On Cylindrical PS-b-PMMA in Moderate and Weak Segregation

Article pubs.acs.org/Macromolecules

On Cylindrical PS‑b‑PMMA in Moderate and Weak Segregation Mark D. Whitmore,*,† Jeffrey D. Vavasour,‡ John G. Spiro,§ and Mitchell A. Winnik§,* †

Department of Physics and Astronomy, University of Manitoba, 186 Dysart Road, Winnipeg, Manitoba R2M 5C6, Canada Code Mystics Inc., 1500-701 West Georgia Street, Vancouver, British Columbia V7Y 1C6, Canada § Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, Ontario M5S 3H6, Canada ‡

ABSTRACT: In recent years, there has been an explosion in the number of papers dealing with “cylindrical” PS-b-PMMA, for nanoscopic template or similar applications. This paper deals with two aspects of these systems. One is the detailed shape of the cores, i.e., the degree to which they are actually cylindrical in the hexagonal unit cell, as well as the nature of the core−corona interphase region. The second one relates to the use of fluorescence techniques to study these interphases, including whether or not one needs to consider energy transfer from one cylinder to neighboring ones. We approached these questions with numerical self-consistent field theory plus simulations of energy transfer based on them, and we examined two systems: one which mimics “typical” experimental systems at weak to moderate segregation, and one chosen to explore the limits of very weak segregation and polymer composition where the effects of interest could be maximal. In addition to providing basic information about the system characteristics, the results provide practical guidance on using the assumptions of cylindrical symmetry and no intercell energy transfer.

1. INTRODUCTION Over the past decade or so, there has been an explosion in the number of papers dealing with cylinder-forming block copolymers, particularly in thin films. This activity is stimulated by the intense interest in such systems for nanopatterning applications or for the creation of nanodots or nanowires in device applications. 1 Some of these papers focus on polystyrene-block-poly(4-vinylpyridine) (PS-b-P4VP),2 PS-bPDMS (PDMS = polydimethylsiloxane)3 or PS-b-PEO (PEO = poly(ethylene oxide)),4 where the two polymers are strongly segregated. Other papers describe cylindrical polystyrene-blockpoly(methyl methacrylate) (PS-b-PMMA), which is more weakly segregated.5 The interest in PS-b-PMMA is driven, at least in part, by the fact that the elimination of the PMMA component can provide a mechanism to transform thin films of this block copolymer into periodic arrays of nanopores or nanocylinders. Nanopores form when PS is the major component, and the thin film consists of a hexagonal array of PMMA cylinders.6 When PMMA is the major component, the film consists of a hexagonal array of PS cylinders that persist as rods or posts after etching away the PMMA phase.7 There are a number of challenges and issues that need to be addressed in these systems for applications that require longrange order. First, block copolymers do not inherently display global ordering. Instead, locally ordered domains tend to nucleate randomly, ultimately producing a material with many grains of random orientation.8 A second challenge is to align the cylinders perpendicular to the substrate.9 A third one is controlling polydispersity and understanding its effects.8,10 In fact, studies have shown that polydispersity can help achieve ordering, but complete removal of defects over macroscopic length scales requires care.8 © XXXX American Chemical Society

A fourth set of issues is particular to PS-b-PMMA and relates to the fact that it forms relatively weakly segregated systems, with diffuse interfaces between the PS and PMMA domains. To the best of our knowledge, the effects of weak segregation have not been discussed in nanoscopic template/nanolithography applications. For some applications, it is necessary to completely remove the PMMA block up to the junction point between PS and PMMA.6a In other applications such as sequential infiltration synthesis, the PMMA domains are infiltrated with an inorganic material.11 Since neither the width nor anisotropy of the junction distributions is known, one cannot now predict how these factors affect the final product. Our overall goal in this paper is to shed light on some of these matters, including the PS-b-PMMA junction distributions, the interface shape and thickness, and how these properties affect fluorescence decay measurements12,13 in fluorescence nonradiative energy transfer (FRET) experiments14 on junction-labeled samples, a technique that can be used to probe these systems.15 We have approached these questions by studying two systems. The first one is directly related to an ongoing experimental program involving the blending of PS-b-PMMA samples. Specifically, we modeled a PS-b-PMMA sample of 79,200 molecular weight (Mn), containing 70 vol % PMMA. This molecular weight, and hence the dimensions of a cylinder, are similar to block copolymers that have frequently been considered as starting points for nanoscopic templates, e.g., refs 1, 6a, 7, 8, 16, and 17, and so the results should be of practical Received: September 6, 2013 Revised: October 28, 2013

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dx.doi.org/10.1021/ma401862w | Macromolecules XXXX, XXX, XXX−XXX

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the UCA. For the UCA-based decay profiles, we largely used techniques we developed earlier. The new techniques for the anisotropic profiles are presented in section 4 and the Appendices. The results are presented in section 5. They include estimates of quantum efficiencies of energy transfer, ΦET, and a discussion of the precision of the Monte Carlo simulations. We summarize our conclusions in the final section.

interest. As we will see below, in these systems, when labeled with fluorescent dyes, virtually all the energy transfer occurs within a given cell. The second system is a hypothetical one designed to explore the limits of very weak segregation, a large core diameter relative to the overall unit cell size, and maximal value for the Förster radius18 which enters the energy transfer process and hence fluorescence decay. These choices create conditions where the interface is expected to be broad and most anisotropic, and where there is the greatest possible intercell energy transfer. Although the system is hypothetical, it is not very different from real systems, such as the cylindrical PS-b-PMMA with Mw as low as 39 000 studied by ThurnAlbrecht, DeRouchey, and Russell.19 For each of these systems, we used numerical self-consistent field (NSCF) theory to calculate the density profiles, junction distributions and cylinder dimensions, in two ways: (i) using the full symmetry of the hexagonal unit cell, and (ii) approximating this unit cell by a simple cylinder, i.e, the “unit cell approximation”, or UCA.20,21 We then used the results of each set of calculations to simulate fluorescence decay and energy transfer among junction-labeled block copolymers, with and without the possibility of energy transfer from one cell to neighboring ones, using a Monte Carlo technique. The approach extends earlier work13 in which we employed Monte Carlo simulations to analyze experimental fluorescence decay data in FRET experiments on cylindrical polyisoprene-bpoly(methyl methacrylate) (PI-b-PMMA) samples. That system is strongly segregated, the junction distributions and the interfaces between the blocks are narrow and essentially isotropic, and the interfaces in one cell are well separated from those in neighboring cells. For these reasons, we used the cylindrical profiles based on Helfand−Tagami theory,22 and considered only energy transfer within each cylinder. However, this approach may not be appropriate for low-molecular-weight, relatively weakly segregated samples, for two reasons. First, the junction distributions in each cell could be sufficiently broad that they overlap with, or come sufficiently close to, ones in neighboring cells so that Förster energy transfer18 between cells could occur. Second, since the core−corona interphases are broader and closer to the cell boundaries, the cores and junction distributions could be more anisotropic, in which case it is not clear how well cylindrically symmetric distributions can be used to approximate the energy transfer into horizontally neighboring cells. The UCA, first suggested by Helfand and Wasserman,20 was necessary in the early days of NSCF theory, although it is no longer so with modern computers. Nonetheless, it remains interesting to examine how close to cylindrical the cores actually are, and how accurate the UCA is, in real systems; this is one of the goals of this paper. A second one is to introduce a method of calculating decay curves from the anisotropic density profiles, which is the primary theoretical extension in this paper. The final goal is to determine if using the UCA affects the interpretation of fluorescence decay measurements. In section 2, we describe the characteristics of each system, and the related input parameters for the NSCF and fluorescence calculations. We begin our investigation of both systems in section 3, by examining the shape of the “cylindrical” cores and the accuracy of the unit cell approximation, by performing and comparing NSCF calculations with and without the UCA for each system. Sections 4 and 5 describe and present the Monte Carlo simulations of the fluorescence decay for each system, and for NSCF calculations with and without

2. SYSTEM CHARACTERISTICS AND INPUT DATA As noted in the Introduction, the Mn = 79 200 PS210-bPMMA575 sample, which we refer to as “System 1” in this paper (see structure in Scheme 1 below), was one of two cylinderScheme 1. Structure of PS-b-PMMA Showing Attachment of Phenanthrene (1-D) as a Donor Dye, Anthracene (1-A) as an Acceptor Dye, or a Phenyl Group (1-Ph) in the Polymer Lacking a Dye at the Junctiona

a

In system 1, m = NPS = 210, n = NPMMA = 575. The simulations reported here do not take into account the detailed structure of the connection between the PS and PMMA polymers.

forming diblock copolymers that we had designed for blending studies. We have reported some details of this design elsewhere.23 For present purposes, it suffices to note that the PS and PMMA block degrees of polymerization are NPS = 210 and NPMMA = 575. However, the pure component number densities in monomers per unit volume, ρ, statistical segment lengths, b, and the S-MMA Flory−Huggins χ parameter, using the PS density for the reference value, also needed to be specified. These values are ρPS = 5.795 nm−3,24 ρPMMA = 6.796 nm−3,25 bPS = 0.65 nm,26 bPMMA = 0.652 nm,26 and χ = 0.02875,12,27 taken from the references as indicated and interpolated to an annealing temperature of 140 °C. For this system χN ∼ 22−23, which is relatively weak segregation. “System 2” is the hypothetical system with design motivated as described in the Introduction. We chose conformational symmetry for simplicity, using equal values of densities and statistical segment lengths for the two blocks. For the energy transfer calculations, we needed to specify actual values, and used those for PS, i.e., ρA = ρB = ρPS and bA = bB = bPS, and we used the same value of χ as above. In order to ensure broad interphases and maximal overlap of junction distributions associated with neighboring cells, we chose the volume fraction of the minority component to be f = 0.4, and χN = 15. These choices imply N = 522, which is a reasonable value for a system like this, and that the system is at or beyond the limits of stability of the cylindrical morphology, especially when allowing for the possible existence of a gyroid phase,28 or allowing for shifts in the stability limits due to conformational asymmetry.29,30 This choice was deliberate, as it ensures that we are probing the limiting case. With one exception, the photophysical and related parameters we employed in the fluorescence decay simulations were taken from ref 13. For both systems, we used an unquenched decay time, τD, of 43.3 ns, and a preaveraged orientation factor, ⟨|κ|⟩2, of 0.476.31 The time interval for which we computed fluorescence decay was approximately 232 ns B


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(256 channels at a time scale of 0.907 ns/channel). For system 1, we took the Förster radius from ref 13, with value R0 = 2.28 nm. However, again in keeping with our goal of exploring the limits of maximum energy transfer between cells, we chose R0 = 5.0 nm for system 2, likely the largest realistic value.32 In our simulations we assumed that all junctions are labeled, with a 2:3 donor:acceptor ratio.

Table 1. NSCF Results for the Equilibrium Unit Cell Dimensions HEX model

system 1 system 2

3. CORE SHAPES AND THE UNIT CELL APPROXIMATION We first examined the degree to which the cores are actually cylindrical, and whether the hexagonal unit cell can be approximated by a cylinder. All calculations using the hexagonal unit cell, i.e., the HEX model, were based on barycentric coordinates and the triangular grid shown schematically in Figure 1.33 Each triangle is discretized by dividing each side into

UCA model

unit cell parameter (side of triangle), nm

area of cell, nm2

unit cell parameter (cylinder radius), nm

area of cell, nm2

17.57 14.60

802.0 553.8

16.04 13.36

808.3 560.9

junction distribution appear to be very close to circular, with no discernible anisotropy. Figure 3 juxtaposes the core profiles as calculated with and without the UCA: very small differences are just visible on this figure. Finally, Figure 4 compares the junction distributions calculated the two different ways, illustrated for two directions in the HEX model of system 1. They are virtually indistinguishable except at the outer edges. Overall, we conclude that the core is very close to cylindrical, and well represented using the UCA. Figure 5 shows the density and junction color maps for the very weakly segregated, “limiting” case, System 2. Again, the color maps indicate that the density and junction distributions are very similar in the two methods, with no visible anisotropy in the inner region. The main conclusion from this section is that the UCA is a reasonable assumption for analyzing typical systems of hexagonal symmetry that might be formed by block copolymer self-assembly.

4. FLUORESCENCE DECAY SIMULATION TECHNIQUES In FRET experiments designed to measure the width of the interface for block copolymer melts, one employs block copolymers with a single fluorescent dye at the junction between the A-block and the B-block. One prepares mixtures of two block copolymers of (nearly) identical composition that differ in the dye attached to the junction point. One of the dyes (the donor dye D) is selectively excited in a fluorescence decay measurement to generate electronically excited donors D*. The other dye (the acceptor dye A) can accept energy from D* in competition with radiative decay (fluorescence) from D*. As a consequence, the measured fluorescence decay rate (ID(t)) of the excited donor is shortened. It is a synthesis challenge to prepare these two dye-labeled polymers with nearly identical lengths and compositions. As we will see below, it is sometimes necessary to prepare a third polymer, identical in length and composition to the first two polymers, but without a dye at the junction. This polymer can serve to dilute the sample, increasing the mean D/A separation in samples where the FRET efficiency is so high that the donor fluorescence is hard to detect. The structures of the polymers that we simulate in this paper are presented in Scheme 1. 4.1. Hexagonal Symmetry and Anisotropic Junction Distributions. We first describe the technique we have developed for calculating the decay curves from the NSCF solutions, using the full symmetry of the hexagonal unit cell. The key input into these calculations is the anisotropic junction distribution probabilities throughout this unit cell. Because of symmetry, we need only one-sixth of it, which is the triangle described above and in Figure 1. The first step is to specify the unit cell dimensions. The dimensions in the plane are those given in Table 1. In principle, the unit cell is infinitely long. For the decay simulations, however, we needed to choose a finite length with suitable boundary conditions. Here, we chose it such that each cell

Figure 1. Schematic illustration of the grid, based on barycentric coordinates, used for the hexagonal symmetry NSCF computations and junction distributions. This figure represents one-sixth of the hexagonal cell; the remainder is obtained by symmetry. The actual computations were carried out over a 10-fold finer mesh, i.e., each side was divided into 100 equal distances.

one hundred equal distances, and then the grid lines, similar to those shown, were constructed. This generated 5151 vertices and 10 000 small equilateral triangles. All calculations using the UCA model are based on a discretization of the cylindrical unit cell into 100 equally spaced concentric shells. These discretizations produced very similar mesh resolution in the two approaches. The overall methodology is the same in both approaches: for a given system, NSCF calculations were performed for a variety of unit cell parameters, and the equilibrium value was identified as the one with the minimum free energy. Table 1 shows the calculated unit cell dimensions for each system in each approach. Given the difference in unit cell shapes in the HEX and UCA methods, the best way to compare the unit cells is by using their predicted areas, and these are included in Table 1. For each system, these areas agree very closely; they are approximately 1% larger in the UCA model. Figures 2−4 show the density profiles and junction distributions for system 1, the “typical” one. Figure 2 shows color maps of the density profiles of the PS and PMMA blocks and the junction distributions: visually, the core profile and C


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Figure 2. Density maps of the PS, PMMA, and junction number distributions in system 1, obtained by NSCF calculations based on the hexagonal unit cell. Green indicates the segment density of the PS block, red the segment density of the PMMA block, and blue the junction distribution. The brightness increases with the density.

Figure 3. Comparison of the HEX and UCA solutions for system 1. The HEX solution is shown on the left, and the UCA solution on the right.

Figure 4. NSCF junction number distributions J(r) in units of nm−3. The curves labeled “a” and “c” are for the HEX solution along the lines from the center to the points P and V shown on Figure 3. Curve “b” is for the UCA solution.

contained 2000 molecules, and we applied periodic boundary conditions at the ends. These conditions allowed us to incorporate energy transfer beyond the ends of each cylinder. We needed to discretize the HEX unit cell, and we used the same mesh for this as in the NSCF calculations. Our testing if this is fine enough for the decay simulations is described in Appendix I. The resulting triangles and the length of the cylinder defined 10 000 trigonal prisms, each of cross section 1/ 60 000 that of the hexagon, and of length that spans our model unit cell. Each prism has the same volume in this construction, which we label Vi. Both the central column and, when they are included, each of the six surrounding hexagonal columns consist of 60 000 of these trigonal prisms. Once we specified the unit cells, we needed to place the dyes within them, and calculate the resulting intensity curves. We did

not, of course, know exactly where the dyes would be, and so we created many different sets of placements, i.e. replications of the system, calculated the decay curve for each one, and then averaged the results for our final decay curve. In each replication, there were approximately 2000 dyes per cell, and their placements reflected the junction probability distributions produced by the NSCF calculations. The algorithms used to make these placements are based on the heterogeneous Poisson process, a technique which is sometimes employed by workers in the spatial analysis field.34 We first calculated a quantity, called the intensity of the Poisson process, which we label λi, for each prism. It is the junction distribution function in that prism, normalized so that λi is the mean number of dyes per unit D


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Figure 5. Density maps of the core and corona blocks, and junction number distributions in system 2, obtained by NSCF calculations based on the hexagonal unit cell. Green indicates the segment density of the core block, red the segment density of the corona block, and blue the junction distribution. The brightness increases with the density.

volume in prism i, and λiVi is the mean number of dyes in the prism. By definition, Σi λiVi = 2000. The NSCF calculations provided the junction distributions at the vertices of each triangle, from which we calculated the λi at the center of each triangle. This technical point is described in Appendix II. However, the actual number of dyes in a prism is neither fixed nor equal to λiVi. Instead, it is some integral number, k = 0, 1, 2, .... This number was generated by a weighted randomization process, with weighting given by a Poisson random variates generator35 (λ V )k f (k , λi , Vi ) = i i e−λiVi k!

probability. Finally, dye donor and acceptor placements were extended beyond the ends of the model cell by the periodic boundary conditions. After specifying the donor and acceptor positions, we calculated all the individual donor−acceptor distances, rjk, and the corresponding Förster decay function 6 2 3 κjk ⎛⎜ R 0 ⎞⎟ w(rjk) = 2 τD ⎜⎝ rjk ⎟⎠

(2)

for energy transfer from the jth donor to the kth acceptor. From these, we computed the fluorescence decay at any time t from the expression13 originally developed by Blumen and Manz36

(1)

which has mean value λiVi. This process generates an integer number of dyes in each prism, with the probability of there being k dyes given by f(k, λi, Vi). There are, on average, 2000 dyes per unit cell, but the actual number can vary slightly in each replication. Since each cell was divided into 60 000 prisms but contained only about 2000 dyes, most prisms were empty, and very few had more than one dye. Once the number of dyes in each prism was calculated, they all needed to be placed somewhere within them: each was placed randomly along the length of the unit cell, and at the center of the prism’s cross section. Next, there was a third randomization procedure, which designated each dye as a donor with 40% probability, or an acceptor with 60%

ID(t ) =

NA ⎛ t ⎞ ND 1 exp⎜ − ⎟ ∑ exp[−t ∑ w(rjk)] ND ⎝ τD ⎠ j = 1 k=1

(3)

where ID(t) is the fluorescence intensity resulting from a delta function excitation, normalized to unity at zero time, ND and NA are the numbers of donors and acceptors, t is time (ns), and τD is the lifetime of donors in the absence of acceptors. In practice, w(rjk) is a rapidly decreasing function of separation, so we did not include energy transfer over distances larger than 5R0, and thus discarded all rjk greater than 5R0. As noted in section 2, we always employed a preaveraged orientation factor: κjk2 = ⟨|κ|⟩2 for all (j,k) combinations.37 E


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then used to calculate the Förster decay functions and fluorescence intensities from eqs 2 and 3. We also explored whether the intercell energy transfer could be modeled simply by extrapolating the UCA junction distributions into the neighboring cells. This extrapolation was done simply by reflecting the junction distributions across r = R, to 1.2R or 1.4R, and then multiplying by the area of a shell at each distance. We also discretized the new regions, adding additional quadrilateral prisms of the same size in each shell of the expanded region.

These steps completed the calculation for one replication; they were repeated for all the other ones. For each replication, we used the same set of values λi and Vi . However, the Poisson random variates generator returned a different set of numbers of dyes in each prism, the second randomization procedure placed each of them at new positions along the cell axes, and the third randomization resulted in different ones being donors and acceptors. From these new positions, we then calculated all the new rjk, and corresponding w(rjk), and then a new decay curve ID(t). To obtain our final result, we did all this for 1000 different replications, and then averaged. In some cases, we also included intercell energy transfer from one central cell to the six neighboring ones. Because of the distances involved, it was not necessary to go beyond the immediately adjacent cells, so there were seven unit cells in total to consider. Each of these cells was modeled by 60 000 prisms as described above. The normalized junction distributions, λi, in each prism in the neighboring cells were, of course, the same as those in equivalent positions (by symmetry) in the central cell. The actual numbers of dyes in each prism were then generated independently through Poisson processes as described above, and then their placement along the cell axis, and their identification as donors or acceptors, were all generated by additional independent random processes. Thus, there were a total of approximately 14 000 independently positioned dyes throughout the seven columns, each independently identified as a donor or acceptor with a 40% probability of being a donor. In addition, these dye distributions were extended a distance 5R0 beyond the ends of each column. 4.2. Cylindrical Symmetry and Isotropic Junction Distributions: UCA and Extrapolated UCA Models. The overall procedure for the UCA-based cylindrical approximation was much as described for the full hexagonal symmetry. Again, we carried out averages over 1000 replications of the system. In each one, the dye placements in the basal plane were governed by the junction distributions and Poisson processes. Their placements along the cell direction, and the identification of each as a donor or acceptor, were determined by additional random processes. The unit cell parameter for each system is again in Table 1, the cylinder length was again chosen so as to contain 2000 molecules, and periodic boundary conditions were applied at the ends. The differences arise because of the cylindrical symmetry, with the hexagonal unit cell replaced by a cylinder of radius R. The junction distributions, K(r), are isotropic, and defined at 101 discrete, equally spaced radial coordinates r ε [0,R]. The areal junction distributions which we needed were obtained by normalizing these, as described in Appendix II. The discrete coordinates defined 100 concentric shells, each of which was subdivided into quadrilaterals (except for the central shell whose subdivisions were three-sided) in a particular way: the center disk was divided into six equal parts, the first shell into 18 parts, the next one into 30 parts, etc. This resulted in the same number and the same area of cells as in the HEX model, minimizing any numerical inaccuracies due to different grid sizes in the two models. The volume elements, Vi, are now the columns of quadrilateral prisms spanning the length of the cylinder. Equation 1 for the intensity of the Poisson process still applied, but with these new Vi, and the λi calculated by appropriately normalizing the isotropic junction distribution function, and linear interpolation within each shell. Again, more details are given in Appendix II. As before, the placement of the dyes was

5. FLUORESCENCE DECAY SIMULATION RESULTS 5.1. System 1, the M n = 79 200 PS-b-PMMA Copolymer. We now turn to whether the small differences between the UCA and full HEX solutions would quantitatively affect the predicted energy transfer, and whether there is energy transfer from one cell to neighboring ones, beginning with system 1.6 Figure 6 shows, in principle, three curves related to system 1, calculated using the HEX model: the decay which would occur

Figure 6. Fluorescence decay curves for system 1, the Mn = 79 200 copolymer, using the HEX model. The upper curve is the decay with no energy transfer, and the lower curve is the decay with ET. The results with and without ET into neighboring cells are indistinguishable on this scale.

in the absence of any energy transfer, the decay if there is ET but only within the central unit cell, and the decay with ET both within the central cell and into the neighboring ones. Clearly, ET is significant. However, ET within the central cell accounts for virtually all of it. In fact, the latter two curves are indistinguishable on the scale of this diagram. Quantitatively, we found that including ET into neighboring cells reduces the intensity by less than 1%. We also calculated the quantum efficiencies for the decay curves, defined as ∞

ΦET = 1 −

∫0 ID(t ) dt ∞

∫0 ID0 (t ) dt

(5)

I0D(t)

where ID(t) and refer to the donor decay at time t, following δ-pulse excitation, for experiments carried out in the presence and absence of acceptors, respectively. For this case, we obtained ΦET = 0.34, which is very convenient for experimental conditions.38 F


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In principle, we could also calculate the decay curves and quantum efficiency for the UCA model and compare with these HEX results. However, given the degree of agreement in the junction distributions between these models for this system, this was unnecessary. This will be further illustrated when we examine system 2. A technical question to consider here is if the variance of the Monte Carlo simulations was small enough for meaningful comparisons between different model assumptions. This is addressed in Figure 7 which, like Figure 6, is for the HEX

Figure 8. Fluorescence decay curves for system 2 using the HEX and UCA models, with energy transfer restricted to the central cell. The upper curve is the decay with no energy transfer, and the lower curve is the decay with ET. The results for the two models are indistinguishable on this scale.

We begin with Figure 8, which shows the decay curves obtained for the HEX and UCA models, but restricting ET to a central column. The most interesting aspect of this figure is that the UCA and HEX results are virtually indistinguishable. This implies that, at least in the case of no ET to neighboring cells, the assumption of perfect cylinders is good even for this system. Figure 9 probes the existence of ET to neighboring cells by showing the decay curves with and without it, calculated using

Figure 7. Variations among the 1000 replications used to construct the decay curves for system 1 for the embedded column HEX model calculations. The curve labeled “mean” is the average of the 1000 replications, and is the result shown in Figure 6. The curves labeled “maxima” and “minima” are the ones which gave the maximum and minimum intensities, respectively. Also shown, using the right-hand scale, is the standard deviation (σ) of the values of log ID(t) at each time.

model with ET within the central column and into the six surrounding columns. It shows the curve obtained from averaging over the 1000 replications, the two extreme curves showing the maximum and minimum decays, and the standard deviation in the calculated curves of log ID(t). This standard deviation is almost always at least 3 orders of magnitude smaller than log ID(t), implying that the variance is, indeed, small. Overall, our results suggest that the profiles and interface thicknesses of PS-b-PMMA samples that have been frequently considered for nanoscopic template applicationssimilar in molecular weight to our PS-b-PMMA block copolymercan be readily determined by fluorescence decay measurements without including ET between neighboring cylinders or using the anisotropic hexagonal unit cell. 5.2. System 2, the χN = 15 Copolymer. We now turn to the results for system 2, where we anticipated that the anisotropy and intercell ET would be maximal. Figures 8−10 show decay curves for this system, under various assumptions. In all of them, the quenching is much more rapid than it was in system 1 for two main reasons. First, the Förster radius is larger, and so ET can occur over greater distances. Second, because the molecules are smaller but they still each have one dye attached, the dyes are, on average, closer to each other. Because of this very strong quenching, the time scales in these figures are shorter than in the previous case.

Figure 9. Fluorescence decay curves for system 2 using the HEX model. The upper curve is for energy transfer restricted to the central cell, while the lower one includes ET to neighboring cells.

the full HEX model. The upper curve is the same as in Figure 8. In this case, it is clear that the majority of the quenching is done within the central column, but there is a non-negligible contribution from the neighboring cells. We shall return to this difference below. The possibility of understanding the decay curves simply by extrapolating the UCA distributions is probed in Figure 10, which shows the decay curves calculated from the UCA distributions and compares them with the full HEX model results. The agreement is quite good if the extrapolation is extended to 1.4R. This suggests that the ET to neighboring cells is an important factor but, consistent with what we found above, anisotropy is not important even for this system. G


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Table 2. Quantum Efficiencies of Energy Transfer (ΦET) for System 2, Based on Simulated Experimental Decaysa ΦET, % (replicate analyses) model HEX, embedded column HEX, central column only UCA, central column of width R UCA, central column of width 1.2R UCA, central column of width 1.4R

94.2 90.7 90.7 92.9 94.1

94.3 90.9 90.5 93.0 94.0

94.0 90.6 90.7 93.2 93.6

94.3 90.7 90.6 93.2 b

For each model, the four values of ΦET correspond to four different replications of the noise. The fact that the deconvolution failed in one case illustrates the non-trivial nature of this calculation. bDeconvolution failed. a

Figure 10. Fluorescence decay curves for system 2, using the full embedded HEX model and the “extrapolated UCA model”. The upper curve is for energy transfer restricted to the central cell, the next two are for extrapolations to 1.2R and 1.4R, and the final one is the embedded HEX model.

conclusions of the previous paragraph are independent of the details of the more involved procedure. The high value of ΦET to emerge from the simulations of system 2, and the corresponding rapid and very weak decay of donor fluorescence suggest experiments on binary mixtures of D- and A-labeled polymers, as shown in Scheme 1 would be difficult. There are two mitigating factors that make this experiment a possibility. First, to achieve a value of R0 = 5, one needs a different donor−acceptor pair. We have previously shown32 that coumarin-3 as a donor and disperse red as acceptor have R0 = 5. This donor dye has an intense emission and an unquenched lifetime of 2.0 ns; thus fluorescence will compete much more effectively with FRET than in the case of phenanthrene, with its 43 ns lifetime. In addition, if high values of ΦET still make ID(t) measurements difficult, one can study ternary blends with an unlabeled block copolymer serving to increase the mean separation of D and A groups in the interface.

We conclude this section by probing the possible effects of anisotropy and intercell ET on the quantum efficiency. We did not deem it prudent to calculate ΦET directly from the ID(t) curves, because the decay is so fast, with very substantial decay occurring from one channel to the next, rendering a numerical integration inaccurate. We also felt that this rapid decay warranted the incorporation of the finite width of the pump pulse in a real experiment, instead of a δ function excitation. Accordingly, we proceeded as follows: (1) Convolved the calculated ID(t) with a lamp function, taken from the experiments of Yang et al.6 (2) Added Poisson noise to create a noisy decay curve. (3) Fit the noisy decay from step 2 to a stretched exponential form39 (deconvolution40).

6. CONCLUSIONS We have studied cylinder-forming PS-b-PMMA at moderate and very weak segregation, in particular the degree to which the “cylinders” are actually cylindrical rather than hexagonal, the effects of any such anisotropy on fluorescence decay for dyelabeled samples, and the degree to which energy transfer from one cylinder to neighboring ones needs to be incorporated to fully account for the decay. We considered two systems, one which is “typical” of such polymers, and one in which the effects of a broad and diffuse interface, and relatively large minority component, should lead to maximal anisotropy in the core shapes and maximal energy transfer to neighboring cells. We found that the effects of anisotropy are negligible in both systems: the density profiles and junction distributions were very similar in the UCA and full HEX solutions. For the “typical” system, we also found that there was negligible energy transfer from one cell to neighboring ones. For the second system, which was tuned to maximize any intercell energy transfer, we found that there was some, but it was still small: in spite of the small molecular weight, very diffuse interface, and the large Fö rster radius used, intercell energy transfer accounted for less than 5% of the quenching. Overall, these results imply that, when interpreting fluorescence decay measurements, the anisotropic hexagonal symmetry always has negligible effect, and energy transfer to neighboring cells is small, but not necessarily, negligible.

⎡ ⎛ t ⎞1/2 ⎤ t − 2γ ⎜ ⎟ ⎥ + B2 ID(t ) = B1 exp⎢ − ⎢⎣ τD ⎝ τD ⎠ ⎥⎦ ⎛ t ⎞ exp⎜ − ⎟ ⎝ τD ⎠

(6)

(4) Integrated the fitted curve as in eq (5) to calculate ΦET. The Poisson noise introduces a randomization, and so we did this four times for each model. Table 2 shows the results of the analysis for each model of system 2. The first point to notice is that, for each one, the different replications gave very similar values of ΦET. Beyond that, the significant conclusions are consistent with Figures 8−10, namely that the central column UCA and HEX models account for most of the decay, and they agree well with each other, but that there is energy transfer into neighboring cells. Extrapolation of the UCA model to a distance of 1.4R can be used to account for this transfer, again indicating that the anisotropy of the junction distributions does not contribute significantly to ΦET. Finally, it is worth pointing out that, although the original decay curves of Figure 8 and 9 could not be integrated accurately because of the rapid initial variation in ID(t), these integrals could be estimated. The results agreed with the results in Table 2 to within a few percent, implying that the qualitative H


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faster decay. The results using the two finer meshes, the 0.16885 and 0.08425 nm grids, agreed even more closely: they were identical within the precision of the Monte Carlo simulations. We conclude that the mesh densities corresponding to a discretization of 0.16885 nm are sufficiently fine, and this is essentially what we used in our calculations.

APPENDIX I. DISCRETIZATION TEST All the numerical work requires a discretization of space and associated mesh size. We tested our discretization by using a simple procedure but for different discretizations, and comparing the results. This was done by conducting decay simulations for a system studied earlier, polystyrene-b-poly(butyl methacrylate),41 with the appreciably higher molecular weight of 246,000, and a similar Flory−Huggins χ parameter of 0.030. We used the same 100% efficiency for the junction labeling in a 2:3 donor: acceptor ratio. However, we averaged over 10 separate replications, rather than 1,000 as in the full calculations. The tests were done using the UCA model, which is sufficiently precise for these larger molecules. The unit cell radius for this system is 33.77 nm, and we used the Förster radius of 2.28 nm. We again used cylinders of finite length, long enough to hold 2,000 molecules, and applied periodic boundary conditions at the ends. In this case, we subdivided the unit cell into a set of concentric cylindrical shells of equal thickness, and placed the junctions on the surface of each of these shells. The number of dyes on each surface and their placement on it were determined by the same algorithm described in section 4, except that eq 1 was replaced by g (k , μi , Ai ) =

(μi Ai )k k!

e−μi A i

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APPENDIX II. CALCULATION OF THE JUNCTION (DYE) PROBABILITY DENSITIES AT THE PRISM CENTERS In each of our models, we needed to calculate the λi corresponding to the center of each prism by interpolating the NSCF results which gave the junction distributions at each prism’s vertices. This Appendix describes this interpolation for each model. II.1. HEX Model

To calculate the intensities λi for each of the small prisms as described in section 4.1, we needed to compute the concentration of junctions in each of 10 000 small triangles, similar to those shown in Figure 1. To do so, we assumed the distribution is linear within each grid cell, and employed the integration technique appropriate to barycentric coordinates. This resulted in the simple formula 3

λi =

(I-1)

where, now, g(k, μi, Ai) is the probability that cylinder i, which has surface area Ai, (outer surface of shell i) has k dyes on it. We employed three different grids in these tests, and calculated and compared the intensity decay curves, ID(t), from each one. For the first grid, we divided the unit cell into 100 equally spaced concentric cylindrical shells, which each had a thickness 0.3377 nm. In the second simulation, we doubled the spatial resolution by using 200 shells, each with one-half this thickness, and interpolated the NSCF junction distribution. In the third simulation, we doubled the resolution again, using 400 concentric shells. The resulting intensity decay curves are shown in Figure 11. The curves for the two coarser meshes are shown in the figure, but are virtually indistinguishable on this scale. Numerically, they differ by about 1%, with the finer mesh giving the slightly

3

∑ λil /3 =

∑ J(i , l)/3

l=1

l=1

(II-1)

Here, J(i,l) is the block junction density in nm−3 for the l-th vertex of the i-th triangle. II.2. UCA Model for Grid Cells Based on Results for the χN = 15 Copolymer

We needed the number density of block junctions in a central disk of radius R/100 and in 99 cylindrical shells of radial thickness R/100. The NSCF calculations provided junction distributions K(r) which are normalized to satisfy

∫0

R

2πrK (r ) dr = R2π

(II-2)

The areal junction number densities which we need are then given by d(r) = K(r)/πR2 = 1.78274 × 10−3 × K(r) nm−2. The number of junctions in the central disk or a shell is given by the integral 2π

N (r1 , r2) =

∫0 ∫r

1

r2

d(r )r dr dθ = 2π

∫r

r2

d (r )r d r

1

(II-3)

where r1 and r2 are the inner and outer radii for the shell under consideration, with r1 = 0 for the central disk. However, the NSCF results only give the distributions on each shell: as for the HEX model, the junction density at a value r was obtained by linear interpolation: r − r1 d(r ) = d(r1) + [d(r2) − d(r1)] r2 − r1 (II-4) The distributions are normalized to add up to one junction per molecule. For a column of 2,000 molecules we get mean junction (dye) numbers 2,000 × 1.78274 × 10−3 × N(r1,r2), to be divided, as mentioned in section 4.2, by the number of prisms per shell, 6, 18, ..., leading to the intensities of the heterogeneous Poisson process.

Figure 11. Fluorescence decay curves based on different mesh sizes, as labeled. The results using the 0.084425 nm mesh were identical to those using the 0.16885 nm mesh within the precision of the computations. I


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AUTHOR INFORMATION

Corresponding Authors

*E-mail: (M.D.W.) [email protected]. *E-mail: (M.A.W.) [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Mr. Markus Jonsson of COMSOL Inc. for assistance in validating our computer codes, and Drs. Gérald Guerin and Nicolas Illy for assistance with constructing the figures. The work was supported by the Natural Sciences and Engineering Research Council of Canada.

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On Cylindrical PS-b-PMMA in Moderate and Weak Segregation

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