET distribution model, we are able to interpret DET experiments in terms of the thickness of the interface between the PI and PMMA blocks. Since the KB equation has been widely used to fit fluorescence decay curves for DET experiments involving dyes in restricted domains, we simulated data to investigate the connection between the fitting parameters for this model and the “true” structure of the interface. One of the fitting parameters in the KB model is an “apparent dimension” which often takes a fractional value in this analysis. As pointed out by El-Sayed,40 the apparent dimension d obtained when using the KB model to fit fluorescence decay curves of systems with dyes in restricted geometry depends on the details of the dye distribution.

Farinha et al. TABLE 1: Characteristics of the Poly(isoprene-b-methyl methacrylate) Diblock Copolymers chromophore phenanthrene anthracene no label

labeling efficiency Mna (PI/PMMA) Mw/Mnb NPI-NPMMAc 98% 100%

10.2 K/46.6 K 10.3 K/54.9 K 8.4 K/48.7 K

1.11 1.09 1.17

150:470 150:550 125:485

a Number-averaged molecular weight of the two blocks obtained from gel permeation chromatography (GPC). b Polydispersity index, where Mw and Mn are the weight- and number-averaged molecular weight obtained from GPC. c Calculated number of isoprene and methyl methacrylate monomers per polymer chain.

In this paper, we are interested in finding out if there is any systematic change of d with changes in the thickness of the interfacial domains where the dyes are distributed. Using the new distribution model of energy transfer, we carried out simulations of fluorescence decay curves of dyes distributed in the core-corona interface of block copolymer micelles and then analyzed these decay curves with the KB equation. The apparent dimension d fitted in this way was found to increase with an increase in the magnitude of the interface thickness δ. From this relationship and from the values of d obtained from the fitting of the experimental decay curves with the KB equation, we show that it is possible to deduce with good precision the thickness of the interface between the core and the corona in these block copolymer micelles. The paper is organized as follows. We begin with a brief description of the experimental work. We then present the models which serve as the basis for this paper, the structural (Helfand-Tagami) model for the distribution of junctions across the block copolymer interface, and the two models employed for describing energy transfer kinetics (the KB equation and the DET distribution model). This presentation is followed by (i) an analysis of the experimental energy transfer data using the KB equation (dyes on the surface of a sphere), (ii) the simulation of fluorescence decay curves using the DET distribution model, and (iii) the analysis of the experimental data using the DET distribution model. Finally, we compare the KB and the DET distribution models for the analysis of experimental results and data simulated for energy transfer between junction points in the block copolymer interface. We end with our conclusions. Experimental Section All the experimental data reported here has previously been published.29 In this paper, we fit these data to different models. The PI and PMMA diblock copolymers (labeled with either phenanthrene or anthracene at the block junction) were synthesized as previously described.28,41 Unlabeled PI-PMMA copolymers of similar molecular weight and block lengths were also prepared. The characteristics of the polymers are summarized in Table 1. The labeling efficiency for phenanthrene was 100% and for anthracene 98%. Acetonitrile and dichloromethane used in the sample preparation were of spectroscopic grade (Aldrich). The micelle solutions were prepared by first dissolving the mixture of donor- and acceptor-labeled block copolymers in the chosen ratio in dichloromethane, a good solvent for both blocks.31 This solution was then evaporated and dried under vacuum at 50 °C for 8 h. Acetonitrile was then added to dissolve the block copolymers, which self-assembled into micelles with a core of PI and a corona of PMMA. These micelles contain a mixture of polymers labeled with donors and acceptors at the interface between blocks. The total polymer concentration was

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kept at 0.25 wt %. The dynamic light scattering measurements referred to here were described in detail in a previous publication.29 For the fluorescent measurements, the micelle solutions were deoxygenated with argon and kept at 22-25 °C. Steady-state fluorescence measurements were carried out with a SPEX Fluorolog 212 fluorescence spectrometer. Fluorescence decay measurements were obtained by the single photon timing technique.42 Excitation with a deuterium-filled coaxial flash lamp give a nanosecond resolution. Phenanthrene was excited at 300 nm, and the emission was collected at 348 nm, using a cutoff filter in front of the detector in order to decrease the scattered light. The instrumental response function was obtained by the mimic lamp method,43 from the decay of a solution of p-terphenyl in aerated cyclohexane (τ ) 0.96 ns). Correction of the scattered light contamination of the decays was done according to Martinho et al.44 Some scatter of the excitation light is inevitable in micellar solutions. One aims to achieve a balance between minimizing the contribution of the scattered light against the cost of a decrease in the signal acquisition rate. Measurements at very low count rates need longer acquisition times and are more susceptible to possible excitation lamp instability.

Figure 1. Radial distribution of junction points PJ(r) and probability of finding the junction points at a distance r from the center of the micelle 4πr2PJ(r), for two micelles with core radii Rcore ) 2δ (s) and Rcore ) 4δ (‚‚‚). The centers of the micelle core-corona interface are marked with vertical lines.

Results and Discussion Structure of the Polymer Interface. In block copolymer melts with a lamellar structure, the distribution of junctions across the interface is described by an inverse hyperbolic cosine function, first derived for strongly segregated systems by Helfand and Tagami (HT).38 This function is symmetric with respect to the center of the interface. It is less peaked and has broader wings than a Gaussian distribution with the same width at half-maximum. In modeling DET in block copolymer micelles in a selective solvent, we will assume that the junctions have an HT distribution across the interface, even when the swelling of the micelle corona or core-corona interface leads to broadening of the interface. To proceed, we modify the HT expression to account for the spherical shell geometry of the micelle core-corona interface as

PJ(r) )

1 nJ cosh[2(r - Rcore)/δ]

(1a)

2

∫0∞cosh[2(r r- R

core)/δ]

dr

A somewhat different junction distribution function was proposed by Semenov for the case of an isolated spherical block copolymer micelle39

SJ(r) )

where δ is the interface width defined in the same way as in the original HT model. Rcore is the radius of the micelle core, and nJ is a normalization constant. Integration over the micelle volume to account for the curvature of the core-shell interface yields

nJ ) 4π

Figure 2. Probability of finding the junction points at a distance r from the center of the micelle 4πr2PJ(r), for a micelle with core radii Rs ) 8.5δ (corresponding to the approximate conditions of our PIPMMA micelles in acetonitrile). The spherical shell volumes containing 60%, 90%, 95%, and 99% of all the junctions have total widths ∆T ) δ, 2.5δ, 3δ, and 5δ, respectively.

(1b)

The distribution function of block junctions defined by eq 1 implicitly describes the effect of increasing concentration toward the center of the micelle, caused by the spherical shell shape of the interface. In the context of the assumptions made, it provides a good approximation for spherical micelle systems, even when the core radius is comparable in size to the interfacial region. The shape of the junction distribution obtained in this way is similar to that calculated by Whitmore et al. from the simulation of a block copolymer micelle in a nonsolvent for the core and a θ-solvent for the corona chains.37

2

4π

Rcore2δ

1 cosh[2(r - Rcore)/δ]

(2)

This expression, however, is only valid when the interface region is much smaller than the micelle core. If the core radius is comparable in size to the interface, the distribution describing eq 2 is no longer normalized and cannot be used to calculate the junction density profile across the interface. Therefore, we will use our modification of the HT expression (eq 1) in the calculations described below, instead of the Semenov expression (eq 2). In Figure 1 (top), we plot the radial distribution of junction points calculated with eq 1, for micelles with core radii Rcore ) 2δ and Rcore ) 4δ, where δ is the interface thickness as defined in eq 1. For the micelle with the smaller core radius, the range of the distribution is larger than the core dimensions. If, however, we consider the probability of finding the junction points at a distance r from the center of the micelle, 4πr2PJ(r), the probability of having a block junction at the center of the micelle core vanishes (Figure 1, bottom). In this case, the spherical shell with the highest probability for the junction points

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Farinha et al.

(the maximum of the radial distribution) is shifted to the outside of the center of the interface. In Figure 2, we plot the probability of finding junction points at a distance r from the center of a micelle with a core radius Rcore ) 8.5δ. (These parameters approximate the conditions of our experiments on PI-PMMA micelles in acetonitrile.) The widths of the spherical shells that contain 99%, 95%, 90%, and 60% of all the junction points, are approximately 5δ, 3δ, 2.5δ, and δ. This means that, when we consider a interface of thickness δ, about 40% of all the block junction points lie outside the spherical shell of width δ centered at the corecorona interface. Energy Transfer Kinetics. For a dipole-dipole coupling mechanism, Fo¨rster45,46 showed that the rate of energy transfer w(r) between a donor and an acceptor depends on their separation distance r

w(r) )

R r6

(3a)

3R06κ2 R) 2τD

(3b)

where τD is donor fluorescence lifetime in micelles labeled with donor only. R0 is the critical Fo¨rster distance, with R0 ) (2.3 ( 0.1) nm for the present system,32 and κ2 is a dimensionless parameter related to the relative orientation of the donor and acceptor transition dipole moments. Since the micelle core is in the molten state and the corona is flexible, we assume that the dipoles are randomly oriented and rotate freely during the transfer time. In this case, we can use the value κ2 ) 2/3.47 According to eq 3, if the donors and acceptors are homogeneously distributed in an infinite volume with a constant separation r between each pair, the donor decay function is still exponential but faster than the decay of the donor alone

( )

ID(t) ) exp -

t exp[-w(r)t] τD

(4)

On the other hand, if the donors transfer energy to a random distribution of acceptors, the donor decay function will have a stretched exponential form

( ) [ ( )]

ID(t) ) exp -

t t exp -P τD τD

β

(5)

where β ) d/6. In eq 5, d is the Euclidean dimension of the space in which the chromophores are distributed, and P is a parameter proportional to the local concentration of acceptors. This expression was first derived by Fo¨rster45,46 for DET in three dimensions (d ) 3) and later extended to one (d ) 1) and two (d ) 2) dimensions by Hauser et al.48 Klafter and Blumen (KB) showed that eq 5 also describes DET for donors and acceptors embedded in an infinite fractal lattice. Here d is equal to the fractal (Hausdorff) dimension of the lattice.33 The parameter P in eq 5 also depends on the averaged relative orientation of the donor and acceptor dipole moments κ2

( )

P ) c∆

3κ2 β Γ(1 - β) 2

(6)

where c∆ is the number of acceptors in a ∆-dimensional sphere of radius R0, and Γ is the gamma function. For one, two, and three dimensions, ∆ equals the Euclidean dimension d of the

space where the chromophores are distributed. In the case of DET on a fractal lattice, some assumptions have to be made in order to use the information contained in c∆. When donors and acceptors are distributed in a restricted volume in which all donors occupy equivalent positions, Blumen, Klafter and Zumofen derived a general expression,49

( )

t φ(t) τD

(7a)

∫V{1 - exp[-w(r)t]}F(r) dV)

(7b)

ID(t) ) exp φ(t) ) exp(-p

where φ(t) is the donor survival probability with respect to DET, V is the volume of the microdomain, p is proportional to the acceptor concentration, F(r) is proportional to the probability distribution of finding a donor-acceptor pair separated by a distance r, and w(r) is the rate of energy transfer given by eq 3. This expression can be integrated for a number of interesting cases, including D and A distributed in the surface of a sphere, for which the donor survival probability with respect to DET, φ(t), is given by50,51

φ(t) ) exp[-g(t)]

(8a)

∫02R r{1 - exp[-w(r)t]} dr

g(t) ) 2πCA

s

(8b)

where CA is the acceptor surface density, and Rs is the radius of the sphere (Rs ) Rcore for the labeled PI-PMMA micelles treated in this paper). Equation 8b can be expressed as35

[

( )( )

g(t) ) g2D 1 + 0.0231

R0 Rs

4

t tD

2/3

-

7.21 × 10-5

( )( ) ] R0 Rs

10

t τD

5/3

+ ... (9)

where g2D ) P(t/τD)1/3 is the exponent in eq 5 for an infinitely extended two-dimensional space. Because of the (R0/Rs)4 factor in the second term on the right-hand side of eq 9, this term and higher terms become insignificant for values of Rs > R0. When this term is negligible, DET for donors and acceptors homogeneously distributed in the surface of a sphere can be described by eq 5 with β ) 1/3. For more complex systems of restricted geometry, particularly those in which there is a distribution of nonequivalent donor positions, eq 5 has been called the KB equation and employed as a phenomenological expression for data analysis. Under these circumstances, β loses its meaning in terms of a dimensionality of space and appears to be sensitive primarily to edge effects of the confining geometry.40 Some information can usually be obtained from P through eq 6, where c∆ has been calculated assuming that ∆ represents the average three-dimensional concentration of acceptors in a sphere of radius R0 surrounding an average donor.28 Within this approximation, all donors are treated as though they are situated in equivalent environments. The analysis assumes that any sphere of radius R0 around a donor is representative of the entire interface. The volume of this sphere should then equal the product of the number of block copolymer chains inside the sphere and the volume occupied by each junction. In this context, the interface volume occupied per junction equals VJ ) fAV∆/c∆, where fA is the number fraction of acceptor molecules in the micelle, and V∆ is the volume of the ∆-dimensional sphere of radius R0. The interface volume

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occupied per junction can then be calculated as

VJ )

fA 4 πR 3 c∆ 3 0

(

)

(10)

Within this approximation, DET data has been fitted to eq 5 for junction-labeled block copolymers in the bulk state. Interface thicknesses ranging from 2.5 nm for PI-PMMA28 to 5 nm for PS-PMMA41 were inferred from this phenomenological analysis of the decay curves. These values of δ are only approximate because the analysis neglects the details of the donor and acceptor distribution across the restricted geometry of the interface. There is an interesting paradox associated with fitting donor decay profiles from block copolymer micelles to eq 5. If the core-corona interface is sharp, the donors and acceptors are distributed on the surface of a sphere. If this sphere is large compared to R0, eq 5 with β ) 1/3 is rigorously correct. If, however, the interface has a finite thickness, incursion of the third dimension makes analysis of the data via eq 5 phenomenological. The following sections in this paper are devoted to examination of this crossover from a two-dimensional-like spherical surface to a three-dimensional system with a thin but finite interface. Diblock copolymer micelles made up of junction-labeled polymers represent a system far more complex than the situations previously considered by Blumen, Klafter, and Zumofen.49 In the present system, there is a distribution of nonequivalent donor positions and the fluorescence decay rate depends on the donor and acceptor distributions at the interface. According to the distribution model for energy transfer in spherical systems,34-36 the donor decay function for a δ-pulse excitation will be given by

( )∫

ID(t) ) exp φ(t,rD) ) 2π exp rD

(

t τD

C (r )φ(t,rD)rD2 Vs D D

drD

(11a)

)

+r CA(rA)rA drA]r dr ∫R∞ {1 - exp[-w(r)t]}[∫|rr -r| D

e

D

(11b) where Vs is the micelle volume, τD is the intrinsic fluorescence lifetime of the excited donor, and CD(r) and CA(r) are the concentration profiles of donor and acceptor, described by eq 1. The donor concentration profile CD(r) can have any convenient concentration units, so that we can use CD(r) ) PJ(r). The acceptor concentration profile CA(r) must have units of number density, and therefore is given by CA(r) ) nAPJ(r), where nA is the total number of acceptor molecules in the micelle. The encounter radius, Re, is the minimum distance between donor and acceptor at which the effect of dipolar energy transfer can still be observed in the donor fluorescence. It is usually set equal to the sum of the donor and acceptor van der Waals radii. However, the results here were found to be insensitive to a change of this cutoff parameter over a significant range of values (see below). Some comments are in order about the numerical evaluation of eq 11 using the junction point distribution, eq 1. First, the integration over r in eq 11b need only to be calculated from Re to about 3 R0, since w(r) is a very sharply peaked function of r for all accessible experimental times. Next, the junction point distribution, eq 1, is defined in terms of an interface thickness δ but expands much further than 0.5δ on each side of Rcore. To

Figure 3. P parameter values obtained by fitting the experimental decay profiles of labeled PI-PMMA block copolymer micelles in acetonitrile with eq 5, for different acceptor mole fractions.

simplify the numerical calculations, it is convenient to set up a range for this function. For the present system, we truncate the distribution at Rcore ( 2.5δ, which corresponds to a region including 99% of all the junction points (Figure 2). Third, a simple trapezoidal quadrature is used in the evaluation of all the integrals since the more sophisticated adaptive quadrature routines are unstable with respect to multiple integral evaluation.52 Dyes on the Surface of a Sphere. The fluorescence decay profile of a micelle solution containing only phenanthrenelabeled block copolymer exhibits an exponential decay from which the donor lifetime was determined to be τD ) 45.5 ns. Fits of the experimental donor decay curves to eq 5 for micelles containing different ratios of donor- and acceptor-labeled polymer yielded P values that increase linearly with the acceptor-to-donor ratio (Figure 3). From these fits, values of β ) 0.35 ( 0.03 were obtained, which correspond to an apparent dimension d ) 2.1 ( 0.2. This result implies that the interface is sharp with respect to R0. We can obtain information about micelle structure from the P values by assuming that the interface is a two-dimensional spherical surface (d ) 2) and that the micelle core is composed of PI only31 with MnPI ) (10.3 ( 0.6) × 103 and FPI ) 0.913 g/cm3. We calculate the micelle core radius Rcore ) (7.6 ( 0.8) nm, and the number-averaged micelle aggregation number Nnagg ) 98 ( 22.29 The core radius is about 3 times the value of R0 for this system, and therefore there is no DET across the center of the micelle. Furthermore, since Rcore . R0, the KB model, eq 5, is a good approximation of eq 8 for donors and acceptors homogeneously distributed in the surface of a sphere. The calculated surface area per chain is 7.4 nm2. This value is equal to the value obtained from light scattering measurements (Appendix). Note that both calculations assume a dense core consisting only of PI. Within the approximations of eq 9, the volume per junction calculated from the P parameters obtained at several acceptor molar fractions is VJ ) (22.7 ( 0.6) nm3. Two different averages of the aggregation number are measured with the DET and SLS experiments, Nnagg and Nwagg, respectively. The ratio Nwagg/Nnagg ) 1.3 may be interpreted as a polydispersity index of the micelles. DET occurs between a specific number of dyes, and the P value is directly related to the local number density of acceptors. SLS measurements, on the other hand, give the weight-average of the micelle molecular weight.29 The aggregation number ratio Nwagg/Nnagg ) 1.3 obtained from SLS and DET results is in good agreement with the micelle polydispersity index from DLS Mw/Mn ) 1.1-1.3 (Appendix). It may then be concluded that the aggregation number obtained from analysis of the fluorescence decay curves

2492 J. Phys. Chem. B, Vol. 103, No. 13, 1999

Farinha et al. number obtained from the analysis of the light scattering measurements, Rcore ) 8.7 nm and Nwagg ) 127. To compare simulations with experiment, the noise-free response functions have to be modified to introduce the features characteristic of real experiments. We generated simulated donor fluorescence decay profiles using experimental response functions as the excitation source, using the protocol described previously.54 The donor decay functions obtained using eqs 1 and 11 were convoluted with experimental response functions L(t),

∫0t L(s)ID(t - s) ds

IDconv(t) ) a

Figure 4. Fluorescence decay functions of labeled PI-PMMA block copolymer micelles in acetonitrile simulated with the distribution model of energy transfer, eqs 1 and 11, for interface thickness values of (1) 0 nm, (2) 0.6 nm, (3) 0.9 nm, (4) 1.2 nm, and (5) 2.4 nm. In the insert, we present the corresponding junction point distribution function calculated with eq 1.

in terms of donor and acceptor groups distributed in the surface of a sphere correlate well with the results obtained by light scattering for PI-PMMA micelles in CH3CN. Simulation Procedure. Simulations allow one to examine how changes in the morphology of the micelle affect the nature of energy transfer between donor and acceptor dyes attached to the block copolymer junction. In Figure 4, we plot some of the curves of a series of donor decay functions calculated according to eqs 1 and 11 for block copolymer micelles with interfaces ranging in thickness from 0 to 2.4 nm.53 We used κ2 ) 2/3, R0 ) 2.3 nm, τD ) 45.5 ns, and a cutoff distance Re ) 0.5 nm as spectroscopic parameters. In these calculations, the micelle itself had 98 polymer molecules, of which 74% were labeled with the acceptor chromophore (Nnagg ) 98, nA ) 73). These micelles, with a dense PI core, have a core radius Rcore ) 7.6 nm. One sees in Figure 4 that, the more narrow the interface, the steeper the decay profile, particularly at early times. This behavior makes sense because a narrow interface concentrates the chromophores bound to the polymer junctions, decreasing their separation and consequently increasing the overall energy transfer rate. In the insert in the upper right corner of Figure 4, we show the calculated distribution of junctions for interfaces characterized by δ ) 0.6, 0.9, 1.2, and 2.4 nm. As the interface thickness increases, an increasing fraction of the junctions lie in the “wings” of the distribution. Donor dyes located in the wings have a lower probability of energy transfer than those toward the center of the distribution. It is no accident that the long-time decays for curves 1-4 in Figure 4 have similar slopes. This result implies that each of these systems contains chromophores that emit with decay times close to the unquenched lifetime of τD ) 45.5 ns and that, as the interface broadens, the fraction of such groups increases. It is important to realize that the data in Figure 4 are “noisefree” data, which allow one to examine the behavior of the theoretical response functions of the system as the input parameters are varied. In this context, other sets of curves were generated, using the same parameters but changing the cutoff radius Re between 0.3 and 0.6 nm, to investigate the effect of this parameter in the analysis results. Finally, to study the sensitivity of the calculated parameters to the value of the micelle core radius, we kept all other parameters constant (with Re ) 0.5 nm) and used the values of core radius and aggregation

(12)

Using the parameter a, the number of counts on the decays were adjusted to give 20 000 counts in the most populated channel. Since the experimental fluorescence intensities are subject to random errors that obey Poisson statistics, we added Poisson noise to each channel of the convoluted decay curves. For channels with fewer than 100 counts, Poisson noise was generated using an inverse transformation algorithm.55 For the other channels, the central limit theorem is applicable and we used normal distributed noise generated by a polar variation of the Box-Mueller algorithm.55 A pseudo random number generator based on the multiplicative generator55 was used to obtain pseudo random numbers uniformly distributed between 0 and 1. Each donor decay function was convoluted with three different experimental response functions and then about 30 different decay profiles were generated using different pseudo random numbers. DET Distribution Model Analysis. In this section, we fit experimental fluorescence decay profiles for PI-PMMA block copolymer micelles in acetonitrile to simulated decays. We refer to the experimental decay profiles as IDexp(t) and the simulated decays convoluted with real lamp excitation profiles as IDconv(t). Both decays correspond to a system with an acceptor mole fraction of 0.74. The IDexp(t) profiles were fitted to each of the IDconv(t) curves, using a linear reconvolution algorithm. The fitting equation is

IDexp(t) ) aNIDconv(t) + aLL(t)

(13)

where the fitting parameters are the decay intensity normalization factor aN and the light scattering correction aL. To evaluate the quality of the fitting results, we calculated the reduced χ2, the weighted residuals, and the autocorrelation of residuals. The fitted decays for interface thicknesses of 0, 0.6, 0.9, and 1.1 nm are shown in Figure 5. A plot of the reduced χ2 versus interface thickness δ (Figure 6) shows that only for interface thickness values of 0.8 to 1.0 nm is it possible to obtain χ2 < 1.5 and to obtain an evenly distributed autocorrelation of the residuals. The decay intensity normalization and the light scattering correction parameters, aN and aL, do not change significantly with the variation the interface thickness value. From examination of the autocorrelation of residuals, we can conclude that only the decay profile that corresponds to a 0.9 nm interface yields a good fit. For thicknesses smaller than 0.8 nm, the reduced χ2 are not very high (χ2 ≈ 2) but the autocorrelation functions are biased, indicating the inadequacy of the distribution. On the other hand, for thickness values greater than 1.0 nm, good fits are not possible, and the reduced χ2 values increase steadily up to a value of 54 at δ ) 2.4 nm. The invariance of the fitting results for very sharp interfaces can be explained by the fact that when the interface is smaller than R0 the donor senses an ap-

Interfaces in Self-Assembling Diblock Copolymers

Figure 5. Experimental instrument response function (i), decays simulated using eqs 1 and 11, for interface thickness of 0, 0.6, 0.9, and 1.1 nm, and experimental donor decay profile of labeled PI-PMMA block copolymer micelles in acetonitrile (e) fitted with the simulated decays. Only the autocorrelation relative to the fitting of the curve simulated with interface thickness 0.9 nm is well distributed.

Figure 6. Plot of the reduced χ2 parameter versus interface thickness δ, obtained for the best fit of an experimental fluorescence decay profile of PI-PMMA block copolymer micelles in acetonitrile IDexp(t) to simulated decays IDconv(t). Both decays correspond to an acceptor mole fraction of 0.74. The decays IDconv(t) were simulated using eqs 1 and 11, for interface thicknesses δ ranging from 0 to 1.2 nm, with Rcore ) 7.6 nm (0) and Rcore ) 8.7 nm (b). Only for interface thickness values of 0.8-1.0 nm is it possible to obtain acceptable values of χ2 < 1.5.

proximately two-dimensional distribution of acceptors. The energy transfer experiment only starts to be sensitive to the interface thickness above δ ≈ R0/5. This value corresponds to the point for which R0 equals the total width of the spherical shell containing 99% of all the junctions (Figure 2). The analysis of simulated decays with the encounter radius Re ranging from 0.3 to 0.6 nm did not produce any significant changes in the results. Also, the system was found to be insensitive to the change of the core radius from 7.6 to 8.7 nm, as long as the surface area per chain was kept constant at 7.4 nm2 (Figure 6). This result is consistent with the results of previous simulations that indicate that the area per chain and interface thickness were the major factors affecting the donoracceptor energy transfer in the interface.36 With the new DET distribution model for spherical micelles (eq 11) in conjunction with the modified HT equation (eq 1), it is possible to estimate the interface thickness for PI-PMMA diblock copolymer micelles in acetonitrile as δ ) (0.9 ( 0.1) nm. This value is sufficiently small, as compared to R0 ) 2.3 nm, for the analysis of the same data with eq 5 to yield a quasitwo-dimensional interface for this system. Comparison of the KB and Distribution Models. The fluorescence decay profile analysis described above takes

J. Phys. Chem. B, Vol. 103, No. 13, 1999 2493

Figure 7. Optimized values of the β parameter obtained from the fitting of simulated decay curves with eq 5. A series of decay curves was simulated for micelles with 18 different values of the interface thickness δ, ranging from 0 to 2.4 nm. For each value of δ, several decay curves were simulated with different synthetic noise and convoluted with different experimental response functions. Each decay curve was fitted to eq 5 and values of P and β were optimized. In this way, error bars corresponding to 95% confidence intervals were evaluated. Rcore ) 7.6 nm (0) and Rcore ) 8.7 nm (b).

explicit account of the donor and acceptor distributions in the restricted geometry corresponding to the core-corona interface of a block copolymer micelle. This strategy, for interfaces of finite width, is more rigorous than one in which one fits data to eq 5 with P and β as variable parameters. Nevertheless, eq 5, the Klafter-Blumen (KB) equation, is often used as a phenomenological equation to fit experimental fluorescence decay curves obtained for systems with dyes distributed in a restricted geometry. There are two features which make this approach to data analysis attractive. The first is the simplicity of the model. The second is the remarkable ability of eq 5 to give good fits to experimental data, with good precision in the fitting parameters and little or no parameter correlation. With these advantages, it would be useful to put this strategy for data analysis on a firmer basis through comparison of data analysis with eq 5 with the more rigorous analysis of eq 11. We approach this problem by simulating fluorescence decay curves according to eqs 1 and 11, for micelles with interfaces of “known” thickness and fitting the data to eq 5. We simulated a series of fluorescence donor decay profiles for micelles with 18 different values of interface thickness δ ranging from 0 to 2.4 nm (Figure 3). For each value of δ, several decay curves were simulated, with different synthetic noise and convoluted with different experimental response functions. Each decay curve was fitted to eq 5 and the values of P and β were optimized. In this way, error bars corresponding to 95% confidence intervals were evaluated by rejecting 2.5% of the values in each of the extremes of the obtained fitted parameter distributions.54 The KB model produces good fits for all the simulated interface thickness values, with χ2 ) 1.1 ( 0.1 and well distributed residuals and autocorrelation of residuals. In Figure 7, we present the variation of the fitted β parameter values with interface thickness. For interfaces that are thin compared to R0, the value of β corresponds to a surface with a dimension d ) 2. The apparent dimension of the interface starts to increase with increasing interface thickness for δ > R0/5 and reaches a plateau at about δ ≈ 3R0/5. This means that, when the interface thickness is below δ ≈ R0/5 or above δ ≈ 3R0/5, energy transfer, as reflected in the value of β, is not sensitive to changes in the distribution of the chromophores. If we define the total width of the interface

2494 J. Phys. Chem. B, Vol. 103, No. 13, 1999

Figure 8. Plot of the calculated P parameter values obtained by fitting simulated decay curves with eq 5 (corresponding to the data in Figure 7). The right-hand y-axis presents the corresponding inverse of the interface volume VI, defined as the volume of the spherical shell containing 99% of all the junction points. Rs ) 7.6 nm (0) and Rs ) 8.7 nm (b).

∆T as that of the spherical shell of thickness 5δ, containing approximately 99% of all the junction points, the range for which DET is sensitive to the distribution of donors and acceptors can be expressed as R0 < ∆T < 3R0. That is, if 99% of all the donors and acceptors (with an HT distribution) are distributed in a spherical shell thinner than R0, DET will only sense a spherical surface. On the other end, if 99% of the chromophores lie in a spherical shell thicker than 3R0, DET will perceive an approximately homogeneous distribution over the volume of the interface. The value 3R0 coincides with the effective maximum distance for energy transfer that corresponds to the value at which w(r) in eq 11b has fallen bellow 0.1% of its initial value. The results obtained from fitting data to the KB equation were found to be insensitive to the values of Rcore and Re, when these parameters were varied in the range (7.6 nm < Rcore < 8.7 nm) and (0.3 nm < Re < 0.6 nm). For different cut off radii, Re, the results could not be distinguished, whereas for different core radii, Rcore, the error bars overlap (Figure 7). Figure 7 serves as a calibration curve for data analysis with eq 5. When the experimental fluorescence decays IDexp(t) are fitted to eq 5, a value of β ) 0.35 (d ) 2.1) is obtained. As the horizontal line in Figure 7 indicates, this value corresponds to a system with δ in the range 0.8-1.1 nm. It is fortunate that the value of β obtained from IDexp(t) corresponds to the steepest portion of the sigmoidal plot of β vs δ in Figure 7. If the micelles were characterized by a much thinner or a much thicker interface, Figure 7 would be less useful for determining a realistic value of δ and the uncertainty in the interface thickness determination would be much higher. This method only works well when the interface thickness corresponds to the crossover region of the β parameter (R0 < ∆T < 3R0). In Figure 8, we present the variation of the fitted P values with interface thickness corresponding to the data in Figure 7. Again for interface thickness values below δ ≈ R0/5 energy transfer is not sensitive to the increase in the chromophore concentration accompanying the decrease in interface thickness. Above δ ≈ R0/5, the change of P with δ follows the dilution effect predicted from the increase in the total interface volume VI, defined as the volume of the spherical shell containing 99% of all the junctions. From the results of the distribution analysis of energy transfer, we can calculate the volume per block junction VJ by dividing

Farinha et al. the total interface volume VI by the aggregation number. If we consider the total interface volume as the volume containing 99% of the junction points, the volume per junction is VJ ) 34 nm3. However, if we define the total interface volume for a 95% probability to contain the junction points (Figure 2), we obtain a volume per junction that coincides with the one calculated from the P parameter using eqs 6 and 10, VJ ) (22.7 ( 0.6) nm3. Figure 8 can also serve as a calibration curve for inferring values of δ from fits of experimental decay curves with eq 5. Consider the micelle sample with a 0.74 molar fraction of acceptor-labeled block copolymer. When the experimental donor fluorescence decay is fitted to eq 5, one obtains P ) 2.1. From the relation plotted in Figure 8, this value of P corresponds to an interface thickness value of δ ) 0.9-1.0 nm, which again agrees with the value obtained from the fitting with the distribution model. When we change the parameters Rcore and Re in the range (7.6 nm < Rcore < 8.7 nm) and (0.3 nm < Re < 0.6 nm), the differences in the fitted P values are negligible (Figure 8). As a final comment, we point out that for small micelle cores characterized by “thick” interfaces that reach the center of the core, the experimental fluorescence decay curves are no longer sensitive to the interface thickness, but they become sensitive to the core radius. In the present case, for interface thickness values above 2.4 nm, the simulated micelles are very swollen and the probability of having DET across the core of the micelle becomes finite. At the limit of δ ) 2.4 nm, the apparent dimension of the interface is only d ) 2.3 ( 0.1, still far from the value d ) 3 expected for a homogeneous distribution of donors and acceptors in an infinite volume in three dimensions. This effect is due to the restricted geometry of the micelle. When δ is large enough to produce an almost homogeneous distribution of donors and acceptors in the interface, DET becomes only sensitive to the micelle radius. In this case, the systems correspond approximately to the case of donors and acceptors homogeneously distributed inside a sphere of radius Rs (Rs ) Rcore for the PI-PMMA micelles) and the donor fluorescence decay profile for a concentration of acceptors CA is described by35

( )∫ t τD

ID(t) ) exp -

(

φ(t,rD) rD2 drD

(14a)

∫RR -r {1 - exp[-w(r)t]}r2 dr -

φ(t,rD) ) exp -4πCA πCA rD

Rs

0

s

s

D

)

∫RR-r+r {1 - exp[-w(r)t]}r[Rs2 - (rD - r)2] dr s

s

D

D

(14b)

Conclusions We proposed a new model of energy transfer between donors and acceptors distributed across the core-corona interface of block copolymer micelles. The model takes explicit account of the junction distribution across the interface in terms of a modified Helfand-Tagami equation. This model was used successfully to characterize the interface of PI-PMMA diblock copolymer micelles in acetonitrile. An interface thickness of δ ) (0.9 ( 0.1) nm was found which is thinner than the value inferred for PI-PMMA in the bulk state.28 The decrease of the interface thickness with respect to its value for the bulk polymer results from the solvent-swelling of the micelle corona. When the solvent quality decreases with respect to the core polymer, the polymer segments at the core try to avoid contact with the solvent-swollen corona, decreasing the interface thickness.

Interfaces in Self-Assembling Diblock Copolymers From the analysis of experimental data with the new distribution model for energy transfer, we recover a finite interface thickness, δ ) (0.9 ( 0.1) nm. However, the interface is still sufficiently sharp when compared to R0, to allow the fit of the donor fluorescence decays with the equation for donors and acceptors distributed in the surface of a sphere. In this situation, we could calculate the micelle core radius and aggregation number in good agreement with the light scattering results. Simulation results show that the apparent dimension d obtained from fits of the same experimental data to the KB equation (eq 5) is directly related to the interface thickness δ. Comparison of simulated and experimental data can be used to determine δ values from “best-fit” values of the apparent dimension parameter. As a consequence, if one knows the aggregation number and the core radius of a given block copolymer micelle containing appropriate donor and acceptor chromophores, it is possible to determine the core-corona interface thickness. Acknowledgment. The authors thank NSERC Canada and the donors of the Petroleum Research Fund administered by the American Chemical Society for their support of this research. J. P. S. Farinha acknowledges the support of FCT-PRAXIS XXI and K. Schille´n acknowledges support from the Swedish Natural Science Research Council (NFR). Appendix Micelle Size Determination. According to the DLS and viscosity data described in detail in our earlier publication,29 the micelles exhibit the hydrodynamic behavior of impermeable hard spheres. The micelles generated are “kinetically frozen micelles,” in that they do not exchange chains in acetonitrile on a time scale of weeks. The micelle size is somewhat sensitive to sample preparation protocol, and all of the data point to a significant but not very broad distribution of micelle sizes,29 which we comment upon below. The largest number of experiments were carried out on micelles formed from a 1:1 mixture of donor- and acceptor-labeled copolymers. Static light scattering experiments on this sample provide a molecular weight Mw ) (8.45 ( 0.03) × 106, corresponding to a weightaveraged aggregation number Nwagg ) 127 ( 6. Since the micelle core is not significantly swollen,31 we can calculate from the aggregation number and the size of the polyisoprene block (MnPI ) (10.3 ( 0.6) × 103) a core radius of Rs ) (8.7 ( 0.2) nm, which corresponds to a surface area per chain of 7.4 nm2. In DLS experiments on this sample and on micelles prepared from other mixtures of donor- and acceptor-labeled polymer, the intensity correlation functions were single exponential and narrow relaxation time distributions were obtained. Analysis of these decay profiles in terms of a cumulant analysis, assuming a log-normal molecular weight distribution, led to the conclusion of a modest size polydispersity, with Mw/Mn ) 1.1 to 1.3. References and Notes (1) Kwon, G. S.; Kataoca, K. AdV. Drug. DeliVery ReV. 1995, 16, 295. (2) Karymov, M.; Prochazka, K.; Mendenhal, J.; Martin, T. J.; Munk, P.; Webber, S. E. Langmuir 1996, 12, 4748. (3) Riess, G.; Hurtrez, G.; Bahadur, P. In Encyclopedia of Polymer Science and Engineering, 2nd ed.; Mark, H. F., Bikales, N. M., Overberg, C. G., Menges, G., Eds.; Wiley: New York, 1985; Vol. 2, pp 324-436. (4) Xu, R.; Winnik, M. A.; Riess, G.; Chu, B.; Croucher, M. D. Macromolecules 1992, 25, 644. (5) Tuzar, Z.; Kratochvil, P. In Surface and Colloid Science; Matijevic, E., Ed.; Plenum: New York, 1993; Vol. 15, p1. (6) SolVents and Self-Organization of Polymers; Webber, S. E., Munk, P., Tuzar, Z., Eds.; NATO ASI Series E327; Kluwer: Dordrecht, The Netherlands, 1996.

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