Global Analysis of Fluorescence Decays to Probe ... - ACS Publications

Instructional Review pubs.acs.org/Langmuir

Global Analysis of Fluorescence Decays to Probe the Internal Dynamics of Fluorescently Labeled Macromolecules Jean Duhamel Institute for Polymer Research, Waterloo Institute for Nanotechnology, Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

ABSTRACT: The aim of this review is to introduce the reader first to the mathematical complexity associated with the analysis of fluorescence decays acquired with solutions of macromolecules labeled with a fluorophore and its quencher that are capable of interacting with each other via photophysical processes within the macromolecular volume, second to the experimental and mathematical approaches that have been proposed over the years to handle this mathematical complexity, and third to the information that one can expect to retrieve with respect to the internal dynamics of such fluorescently labeled macromolecules. In my view, the ideal fluorophore−quencher pair to use in studying the internal dynamics of fluorescently labeled macromolecules would involve a long-lived fluorophore, a fluorophore and a quencher that do not undergo energy migration, and a photophysical process that results in a change in fluorophore emission upon contact between the excited fluorophore and quencher. Pyrene, with its ability to form an excimer on contact between excited-state and ground-state species, happens to possess all of these properties. Although the concepts described in this review apply to any fluorophore and quencher pair sharing pyrene’s exceptional photophysical properties, this review focuses on the study of pyrene-labeled macromolecules that have been characterized in great detail over the past 40 years and presents the main models that are being used today to analyze the fluorescence decays of pyrene-labeled macromolecules reliably. These models are based on Birks’ scheme, the DMD model, the fluorescence blob model, and the model free analysis. The review also provides a step-by-step protocol that should enable the noneducated user to achieve a successful decay analysis exempt of artifacts. Finally, some examples of studies of pyrene-labeled macromolecules are also presented to illustrate the different types of information that can be retrieved from these fluorescence decay analyses depending on the model that is selected.

■

lifetime τDo of many dyes is usually on the order of a few tens of nanoseconds, an excited dye whose concentration decays exponentially with time by fluorescence has only a temporal window equal to ∼5τDo to undergo a photophysical process before going back to the ground state, at which point it will no longer be detected as its concentration at t = 5τDo equals 0.6% of its original value. Although the temporal window of 5τDo offered by many dyes is sufficiently long to cover the time scale over which the conformational rearrangements of many macromolecules occur, it is also short enough to prevent intermolecular photophysical processes from taking place as long

INTRODUCTION Although fluorescence is well known for its ability to probe intermolecular phenomena happening between macromolecules, this review will consider solely the ability of fluorescence to probe the intramolecular behavior of isolated macromolecules in solution. As it turns out, fluorescence is ideally suited to probing single macromolecules in solution that have been labeled with two small molecules selected for their ability to interact intramolecularly within the macromolecular volume via a number of photophysical processes, such as fluorescence resonance energy transfer (FRET),1 electron transfer,2 exciplex formation,3 and excimer formation.4,5 This assertion is a consequence of the combination of two key factors. The first factor is the time scale set by fluorescence over which a given photophysical process can occur. Because the fluorescence natural © 2013 American Chemical Society

Received: September 25, 2013 Revised: October 30, 2013 Published: October 31, 2013 2307

dx.doi.org/10.1021/la403714u | Langmuir 2014, 30, 2307−2324

Langmuir

Instructional Review

second to present some of the solutions that have been proposed over the years to deal with this complexity.

as the solution is dilute enough to prevent random intermacromolecular encounters via translational diffusion during this time interval. This is where the second key factor comes into play. Thanks to its outstanding sensitivity, fluorescence can probe solutions containing less than 10 mg/L of a fluorescently labeled macromolecule. Because the overlap concentration C* (i.e., the concentration where the entire solution volume is fully occupied by polymer coils) of a well-dissolved polymer is usually in the 1−10 g/L range, fluorescence experiments can be conducted with a concentration of macromolecules that is more than 2 orders of magnitude lower than C*, so low in fact that no intermolecular interactions between any two macromolecules can occur within the time scale defined by 5τDo, ensuring that only intramolecular photophysical processes are being probed. Although some cases exist where a dye binds so strongly to a macromolecule that no covalent attachment between them is required (binding of ethidium bromide to DNA is one example),6 the low concentrations of macromolecules typically used in fluorescence experiments that target isolated macromolecules in solution disfavor the binding of an extrinsic dye to the macromolecule, and the covalent attachment of a dye is preferred to avoid the contamination of the fluorescence signal by free dyes in solution. One of the important macromolecular properties that many scientists are interested in characterizing by fluorescence is the internal dynamics (ID) of a macromolecule,1−5,7−10 which reflects how flexible the macromolecule is on the molecular level. In turn, this information sheds light on the often complex behavior exhibited by solutions of macromolecules on the molecular and macroscopic levels. For instance, information on the ID of a polypeptide in solution can lead to predictions on how it might fold into the 3D-ordered structure of a catalytically active protein.7,8 The shear-induced relaxation of a polymeric network prepared from an aqueous solution of associative thickeners depends in part on the flexibility of the polymeric backbone.9,10 In fact, many physical phenomena involving macromolecules exist only because of the ability of macromolecules to adopt different conformations over a range of time scales, which can be inferred from the characterization of their ID. Theoretically, information about the ID of fluorescently labeled macromolecules can be retrieved by analyzing their fluorescence spectra or decays acquired by steady-state or timeresolved fluorescence, respectively. In practice, the ability of time-resolved fluorescence (TRF) to distinguish quantitatively between all of the fluorescent species present in solution, most importantly, between the dyes that are bound to and thus report on the macromolecule and those that are not, makes TRF a very powerful analytical tool for probing the ID of macromolecules quantitatively on the molecular level. Furthermore, whereas steady-state fluorescence (SSF) experiments provide quantitative information about the effectiveness of a given photophysical process, the characterization of the ID of a macromolecule is best described by parameters expressed in s or s−1 that can be retrieved directly from the analysis of fluorescence decays acquired by TRF. Unfortunately, although being much more informative and much less prone to artifacts than the interpretation of steady-state fluorescence spectra, the analysis of fluorescence decays acquired from fluorescently labeled macromolecules is usually much more complex and involved than that of fluorescence spectra. It is the purpose of this review first to explain the origin of the complexity associated with the fluorescence decay analysis of fluorescently labeled macromolecules to retrieve quantitative information about their ID and

■

MATHEMATICAL COMPLEXITY Whether dealing with FRET, electron transfer, or excimer formation, the same basic reaction scheme applies to the description of any of these photophysical phenomena, and it is depicted in Scheme 1. Upon absorption of a photon of energy hν, the dye D Scheme 1. Quenching of Chromophore D by Quencher Q Covalently Attached to a Polymer with a Quenching Rate Constant of kQ

used to fluorescently label the macromolecule is excited. Excited dye D* can then release its excess energy by either emitting a photon via fluorescence with a natural lifetime of τDo or interacting with a quencher that occur with a rate constant of kQ. On the basis of Scheme 1, an expression of the dye concentration varying as a function of time ([D*](t)) can be obtained through the integration of eq 1, which results in eq 2. According to eq 2, [D*](t) decays monoexponentially with a decay time τ equal to (kQ + (1/τDo))−1. Consequently, kQ equals 1/τ − 1/τDo. The quenching rate constant kQ expressed in s−1 reflects the mobility of the dye and quencher, and because both dye and quencher are covalently attached to the macromolecule, kQ describes the ID of the macromolecule. ⎛ d[D*] 1⎞ = −⎜k Q + o ⎟[D*] dt τD ⎠ ⎝

(1)

⎡ ⎛ 1 ⎞⎤ [D*](t ) = [D*]o exp⎢ −⎜k Q + o 1⎟t ⎥ ⎢⎣ ⎝ τD ⎠ ⎥⎦

(2)

The expression kQ = 1/τ − suggests that using an infinitely long-lived dye for which 1/τDo approaches zero will yield the largest possible dynamic range of kQ values that could be studied. However, one must keep in mind that a long-lived dye will also allow the fluorescently labeled macromolecule to undergo large translational displacements that can lead to unwanted intermacromolecular encounters. Thus, a compromise needs to be found. To this end, eq 3 can be used as a first approximation to relate the temporal window given by 5τDo over which a fluorescence experiment is being conducted with the translational diffusion coefficient of the macromolecule Dt, its molar mass M, and its mass concentration c used in the experiment. Equation 3 yields the longest lifetime τDo that can be used for a fluorescent label while still preventing intermacromolecular encounters at mass concentration c. c 1 = 3 4 M π D × 5τ o N 1/τDo

3

t

D

(3)

A

−7

2 −1

Using a diffusion coefficient Dt of 10 cm ·s , a molar mass M of 105 g mol−1, and a mass concentration c of 10 mg·L−1 where c is at least 2 orders of magnitude smaller than the overlap concentration C* of many macromolecules, eq 3 predicts that the experiment depicted in Scheme 1 can be conducted on individual fluorescently 2308


Langmuir


labeled macromolecules in solution as long as τDo is shorter than 500 μs. Because fluorescence occurs on time scales given by 5τDo that are usually much shorter than 5 μs, these considerations confirm that fluorescence is ideally suited to probing the internal dynamics of isolated macromolecules in solution. However, it becomes clear from eq 3 that using much longer-lived dyes that, for instance, would phosphoresce instead of fluorescing might lead to unwanted intermacromolecular encounters if their τDo value is greater than 500 μs in the example at hand. Over the years, a number of protocols have been implemented to covalently attach dyes and quenchers onto a macromolecule and they all fall into two general categories depending on whether the macromolecule can be labeled at specific or random positions with the dye and its quencher. Labeling at specific positions typically represents the labeling of the chain ends of a linear polymer1−4,7,8 or dendrimer.11,12 This is usually done for macromolecules that exhibit reactive groups at the chain ends. These reactive groups can naturally be present in the macromolecule such as for dendrimers that are synthesized by successive additions onto their end groups and some linear polymers prepared by condensation or chain growth polymerization such as peptides and poly(ethylene glycol), respectively. The reacting groups can also be introduced to terminate the polymerization, for instance, by capping the propagating ends of a polymer prepared by anionic polymerization with carbon dioxide or ethylene oxide to introduce a carboxylic acid or hydroxyl functionality at the chain ends, respectively.13 Labeling a polymer at specific positions is usually more synthetically challenging because it requires the use of controlled polymerization. By contrast, random labeling of a macromolecule is usually easier from a synthesis viewpoint, and it can be executed in a number of ways. The monomer constituting a polymer can be copolymerized with either a fluorescently labeled monomer14 or a dormant monomer bearing a reacting group that can be activated after polymerization to react with the desired fluorescent labels.15 Another possibility consists of grafting reactive groups onto a polymer and reacting these groups with a dye derivative16 or grafting a dye derivative directly onto a polymer.17 Having fluorescently labeled the macromolecule at specific or random positions with the appropriate dye and quencher, we need to consider some theoretical aspects of the kinetics describing their interactions. A theoretical study by Wilemski and Fixman,18,19 confirmed later by numerous experiments,1−8,20−22 has established that kQ in Scheme 1 depends strongly on the chain length spanning D and Q. This result implies that the monoexponential decay introduced in eq 2 will be obtained only for polymeric constructs bearing a single dye and a single quencher separated by a single chain length. In turn, this stringent requirement leads to the conclusion that the study of the vast majority of macromolecules will involve the analysis of complex multiexponential decays because their synthesis does not lend itself to the easy incorporation of a single dye and a single quencher at two specific positions within the macromolecule. This complexity is clearly illustrated in Figure 1 by considering macromolecules randomly labeled with a dye D and its quencher Q. Because the excitation source used in fluorescence experiments does not generate a flow of photons intense enough to excite more than one dye per macromolecule, the fluorescently labeled macromolecule can be viewed as possessing a single excited dye D* and a random number nQ of quenchers Q, even though it might also contain many dyes in the ground state. Because a fluorescence experiment probes only the excited dyes, the many dyes in the

Figure 1. Distribution of rate constants resulting from the random incorporation of quenchers in a macromolecule.

ground state remain invisible. Assuming that kQ depends solely on the chain length separating D* and Q, the same kQ value is expected whether D* and Q are at positions i and j or j and i. Another assumption made in Figure 1 is that all macromolecules bearing nQ quenchers exhibit the same distribution of ki,nQ quenching rate constants. It is clear from Figure 1 that a chain Q with nQ quenchers will generate ∑ni=1 i = nQ(nQ + 1)/2 rate constants ki,nQ. Because each rate constant results in an exponential decay similar to that given in eq 2, the fluorescence decay of D* should be well represented by eq 4. ∞

[D*](t ) =

nQ

∑ ∑ [D*i ,n nQ = 1 i = 1

⎡ ⎛ 1⎞⎤ ⎢ − ⎜k i , n + o ⎟t ⎥ ] exp o Q ⎢⎣ ⎝ Q τD ⎠ ⎥⎦

(4)

In eq 4, [D*i,nQ]o represents the concentration of chains having nQ attachment points for quenchers and ki,nQ is the ith quenching rate constant of a chain having nQ quenchers. Equation 4 is already quite complex, but it is a very simplified version of what [D*](t) actually is because it assumes that each chain with nQ quenchers has the same distribution of ki,nQ quenching rate 2309


Langmuir


processes undergone by the fluorophores covalently attached to the macromolecule.12,25 Although these improvements in the global analysis of fluorescence decays are recommendable, the resolution of more than three different exponentials from two independent fluorescence decays remains challenging. In any case, the resolution of the large number of exponentials used in eq 4 is currently out of reach, and simplifications need to be introduced, as will be explained hereafter. Strategy 2: Selection of a Simple Quenching Mechanism. The two best known photophysical phenomena used to probe the internal dynamics of macromolecules via two independent fluorescence channels are FRET1,11,24 and excimer formation.4,5,12,20−22,25−27 In the case of FRET, the macromolecule needs to be labeled with an energy donor (D) and an energy acceptor (A). Because the rate constant for FRET depends on the through-space distance separating D* from A, it decreases over time as the average distance between D* and A increases. This behavior follows from the fact that the D*−A pairs separated by a short distance undergo FRET more efficiently than the D*−A pairs separated by a longer distance. The through-space distance dependency of FRET that evolves with time requires the fluorescence decay analysis to ensemble average the remaining D*−A pairs at each time interval.1,28 This represents an additional complication that has been accounted for successfully in the case of monodisperse macromolecules such as end-labeled peptides28 where the distribution of end-toend distances is known to obey Gaussian statistics. However, its applicability to randomly labeled macromolecules is more challenging, and such a study has not yet been reported. Quenching mechanisms that occur on contact are much simpler to deal with mathematically because they do not require an ensemble average of all fluorophore and quencher pairs over time. There exist numerous fluorophores that are quenched on contact upon encountering a small molecular inhibitor, but very few of these fluorophore−quencher encounters result in an emission that is distinct from that of the fluorophore. A FRET mechanism with a Förster radius R0 equal to the encounter radius between the fluorophore and the acceptor, also referred to as Dexter energy-transfer mechanism, would result in the desired effect because FRET would take place only on contact and transfer of the excess energy of the donor to the acceptor would provide a second independent emission channel.28 However, such donor−acceptor pairs have not been widely used in the literature because large R0 values are typically preferred for conducting FRET experiments. Beside FRET, excimer formation between an excited-state and a ground-state monomer constitutes a second quenching mechanism resulting in two independent spectroscopic channels that can be used to acquire the monomer and excimer fluorescence decays at wavelengths λM and λE, respectively. However, many fluorophores capable of forming an excimer such as naphthalene or perylene exhibit overlapping absorption and emission spectra, which leads to energy hopping via FRET29,30 when the through-space distance separating an excited-state and a ground-state fluorophore becomes smaller than R0, as is usually the case when the fluorophores are attached to the same macromolecule. Energy hopping represents an artifact for the study of ID because the energy travels through space rapidly between fluorophores via FRET instead of being localized on an excited fluorophore whose diffusion is controlled by the much slower ID of the macromolecule until it encounters a groundstate fluorophore to form an excimer.

constants. This is highly unlikely because the nQ attachment points for quenchers will be separated by different chain lengths on different macromolecules and thus will result in different distributions of ki,nQ rate constants. In other words, each macromolecule with nQ quenchers will have its own ki,nQ distribution, implying that eq 4 actually underestimates the number of exponentials that needs to be applied. Also, the rate constants ki,nQ for a given chain length separating D* and Q are assumed to be independent of the through-space distance separating D* and Q, which is true for collisional quenching but not if FRET1 and to a lesser extent electron transfer2,8 are taking place between D* and Q. Nevertheless and regardless of these additional complications, the simplified version of eq 4 captures the strong multiexponential nature of the fluorescence decays that are expected for a randomly labeled macromolecule. Considering that the analysis of fluorescence decays is notoriously unreliable for decays that are sums of more than three exponentials,23 strategies needed to be developed to deal with the inherent distribution of rate constants resulting from the random labeling of a macromolecule.

■

DEALING WITH MATHEMATICAL COMPLEXITY Strategy 1: Improvements in Resolution. The accurate retrieval of individual rate constants associated with a multiexponential fluorescence decay can be substantially improved if two independent fluorescence channels are available to probe the distribution of rate constants, as is the case in experiments that use FRET1,11,24 or excimer formation.4,5,12,25 In a FRET experiment, an excited donor D* that emits fluorescence at a wavelength λD transfers its excess energy to a ground-state acceptor A that becomes the excited species A* that emits fluorescence at a specific wavelength λA ≠ λD.1,11,24 The fluorescence decay of D* acquired at λD shows an exponential decay that reflects the distribution of rate constants for energy transfer ET o −1 kET i through a distribution of decay times τi = (ki + 1/τD ) , and the fluorescence decay of A* acquired at λA shows a rise time associated with the same decay times τi. Consequently, the donor and acceptor decays contain the same information about the kinetics of the FRET process expressed with two different equations that reflect the disappearance of donor D* and the creation of acceptor A*. Global analysis conducted by optimizing the same τi’s in the different expressions used to represent the donor and acceptor decays enables 25 a better resolution for the retrieval of rate constants kET i . A similar situation is obtained for macromolecules labeled with a chromophore that can form an excimer resulting from the encounter between an excited-state and a ground-state chromophore. The monomer and excimer fluoresce at two distinct wavelengths λM and λE, respectively. The fluorescence decay of the monomer acquired at λM shows a decrease in fluorescence intensity due to the distribution of rate constants kEX describing excimer formation whereas the excimer i decay shows an increase in fluorescence intensity that is described by the same distribution of rate constants. As for the FRET experiments, global analysis of the fluorescence decays of the monomer and excimer improves the accuracy of the pre-exponential factors and o −1 12,25 decay times τi = (kEX i + 1/τM ) . Global analysis of fluorescence decays usually refers to analysis programs where the same decay times are being used to fit different fluorescence decays and the pre-exponential factors are free to float.26,27 Work in this laboratory has demonstrated that the accuracy of the retrieved pre-exponential factors and decay times can be further enhanced if the pre-exponential factors and the decay times can be expressed and then optimized as a function of the kinetic parameters describing the photophysical 2310


Langmuir


Going through the list of excimer-forming fluorophores and looking for those having nonoverlapping absorption and fluorescence spectra, the pyrene fluorophore stands out. Its 0−0 transition that is responsible for energy migration for many other excimer-forming fluorophores is symmetry-forbidden,31 which prevents energy hopping from taking place between an excited pyrene and a ground-state pyrene. Consequently, pyrene is one of a few excimer-forming fluorophores that does not undergo energy migration when an excited pyrene approaches a ground-state pyrene to form an excimer. Because the lifetime of a fluorophore is inversely proportional to the average molar absorbance coefficient of the fluorophore in the S0 → S1 transition,32 which is symmetrically forbidden for pyrene,31 pyrene has a very long natural lifetime that can reach 360 ns for 1-pyrenemethylsuccinimide in hexane,33 thus offering an extended temporal window for probing all but the slowest ID of macromolecules. Certainly the better-known effect resulting from the symmetry-forbidden 0−0 transition of pyrene is the sensitivity of the fluorescence spectrum of pyrene to solvent polarity. The fluorescence intensity of the first peak (I1) is strongly reduced in low-polarity solvents when compared to the fluorescence intensity of the third peak (I3). However, in more polar solvents, the 0−0 transition is partially restored. Consequently, the I1/I3 ratio takes a much lower value in cyclohexane than in water. Although these effects are well established for molecular pyrene and have been reviewed numerous times (e.g., ref 12), it is worth considering how they affect the fluorescence spectrum of pyrenelabeled macromolecules because the substituent of the pyrene derivatives used to label a macromolecule inherently destroys the symmetry of pyrene by its very presence. Indeed, 1-pyrenebutyl derivatives are hardly sensitive to the polarity of the solvent.20 By contrast, replacing the methylene unit in the β position of the 1-pyrene derivative with a heteroatom such as nitrogen or oxygen partially restores the sensitivity of the fluorescence spectrum of the pyrene derivative to the solvent polarity.20,22 These considerations matter if one is interested in using a pyrene derivative to probe the polarity of microdomains generated by a macromolecule. Other advantages associated with pyrene excimer formation include the ease of labeling a macromolecule, which can be conducted in a single step because a single chromophore (i.e., pyrene) is involved, the large molar absorption coefficient at 344 nm for the S0 → S2 transition of about 40 000 cm−1·M−1 for most pyrene derivatives, and the small and usually negligible rate constant of pyrene excimer dissociation at temperatures below 40 °C as a result of the strong excimer binding energy afforded by the four fused benzene rings constituting pyrene. For these reasons, the remainder of this review will focus mainly on the use of excimer formation to characterize the ID of pyrene-labeled macromolecules that have been studied extensively over the past 40 years. However, it is important to note that the following discussion would also apply to any other fluorophore sharing pyrene’s remarkable photophysical properties or any donor− acceptor pair in a FRET experiment where the donor is longlived and the donor and acceptor do not form an excimer, do not undergo energy migration, and have an R0 value that approaches their encounter radius. Strategy 3: Reducing the Width of the Distribution of Rate Constants. Macromolecules prepared by controlled living polymerization that incorporate two reactive sites separated by a chain length having a narrow molecular weight distribution can be covalently end-labeled with a pyrene derivative. Numerous

monodisperse linear chains1−8,20−22 and one example of a dendrimer34 have been prepared in this fashion. Because the weak excitation sources used in fluorescence experiments can excite only a single pyrene label in a macromolecule strictly bearing two pyrenes, the second pyrene is in the ground state and acts as the quencher. In this case, nQ equals 1, and because the two pyrenes are separated by a single chain length, Wilemski and Fixman’s theoretical work18,19 dictates that the excimer is formed with a single rate constant and eq 4 reduces to eq 1. Although a monoexponential decay is expected, a slightly biexponential monomer decay is usually obtained as a result of residual excimer dissociation. Such kinetics are well described by Birks’ scheme,4,20−22,35 which is presented in the following section that reviews the different models used to fit the fluorescence decays of pyrene-labeled macromolecules. Although the analysis of the fluorescence decays acquired with macromolecules bearing strictly two pyrene labels separated by a single chain length is simplified by the fact that the fluorescence decay can be described by a single exponential, a review of the work conducted over the past 35 years indicates that this type of study is mostly restricted to the characterization of short endlabeled oligomers. As it turns out, the synthesis of such polymeric constructs is usually too demanding and is not amenable to the vast majority of macromolecules. Also, the rate constant of end-to-end cyclization kcy plummets with increasing chain length as kcy scales as N−α, where N is the degree of polymerization and the scaling exponent α has been found to take values between 0.9 and 1.9.4,20−22,25 In practice, this scaling relationship implies that almost no excimer is being formed for most common polymers having an N value greater than 100. This observation rationalizes why only short chains end-labeled with pyrene are being characterized in the scientific literature. Random labeling of a polymer with pyrene generates excimer much more efficiently,20 but this labeling scheme leads to a distribution of rate constants resulting in multiexponential fluorescence decays that are better described by eq 4. Because the exponentials constituting eq 4 cannot be resolved individually, the distribution of rate constants used in eq 4 needs to be simplified, as explained in the following section. Strategy 4: Simplification of the Distribution of Rate Constants. One important aspect of the scaling relationship kcy ∼ N−α is that excimer formation in a macromolecule randomly labeled with pyrene is dominated by those pyrene pairs that are separated by a short chain length. In fact, excimer formation occurs locally within the pyrene-labeled macromolecule, and the infinite distribution of rate constants inferred from eq 4 can be properly described by a much smaller subset of parameters. Two models have been derived to date that take advantage of this simplification, namely, the fluorescence blob model (FBM)5,16,20−22,25 and the model free analysis (MFA).12,24,25,36,37 Their derivation is described at the end of the following section.

■

MODELS USED TO DESCRIBE PYRENE-LABELED MACROMOLECULES The complexity of the models that have been introduced to fit the fluorescence decays of pyrene-labeled macromolecules parallels very closely the complexity of macromolecular architecture and labeling protocols introduced by synthesis chemists. The demonstration by Zachariasse and Kühnle in 197638 that excimer formation between two pyrene labels covalently attached to the ends of an alkyl chain decreased with increasing chain length encouraged polymer chemists to use the 2311


Langmuir


same architecture to probe the end-to-end cyclization of monodisperse linear chains first with poly(ethylene oxide) (PEO)39 and second with polystyrene (PS).40 In 1980, Winnik showed that Birks’ scheme analysis of the monomer and excimer fluorescence decays with two coupled exponentials satisfyingly described the kinetics of pyrene excimer formation for a series of pyrene end-labeled polystyrenes. This study was the first to demonstrate that kcy scaled as N−1.6 for polystyrene in cyclohexane at 34.5 °C, a θ solvent for polystyrene.4,40 Although the applicability of Birks’ scheme to fit the fluorescence decays of pyrene end-labeled monodisperse polymers has been confirmed for polymeric backbones other than PEO and PS,21 numerous examples with other pyrene-labeled constructs were also found where Birks’ scheme did not apply. In particular, three coupled exponentials were required to fit the monomer and excimer decays of 1,3-di(1-pyrenyl)propane where the short alkyl chain induces conformational constraints between the pyrene labels that lead to the formation of two excimers.26 To account for these complications, Zachariasse et al. introduced the DMD model and a number of variants in 1986. At about the same time, Winnik reported on at least two occasions that macromolecules containing more than two pyrenes yield fluorescence decays that could not be fitted according to Birks’ scheme as a result of the distribution of rate constants inferred from Figure 1.41,42 It was only about 15 years later that the fluorescence blob model (FBM)5,16 was introduced to deal with macromolecules randomly labeled with pyrenes, and the model free analysis (MFA)12,25,36 was implemented six years ago to provide quantitative information about the kinetics of excimer formation for any type of pyrenelabeled macromolecule. Birks’ scheme, the DMD model, the FBM, and the MFA are described hereafter. Birks’ Scheme. Birks’ scheme applied to pyrene end-labeled monodisperse chains4 is depicted in Scheme 2. Upon absorption

[E*](t ) =

⎛ ⎛ t⎞ ⎛ t ⎞⎞ ⎜⎜− exp⎜− ⎟ + exp⎜− ⎟⎟⎟ ⎝ τ1 ⎠ ⎝ τ2 ⎠⎠ (X − Y ) + 4k1k −1 ⎝ kcy[Py*diff ]o 2

(8)

Equations 7 and 8 indicate that the decays of the pyrene monomer and excimer are biexponential with the same two decay times τ1 and τ2 whose expressions are given in eqs 9 and 10. Parameters X and Y equal k1 + τM−1 and k−1 + τE0−1, respectively. τ1−1 =

τ2−1 =

X+Y+

(X − Y )2 + 4k1k −1 2

X+Y−

(9)

(X − Y )2 + 4k1k −1 2

(10)

In terms of the characterization of the ID of a macromolecule, the rate constant k1 is the most important parameter retrieved from such an analysis because it provides a quantitative measure of the rate constant of end-to-end cyclization that depends on both the backbone flexibility and the polymer chain length. DMD Model. The DMD model was the first reaction scheme introduced to handle pyrene excimer kinetics that was not described by Birks’ scheme. It was first applied to describe the kinetics of pyrene excimer formation in 1,3-di(1-pyrenyl)propane.26 It has also been applied to study some pyrene-labeled polymers,43 but its applicability to the study of large macromolecules labeled with pyrene has been questioned.44 In the DMD model, the excited pyrene monomer can form two types of excimers referred to as E1* and E2* in Scheme 3. Diffusive encounters between Pydiff * and a ground-state pyrene result in the formation of excimers E1* and E2* with rate constants of k1 and k2, respectively. Excimers E1* and E2* can fluoresce with their natural lifetimes τE1 and τE2 or dissociate with dissociation rate constants of k−1 and k−2, respectively. The three coupled differential equations shown in eqs 11−13 describe the kinetics of pyrene excimer formation presented in Scheme 3.

Scheme 2. Birks’ Scheme Used to Describe Pyrene Excimer Formation for End-Labeled Monodisperse Linear Chains

Scheme 3. DMD Scheme Used to Describe Pyrene Excimer Formation in End-Labeled Short Alkyl Chains Being Conformationally Strained of a photon, the excited pyrene monomer can emit fluorescence with its natural lifetime τPyo, or it can encounter a ground-state pyrene via diffusion to form an excimer with a rate constant of k1. The excimer can fluoresce with its natural lifetime τE0o or dissociate with a rate constant k−1. The kinetics of pyrene excimer formation is described by two coupled differential equations given in eqs 5 and 6. d[Py *diff ] dt

⎛ 1 ⎞ = −⎜⎜ o + k1⎟⎟[Py *diff ] + k −1[E0*] ⎝ τPy ⎠

⎛ 1 ⎞ d[E0*] = k1[Py *diff ] − ⎜ o + k −1⎟[E0*] dt ⎝ τE0 ⎠

d[Py*diff ] dt

(5)

(6)

The integration of eqs 5 and 6 yields the expressions for the concentrations [Pydiff * ](t) and [E0*](t) given in eqs 7 and 8. [Py*diff ](t ) =

⎛ ⎛ t⎞ ⎜⎜(X − τ2−1) exp⎜− ⎟ 2 ⎝ τ1 ⎠ ⎝ (X − Y ) + 4k1k −1 [Py*diff ]o

⎛ t ⎞⎞ − (X − τ1−1) exp⎜− ⎟⎟⎟ ⎝ τ2 ⎠⎠

⎛ 1 ⎞ = − ⎜⎜k1 + k 2 + o ⎟⎟[Py *diff ] + k −1[E1*] + k −2[E2*] τPy ⎠ ⎝

(11)

⎛ d[E1*] 1 ⎞ = k1[Py *diff ] − ⎜k −1 + o ⎟[E1*] τE1 ⎠ dt ⎝

(12)

⎛ d[E2*] 1 ⎞ = k 2[Py *diff ] − ⎜k −2 + o ⎟[E2*] τE2 ⎠ dt ⎝

(13)

This set of three coupled differential equations can be solved by using a matrix-based formalism as shown in eq 14.

(7) 2312


Langmuir


⎞ ⎛ ⎛ ⎞ ⎟ ⎜− ⎜k + k + 1 ⎟ k −1 k −2 2 ⎟ ⎜ ⎜ 1 τoPy ⎟⎠ ⎝ ⎟ ⎛[Py * ]⎞ ⎛[Py * ]⎞ ⎜ ⎟ ⎜ diff ⎟ ⎜ diff ⎟ ⎜ d⎜ ⎛ ⎞ 1 ⎟ × ⎜ [E1*] ⎟ [E1*] ⎟ = ⎜ 0 k k − + ⎜ ⎟ 1 −1 o ⎟ ⎜ ⎟ ⎟ ⎜ dt ⎜ τ ⎝ ⎠ ⎟ ⎜ E1 ⎟ ⎜⎝ [E2*] ⎟⎠ ⎝ [E2*] ⎠ ⎜ ⎟ ⎜ ⎛ 1 ⎞ ⎜ 0 k2 − ⎜k −2 + o ⎟ ⎟⎟ ⎜ τE2 ⎠ ⎠ ⎝ ⎝

According to eq 14, concentrations [Pydiff * ], [E1*], and [E2*] should be well described by a sum of three exponentials sharing the same decay times τ1, τ2, and τ3 as shown in eqs 15−17.

and excimer decays, resulting in a total of nine independent parameters. Because a minimum of nine independent equations are needed to retrieve nine independent unknowns, linear combinations of pre-exponential factors Aij used in eqs 15−17 provide the additional six missing relationships that are required. An analysis based on the DMD model proceeds in two consecutive steps. First, the monomer and excimer decays are fitted with two sums of three exponentials that share the same decay times but with the pre-exponential factors being free to float. Second, the pre-exponential factors and decay times are combined into a set of linear equations from which the kinetic parameters are extracted. Consequently, the second step in the optimization protocol depends critically on the accuracy of the pre-exponential factors and decay times obtained in the first step. When the decay times describing the kinetics of pyrene excimer formation are well separated such as for short alkyl chains terminated at both ends with a pyrenyl group, the second step proceeds smoothly and the rate constants k1 and k2 in Scheme 3 describe the ID of these molecular constructs.26 The situation is more complex in the case of pyrene-labeled macromolecules where the decay times are usually poorly separated from one another because of the distribution of rate constants for excimer formation as seen in Figure 1 and eq 4. In this case, the applicability of the DMD model is more challenging. Fluorescence Blob Model (FBM). The FBM was introduced in 1999 to analyze the fluorescence decays acquired with macromolecules randomly labeled with pyrene.16 The FBM is based on the assumption that an excited pyrene can probe a finite volume called a blob only while it remains excited. The blob size depends on the ID of the macromolecule, the solvent viscosity, and the natural lifetime of the pyrene derivative used to label the macromolecule as a more flexible backbone, a less viscous solvent, and a longer-lived pyrene derivative result in a larger blob. The blob is viewed as a unit volume used to divide the entire macromolecular volume into a cluster of blobs among which the pyrene labels are randomly distributed as a result of the random labeling of the macromolecule. The FBM has been depicted for a linear chain randomly labeled with pyrene in Figure 2. In its most advanced form, excimer formation based on the FBM is assumed to occur in a sequential manner whereby the slow ID of the macromolecule brings two segments, each bearing a pyrene label, into contact according to a timedependent rate constant, f(t), followed by the rapid rearrangement of the pyrene pendants to form an excimer with a rate constant of k2.21,45 The kinetics describing pyrene excimer formation according to the FBM is shown in Scheme 4, and it can be handled by three differential equations given as eqs 21−23.

⎛ t ⎞ ⎛ t⎞ [Py *diff ] = A11 exp⎜ − ⎟ + A 21 exp⎜ − ⎟ ⎝ τ2 ⎠ ⎝ τ1 ⎠ ⎛ t ⎞ + A31 exp⎜ − ⎟ ⎝ τ3 ⎠

(15)

⎛ t ⎞ ⎛ t ⎞ ⎛ t⎞ [E1*] = A12 exp⎜ − ⎟ + A 22 exp⎜ − ⎟ + A32 exp⎜ − ⎟ ⎝ τ2 ⎠ ⎝ τ1 ⎠ ⎝ τ3 ⎠ (16)

⎛ t ⎞ ⎛ t ⎞ ⎛ t⎞ [E2*] = A13 exp⎜ − ⎟ + A 23 exp⎜ − ⎟ + A33 exp⎜ − ⎟ ⎝ τ2 ⎠ ⎝ τ1 ⎠ ⎝ τ3 ⎠ (17)

Experimentally, the monomer decay is fitted with eq 15 and the excimer decay is fitted with the sum [E1*] + [E2*] obtained from eqs 16 and 17. According to the matrix-based formalism used in eq 14, the inverses of decay times τ1, τ2, and τ3 are the eigenvalues of the 3 × 3 matrix and they obey the relationships listed in eqs 18−20. 1 1 1 1 1 1 + + = k1 + k 2 + + k −1 + + k −2 + τ1 τ2 τ3 τM τE1 τE2 (18)

⎛ 1 1 1 1 ⎞⎛ 1 ⎞ + + = ⎜k 1 + k 2 + ⎟ ⎟⎜k − 1 + τ1τ2 τ2τ3 τ2τ3 ⎝ τM ⎠⎝ τE1 ⎠ ⎛ 1 ⎞⎛ 1 ⎞ + ⎜k 1 + k 2 + ⎟ ⎟⎜k − 2 + τM ⎠⎝ τE2 ⎠ ⎝

−

(19)

⎛ 1 1 ⎞⎛ 1 ⎞⎛ 1 ⎞ = − ⎜k 1 + k 2 + ⎟⎜k − 2 + ⎟ ⎟⎜ k − 1 + τ1τ2τ3 τM ⎠⎝ τE1 ⎠⎝ τE2 ⎠ ⎝ ⎛ ⎛ 1 ⎞ 1 ⎞ + k 2k −2⎜k −1 + ⎟ + k1k −1⎜k −2 + ⎟ τE1 ⎠ τE2 ⎠ ⎝ ⎝

(14)

(20)

Fitting the monomer and excimer decays with the sums of three exponentials yields decay times τ1, τ2, and τ3, which can be combined to provide the three relationships given by eqs 18−20 that link the six kinetic parameters k1, k2, k−1, k−2, τE1, and τE2. In addition to the kinetic parameters, the fluorescence decay analysis is also expected to retrieve the molar fractions fdiff, f E1, and f E2 of the chromophores Pydiff * , E1*, and E2* present in solution. Molar fractions fdiff, f E1, and f E2 represent two additional independent parameters because f E2 = 1 − fdiff − f E1. Finally, a normalization factor needs to be introduced to maintain the same ratio between fdiff and f E1 in the monomer 2313


Langmuir d[Py *diff ] dt d[Py *k ] 2

dt


= − f (t ) −

1 * o [Py diff ] τPy

⎛ 1 ⎞ = f (t ) − ⎜⎜k 2 + o ⎟⎟[Py *k ] 2 τPy ⎠ ⎝

d[E0*] 1 = k 2[Py *k ] − [E0*] 2 dt τE0

Scheme 4. Pyrene Excimer Formation in a Macromolecule Randomly Labeled with Pyrene According to the FBM (21)

(22) ∞ ⎛ A i A 2 + iA4 ⎞ ⎟⎟ [E*](t ) = k 2⎜⎜[Py*k ]o + [Py *diff ]o e−A3 ∑ 3 2 ⎝ i = 0 i! A 2 + iA4 − k 2 ⎠

(23)

Species Py*k2 represents those pyrenes that have been brought close to each other intramolecularly via slow diffusive motions of the macromolecule. They rearrange on a fast time scale to form an excimer with a rate constant of k2. The function f(t) used to represent the slow diffusive motions of the macromolecule is determined by using a mathematical treatment applied to study the kinetics of quenching between dyes and quencher randomly distributed among surfactant micelles.16 After the integration of eqs 21 and 22, expressions of the concentration of the pyrene monomer and excimer are obtained. They are given in eqs 24 and 25, respectively. Equations 24 and 25 might look complicated, but they are functions of nine parameters that can be resolved through the global analysis of the fluorescence decays of the pyrene monomer and excimer.

∞ i=0

− [Py *diff ]o e−A3 ∑ i=0

⎜

1 o τPy

−

2

+

1 τE0

A3i A 2 + iA4 i! A 2 + iA4 − k 2

1 ⎞ ⎞ o ⎟t ⎟ τPy ⎠⎠

+ [Py*diff ]o e−A3

⎛ ⎛ exp⎜− ⎜A 2 + iA4 + ⎝ ⎝

1 ⎞ ⎞ o ⎟t ⎟ τPy ⎠⎠

A 2 + iA4 +

⎛ t ⎞ + [E0*]o exp⎜− o ⎟ ⎝ τE0 ⎠

1 o τPy

−

⎞

( ) ⎟⎟ t

− exp − τ o

E0

1 o τE0

⎟ ⎟ ⎠

(25)

The expressions of parameters A2, A3, and A4 are given in eq 26. They are a function of kblob, the rate constant of encounter between two segments bearing a pyrene label located in the same blob, ke[blob], the product of the rate constant ke describing the exchange of pyrene labels from one blob to another blob by the local blob concentration inside the polymer coil, and ⟨n⟩, the average number of pyrene pendants located inside a blob. A 2 = ⟨n⟩

k blobke[blob] k blob + ke[blob]

A3 = ⟨n⟩

k blob2 (k blob + ke[blob])2

A4 = k blob + ke[blob]

⎞⎞ A3i A 2 + iA4 ⎞ ⎛⎜ ⎛ ⎟exp − ⎜k 2 + 1o ⎟⎟t ⎟ i! A 2 + iA4 − k 2 ⎟⎠ ⎜⎝ ⎜⎝ τPy ⎠ ⎟⎠ ∞

∑ i=0

⎞ ⎛ − A3(1 − exp( − A4 t ))⎟⎟ + ⎜⎜[Py *k ]o + [Py*diff ]o e−A3 2 ⎠ ⎝

∑

k2 + ∞

⎛ ⎛ 1 ⎞ [Py*](t ) = [Py *diff ](t) + [Py *k ](t) = [Py *diff ]o exp⎜⎜ − ⎜⎜A 2 + o ⎟⎟t 2 τPy ⎠ ⎝ ⎝

×

t

E0

×

×

⎛

( ) − exp⎜⎝−⎛⎝k

exp − τ

(26)

In terms of ID characterization, the parameters of interest retrieved from the FBM analysis are ⟨n⟩ and kblob. The blob size is obtained by using eq 27, where λPy is the pyrene content of the polymer expressed in mol·g−1, f Mfree is the fraction of pyrene monomers in the monomer decay that do not form excimers and emit as if they were free in solution, x is the molar faction of pyrene-labeled monomers in the macromolecule, and MPy and M are the molar masses of the pyrene-labeled monomer and unlabeled monomer, respectively.

⎞ ⎛ ⎛ A 2 + iA4 1 ⎞ exp⎜⎜ − ⎜⎜A 2 + iA4 + o ⎟⎟t ⎟⎟ τPy ⎠ ⎠ i! A 2 + iA4 − k 2 ⎝ ⎝

A3i

(24)

Nblob =

1 − fM λPy

free

n xMPy + (1 − x)M

(27)

Nblob increases with the volume of the blob when the solvent viscosity decreases,20,21,45 the pyrene lifetime increases,46 or the flexibility of the polymer backbone increases.47 Most interestingly, the product kblob × Nblob expressed in s−1 has been found to provide a direct measure of the ID of a macromolecule in solution that is independent of the pyrene content and polymer chain length.20,21,45 Model Free Analysis (MFA). The MFA was introduced to deal with complex macromolecular architectures that were not randomly labeled with pyrene, such as pyrene end-labeled dendrimers.12,24,25,37,44 Such macromolecules contain more than two pyrenes that are located at specific positions in the macromolecule (the chain ends). Consequently, neither Birks’ scheme nor FBM is well suited to describe the kinetics of pyrene excimer formation in these constructs. In the MFA, the pyrene-

Figure 2. Coil of a polymer randomly labeled with pyrene divided into a cluster of blobs according to the fluorescence blob model (FBM). 2314


Langmuir


polymers.44 Consequently, the main appeal of the MFA is that it applies to a broad variety of pyrene-labeled macromolecules and that its modular design allows it to adapt to the complexity of the decays at hand simply by increasing the number of exponentials required to fit a given decay.

Scheme 5. Excimer Formation between Pyrenyl Groups Covalently Attached to a Macromolecule According to the MFA

■

FLUORESCENCE DECAY ANALYSIS: MODEL SELECTION Now that Birks’ scheme, the DMD model, the FBM, and the MFA have been reviewed, the most appropriate model to fit the monomer and excimer decays obtained for a given macromolecule needs to be selected. This selection proceeds through a series of steps that are taken in a rational order as described hereafter. Determination of the Natural Lifetime of the Pyrene Monomer τPyo. Close attention must be paid to the determination of τPyo because it represents the reference against which all other photophysical processes undergone by the excited pyrene monomer must be compared. Because τPyo is the longest photophysical process taking place in the solution, τPyo defines the temporal window over which the kinetics of pyrene excimer formation will be monitored. All models described in the previous section use rate constants ki of excimer formation that appear in the form of ki + (1/τPyo) in the equations describing the concentrations of the pyrene monomer and excimer. Consequently, underestimating τPyo can actually result in a negative and physically impossible rate constant ki for small ki values. For this not to happen, a proper model compound needs to be selected in order to determine τPyo. Because pyrene derivatives such as 1-pyrenemethanol, 1-pyrenemethylamine, 1-pyrenebutanol, and 1-pyrenebutyric acid are commercially available and are often used to label the macromolecule, it is best to label the macromolecule with the same pyrenyl group used for the fluorescence study at a single position or, if this is not possible, with a small amount of pyrene to ensure that most pyrene labels covalently attached to the macromolecule are isolated on the macromolecule, do not form excimer, and emit with a lifetime τPyo. The fluorescence spectrum of this macromolecule labeled with one or a very small amount of pyrene should exhibit hardly any excimer emission in its fluorescence spectrum at the low polymer concentrations typically used in these experiments ([Py] ≈ 2.5 × 10−6 M corresponding to an absorbance at 344 nm of 0.1). As a result, the fluorescence decay of the pyrene monomer acquired with the macromolecule prepared with a low pyrene content should be at most biexponential with a strong contribution representing more than 75% of the pre-exponential weight of pyrenes that do not form an excimer and emit as if they were free in solution. This species has been referred to as Pyfree * in this laboratory. The decay time associated with the Py*free species is then taken as τPyo. As mentioned earlier, pyrene is a long-lived fluorophore; consequently, its lifetime is quite sensitive to the presence of minute amounts of quencher, in particular, oxygen that is a potent quencher of pyrene in organic solvents. To maximize the time window over which the fluorescence of pyrene can be detected, pyrene solutions in organic solvents need to be thoroughly outgassed with nitrogen to displace oxygen from the solvent. In turn, the sensitivity of pyrene to oxygen quenching has been harvested to probe the accessibility of the pyrene label to the solution when hydrophobic microdomains are generated in nondegassed aqueous solutions of pyrene-labeled water-soluble polymers.15 Multiexponential Analysis of the Fluorescence Decays. After having determined τPyo, the monomer and excimer

labeled macromolecule is viewed as a shapeless and deformable object that can morph into different conformations with a timedependent rate constant f(t) as shown in Scheme 5. On the basis of Scheme 5, the kinetics of pyrene excimer formation can be described by two differential equations given in eqs 28 and 29. d[Py *diff ] dt

= − f (t ) −

1 * o [Py diff ] τPy

(28)

d[E0*] 1 = f (t ) − [E0*] dt τE0

(29)

Because the concentration of the pyrene monomer [Pydiff * ](t) can always be approximated by a sum of exponentials such as that shown in eq 30, eq 28 can be rearranged to yield f(t), which can be used to integrate eq 29 to obtain the excimer concentration [E0*] as shown in eq 31.12,24,25,37,44 n ⎛ t⎞ [Py *diff ](t) = [Py *diff ]o ∑ ai exp⎜ − ⎟ ⎝ τi ⎠ i=1 n

1 τi

[E0*](t) = − [Py *diff ]o ∑ ai 1 i=1

τi

−

1 o τPy

−

1 o τE0

1 ⎛ n − τi ⎜ + ⎜[E0*]o + [Py *diff ]o ∑ ai 1 ⎜ − i=1 τi ⎝

(30)

⎛ t⎞ exp⎜ − ⎟ ⎝ τi ⎠ ⎞ ⎛ t ⎞ ⎟ exp⎜ − o ⎟ 1 ⎟ ⎟ ⎝ τE0 ⎠ o τE0 ⎠

1 o τPy

(31)

In eqs 30 and 31, the pre-exponential factors ai are normalized so that their sum equals unity and n represents the number of exponentials used in eq 30. To date, no set of monomer and excimer fluorescence decays handled by this laboratory has required the use of more than three exponentials (i.e., n ≤ 3) in eq 30 to describe the pyrene labels forming excimer by diffusion. Information about the ID of a pyrene-labeled macromolecule whose fluorescence decays have been fitted according to the MFA is obtained through the average rate constant of excimer formation ⟨k⟩, whose expression is given in eq 32. In eq 32, the number-average lifetime ⟨τ⟩ equals ∑ni=1aiτi because the preexponential factors ai are normalized (∑ni=1ai = 1). 1 1 ⟨k⟩ = − o τ τPy (32) A comparative study dealing with end-labeled linear chains, randomly labeled polymers, and end-labeled dendrimers showed that ⟨k⟩ obtained via the MFA yields comparable results to what would have been obtained through an analysis based on Birks’ scheme for end-labeled polymers and FBM for randomly labeled 2315


Langmuir


scales as Mn−1.2 as shown in Figure 4B.22 Although IE/IM should in theory behave in a manner similar to kcy, the presence * in the polymer samples with of increasing quantities of Pyfree increasing Mn induces a steeper decrease of IE/IM with increasing Mn, resulting in an artificially large scaling exponent for the relationship IE/IM ≈ M−α. The fluorescence decay analysis being able to distinguish between the different species present in solution can separate the contribution of Pyfree * from that of the * species forming excimer by diffusion, which results in the Pydiff more accurate kcy ∼ M−α trend. The second feature to look for in the multiexponential analysis of the fluorescence decays is whether some of the preexponential factors obtained for the fit of the excimer fluorescence decay are negative to reflect the rise time expected in the excimer fluorescence decay. (See the excimer rise time in Figure 3B between 50 and 100 ns.) Ideally, the ratio of the sum of the negative pre-exponential factors divided by the sum of the positive pre-exponential factors, namely, the AE−/AE+ ratio, should equal −1.0 if excimer formation occurs solely via diffusion. This comes from the fact that if the excimer concentration can be approximated by a sum of exponentials as [E0*] = ∑ni=1ai exp(−t/τi) then the formation of excimer by diffusion dictates that the excimer does not exist at the initial time (t = 0) so that [E0*]o = 0 = ∑ni=1ai. For this to happen, some of the pre-exponential factors ai must be negative and the AE−/AE+ ratio must equal −1.0. Deviation of the AE−/AE+ ratio from −1.0 implies that some excimer is generated instantaneously by direct excitation of a ground-state dimer. This is usually the case when pyrene is randomly attached to the macromolecule because a nonzero probability of incorporating two pyrene labels close to each other on the same macromolecule always exists. Birks’ scheme, the DMD model, the FBM, and the MFA are all based on the idea that the excimer is formed by diffusion and that the decay of the monomer is paralleled by the formation of the excimer. In the extreme case where no excimer is being formed by diffusion (AE−/AE+ ≈ 0), it can be assumed that the kinetics leading to the decay of the pyrene monomer and the formation of the excimer are uncoupled and the models described earlier no longer apply. For most pyrene-labeled macromolecules, it is usually the case that AE−/AE+ takes more positive values than −1.0. The models presented earlier account for this by yielding nonzero initial concentrations for the excimer species present in solution. Two types of ground-state dimers are often encountered when dealing with a pyrene-labeled macromolecule: (1) those constiting of either two well- or poorly stacked pyrene moieties that yield either a well-behaved excimer that emits with the lifetime τE0o, that is usually between 40 and 60 ns depending on the solvent, and the pyrene derivative used to label the macromolecule or (2) a long-lived excited dimer that emits with a lifetime τDo usually between 80 and 130 ns. The existence of long-lived excited pyrene dimers is exacerbated by the use of a rigid linker to connect the pyrene derivative to the macromolecule because it reduces the conformational freedom of the pyrene labels leading to poorly stacked pyrene dimers. Consequently, pyrene derivatives such as 1-pyrenebutanol and 1-pyrenebutyric acid yield fewer ground-state dimers than does 1-pyrenemethanol or 1-pyrenemethylamine when they are used to label a macromolecule. They are, however, shorter-lived by about 60 ns compared to the latter pyrene derivatives, which shortens the temporal window to probe the ID of macromolecules.20,45 Ground-state dimers are rarely encountered when dealing with monodisperse linear polymers end-labeled with pyrene because the pyrene groups are held far apart from each other. Thus,

fluorescence decays need to be analyzed independently with a sum of exponentials where all pre-exponential factors and decay times are free to float. When doing this analysis, the number of exponentials is increased one exponential at a time until a good fit of the decays is obtained. The quality of the fit is based on a χ2 value close to unity and the random distribution of the residuals and autocorrelation of the residuals. After a satisfying fit has been achieved, three features need to be assessed from this multiexponential analysis of the fluorescence decays before deciding whether to pursue a global analysis based on Birks’ scheme, the DMD model, the FBM, or the MFA. The first feature to look for is whether a long decay time is present in the monomer decay that matches τPyo. If this is observed, then it suggests that a fraction of the pyrene labels does not form excimer and that these pyrenyl pendants emit with a lifetime τPyo as if they were free in solution. The main implication * ]o exp(−t/τoPy) needs to of that observation is that the term [Pyfree be added to eqs 7, 15, 24, and 30 that describe the pyrene monomer concentration according to Birks’ scheme, the DMD model, the FBM, or the MFA, respectively. This term accounts for the presence of species Pyfree * in the sample. When conducting the analysis, the lifetime τPyo is fixed to its value to reduce the number of floating parameters. This procedure enables one to isolate the contribution due to the Pyfree * species from the multiexponential decay. The presence of the Py*free species in a polymer sample is usually quite obvious to observe as shown in Figure 3. The monomer decay shown in Figure 3A corresponds to that of a pyrene end-labeled poly(ethylene oxide) sample with a number-average molecular weight Mn of 2K that contains no Py*free. This sample is referred to as PEO(2K)-MPy2 because it was terminated with a 1-pyrenemethoxy group at both ends and its chemical structure is shown in Table 1. In the presence of 5 mol% of monolabeled chain, the decay changes dramatically as shown in Figure 3C as a result of the introduction of the [Py*free]o exp(−t/τoPy) term.48 Because the fluorescence decay of Py*free with its τPyo lifetime is the longest photophysical process happening in the solution, it corresponds to the long-decay component of the fluorescence decay stretching across Figure 3C from 600 to 1000 ns. Adding the [Py*free]o exp(−t/τoPy) term to the expression of eqs 7, 15, 24, and 30 and fixing the decay time in the analysis program to equal τPyo enables one to isolate the unwanted contribution of Py*free and retrieve reliable parameters to describe the behavior of those pyrenes that form excimer by diffusion * ). Because the presence of Pyfree * does not affect excimer (Pydiff formation at the dilute concentrations used in these experiments, the excimer fluorescence decay is identical whether 5 mol% Py*free is present in the solution (cf. Figure 3B,D).48 The presence of Pyfree * in a sample of pyrene-labeled macromolecules represents a very potent artifact. In fact, the * population in the solution represents a fluorescent impurity Pyfree in terms of pyrene excimer formation, and its presence can result in an erroneous analysis of the fluorescence spectrum of the pyrene-labeled macromolecule. For example, Figure 4A shows the fluorescence spectra of four pyrene end-labeled poly(ethylene oxide) samples. Each sample is contaminated with a relatively small (3−20 mol%) fraction of Py*free that increases with increasing chain length as a result of the difficulty of efficiently labeling longer chains and because longer chains form excimer less efficiently by diffusion.22 Consequently, IE/IM, which is the fluorescence intensity of the excimer divided by the fluorescence intensity of the monomer, scales as Mn−1.6 whereas kcy obtained by Birks’ scheme analysis of the fluorescence decay 2316


Langmuir


Figure 3. Simulated fluorescence decays of the pyrene monomer (A, C) and excimer (B, D) of a PEO(2K)-MPy2 sample in THF containing 0 mol% (f free = 0.00; A, B) and 5 mol% ( f free = 0.05; C, D) monolabeled chains. Parameters used for the simulations are k1 = 1.33 × 107 s−1, k−1 = 2.26 × 106 s−1, τM = 258 ns, and τE0 = 48 ns.

The third and last feature consists of determining the number and value of the common decay times that are found in the analysis of both the monomer and excimer fluorescence decays. Any model used to fit the fluorescence decays of the pyrene monomer and excimer, be it the Birks’ scheme, the DMD model, the FBM, or the MFA, requires that the same decay times

Birks’ scheme is usually not affected by them. The DMD model accounts for them because it deals with two different excimer species. The presence of ground-state pyrene dimers emitting with a lifetime of τDo is taken into account in the FBM20 and MFA44 by adding the term [D*]o exp(−t/τoD) to the expression of the excimer concentration in eqs 25 and 31, respectively. 2317


Langmuir


Table 1. Pyrene Contents Expressed as the Molar Fraction x in mol% Pyrene-Labeled Monomer and λPy in μmol·g−1, NumberAverage Molecular Weights, and PDIs for Some of the Linear Polymers Studied in This Laboratory

Figure 4. (Left) Fluorescence spectra normalized at 375 nm for poly(ethylene oxide) end-labeled with 1-pyrenemethoxide in tetrahydrofuran. (Top to bottom) PEO2K-MPy2, PEO5K-MPy2, PEO10K-MPy2, and PEO16.5K-MPy2. λex = 344 nm. (Right) Plot of IE/IM ∼ Mn−1.6 (■) and kcy ∼ Mn−1.2 (□) as a function of Mn = 2K, 5K, 10K, and 16.5K.

and that at least one of the models described earlier will apply. The question is to determine which one will be the most appropriate. Because decay analysis is a time-consuming process, rapid identification of the better-suited model can save a lot of time down the road, leading to a full description of the kinetics of

be obtained, as can be readily seen from the expressions of the monomer and excimer concentrations. The observation that several decay times should be the same from a multiexponential analysis of the monomer and excimer decays provides a nice confirmation that the kinetics of excimer formation are coupled 2318


Langmuir


pyrene excimer formation in the pyrene-labeled macromolecule of interest. The difference between a decay whose fit requires one or two exponentials or three or four exponentials should be obvious through a visual inspection of the decay. One can appreciate the difference if the monomer decay of PEO(2K)-MPy2 in Figure 3A is compared to that given in Figure 5A of a polystyrene sample

Table 2. Pyrene Contents Expressed as the Molar Fraction x in mol% of Pyrene-Labeled Monomer and λPy in μmol·g−1, Number-Average Molecular Weights, and PDIs for Some of the Pyrene-Labeled Dendrons Studied in This Laboratory

end-labeled polystyrenes,40 pyrene end-labeled monodisperse polymers represent the family of pyrene-labeled macromolecules that has certainly been the most thoroughly and carefully studied.4,20−22,44,48 The demonstration by Winnik et al. that the kinetics of excimer formation between the two pyrene labels attached to the ends of a linear monodisperse chain could be satisfyingly handled by Birks’ scheme was followed by numerous studies that were carried out with a variety of polymeric backbones end-labeled with pyrene21,22 or other dye/ quencher pairs.1−3,7,8,28 All have confirmed Winnik’s early insight showing a strong decrease in kcy with increasing Mn. A typical experiment with pyrene end-labeled linear chains consists of preparing a series of samples with different molecular weights to establish the kcy ∼ Mn−α relationship (Figures 4B and 6B).4,40 Because kcy depends on both the chain length and backbone flexibility, establishing the kcy ∼ Mn−α scaling law enables one to isolate the effect that chain length has on kcy, leaving backbone flexibility as the last remaining factor affecting kcy. Considering the many studies that have been conducted with different polymer backbones over the past 35 years, it might be a bit disappointing that a relationship relating kcy to backbone flexibility, as expected from the known chemical composition of a polymer, has not been established. Such a relationship would be very valuable because it would enable one to predict quantitatively how a polymer chemical composition might affect its ID. One reason for the absence of such a relationship might be the strong decrease in pyrene excimer formation that takes place with both polymer chain lengths as a result of the kcy ∼ Mn−α scaling law (see Figure 6A to appreciate the small amount of excimer generated by the polystyrene sample with an Mn value of 27K) and backbone stiffness. As the backbone stiffness increases, the range of molecular weights available to establish the kcy ∼ Mn−α scaling law reduces to the point where not enough samples are available and the scaling law can no longer be determined. In that respect, the use of longer-lived dyes to study end-to-end cyclization with monodisperse linear chains would enable stiffer backbones to be probed.45 Short-Range Excimer Formation. As was described in the previous section, Zachariasse and Kühnle’s demonstration in 1976 that the end-to-end cyclization of linear chains could be probed by pyrene excimer formation for a series of α,ω(1-pyrenyl)alkanes38 was quickly applied to the study of pyrene end-labeled polymers.39,40 It also led to a number of investigations that looked at the effect that the alkyl chain length had on the rate constant of pyrene excimer formation in different series of pyrene end-labeled low-molecular-weight alkanes.50 As mentioned earlier, Birks’ scheme was found to apply to many

Figure 5. Simulated fluorescence decays for a polystyrene sample in tetrahydrofuran prepared by copolymerizing styrene with 6.4 mol% 1-pyrenemethoxymethylstyrene. (A) Monomer and (B) excimer.

labeled with 6.4 mol% 1-pyrenemethoxymethylstyrene and referred to as CoEt-PS-MPy(6.4) in Table 1. The monomer decay of the CoEt-PS-MPy(6.4) sample requires a minimum of three exponentials to obtain a good fit. Four exponentials would be necessary to obtain a good fit if the Py*free contribution was taken into account by setting one of the decay times equal to τPyo in the sum of exponentials. By contrast, no more than two exponentials are needed to fit the fluorescence decay of PEO(2K)-MPy2 in Figure 3A. Birks’ scheme and the DMD model work best with fluorescence decays of the pyrene monomer that are described by a small number of exponentials. Strongly multiexponential monomer decays are best handled with the FBM. With its modular design, the MFA applies to all types of decays because one needs only to increase the number n of exponentials in eqs 30 and 31 to allow the equations to adapt to the complexity of the decay, with a more curved decay requiring a larger number of exponentials. The major families of macromolecules that have been studied with these models and the information retrieved from these studies are described hereafter.

■

INFORMATION RETRIEVED FROM FLUORESCENCE DECAY ANALYSIS Over the years, this laboratory has dealt with a wide range of pyrene-labeled macromolecules. To achieve this, a large library of fluorescence decay analysis routines has been implemented on the basis of the models that were discussed in the previous section. The following discussion will focus mainly on the many samples that have been investigated by this laboratory, whose chemical structures are listed in Tables 1 and 2. Pyrene End-Labeled Linear Chains. Since the pioneering work by Perico and Cuniberti in 1977 using SSF to study a series of pyrene end-labeled poly(ethylene oxide)s39 and by Winnik et al. in 1980 using TRF to determine kcy for a series of pyrene 2319


Langmuir


the introduction of bulky pyrene groups affects the ID of a polymer backbone. However, no indication of such a problem has ever been reported by this laboratory since it started studying pyrene-labeled macromolecules in 1996. One reason for this might reside in the decoupling depicted in Scheme 4 that exists between the slow diffusive motions of the polymer backbone described by the function f(t) and the rapid rearrangement of the pyrene labels to form an excimer with a rate constant of k2. Backbone motions of the polymers studied so far with the FBM appear to be sufficiently slow that they remain unaffected by the presence of a small number of pyrene labels along the chain. Furthermore, the increase in pyrene content from 1 to 8 mol% should affect the kinetic parameters kblob and ke[blob], which reflect backbone motion. This is not observed when the fluorescence decays are analyzed with eqs 24 and 25. The main effect associated with an increase in pyrene content, referred to as λPy by this laboratory, is to increase the average number of pyrenes per blob, ⟨n⟩. In turn, the application of eq 27 that divides ⟨n⟩ by λPy yields Nblob, the parameter representing the number of monomer units encompassed within a blob. Within experimental error, Nblob remains constant with pyrene content. The other parameters that depend on λPy are the molar fractions f free and f E0 representing species Py*free and E0*, respectively. As λPy increases, more pyrenes are added along the chain, f free decreases down to values close to zero for pyrene contents higher than 3 mol%, and f E0 increases because more pyrene labels are incorporated close to each other along the polymer backbone. The blob size given by Nblob reflects the polymer chain dynamics as a more flexible polymer enables an excited pyrene to probe a larger volume or blob within the polymer coil. However, the excited pyrene can probe the same blob at a different velocity, which is described by kblob expressed in s−1. As it turns out, the product kblob × Nblob also expressed in s−1 appears to be a more useful parameter for describe the ID of a polymer. In two instances, comparison of kblob × Nblob obtained for randomly labeled polystyrene20,45 and poly(N-isopropylacrylamide)21 in different solvents yielded trends as a function of solvent viscosity that were similar to those obtained with kcy for the same polymers end-labeled with pyrene in the same solvents. (See Figure 7 for the polymers whose chemical structure is listed in Table 1.) The similarity in behavior between kblob × Nblob and kcy together with the large enhancement in pyrene excimer formation20 resulting from the random labeling of polymers makes polymers randomly labeled with pyrene good candidates for studying the more rigid backbones that could not be investigated with pyrene endlabeled constructs. As a matter of fact, work being currently done by this laboratory on a series of poly(alkyl methacrylate)s suggests that the product kblob × Nblob can be used to probe the ID of polymers in solution in the same manner that the glasstransition temperature Tg is being used to probe the backbone dynamics of polymers in the solid state. Internal Dynamics of Pyrene End-Labeled Dendrimers. The last family of pyrene-labeled macromolecules to be considered in this review is that of pyrene end-labeled dendrimers for which a substantial amount of work has been done.12 In these experiments, the reactive end-groups of the dendrimers are targeted for pyrene labeling because they are used to grow a dendrimer to a higher generation. Pyrene labeling of the large number of reacting groups concentrated in the small molecular volume of the dendrimer generates a large local pyrene concentration that induces the efficient formation of excimer. Interestingly, the special architecture of pyrene end-labeled dendrimers is not accounted for by Birks’ scheme (only two

Figure 6. (A) Fluorescence spectra of three monodisperse polystyrenes (Mn = 3.9K, 6.2, and 27K) end-labeled with 1-pyrenebutyric acid. (B) Plot of log ⟨kcy⟩ vs log(Mn); ⟨kcy⟩ was calculated from the analysis of the fluorescence decays that yielded the decay time τ = (⟨kcy⟩ + 1/τPyο)−1 (○) and spectra that yielded the IE/IM ratio (□). Reproduced from ref 40.

constructs, but in some cases, three exponentials were needed to fit the excimer decays, an observation that led to the inception of the DMD model.26 In the case of 1,3-di(1-pyrenyl)propane (DPP), the monomer decay was found to be mostly monoexponential with the same decay time found in the excimer decay. The triexponential decay of the excimer led to the hypothesis that two excimer species, E1* and E2*, are present upon excitation of DPP as shown in Scheme 3. The existence of two excimers is certainly a consequence of conformational constraints between the two pyrenyl groups being separated by a short propane linker. Although the applicability of the DMD model is unquestioned for the study of short-range excimer formation between pyrenyl groups separated by a short alkane, the DMD model has not been widely applied to the study of pyrene-labeled polymers resulting in triexponential decays, particularly in the case of randomly labeled polymers. Studies conducted so far with randomly labeled macromolecules suggest that analyses based on the FBM or MFA seem to be better suited. Macromolecules Randomly Labeled with Pyrene. The synthesis challenges associated with the preparation of pyrene end-labeled monodisperse linear chains led to the pursuit of alternate routes to preparing pyrene-labeled macromolecules. Polymers randomly labeled with pyrene are certainly much easier to synthesize, but they are subject to the complex kinetics of pyrene excimer formation that involves an infinite distribution of rate constants as illustrated in Figure 1 and eq 4. Compared to pyrene end-labeled monodisperse linear chains, many fewer studies have been conducted on randomly labeled polymers, in part as a result of the noncommercial availability of the analysis programs based on eqs 24 and 25. This problem is being addressed by this laboratory, which is implementing a Web site where users will be able to upload their fluorescence decays and process them remotely through programs globmis90bbg and globmis90cbg. These two programs are the routines representing the most advanced version of the FBM as described in eqs 24 and 25, and they are now most commonly used by this laboratory to analyze the fluorescence decays of polymers randomly labeled with pyrene according to the FBM.21,45 Studies dealing with randomly labeled polymers require the preparation of a series of polymers with pyrene contents ranging from 1 to 8 mol%. The pyrene content is kept relatively small to minimize its effect on the ID of the polymer and to ensure that the pyrene-labeled constructs faithfully represent the polymer under study. Indeed, a common concern among scientists is that 2320


Langmuir


Figure 7. Plots of ⟨kblob × Nblob⟩ and ⟨kcy × N⟩ vs the inverse of solvent viscosity for polymer randomly or end-labeled with pyrene, respectively. (A) CoEt-PS-MPy (◇), CoEs-PS-BuPy (○), CoAm-PS-MPy (□), and PS-BuPy2 (▲) with γ = 0.53, 0.86, 1.00, and 2.9 for CoEt-PS-MPy, CoEs-PS-BuPy, CoAm-PS-MPy, and PS-BuPy2, respectively. (B) CoAm-PNIPAM-BuPy (○) and PNIPAM-BuPy2 (▲) with γ = 1.8.

pyrenes located at two specific positions),4 the DMD model (only two rate constants for pyrene excimer formation),26 or the FBM (the pyrenes are randomly distributed throughout the macromolecule).5 The MFA that can be applied to any type of pyrene-labeled macromolecule12,25 is ideally suited to characterize the complex kinetics of pyrene excimer formation encountered in pyrene end-labeled dendrimers. These experiments require the preparation of a series of fully end-labeled dendrimers of increasing generation number whose monomer and excimer fluorescence decays are fitted according to the MFA that yields the average rate constant of excimer formation for pyrene-labeled dendrimers ⟨k⟩ given in eq 32, which has been shown to take the largest values found among all pyrene-labeled macromolecules prepared to date.44 Furthermore, ⟨k⟩ has been found to increase linearly with increasing generation number.37 The MFA of fluorescence decays acquired with pyrene-labeled dendrimers has also identified an important artifact that affects to a great extent the analysis of steady-state fluorescence spectra due to the residual presence of the Py*free species. Because the pyrenes form excimer by diffusion very efficiently within a dendrimer, the fluorescence of the pyrene monomer forming excimer Pydiff * is strongly diminished, so much so that it becomes comparable to the fluorescence emitted by residual quantities of pyrene labels that are free in solution and were not properly removed during sample purification. In turn, the presence of the Pyfree * monomers affects the determination of the IE/IM ratio. In one dramatic example, the presence of 3 mol% of Py*free monomers reduced the IE/IM ratio of a pyrene end-labeled dendrimer by 75%!37 This artifact is believed to be the source of a number of surprising results that have been reported about dendrimers in the scientific literature.12 The rate constant ⟨k⟩ is a pseudounimolecular rate constant that is the product of the bimolecular rate constant kdiff for excimer formation between an excited-state and a groundstate pyrene both located inside a same dendrimer and the local concentration of pyrene [Py]loc as described in eq 33. ⟨k⟩ = kdiff [Py]loc

groups attached to the dendrimer divided by the dendrimer volume (Vden). Consequently, if each terminal group is increased x-fold after each generation increase, then [Py]loc is given by eq 34. [Py]loc =

xn − 1 Vden

(34)

In eq 34, the number of pyrenes in the dendrimer xn is reduced by 1 to account for the excited pyrene that leads to excimer formation. Combining eqs 33 and 34 yields a measure of the dendrimer volume as shown in eq 35. Vden =

kdiff n (x − 1) k

(35)

Because kdiff can be assumed to be constant with generation number, eq 35 predicts that the dendrimer volume scales as (xn − 1)/⟨k⟩. The expected enhancement in [Py]loc as a function of increasing generation number can also be captured by evaluating the mean-squared end-to-end distance for a dendrimer of generation n (r2n) separating the excited pyrene from a groundstate pyrene located at another end of the dendrimer by assuming that the pyrene-labeled terminals are connected via freely jointed polymer segments obeying Gaussian statistics. In this derivation, x is set to equal 2 and the two pyrenes of generation 1 (n = 1) are assumed to be separated by N polymeric segments of length l. According to Figure 8, r2n is given by eq 36. n

rn2

=

∑i = 1 2i − 1 ∫

∞

r=0

n

∑i = 1 2i − 1 ∫

∞

r=0

3/2 3 2Nil 2

( ) ( )

3/2 3 2 2Nil

3r 2 2Nil 2

4

dr

3r 2 2Nil 2

2

dr

( )4πr exp(− )4πr

exp −

n

= Nl

2

∑i = 1 i × 2i − 1 n

∑i = 1 2i − 1

(36)

Using (r2n)3/2 as a measure of Vden in eq 34 yields [Py]loc, and a plot of (2n − 1)/(r2n)3/2 as a function of generation number is shown in Figure 9. Despite the rough approximations made to derive eq 36, relatively good agreement between experiment and eq 36 was found for a series of four-generation dendrons based on a bis(hydroxymethyl)propionic acid backbone (PP-GnBuPy2n) whose structure is shown in Table 2. The trend for [Py]loc shown in Figure 9 assumes that Vden scales as (r2n)3/2 where r2n was calculated with eq 36. Accounting

(33)

NMR experiments have shown that the relaxation time of the terminal groups in dendrimers does not change much as a function of the generation number, which suggests that kdiff in eq 33, describing excimer formation between pyrene labels covalently attached to the dendrimer end-groups, remains constant with generation number.37 Furthermore, [Py]loc, the concentration of pyrene groups inside a dendrimer, equals the number of pyrene 2321


Langmuir


and the mathematical solutions that have been proposed over the years to deal with this complexity have been presented. Compared to the early beginnings of the field that started from the study of pyrene end-labeled monodisperse linear chains, the relatively recent resolution of several mathematical hurdles has led to the creation of mathematical models and the implementation of a number of new analysis programs that enable the characterization of the ID of macromolecules having increasingly complex architectures and using a variety of pyrene labeling schemes. To this date, it has been the experience of this laboratory that the fluorescence decays of any pyrene-labeled macromolecule can be quantitatively analyzed by at least one of the four models described in this review. Many types of macromolecules were discussed in the review, but others currently under investigation by this laboratory include pyrene-labeled thermoresponsive polymers and polymeric bottlebrushes. It is my hope that the recent analytical improvements that have been presented will open new research venues to characterizing the ID of macromolecules on the molecular level in solution.

Figure 8. Number of pyrenes located at the same average end-to-end distance from the excited pyrene.

■

for the branched nature of the dendrimer by assuming that Vden scales as (r2n)3/2 reproduces the increase in ⟨k⟩ with increasing generation number relatively well. A similar increase in ⟨k⟩ with generation number was observed for two pyrene-labeled polyaryl Fréchet-type dendrons (PA-GnBuPy2n) whose chemical structure is also shown in Table 2. The ⟨k⟩ values obtained for the polyaryl dendrons were lower for a given generation than those obtained for the PP-Gn-BuPy2n series as a result of the more rigid interior of the former dendrons. The construction shown in Figure 8 also suggests

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest. Biography

Jean Duhamel obtained his Ph.D. in 1989 from the Institut National Polytechnique de Lorraine in Nancy (France) under the supervision of Dr. J.-C. André. His graduate work focused on the study of transient effects observed in the kinetics of excimer formation in viscous solutions. He then joined the laboratory of Prof. M. A. Winnik in 1990 at the University of Toronto (Canada), where he applied pyrene excimer formation to characterize the behavior of macromolecules. In 1993, he moved to the laboratory of Prof. P. Lu at the University of Pennsylvania (USA) to study oligonucleotides by fluorescence anisotropy. After joining the University of Waterloo (Canada) in 1996, he expanded the use of time-resolved fluorescence to the point where it can now be applied to probe the internal dynamics of any type of pyrene-labeled macromolecular construct in solution.

Figure 9. ⟨k⟩ for the PP-Gn-BuPy2n series (×) and (2n − 1)/(r2n)3/2 (□) vs generation number.

that the trend shown in Figure 9 should depend on the architecture of the dendron. In particular, a Newkome-type polyamide dendron showing a 3n increase in branching points compared to the 2n increase obtained for the two types of pyrene-labeled dendrons shown in Table 2 would be expected to show a much steeper increase in ⟨k⟩ with increasing generation number.

■

■

CONCLUSIONS This review has provided an overview of the different approaches available today to characterize the ID of pyrene-labeled macromolecules in solution. The mathematical complexity that is associated with the analysis of fluorescence decays acquired with a number of fluorescently labeled macromolecules was described,

ACKNOWLEDGMENTS

This review cites the work conducted in the Duhamel laboratory by numerous graduate students and postdoctoral fellows. Their dedication and hard work is duly acknowledged. The bulk of their research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC), whose financial support was critical to conducting these experiments. 2322


Langmuir

■


(23) James, D. R.; Ware, W. R. A Fallacy in the Interpretation of Fluorescence Decay Parameters. Chem. Phys. Lett. 1985, 120, 455−459. (24) Zaragoza-Galán, G.; Fowler, M.; Duhamel, J.; Rein, R.; Solladié, N.; Rivera, E. Synthesis and Characterization of Novel PyreneDendronized Porphyrins Exhibiting Efficient Fluorescence Resonance Energy Transfer (FRET): Optical and Photophysical Properties. Langmuir 2012, 28, 11195−11205. (25) Duhamel, J. New Insights in the Study of Pyrene Excimer Fluorescence to Characterize Macromolecules and their Supramolecular Assemblies in Solution. Langmuir 2012, 28, 6527−6538. (26) Zachariasse, K. A.; Busse, R.; Duveneck, G.; Kühnle, W. Intramolecular Monomer and Excimer Fluorescence with Dipyrenylpropanes: Double-Exponential versus Triple-Exponential Decays. J. Photochem. 1985, 28, 237−253. (27) Seixas de Melo, J.; Costa, T.; Francisco, A.; Maçanita, A. L.; Gago, S.; Gonçalves, I. S. Dynamics of Short as Compared with Long Poly(acrylic acid) Chains Hydrophobically Modified with Pyrene, as Followed by Fluorescence Techniques. Phys. Chem. Chem. Phys. 2007, 9, 1370−1385. (28) Jacob, M. H.; Dsouza, R. N.; Ghosh, I.; Norouzy, A.; Schwazlose, T.; Nau, W. M. Diffusion-Enhanced Förster Resonance Energy Transfer and the Effects of External Quenchers and the Donor Quantum Yield. J. Phys. Chem. B 2013, 117, 185−198. (29) Nowakowska, M.; White, B.; Guillet, J. E. Studies of the Antenna Effect in Polymer Molecules. 12. Photochemical Reactions of Several Polynuclear Aromatic Compounds Solubilized in Aqueous Solutions of Poly(sodium styrenesulfonate-co-2-vinylnaphthalene). Macromolecules 1989, 22, 2317−2324. (30) Jones, A.; Dickson, T. J.; Wilson, B. E.; Duhamel, J. Fluorescence Properties of Poly(Ethylene Terephthalate-co-2,6-Naphthalene Dicarboxylate) with Naphthalene Contents Ranging from 0.01 to 100 mol%. Macromolecules 1999, 32, 2956−2961. (31) Nakajima, A. Fluorescence Spectra of Pyrene in Chlorinated Aromatic Solvents. J. Lumin. 1976, 11, 429−432. (32) Strickler, S. J.; Berg, R. A. Relationship between Absorption Intensity and Fluorescence Lifetime of Molecules. J. Chem. Phys. 1962, 37, 814−822. (33) Mathew, A. K.; Duhamel, J.; Gao, J. Maleic Anhydride Modified Oligo(isobutylene) I: Microstructure Characterization by Fluorescence Spectroscopy. Macromolecules 2001, 34, 1454−1469. (34) Wilken, R.; Adams, J. End-Group Dynamics of Fluorescently Labelled Dendrimers. Macromol. Rapid Commun. 1997, 18, 659−665. (35) Birks, J. B.; Dyson, D. J.; Munro, I. H. ‘Excimer’ Fluorescence. II. Lifetime Studies of Pyrene Solutions. Proc. R. Soc. A 1963, 275, 575− 588. (36) Siu, H.; Duhamel, J. Comparison of the Association Level of a Hydrophobically Modified Associative Polymer Obtained from an Analysis Based on Two Different Models. J. Phys. Chem. B 2005, 109, 1770−1780. (37) Yip, J.; Duhamel, J.; Bahun, G. J.; Adronov, A. A Study of the Branch Ends of a Series of Pyrene-Labeled Dendrimers Based on Pyrene Excimer Formation. J. Phys. Chem. B 2010, 114, 10254−10265. (38) Zachariasse, K. A.; Kühnle, W. Intramolecular Excimers with α,ωDiarylalkanes. Z. Phys. Chem. 1976, 101, 267−276. (39) Cuniberti, C.; Perico, A. Intramolecular Excimers and Microbrownian Motion of Flexible Polymer Molecules in Solution. Eur. Polym. J. 1977, 13, 369−374. (40) Winnik, M. A.; Redpath, T.; Richards, D. H. The Dynamics of End-to-End Cyclization in Polystyrene Probed by Pyrene Excimer Formation. Macromolecules 1980, 13, 328−335. (41) Winnik, M. A.; Li, X.-B.; Guillet, J. E. Cyclization Dynamics of Polymers. 13. Effects of Added Polymer on the Conformation and Dynamics of Polystyrene Containing Evenly Spaced Pyrene Groups. Macromolecules 1984, 17, 699−702. (42) Winnik, M. A.; Egan, L. S.; Tencer, M.; Croucher, M. D. Luminescence Studies on Sterically Stabilized Polymer Colloid Particles: Pyrene Excimer Formation. Polymer 1987, 28, 1553−1560. (43) Seixas de Melo, J.; Costa, T.; Francisco, A.; Maçanita, A. L.; Gago, S.; Gonçalves, I. S. Dynamics of Short as Compared with Long

REFERENCES

(1) Möglich, A.; Joder, K.; Kiefhaber, T. End-to-End Distance Distributions and Intrachain Diffusion Constants in Unfolded Polypeptide Chains Indicate Intramolecular Hydrogen Bond Formation. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 12394−12399. (2) Goldberg, J. M.; Speight, L. C.; Fegley, M. W.; Petersson, E. J. Minimalist Probes for Studying Protein Dynamics: Thioamide Quenching of Selectively Excitable Fluorescent Amino Acids. J. Am. Chem. Soc. 2012, 134, 6088−6091. (3) Hudgins, R. R.; Huang, F.; Gramlich, G.; Nau, W. M. A Fluorescence-Based Method for Direct Measurement of Submicrosecond Intramolecular Contact Formation in Biopolymers: An Exploratory Study with Polypeptides. J. Am. Chem. Soc. 2002, 124, 556−564. (4) Winnik, M. A. End-to-End Cyclization of Polymer Chains. Acc. Chem. Res. 1985, 18, 73−79. (5) Duhamel, J. Polymer Chain Dynamics in Solution Probed with a Fluorescence Blob Model. Acc. Chem. Res. 2006, 39, 953−960. (6) Cantor, C. R.; Schimmel, P. R. In Biophysical Chemistry II: Techniques for the Study of Biological Structure and Function; Freeman and Company: New York, 1980; pp 444−448. (7) Fierz, B.; Satzger, H.; Root, C.; Gilch, P.; Zinth, W.; Kiefhaber, T. Loop Formation in Unfolded Polypeptide Chains on the Picoseconds to Microseconds Time Scale. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 2163− 2168. (8) Huang, F.; Hudgins, R. R.; Nau, W. M. Primary and Secondary Structure Dependence of Peptide Flexibility Assessed by FluorescenceBased Meassurement of End-to-End Collision Rates. J. Am. Chem. Soc. 2004, 126, 16665−16675. (9) Annable, T.; Buscall, R.; Ettelaie, R.; Whittlestone, D. The Rheology of Solutions of Associating Polymers: Comparison of Experimental Behavior with Transient Network Theory. J. Rheol. 1993, 37, 695−726. (10) Leibler, L.; Rubinstein, M.; Colby, R. H. Dynamics of Reversible Networks. Macromolecules 1991, 24, 4701−4707. (11) Adronov, A.; Fréchet, J. M. J. Light Harvesting Dendrimers. Chem. Commun. 2000, 1701−1710. (12) Duhamel, J. Internal Dynamics of Dendritic Molecules Probed by Pyrene Excimer Formation. Polymers 2012, 4, 211−239. (13) Odian, G. Principles of Polymerization, 3rd ed.; Wiley: New York, 1991; p 427. (14) Mizusaki, M.; Morishima, Y.; Winnik, F. M. Hydrophobically Modified Poly(sodium 2-acrylamido-2-methylpropanesulfonate)s Bearing Octadecyl Groups: A Fluorescence Study of their Solution Properties in Water. Macromolecules 1999, 32, 4317−4326. (15) Barros, T. C.; Adronov, A.; Winnik, F. M.; Bohne, C. Quenching Studies of Hydrophobically-Modified Poly(N-isopropylacrylamides). Langmuir 1997, 13, 6089−6094. (16) Mathew, A.; Siu, H.; Duhamel, J. A blob Model to Study Chain Folding by Fluorescence. Macromolecules 1999, 32, 7100−7108. (17) Nishikawa, K.; Yekta, A.; Pham, H. H.; Winnik, M. A. Fluorescence Studies of Hydrophobically Modified Hydroxyethylcellulose (HMHEC) and Pyrene-Labeled HMHEC. Langmuir 1998, 14, 7119−7129. (18) Wilemski, G.; Fixman, M. Diffusion-Controlled Intrachain Reactions of Polymers. I Theory. J. Chem. Phys. 1974, 60, 866−877. (19) Wilemski, G.; Fixman, M. Diffusion−Controlled Intrachain Reactions of Polymers. II Results for a Pair of Terminal Reactive Groups. J. Chem. Phys. 1974, 60, 878−890. (20) Ingratta, M.; Hollinger, J.; Duhamel, J. A Case for Using Randomly Labeled Polymers to Study Long-Range Polymer Chain Dynamics by Fluorescence. J. Am. Chem. Soc. 2008, 130, 9420−9428. (21) Yip, J.; Duhamel, J.; Qiu, X. P.; Winnik, F. M. Long-Range Polymer Chain Dynam ics of Pyrene-Labe lle d Poly(Nisopropylacrylamide)s Studied by Fluorescence. Macromolecules 2011, 44, 5363−5372. (22) Chen, S.; Duhamel, J.; Winnik, M. A. Probing End-to-End Cyclization Beyond Willemski and Fixmann. J. Phys. Chem. B 2011, 115, 3289−3302. 2323


Langmuir


Poly(acrylic acid) Chains Hydrophobically Modified with Pyrene, as Followed by Fluorescence Techniques. Phys. Chem. Chem. Phys. 2007, 9, 1370−1385. (44) Fowler, M. A.; Duhamel, J.; Bahun, G. J.; Adronov, A.; ZaragozaGalán, G.; Rivera, E. Studying Pyrene-Labeled Macromolecules with the Model Free Analysis. J. Phys. Chem. B 2012, 116, 14689−14699. (45) Ingratta, M.; Mathew, M.; Duhamel, J. How Switching the Substituent of a Pyrene Derivative from a Methyl to a Butyl Affects the Fluorescence Response of Polystyrene Randomly Labeled with Pyrene. Can. J. Chem. 2010, 88, 217−227. (46) Kanagalingam, S.; Spartalis, J.; Cao, T.-C.; Duhamel, J. Scaling Relations Related to the Kinetics of Excimer Formation between Pyrene Groups Attached onto Poly(N,N-dimethylacrylamide)s. Macromolecules 2002, 35, 8571−8577. (47) Teertstra, S. J.; Lin, W. Y.; Gauthier, M.; Ingratta, M.; Duhamel, J. Comparison of the Long Range Polymer Chain Dynamics of Polystyrene and cis-Polyisoprene Using Polymers Randomly Labeled with Pyrene. Polymer 2009, 50, 5456−5466. (48) Chen, S.; Duhamel, J.; Bahun, G.; Adronov, A. Effect of Fluorescent Impurities in the Study of Pyrene-Labeled Macromolecules by Fluorescence. J. Phys. Chem. B 2011, 115, 9921−9929. (49) Nau, W. M.; Huang, F.; Wang, X.; Bakirci, H.; Gramlich, G.; Marquez, C. Exploiting Long-Lived Molecular Fluorescence. Chimia 2003, 57, 161−167. (50) Zachariasse, K. A.; Maçanita, A. L.; Kühnle, W. Chain Length Dependence of Intramolecular Excimer Formation with 1,n-Bis(1pyrenylcarboxy)alkanes for n = 1−16, 22, and 32. J. Phys. Chem. B 1999, 103, 9356−9365.

2324


Global Analysis of Fluorescence Decays to Probe ... - ACS Publications

Recommend Documents