7114
J. Phys. Chem. 1996, 100, 7114-7121
Energy Transfer in Restricted Geometry: Polyisoprene-Poly(methyl methacrylate) Block Copolymer Interfaces Olga Tcherkasskaya, John G. Spiro, Shaoru Ni, and Mitchell A. Winnik* Department of Chemistry and Erindale College, UniVersity of Toronto, 80 St. George Street, Toronto, Ontario, Canada M5S 1A1 ReceiVed: July 28, 1995; In Final Form: December 18, 1995X
The kinetics of direct nonradiative energy transfer between dyes confined to the 2.6 nm wide interface region of polyisoprene-poly(methyl methacrylate) block copolymer films are reported. This system differs from restricted geometry systems examined previously because of the diffuse nature of the edges of the confining space. The interface thickness is similar in magnitude to the characteristic distance for energy transfer (R0 ) 2.3 nm) for the donor-acceptor dye pair (phenanthrene-anthracene) employed here. Samples were prepared from matched pairs of block copolymers, one containing a donor dye and the other an acceptor dye, at the PI-PMMA junction. Donor fluorescence decay profiles were fitted to the Klafter-Blumen expression [ID(t) ) A1 exp{-(t/τD) - P(t/τD)β} + A2 exp(-t/τD)] containing the additional A2 term to account for donors (ca. 3%) outside the interface. The parameters obtained followed the predicted behavior, namely, that the preexponential term P was proportional to the acceptor concentration, whereas the stretched-exponential parameter β was independent of the global acceptor concentration CA for acceptor-to-donor ratios CA/CD > 1. One of the most unusual features of the data is a crossover in β observed as a function of a global acceptor concentration CA for a certain range of donor-acceptor composition, CA/CD < 1.
Introduction In this paper we examine block copolymers as systems of restricted geometry for carrying out direct nonradiative energy transfer (DET) experiments.1-4 A restricted geometry is one in which the kinetics of DET is affected by edge effects.5 Donor-acceptor pairs at the edges of the confining space have a different pair distribution than those in bulk. If the confining space has at least one dimension not much larger than the characteristic distance for donor-acceptor interaction, then edge effects will make a significant contribution to the measured DET kinetics. While studies of a number of systems have been reported in the literature,1,6-9 the topic is sufficiently new, and our understanding, sufficiently limited, that new systems, particularly those exhibiting unique features, needs to be investigated. Block copolymer systems offer the attraction of providing microdomains of consistent size, whose dimensions can be varied by selecting the chemical composition or the chain length of the constituent polymer blocks.10 In the bulk state, block copolymers form periodic structures if the chemical dissimilarity between the two components is sufficiently large. For example, if the A-polymer and the B-polymer are comparable in length, the system self-assembles into lamellar phases with a total period DT ) hA + hB, where hA and hB refer to the lengths of the A-polymer and B-polymer phases, respectively.10,11 When the chain lengths differ, cylindrical, spherical, or, in some instances, more complex bicontinuous structures form. As our knowledge of these phases increases, the kineticist interested in DET in restricted geometries has a richer set of structures from which to choose. In Figure 1 we depict the lamellar structure of a diblock copolymer system. In addition to the two characteristic lengths hA and hB, the discrete A-polymer and B-polymer phases are separated by an interphase of finite thickness δ. The chemical dissimilarity between the polymers is normally quantified in X
Abstract published in AdVance ACS Abstracts, March 1, 1996.
0022-3654/96/20100-7114$12.00/0
Figure 1. Block copolymers in a lamellar mesophase. The thickness of the A and B lamellae are hA and hB, respectively. The width of the interface between the A and B regions is δ.
terms of the Flory-Huggins interaction parameter χAB, a measure of the cost in energy per segment of replacing A/A and B/B contacts with A/B contacts. The driving force for phase separation depends upon χABN, the product of this parameter with the overall chain length N of the polymer.12-14 When χABN is large, the system is said to be in the “strong segregation” limit. Under these conditions DT increases as N2/3. According to the theory of block copolymer structure, the magnitude of δ depends primarily on χAB, and in the strong segregation limit, δ has only a weak dependence on N; i.e., its magnitude is determined primarily by local interactions in the interphase.14c There are various ways in which one could employ periodic block copolymer structures as templates for restricted geometry experiments. In our mind, one of the most interesting is to attach donor or acceptor dyes to the junction between the Aand B-polymers. In this way the dyes would be confined to the interface, a lamellar structure of thickness δ. Values of δ can be determined by experiments employing X-rays15 or neutrons16,17 and have established that for poly(isoprene-bstyrene) (PI-PS) δ ) 2.0 nm10b,15 and for poly(styrene-b-methyl methacrylate) (PS-PMMA) δ ) 5.0 nm.10b,17 We have used DET measurements18,19 of the sort reported here to determine that in PS-PMMA the thickness of the interphase δ ) 5.1 nm, and in poly(isoprene-b-methyl methacrylate) (PI-PMMA) δ ) 2.6 nm. These values are comparable in magnitude to the characteristic distance R0 for energy transfer by the dipolecoupling (Fo¨rster) mechanism. Under these circumstances, energy transfer between dye pairs confined to the interface © 1996 American Chemical Society
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J. Phys. Chem., Vol. 100, No. 17, 1996 7115
Figure 2. Density profile through the interface for the single block of diblock copolymer.12
would be influenced by edge effects associated with the restricted geometry. It is important to note one essential difference between the block copolymer systems under consideration here and various systems examined previously. Most systems examined in the past were characterized by, or at least analyzed in terms of, geometries with rigid walls. The block copolymer interface is a system with diffuse edges. According to the theory of Helfand,12 the distribution of the joints, i.e., the probability density to find the joint at x, is governed by an expression
PJ(x) ) [(6χAB)1/2/πb] sech[(6χAB)1/2x/b]
(1)
sometimes referred to as a Tagami function. This function has somewhat broader tails than a Gaussian function of similar width at half-height. The tails of the Tagami function reflect the existence of the junctions located outside the interface. The interface thickness δ is defined in terms of the segment density profiles of the two polymers, A and B, determined by the potential which modifies the chain configurational statistics in the interfacial region. The characteristic interface thickness introduced by Helfand is an estimate of the distance across the interface over which the density of A-chain segments FA increase to their density F0 in the A-rich phase (Figure 2). In the strong segregation limit, the density profile increases from 0 to 1 across the interface, and δ is defined through the tangent to this plot at the inflection point as the distance separating the intersections of the tangent with the axes at FA/F0 ) 0 and 1.12 Then, for polymers with a statistical monomer length b,
δ ) 2b/(6χAB)1/2
(2)
The diffuseness of the interface introduces complications into the data analysis which we are not yet able to handle. A common strategy for analyzing fluorescence decay profiles from complex systems7-9 is to fit the data to the Klafter-Blumen2 equation, which applies rigorously to infinite systems embedded on a fractal lattice:
ID(t) ) A exp{-(t/τD) - P(t/τD)β}
(3)
In this equation, A represents the fluorescence intensity at zero time and τD is the unquenched donor lifetime. P is a parameter proportional to the probability that an acceptor resides within a distance R0 of an excited donor and is thus proportional as well to the bulk concentration of acceptor in the system. The exponent β is a concentration-independent parameter. For DET by the dipole-dipole mechanism on a fractal lattice β is equal to ∆/6, where ∆ is the fractal dimension. In restricted geometries, ∆ loses this physical meaning. Sometimes ∆ is referred to as the “apparent dimension” of a system. A signature
of the influence of restricted geometry on an experiment is a value of ∆ < 3, i.e., β < 0.5. Simulations suggest that in systems of constant geometry, such as spheres containing a uniform distribution of donors and acceptors, the magnitude of β decreases as edge effects become more important, for example, as the radius of the sphere approaches R0. Sometimes the magnitude of β is determined by a spatial or temporal crossover in the system: for energy transfer between donors and acceptors confined to a cylinder, DET at early times occurs between close pairs with a three-dimensional local distribution.7 At later times, the only surviving excited donors are far removed from acceptors, and the pairs have a distribution that is essentially one-dimensional. The experiments reported here refer to pairs of PI-PMMA block copolymers matched in length and composition. One member of the pair is substituted at the PI-PMMA junction with a single 9-phenanthryl group (the donor); its partner is substituted with a 2-anthryl group (the acceptor). Since the block copolymer structure is determined only by the chain length and composition of the polymers, mixtures of the donor- and acceptor-labeled pairs conserve the period length and interface thickness. This allows us to vary the donor-acceptor ratio, and thus the acceptor concentration, by varying the ratio of the two polymers. Some of the data presented here have previously been reported and analyzed in an attempt to infer values for δ and for the χAB parameter characterizing PI-PMMA. Here we look more deeply at the actual data analysis in terms of the Klafter-Blumen model. We examine strategies for data analysis, the issue of parameter correlation, and the uniqueness and uncertainties of the values of P and β obtained. One feature of the data which is particularly fascinating for aficionados of restricted geometry problems is a crossover in the magnitude of β which occurs at a certain range of donor-acceptor composition. Experimental Section Polymer Synthesis and Characterization. A series of isoprene-methyl methacrylate diblock polymers were synthesized by anionic polymerization and characterized as described previously.18,19 The polydispersities of all samples are less than 1.2. Block copolymer compositions and the microstructure of the PI-block were determined with a 400 MHz 1H NMR spectrometer to an accuracy of 5 mol %. The PI-block has a specific microstructure (40% 1,2-addition, 55% 3,4-addition, and 5% 1,4-addition) that is typical of PI prepared by anionic polymerization in THF. To each sample, 1 wt % of 2,5-ditert-butyl-4-methylphenol was added as an antioxidant. In these experiments the donor dye was phenanthrene (Phe) attached at the 9-position, and the acceptor dye was anthracene (An) attached at the 2-position. These dyes in PI-PMMA films undergo direct energy transfer by a dipole-coupling mechanism with a characteristic Fo¨rster distance in this system of R0 ) 2.3 nm. We used the same spectroscopic procedure for determination of R0 as that described previously.18 The main characteristics of the block copolymers are listed in Table 1. Volume fractions were calculated by assuming that the densities of the PI and the PMMA blocks are the same as those for the pure homopolymer (FPI ) 0.913 g/cm3 and FPMMA ) 1.188 g/cm3). The efficiency of dye attachment (98-100%) was determined by UV spectroscopy. From the compositions of the samples (between 0.07 and 0.5 vol % PI), we infer that the three of balanced composition form lamellar phases and that the other two samples form spheres of PI in a PMMA matrix. Sample Preparation. Weighed amounts of block copolymer were dissolved in toluene (an approximately nonpreferential
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TABLE 1: Sample Characteristics for PI-Phe(An)-PMMA Diblock Copolymers Labeled at the Junction sample pair
dye
Mn PI/PMMA
Mw / Mn
ΦPIa
fPIb
fdyec
χNd
PI domain
I-1 I-2 I-17 I-12 I-14 I-13 I-36 I-37 I-21 I-22
Phe An Phe An Phe An Phe An Phe An
1K/16.6K 1K/19K 10K/17K 8K/17.5K 10K/22K 10K/22K 19K/25K 19K/28K 10K/47K 10K/55K
1.09 1.07 1.18 1.07 1.06 1.15 1.17 1.17 1.11 1.09
0.07 0.06 0.43 0.37 0.37 0.37 0.50 0.47 0.20 0.19
0.08 0.07 0.46 0.40 0.40 0.40 0.53 0.50 0.23 0.21
100 100 96 98 100 100 90 100 100 98
14
sphere
24
lamella
28
lamella
42
lamella
51
sphere
a ΦPI ) volume fraction of PI block. b fPI ) mole fraction of PI block. fdye ) efficiency of dye attachment (percent of labeled chains). d χ ) 0.077 ( 0.004 was determined from DET data.19 c
solvent) to give a final concentration of 1-6 wt %. A small amount (0.1 mL) of polymer solution was placed on a quartz plate (11 × 25 mm). The solvent was allowed to evaporate slowly over 2 h, and then another aliquot was added. Then the films were dried under vacuum at 50 °C for 8 h. Films were composed of various mixtures of Phe- and An-labeled block copolymers of nearly identical composition and chain length. The acceptor-donor ratio of the mixture was varied from 0 to 3. The thickness of films obtained ranged from 3 to 6 µm. Note that the antioxidant is carried forward in the experiment and has no influence on the fluorescence decays. Removing the antioxidant leads to ketone formation in the PI block, and these groups are effective quenchers of Phe fluorescence. Thus, preventing oxidation is essential to the success of these experiments. To check the influence of the heating on the chemically sensitive PI-block containing the double bonds in every monomer units, experiments were carried out with and without heating of films. In the case of film samples which were not heated, the polymer solutions were dried at room temperature over 24-48 h. Fluorescence Measurements. Fluorescence decay profiles were obtained by the time-correlated single-photon counting technique.20 All experiments involved a pulsed lamp as an excitation source (0.5 atm of D2). Films were excited at the wavelength of maximum absorption of phenanthrene (300 nm). The fluorescence was detected at the maximum of the fluorescence spectrum of phenanthrene (366 nm). Data were collected at right angles at room temperature up to 20 000 counts in the maximum channel. The reference profile, used for the convolution analysis, was obtained by exciting a solution of p-terphenyl in aerated cyclohexane (τ ) 0.96 ns). The convolution function was compared with the experimental decay by a nonlinear leastsquares optimization using the DFCM algorithm21 for singleexponential decays and the Mimic algorithm22 for DET decays. The minimization was of the reduced chi-squared (χ2) function. Data Fitting and Error Analysis. Many features of the data analysis will be discussed in the text. The general procedure was to fit the experimental donor decays from pure PI-PhePMMA films to the monoexponential model. The value of τD determined from this fit was introduced as a known parameter in fitting the DET decays to eqs 3 and 4 (see below). Equation 4 contains four fitting parameters (A1, A2, P, β). Two of them are interdependent (A1 + A2 ) 1.0). To decrease the number of variable parameters and increase the accuracy of the fit, the data were analyzed in two discrete steps. Initially, a value for β was chosen and then arbitrarily fixed, while the computer sought best values of A1, A2, and P. The value of β was then incremented, and the process was continued. Finally, we constructed plots of the dependence χ2 upon β to obtain the
optimum set of parameters corresponding the global χ2 minimum. This set of parameters was chosen for further statistical analysis. In this way we ensure that the program does not terminate at any local minimum of the χ2-β surface. A byproduct of this approach is that we can draw the χ2-β surface for each set of data and estimate the uncertainty of the recovered dimension through this surface. To carry out the fit according to the procedure mentioned above, we have designed a software package to perform the data fitting based upon the Klafter-Blumen model. The reconvolution technique involves linearization of the fitting function and least-squares fitting. The program employs More’s implementation of the Levenberg-Marquardt nonlinear leastsquares algorithm23 and fits three model parameters at a time. It permits one to carry out the error analyses based on the common method of approximating the residuals by an affine model.23,24 The program has a numerous options which are available regarding details of the models. The background and light scattering factors can be held fixed or fitted by least squares. Tests of the light scattering effect for PI-PMMA films have shown that magnitude of the light scattering correction for all samples investigated was less than 0.5% of the absolute value of the signal decay. Several statistical functions were calculated to estimate the reliability of the fit. These functions included the reduced χ2, the weighted residuals, the shape and amplitude of the autocorrelation function of the weighted residuals, and the 95% confidence intervals for the fitting parameters of each individual decay. The value of χ2 of the fit chosen for analysis in most of cases was less than 1.5. The procedure of error estimation was based on the square root of the diagonal element (bjj) of the variance-covariance matrix at the χ2 minimum. In this case, the standard error can be calculated as ((xbjj). This also yields the 95% confidence interval ((2xbjj). This technique produces a symmetric “(” error and predicts the smallest amount of the uncertainty.24 We found that the 95% confidence interval characterizing the uncertainty of the fitting parameters P and β decreases from 15 to 1% upon increasing the acceptor-donor ratio of the samples. For acceptor concentrations discussed in the present work, the uncertainty of P and β for the individual fits is less than 3%. Data and Data Analysis Fluorescence Decay Experiments. The successful application of the direct energy transfer technique has some experimental prerequisites. The most important is the shape of fluorescence decay of the isolated donor, which must be monoexponential in order for the lifetime of the donor to be well defined. We found that the fluorescence decay of Phe attached to the junction point of PI-PMMA in the films has a strictly monoexponential character which is conserved under variation of film formation procedure (concentration of copolymer in toluene, drying time) as long as oxidation of the PI component is prevented (Table 2). The fluorescence lifetime of the Phe does not depend significantly on the block copolymer composition. Only for samples containing the highest concentration of dye (smallest polymerization index) the lifetime of Phe is reduced (42.9 and 43.6 ns vs 45.0 ns in the other samples). This points to a small amount of self-quenching of Phe corresponding to an enhanced local concentration of Phe in the interface of the PI microdomain of the smallest size. Figure 3 shows typical fluorescence decays of the Phe donor and their evolution in samples of increasing acceptor concentration for a series of PI-PMMA films. Increasing the acceptor concentration leads to more pronounced deviations from expo-
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TABLE 2: Influence of Film Formation Procedure on the Fluorescence Decay of Phenanthrene Attached to the Junction of PI-PMMA; Lifetime of Donor τD Is Calculated Using the Monoexponential Model I(t) ) A exp(t/τD) sample I-14
I-36
Csol,a wt %
t,b h
A
τD, ns
χ2
WRc
AcFd
1 1 6 6 1 1 3 3 1a
8 0 8 0 8 0 8 0 8
0.15 0.15 0.15 0.17 0.16 0.15 0.28 0.28 0.15
44.4 44.3 44.9 45.0 44.6 45.0 44.6 44.2 44.5
1.49 1.44 1.36 1.20 1.18 1.44 1.29 1.42 1.39
4.9 4.5 4.3 3.2 2.6 5.0 3.8 5.2 4.4
0.46 0.42 0.28 0.27 0.22 0.37 0.30 0.36 0.32
aC b sol ) polymer concenration in toluene. t ) time of heating under vacuum at 50 °C. c WR ) weighted residuals. d AcF ) amplitude of autocorrelation function. e Antioxidant is carried forward in the experiment.
Figure 3. Fluorescence decay profiles of the Phe (donor) emission in films composed of various mixtures of Phe- and An-labeled copolymers PI-PMMA (samples 14 and 13). Acceptor-to-donor ratio (top-tobottom) ) 0, 0.2, 1.2, and 2.0.
nential behavior. In analyzing decays from samples with finite acceptor concentration, poor fits to eq 3 were obtained with χ2 . 2. We found it necessary to add an additional term to the Klafter-Blumen expression to accommodate donors located in an environment free of the acceptor. This type of behavior has been noted previously.9,18,19,25
ID(t) ) A1 exp{-(t/τD) - P(t/τD)β} + A2 exp(-t/τD)
(4)
Here, the A1 term represents the contribution of DET to the donor decay, and the A2 term represents the independent fluorescence of the donor not involved in energy transfer. The exact physical meaning of the A2 term is not yet known. There are various possible explanations for the existence of an isolated donor population. There may be small amounts of Phe-capped PI homopolymer in the sample that was not removed during the extraction with cyclohexane. Under these circumstances, the magnitude of the A2 term might be expected to vary from
Figure 4. Examples of the recovered χ2-β surfaces from the donor decay curves with different acceptor-to-donor ratio: 0.1 (1), 0.3 (2), 0.7 (3), 1.0 (4), 1.3 (5), 1.9 (6), and 2.7 (7). The apparent dimensions β are indicated by the minimums of these curves. PI-Phe(An)-PMMA, samples 17 and 12.
sample to sample, which is not observed. In fact, the magnitude of the A2 term found here is similar to (and some what smaller than) that found in similarly labeled PS-PMMA diblock copolymer films. Alternatively, there may be (and should be according to the Tagami function) a small amount of block copolymer chains in each sample whose junctions are located outside the interface. For each block copolymer, the magnitude of the A2 term decreases with increasing acceptor-to-donor concentration ratio and levels off at ca. 3-5% of the total signal, when CA/CD > 0.3. This value is very close to the theoretical estimates of the fraction of the junction located outside the interface which can be obtained by the analysis of the Tagami function tails (see Figure 2). To explain the independent fluorescence of the donor attached to the junction and located outside the block copolymer interface, it is necessary to extend the theoretical analysis of the DET process to systems of restricted geometry with diffuse edges. Optimum Set of Fitting Parameters. As described in the Experimental Section, decay traces were fitted to eq 4 in two steps. Values of β were fixed for the computer to optimize values of A1, A2, and P. The β value was then incremented, and the process was repeated. Figure 4 shows recovered χ2-β surfaces when the DET decays were fitted to eq 4. One can see that the uncertainty of the fit is very high for low acceptor concentrations, i.e., when CA/CD ≈ 0.1. The χ2 surface is flat with respect to the dimension β, and there is no clear minimum to the χ2 vs β plot. This means that in spite of the reasonable value of the statistical criterion χ2, a unique value of β cannot be obtained. Here there are many numerical solutions (sets of parameters) which are available to fit the experimental DET decay curves. A local minimum of the curves χ2-β appears when the acceptor concentrations satisfy the condition CA/CD > 0.3. In this case, the approximation has a single solution, which corresponds to the global χ2 minimum. The optimum set of parameters (A1, A2, P, β) corresponding to the minimum in the χ2 dependence on β was chosen for further analysis. Our results for the five samples of PI-PMMA are summarized in Table 3. Parameter Correlation. A very important question in our data analysis concerns the possible correlation of the key model parameters, P and β. We examine this issue by analyzing the 95% confidence ellipse for P-β. Two examples of a 95% confidence ellipse plotted in the P-β coordinates for individual DET decays are presented in Figure 5. Numerous P, β pairs
7118 J. Phys. Chem., Vol. 100, No. 17, 1996
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TABLE 3: Experimental Analysis of the Fluorescence Decay of Phenanthrene Attached to the Junction of PI-PMMA; Parameters (P, β, A1, A2) Are Calculated Using Eq 4 sample I-1, I-2 sphere
I-17, 12 lamella
I-14, I-13 lamella
I-36, I-37 lamella
I-21, I-22 sphere
CA × 105 mol/cm3
P
β
A2/(A1 + A2), %
χ2
1.47a 2.76 3.33 3.74 4.43 0.39 0.87 1.63 2.03a 2.31 2.68 2.98 3.51 0.20 0.58 1.08 1.55a 1.72 1.80 1.85 2.05 2.28 2.54 2.64 0.18 0.20 0.45 0.68 0.79 0.94 1.10 1.15 1.31a 1.39 1.44 1.61 1.62 1.72 1.79 0.21 0.42 0.62 0.79a 0.915 0.92 1.02 1.23 1.39
1.55 ( 0.03 2.87 ( 0.03 3.01 ( 0.02 3.71 ( 0.03 4.20 ( 0.04 2.00 ( 0.10 1.80 ( 0.08 2.86 ( 0.03 2.46 ( 0.05 3.29 ( 0.04 3.69 ( 0.04 4.00 ( 0.04 4.64 ( 0.04 1.76 ( 0.10 2.48 ( 0.06 2.41 ( 0.07 2.75 ( 0.05 2.78 ( 0.04 2.87 ( 0.04 2.93 ( 0.05 3.47 ( 0.04 3.47 ( 0.04 4.07 ( 0.04 4.35 ( 0.04 5.10 ( 0.70 4.90 ( 0.40 3.81 ( 0.20 3.59 ( 0.09 4.31 ( 0.09 3.62 ( 0.06 3.22 ( 0.07 3.90 ( 0.06 2.81 ( 0.06 3.15 ( 0.06 3.31 ( 0.05 3.45 ( 0.06 3.57 ( 0.05 3.78 ( 0.06 3.98 ( 0.06 3.21 ( 0.10 2.76 ( 0.09 2.48 ( 0.08 2.57 ( 0.10 3.07 ( 0.05 2.92 ( 0.06 3.19 ( 0.04 3.48 ( 0.04 4.31 ( 0.04
0.229 ( 0.002 0.235 ( 0.002 0.200 ( 0.002 0.266 ( 0.002 0.262 ( 0.003 0.350 ( 0.009 0.220 ( 0.006 0.215 ( 0.003 0.280 ( 0.005 0.280 ( 0.003 0.268 ( 0.003 0.276 ( 0.003 0.310 ( 0.002 0.330 ( 0.006 0.200 ( 0.004 0.185 ( 0.003 0.205 ( 0.003 0.230 ( 0.004 0.225 ( 0.003 0.220 ( 0.003 0.245 ( 0.003 0.260 ( 0.003 0.280 ( 0.003 0.270 ( 0.003 0.370 ( 0.090 0.350 ( 0.040 0.240 ( 0.003 0.246 ( 0.003 0.225 ( 0.004 0.315 ( 0.004 0.320 ( 0.005 0.380 ( 0.005 0.375 ( 0.005 0.371 ( 0.005 0.372 ( 0.005 0.374 ( 0.005 0.381 ( 0.006 0.380 ( 0.005 0.385 ( 0.005 0.350 ( 0.050 0.160 ( 0.011 0.190 ( 0.003 0.140 ( 0.002 0.239 ( 0.002 0.171 ( 0.003 0.270 ( 0.002 0.265 ( 0.002 0.270 ( 0.002
0.0 0.3 0.0 0.1 0.3 9.2 4.6 0.7 2.1 1.4 1.1 0.9 0.04 31.0 3.3 4.7 3.1 1.8 2.7 2.5 1.1 1.4 1.1 0.7 76.2 29.9 9.3 7.7 11.0 9.5 12.4 8.7 8.5 9.0 7.1 9.4 6.1 7.5 8.6 21.0 7.3 5.7 4.9 3.4 2.3 1.5 2.5 1.2
1.51 1.14 1.27 1.36 1.56 1.26 0.97 1.21 0.99 1.41 1.65 1.52 1.37 0.93 1.39 1.09 1.35 1.38 1.48 1.63 1.96 1.88 1.21 1.50 1.42 1.35 1.15 1.07 1.59 1.56 1.31 1.43 1.31 1.12 1.20 1.20 1.38 1.29 1.42 1.47 1.17 1.58 1.31 1.22 1.57 1.89 1.66 1.21
Figure 5. The 95% confidence ellipses plotted from donor decays with different acceptor concentration (10-5 mol/cm3): 1.08 (1) and 1.80 (2). PI-Phe(An)-PMMA, samples 14 and 13.
Figure 6. Variation of the fitting parameter P versus the global acceptor concentration CA in PI-Phe(An)-PMMA films (liner part). For the samples shown, the block copolymer number-averaged chain length N has the values (left to right) of 656, 544, 367, 305, and 193 (with corresponding mole fractions of PI block of 0.22, 0.51, 0.40, 0.43, and 0.07).
a Beginning of the linear dependence of the parameter P versus global acceptor concentration CA.
can be used to fit the individual DET decay with the same statistical validity. Nevertheless, the range of variation inside the ellipse is rather small, with variation of absolute values of ca. 1%. From the essentially perpendicular ellipse orientation, it follows that there is no correlation between P and β. Typical changes in P and β for a series of fluorescence decays obtained by increasing the global acceptor concentration are presented in Figures 6-8. According to the Klafter-Blumen model,2 only the parameter P should be related to the acceptor concentration. The β parameter is related to the apparent dimension of the DET process and, in the absence of a crossover, should be concentration independent. As one sees in Figure 6 and 8, the parameter P is proportional to the acceptor concentration, and above a certain concentration, β is constant. At low ratios of CA/CD, values of β decrease. In the following sections, we discuss the dependence of P upon CA and then examine the behavior of β.
/
Figure 7. Variation of the fitting parameter P versus the global acceptor concentration CA for wide a range of acceptor concentrations. For the PI-PMMA samples shown, the averaged chain length N has the values 367 (1) and 656 (2).
Dependence of the Parameter P upon CA. The dependence of the fitting parameter P upon the global acceptor concentration CA was examined for a wide range of acceptor concentrations. Over a wide range of concentrations above a certain minimum,
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Figure 8. Dependence of the apparent dimension β upon the acceptorto-donor ratio for PI-PMMA samples shown in Figure 6.
values of P for all samples are proportional to CA, and the extrapolated line at low concentrations passes through the origin (Figure 6). At low ratios of the acceptor to donor, somewhat different behavior is observed. Data in Figure 7 show that for low acceptor concentrations, CA/CD < 0.5, experimental values of P appear to level off and do not change with decreasing CA. The displacement of the experimental points from the linear dependence observed at higher CA is much larger than values predicted by the uncertainty of the fitting parameters determined for the individual fits. It is difficult to interpret these deviations rigorously, particularly since these samples involve the least amount of energy transfer, and the parameters are known with the least amount of confidence. The concentration range where this occurs corresponds to a region that includes a crossover effect on β (see below). Nevertheless, the major conclusion to be drawn is that, over a wide range of sample compositions (e.g., Figure 6), the data follow precisely the predictions of the Klafter-Blumen equation. In the context of this expression, it has been shown18 that the preexponential term P is related to the global acceptor concentration in the film CA by
P ) (4πR03NAv/3λΦ)gβΓ(1 - β)CA
(5)
where NAv is Avogadro’s number, Φ is the volume fraction of the minor phase, g is the orientation factor, Γ(1 - β) is the complete gamma function, and β is defined in eq 3. We analyze our data under the assumption of a random orientation of immobile dyes in the interface volume, leading to g ) 0.711.3c From the point of view of block copolymer morphology, the parameter of greatest interest is λ, related to R/δ, where R is the half-length of minor phase (see Figure 1).18,19 This relationship depends on the geometry of the minor phase with following definitions for λ
λ ) (1 + δ/R)R - 1
(6)
where R ) 1, 2, and 3 for lamellar, cylindrical, and spherical microdomains, respectively. Thus, the product λΦ in eq 5 gives us the volume of interphase. From the dependence of the fitting parameter P upon the global acceptor concentration, one can determine through λ the morphological parameter R/δ. To obtain this dependence, we measure donor fluorescence decays for films with variable acceptor-donor compositions, but fixed block copolymer overall chain length and composition. Under these circumstances, the local acceptor concentration in the interface volume will be the independent variable which is experimentally manipulated. The magnitude of R/δ is determined by the chemical structure of the block copolymer (by the absolute and relative lengths of the blocks) and does not depend on the
Figure 9. Variation of microdomain size R scaled by the interface thickness δ as a function of overall chain length N for the PI-PMMA samples shown in Figure 6. The proportionality constant C has been calculated as described in ref 26.
Figure 10. Fraction of unquenched donor A2/(A2 + A1) versus the global acceptor concentration for PI-PMMA (1) and PS-PMMA (2). Both block copolymers are labeled by Phe (donor) and An (acceptor) at junction.
acceptor-to-donor ratio. This means that the R/δ ratio will be a physical invariant over a series of fluorescence decay experiments carried out on mixtures of two otherwise identical block copolymers, one labeled with donor and the other with acceptor. Finally, for a series of block copolymers of different compositions, we can test the prediction that R/δ varies as N2/3. This prediction arises because in the theory of the strong segregation limit,12-14 the interface thickness δ is predicted to be invariant, and the microdomain size R to increase, with N2/3. The data in Figure 9 present the dependence of the parameter ln(R/δC) upon the natural logarithm (ln N) of the overall chain length N. The slope of the line (0.650) through the four points of highest polymerization index N is very close to the value predicted by theories for strong segregated systems (0.666). We do not fit the line through the point of lowest N because we believe that it represents a sample outside the strong segregation limit.19 In previous publications,18,19 we have shown that these data can either be combined with scattering experiments or interpreted with the help of a theoretical analysis by Ohta and Kawasaki,26 to obtain the interface thickness in the system. Here we obtain δ ) 2.6 nm for PI-PMMA films. Behavior of the A2 Term. According to the discussion above, we tentatively ascribed the importance of the A2 term in eq 4 to the presence of donors in environments where they are unable to transfer energy to acceptors and thus emit with their unquenched lifetime. We have examined the behavior of the PI-PMMA system very carefully and gone back to reanalyze data from the PS-PMMA system reported previously.18,19 In Figure 10, we plot values of A2/(A2 + A1) as a function of the global acceptor concentration CA in the system. We note first
7120 J. Phys. Chem., Vol. 100, No. 17, 1996
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that data from all experiments with PI-PMMA fall on one line and those involving PS-PMMA fall on a second line. These values are independent of the global morphology (spheres, cylinders, lamellae) of the system, presumably because the radius of curvature is much larger than the interface thickness δ in all of the systems. Values of A2/(A2 + A1) are significant when CA is low, but quickly decrease to ca. 0.03 when the ratio CA /CD approaches 0.5. We find it interesting that the system with the wider interface exhibits large values of A2/(A2 + A1) out to larger values of CA. For PI-PMMA we estimate, for example, that the fraction of donors not involved in DET reaches its asymptotic level when the mean spacing between donors and acceptors becomes comparable to the interface thickness, i.e., at 2.8 nm. For PS-PMMA, for which δ ) 5.1 nm, this fraction levels off when the mean donor-acceptor spacing equals 3.5 nm, which represents the maximum distance for DET for a system with R0 ) 2.4 nm. In spite of similarity in behavior of the A2 term noted for block copolymers with different chemical structures; we find that the A2 term asymptotic level is slightly less for PI-PMMA (about 3%) than for PS-PMMA (up to 7%), which is characterized by a smaller value of the Flory-Huggins interaction parameter χAB. The correlation of A2 term asymptotic level with the thermodynamic state of the interface may be an interesting topic for further investigations. A Crossover in β. Inspection of the local minima of the χ2-β surfaces in Figure 4 indicates that the parameter β takes one value when the acceptor concentration is relatively low and a significantly higher value when the acceptor concentration is increased. When β is plotted as a function of CA/CD, our samples show a universal behavior for β for values of CA/CD > 1 (see Figure 8). In this concentration range, the magnitude of β is very similar to that found previously for PS-PMMA block copolymer films. In each sample, at lower values of CA/ CD, there is a pronounced decrease in β. There appears to be a clear crossover, β taking one common value for CA/CD > 1 and a second, much lower value for CA/CD < 1. The magnitude of the low CA/CD value of β varies from sample to sample. We also note two other features of the data. First, at very low values of CA/CD values of β appear larger. In this range of acceptor concentrations the uncertainty in the parameters is so large that no proper conclusion can be drawn. Second, and more important, the proportionality between P and CA persists well into the range of concentrations where β values vary with CA. It is difficult to propose a rigorous explanation for the magnitude of β. Equations 3 and 4 are in a sense phenomenological equations in which the physical meaning of P is clear, whereas that of β is not. In the absence of proper simulations, we can only speculate about the origin of the crossover. For example, it is unlikely that the crossover is related to the onset of donor-donor energy transfer. Fayer3a,3b has argued that this process is unimportant if
(R0D-D/R0D-A)3 , CA/CD
(7)
For the phenanthrene-anthracene pair employed here, R0D-D ) 0.88 nm and R0D-A ) 2.3 nm, and the criterion of eq 7 is met at all concentrations where meaningful data are obtained. One feature of the geometry that may be important in determining the magnitude of β is the relationship between the interface thickness δ (in this context the characteristic dimension of the restricted space) and the maximum distance Rmax sampled in a DET experiment (the characteristic dimension for DET). At high values of CA/CD, the mean separation between donor and acceptor dyes is smaller than δ and also much smaller than
Rmax. Since most donors are located near the center of the interface, they on average see a spherical distribution of acceptors, biased, of course by the detailed shape of the acceptor distribution function. At low acceptor concentrations, the mean donor-acceptor separation becomes comparable to or larger than δ. This means that most donors see acceptors which are far removed and are confined to a space resembling an oblate ellipsoid. In this sense, our results correspond to the simulations of El-Sayed6 which show that the apparent dimension of DET in thin cylindrical pores depends upon the width-to-length ratio of the pores containing the dyes. Conclusion We have examined the kinetics of direct nonradiative energy transfer between dyes confined to the 2.6 nm wide interface region of polyisoprene-poly(methyl methacrylate) block copolymer films. The interface thickness is similar in magnitude to the characteristic distance for energy transfer (R0 ) 2.3 nm) for the donor-acceptor dye pair (phenanthrene-anthracene) employed here. Samples were prepared from matched pairs of block copolymers, one containing a donor dye and the other an acceptor dye, at the PI-PMMA junction. This system differs from restricted geometry systems examined previously because of the diffuse nature of the edges of the confining space. Donor fluorescence decay profiles were fitted to the KlafterBlumen expression, eq 4, and the parameters obtained followed the predicted behavior, namely, that the preexponential term P was proportional to the acceptor concentration, whereas the stretched-exponential parameter β was independent of global acceptor concentration CA for certain acceptor-to-donor ratios CA/CD > 1. Statistical analysis of the data indicates essentially no correlation between these parameters in the fitting process. One of the most unusual features of the data is a crossover in β observed as a function of acceptor concentration for CA/CD < 1. Acknowledgment. The authors thank NSERC Canada and the Petroleum Research Fund, administered by the American Chemical Society, for their support of this research. References and Notes (1) Klafter, J., Drake, J. M., Ed. Molecular Dynamics in Restricted Geometries; Wiley: New York, 1989. (2) (a) Klafter, J.; Blumen, A. J. Phys. Chem. 1984, 80, 875. (b) Klafter, J.; Blumen, A. J. Lumin. 1985, 34, 77. (c) Blumen, A.; Klafter, J.; Zumofen, G. J. Chem. Phys. 1986, 84, 1397. (3) (a) Loring, R. F.; Andersen, H. C.; Fayer, M. D. J. Chem. Phys. 1982, 76, 2015. (b) Miller, R. J. D.; Pierre, M.; Fayer, M. D. J. Chem. Phys. 1983, 78, 538. (c) Ediger, M. D.; Dominque, R. P.; Fayer, M. D. J. Chem. Phys. 1984, 80, 1246. (d) Peterson, K. A.; Fayer, M. D. J. Chem. Phys. 1986, 85, 4702. (e) Baumann, J.; Fayer, M. D. J. Chem. Phys. 1986, 85, 4087. (f) Peterson, K. A.; Stein, A. D.; Fayer, M. D. Macromolecules 1990, 23, 111. (4) Fredricson, G. H. Macromolecules 1986, 19, 441. (5) (a) Yekta, A.; Duhamel, J.; Winnik, M. A. Chem. Phys. Lett. 1995, 235, 119. (b) Duhamel, J.; Yekta, A.; Winnik, M. A. To be published. (6) (a) Yang, C. L.; Evesque, P.; El-Sayed, M. A. J. Phys. Chem. 1985, 89, 3442. (b) Yang, C. L.; El-Sayed, M. A. J. Phys. Chem. 1987, 91, 4440. (7) (a) Levitz, P.; Drake, J. M. Phys. PeV. Lett. 1987, 58, 686. (b) Levitz, P.; Drake, J. M.; Klafter, J. J. Chem. Phys. 1988, 89, 5224. (c) Levitz, P.; Drake, J. M.; Klafter, J. Chem. Phys. Lett. 1988, 148, 557. (8) Drake, J. M.; Klafter, J.; Levitz, P. Science 1991, 251, 1574. (9) (a) Pekcan, O.; Winnik, M. A.; Croucher, M. D. Phys. ReV. Lett. 1988, 61, 641. (b) Pekcan, O.; Croucher, M. D.; Winnik, M. A. Macromolecules 1990, 23, 2673. (c) Pekcan, O.; Egan, L. S.; Winnik, M. A.; Croucher, M. D. Macromolecules 1990, 23, 2210. (d) Pekcan, O.; Winnik, M. A.; Croucher, M. D. Chem. Phys. 1990, 146, 283. (10) (a) Encyclopedia of Polymer Science and Engineering, 2nd ed.; Mark, H. F., Bikales, N. M., Overberg, C. G., Menges, G., Kroschwitz, J. I., Eds.; Wiley: New York, 1985; Vol. 2. (b) Bates, F. S.; Fredrickson, G. H. Annu. ReV. Phys. Chem. 1990, 41, 525. (c) Halperin, A.; Tirrell, M.; Lodge, T. P. AdV. Polym. Sci. 1992, 100, 31.
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