Distribution of Diffusion Times Determined by Fluorescence (Lifetime

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Distribution of Diffusion Times Determined by Fluorescence (Lifetime) Correlation Spectroscopy Jiří Pánek, Lenka Loukotová, Martin Hrubý, and Petr Štěpánek* Institute of Macromolecular Chemistry, Czech Academy of Sciences, Heyrovský Sq. 2, 16206 Prague, Czech Republic S Supporting Information *

ABSTRACT: Fluorescence correlation spectroscopy (FCS) provides most commonly values of diffusion coefficients of fluorescently labeled species in a solution, especially biopolymers. A newly developed procedure for the determination of diffusion coefficient distributions applicable to polydisperse polymers or nanoparticles, based on the well-known CONTIN algorithm, is described and tested on both simulated FCS correlation functions and real experimental data. Good resolution of bimodal distributions is observed, and it is quantitatively established how the resolution depends on the level of experimental noise. Effects of incorrect calibration of the focal volume on the obtained diffusion coefficient and its distribution are described for single-focus FCS. With rapid expansion of the FCS method from biology and biochemistry into the polymer field, and recently into many other disciplines including, e.g., environmental sciences, such a technique was very much needed.

1. INTRODUCTION Fluorescence correlation spectroscopy (FCS) is used since the early 1970s as a very efficient technique to study dynamic properties of soft matter.1−3 The majority of applications are found in investigation of biological systems, but the method was also proven to be very useful in polymer systems.4,5 The technique is often used for determination of transport analysis within cells to determine binding kinetics, molecular interactions, or association/dissociation processes. The study of dynamics of polymers on microscale by FCS is less straightforward than by dynamic light scattering (DLS) due to the necessary presence of a fluorescent moiety in the systems and due to more complicated and demanding instrumentation.6 On the other hand, FCS has a number of advantages;7 in particular, only a very small volume of sample is required, and it is usable for studying complicated mixtures where only the object of interest is fluorescently labeled while the remainder of the mixture does not contribute to the fluorescent signal. The lifetime variation of FCS developed in recent years, fluorescence lifetime correlation spectroscopy (FLCS), brings additional possibilities and benefits to the method.8 For both FCS and DLS the primary output of the measurement is a time correlation function G(t) where t is the delay time. G(t) has different functional forms for the two © XXXX American Chemical Society

techniques (see below), but in both cases its decay can be related to the dynamic properties of the investigated objects, most commonly to their diffusion coefficient, D. If the diffusing objects are identical (monodisperse), the determination of the diffusion coefficient from the measured correlation function is straightforward and is given by the following equations: For DLS: G(t ) = 1 + β[exp(−Dq2t )]2

(1)

where β is a coherence factor, q is the scattering vector and q = 4πnλ−1 sin(θ/2), with θ the scattering angle, n the refractive index, and λ the wavelength of light. For FCS: −1/2 −1 1⎛ t ⎞ ⎛ t ⎞ G (t ) = ⎜1 + ⎟ ⎜1 + 2 ⎟ N⎝ τD ⎠ ⎝ κ τD ⎠

(2)

where N is the average number of fluorescent species in the focal volume and κ = wz/wxy with wxy and wz being the width of Received: October 18, 2017 Revised: March 20, 2018

A

DOI: 10.1021/acs.macromol.7b02158 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. (a) Simulated correlation functions according to eqs 3 and 4 for diffusion time τD = 1 ms for FCS (○) and DLS (●). (b) Distribution function of relaxation times obtained by inverse Laplace transformation (REPES) of the correlation curves in Figure 1a (same symbols); the arrow shows the center of gravity of the FCS peak.

the focal spot in the x−y plane (perpendicular to the optical axis) and along the z-axis. We have also introduced a diffusion time τD = wxy2/4D. Unlike dual-focus FCS,9 where an external length scale is involved and absolute measurements can be therefore performed without any further calibration, in singlefocus FCS κ has to be obtained by calibration using fluorescent species with known diffusion coefficient. In eq 2 and in the rest of this contribution it is assumed that the process observed is a isotropic normal 3D diffusion and that there is no triplet contribution10 to the correlation function. This can be achieved either by selection of a suitable fluorescent dye with no triplet signal, or the triplet can be ignored in the case when it is much faster than the dynamics of interest. If the triplet contribution cannot be ignored, the correlation function has the form

Polymer systems are usually polydisperse to a smaller or larger extent in molecular weight and size, thus also in diffusion coefficient. In this case the correlation function has the form of an integral equation: DLS: ⎡ ⎤2 G(t ) − 1 = β ⎢⎣ A(D) exp( −Dq2t ) dD⎦⎥

∫

(3)

FCS: G (t ) =

1 N

∫

−1/2 −1 ⎛ t ⎞ ⎛ t ⎞ A(D)⎜1 + dD ⎟ ⎜1 + 2 ⎟ τD ⎠ ⎝ κ τD ⎠ ⎝

(4)

where A(D) is the distribution function of diffusion coefficients D. For DLS eq 3 can be converted to the functional form of a Laplace transformation

G DT(t ) = GT(t )G(t )

where DT stands for “diffusion and triplet” and the triplet contribution is10,11

G (t ) − 1 =

⎛ ⎛ t ⎞⎞ T GT(t ) = ⎜⎜1 + exp⎜ − ⎟⎟⎟ 1−T ⎝ τT ⎠⎠ ⎝

β

∫ A(D) exp(−Dq2t ) dD

(5)

linking the distribution of diffusion coefficients A(D) to the measured correlation function G(t). Obtaining A(D) from G(t) means performing an inverse Laplace transformation of the measured G(t) which is, due to measurement noise, an illconditioned mathematical problem,12 of which the solution is generally unphysical (e.g., containing negative amplitudes Ai) or unrealistic. A number of approaches have been designed in the past to address this difficulty and stabilize the inverse Laplace transformation; for a review see, e.g., ref 12. The most efficient and widely used approach is that of regularization represented by the methods CONTIN13−15 (linear approach) and REPES16 (nonlinear approach). In DLS these methods are now successfully used for more than two decades. For FCS a method for inverting eq 4, i.e., for calculating A(D) from the measured correlation function G(t), is not yet readily available. In this contribution we present an approach (FCS-CONTIN) for obtaining A(D) that is based on the program CONTIN; we demonstrate the results on simulated FCS correlation functions and discuss the resolution of the method and the effects of noise and calibration parameters on the data analysis. In this context by resolution we mean the capability of the program to reconstruct in the calculated distribution function two close peaks that are known to exist in the measured or simulated sample. The minimal distance of

with T and τT being the amplitude and decay time of the triplet decay. GT(t) then has to be divided out from the measured GDT(t) to obtain G(t) by using values of T and τT obtained, e.g., from a single or double decay fit with triplet component usually available in a data analysis software package in commercial FCS instruments. For this procedure the residuals of this fit have to be random in the fast delay time region of the correlation function corresponding to the triplet process. Since all the instrumental parameters are known, i.e., q in eq 1 and κ in eq 2, the value of the diffusion coefficient D and the amplitude parameters β in eq 1 and 1/N in eq 2 can be obtained by a simple fitting procedure. If the system contains more than one type of diffusing objects, e.g., objects of different sizes, the correlation function is a sum of contributions of the individual object types with diffusion coefficients Di with amplitudes Ai. Reliable determination of Di and Ai is possible for two, sometimes three components. For larger i the data fitting procedure yields unstable values due to the presence of experimental noise in the measured correlation function. B


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Figure 2. Reconstruction by FCS-CONTIN of a simulated bimodal distribution of diffusion times of equal amplitude using Chosen solution. Smaller diffusion time is fixed at 10 ms, and larger diffusion time varies between 20 and 100 ms as indicated in the middle figure. The level of random noise is 0.5% in (a), 2% in (b), and 5% in (c).

in the algorithm is that for DLS, i.e., K(τ,t) = exp(−t/τ). In the present approach applicable to FCS the kernel was changed to

distinguishable peaks is given by several factors such as signalto-noise ratio, peak width, number of sampling points used in calculations, etc. Although both eqs 3 and 4 represent the decay of a correlation function, they differ noticeably in the mathematical form. This is exemplified in Figure 1 with simulated data for DLS and FCS in the case of a single diffusion process with diffusion time τD = 1 ms. In view of the logarithmic scale on the t-axis, the difference between the DLS and FCS curves in Figure 1a is very significant. An attempt to analyze FCS data using Laplace inversion programs such as CONTIN or REPES leads to completely wrong results, as shown in Figure 1b. The simulated single relaxation time of 1 ms is correctly recovered for the DLS case in the form of a very narrow peak; for the FCS case the resulting distribution is extremely broad with average value shifted by an order of magnitude to longer times. Several researchers tested various approaches for extraction of the distribution of diffusion times from a measured FCS autocorrelation function. In ref 17 the distributions of sizes obtained by the maximum entropy method and Gaussian distribution model from FCS correlation functions are reported, and it is shown that they provide comparable results. Usage of the CONTIN algorithm is reported in ref 18, where it is used to analyze bimodal binding complexes. A detailed testing of the maximum entropy method is provided in ref 19, where it is concluded that the method is convenient for narrow distributions but the results for wide distributions are significantly affected by the rather low signal-to-noise ratio of FCS measurements. In these references details of obtaining the distributions of diffusion times are not indicated.

−1/2 −1 ⎛ t ⎞ ⎛ t ⎞ K (τ , t ) = ⎜ 1 + ⎟ ⎜1 + 2 ⎟ τD ⎠ ⎝ κ τD ⎠ ⎝

In terms of programming this means changing the FUNCTION USERK in CONTIN to a new form, as well as several control variables, as is shown in detail in the Supporting Information. Additionally, the geometrical parameter κ defining the asymmetry of the confocal volume has to be added to the input file as variable RUSER(11); see an example in Supporting Information. As for any other FCS experiment, κ (or wxy and wz) has to be determined independently by experimental calibration. Both CONTIN and FCS-CONTIN are regularization methods where the value of a regularization parameter α that limits the effect of statistical noise in the data on the final distribution of diffusion times is adjustable. In the tests presented below the automatically selected solution called “Chosen” solution is presented. More details on the regularization procedure can be found in the original references.13−15

3. RESULTS AND DISCUSSION: SIMULATIONS Simulated data have been used to study three types of situations that are most representative of experimentally encountered polymer samples. In order to simulate experimental data as close as possible, noise has been added to the simulated correlation functions. If the modeled noiseless correlation function is Gn(t), then the simulated noisy correlation function is given by G(t) = (1 + ε(R − 0.5))Gn(t), where R is a random number between 0 and 1 as provided by a random number generator available in any programming language. Noise generated in this manner is the same in all delay time regions of the correlation function. However, due to the fact that the correlation function G(t) is built by the correlator with a logarithmic delay time axis t the experimental noise at small values of t is higher than that at high values of t. This should be kept in mind when considering the results of the simulations and experiments discussed below in the sense that the described effects of noise have a stronger effect on the calculated distributions of diffusion times at small values of τ

2. FCS-CONTIN CONTIN is a regularization program for inversion of noisy integral equations of the type G (t ) =

∫a

b

K (τ , t )A (τ ) d τ

(7)

(6)

classified as Fredholm integral equation of the first kind. In eq 6 notation analogous to eq 4 is used. G(t) is known (measured), A(τ) has to be determined, and K(τ,t) is the kernel of the equation. Equations of this type appear in many physical models. In the standard version of CONTIN, the kernel coded C


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Figure 3. Reconstruction by FCS-CONTIN of a simulated bimodal distribution of diffusion times with τD,1 = 10 ms and τD,2 = 100 ms and variable amplitude A1 of τD,1 as shown in legend to (b), using Chosen solution. A2 is then A2 = 1 − A1. (a) Noise level 0.5%, (b) noise level 2%, and (c) noise level 5%. Here the arrows indicate to position of peaks with very small amplitude not clearly visible on the distribution curve.

Figure 4. (a) Ratio of calculated (C) to simulated (S) values for diffusion times and amplitudes as extracted from Figure 3, as a function of amplitude A1, for noise level 2%. (b) Ratio τD,1C/τD,1S of calculated to simulated value of the smaller diffusion time (τD,1 = 10 ms) as a function of amplitude A1 for the indicated three levels of noise. (c) Ratio A1C/A1S of calculated to simulated value of the amplitude A1 of the smaller diffusion time (τD,1 = 10 ms) as a function of amplitude A1 for the three levels of noise 0.5%, 2%, and 5%, as indicated.

than at large values of τ. The same situation occurs in DLS. An effort has been made in earlier years20 to address this problem in DLS by introducing two regularization parametersone for the fast and one for the slow part of the correlation function. This approach did not lead to an applicable data treatment scheme because of calculation instabilities due to the interdependence of the two regularization parameters and to unclear location of the border between the fast and slow parts of the correlation function. Instead of that the subtraction technique was developed21,22 as another approach usable in the analysis of correlation functions that is also applicable to address the problem. In the subtraction technique after a first inversion of the correlation function a region of interest in the decay time distribution is selected for a second analysis. In the second analysis, as a first step all information form the regions of the not-considered parts of the correlation function is subtracted, and the second inversion is performed only in the region of interest. For details see the original references.19,20 The subtraction technique is applicable in the same way to correlation functions obtained in FCS and is implemented in FCS-CONTIN. 3.1. Resolution of Peaks of Equal Amplitude. For this test FCS correlation functions were simulated representing a bimodal distribution of diffusion times with three levels of random noise typical for experimental data: ε = 0.5%, 2%, and

5%. In this simulation the two diffusion times were supposed to be monodisperse (i.e., a δ-function was used in the simulations). The ratio of the two diffusion times was varied in the range τD,2/τD,1 = 2 to 10; i.e., τD,1 was fixed at 10 ms, and τD,2 was varied between 20 and 100 ms. The results are shown in Figure 2 for the three levels of noise. It is apparent that using the Chosen solution of FCSCONTIN, two close components can be resolved when the ratio of their diffusion times is larger than 3, for good measurements with noise level of 0.5%. For larger noise levels the resolution capability decreases, two components can be resolved when τD,2/τD,1 > 5 for noise level of 2% or τD,2/τD,1 > 10 for noise level = 5%. When the two peaks are resolved, their relative amplitude corresponds to the simulated one with accuracy of a few percent getting closer to the simulated values with decreasing level of noise. 3.2. Resolution of Peaks of Small Amplitudes. In this section a bimodal distribution of diffusion times was simulated with two fixed diffusion times of τD,1 = 10 ms and τD,2 = 100 ms. The amplitude A1 of τD,1 was varied between 2% and 98%; therefore, the amplitude of τD,2 was A2 = 1 − A1 between 98% and 2% (Figure 3). Intermediate values of A1 are shown in the legend of Figure 3b. It is clear from Figure 3 that as the amplitude of a peak decreases, the corresponding calculated D


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Figure 5. (a) Reconstruction by FCS-CONTIN of a simulated superposition of two log-normal distributions of diffusion times with characteristic times τ0,1 = 3 ms and τ0,2 = 30 ms and constant width of σ0 = 0.5 and variable noise from 1% to 7% (top to bottom at center of a peak). (b) Comparison of simulated (full line) and recovered (dashed line) distribution with 2% noise and σ0 = 0.5 for both components. (c) Same as (b) with larger width σ0 = 1.0 of both components.

3.4. Influence of the Geometrical Parameter κ. The geometrical parameter κ, i.e., the asymmetry of the focal volume wz/wxy, has to be determined by calibration of the experimental setup using fluorescent species with known diffusion coefficient. The parameter κ depends on the experimental conditions such as laser wavelength, refractive index of solvent, type of microscope objective, or temperature. It is of interest to see the effect of using an incorrect value of κ when the FCS setup was not recalibrated after a change of conditions or sample type. Typical values of κ are in the range 3−5. Therefore, we simulated correlation functions with a single decay time of τ0 = 10 ms and κ0 = 5, and we applied the FCS-CONTIN fit with several fixed values of κ in the range from 1.2 to 20. The results are shown in Figure 6 in the form of τ/τ0 as a function of κ/κ0.

diffusion time deviates increasingly from the simulated value, the more so that the noise in the data is larger. More quantitatively, values extracted from data in Figure 3 are summarized in Figure 4 for the three levels of noise. In order to present all four parameters in a comparable way, Figure 4 presents the diffusion times and amplitudes in a normalized way, i.e., as a ratio of the calculated value to the simulated value. It can be seen in Figure 4a that when the amplitude of the smaller peak decreases, e.g., below 0.2, the discrepancies between the calculated and simulated values of the diffusion times and amplitudes significantly increase. Figure 4a represents data obtained with the level of noise 2%. Figures 4b and 4c present in more detail the amplitudes and diffusion times of the faster component for the three levels of noise. It is clear from Figures 4b and 4c that reducing the noise to or below 0.5% (still achievable experimentally) dramatically increases the reliability of characterization of the weaker component in the diffusion time distribution. 3.3. Resolution of Peaks with Wide Distributions. Polymers produced by synthetic routes have almost always a non-negligible distribution of molecular weights (“polydispersity”) so as a consequence also of diffusion times when dissolved in dilute solution. Therefore, it is of interest to examine how does a possible width of the distribution functions influence the performance of the method. FCS correlation functions were simulated using a log-normal distribution of diffusion times with variable width and for several values of noise. Figure 5a shows the effect of noise on calculated distributions of diffusion times modeled with two log-normal distributions of width σ0 = 0.5 and ratio of characteristic times τ01/τ02 = 10 for various levels of noise in the range 0% to 7%. It can be seen that for noise levels up to 5% the two original components can be properly recovered; for higher levels of noise of 6 and 7% they are overlapping and almost merged. Figures 5a and 5b show another perspective and compare the shape of the simulated distributions at constant noise level of 2% for the width of each component σ0 = 0.5 in Figure 5b and σ0 = 1.0 in Figure 5c. In the latter case the simulated distributions already overlap significantly, and this is reproduced also in the calculated distribution. For very wide distributions and depending on the level of noise the calculation procedure may have a tendency to yield two or three peaks instead of a very broad one. In such a case the parsimony principle will help in selecting a solution close to the “chosen” solution but satisfying our prior knowledge about the investigated polymer system.

Figure 6. Dependence of τ/τ0 on κ/κ0 for simulated values of τ0 = 10 ms and κ0 = 5 with noise level of 2%. The open point (○) corresponds to calculation with the correct value of κ.

It is interesting to observe that if values of κ larger than the correct κ0 are used in the calculation, the calculated value of diffusion time deviates from the simulated one by no more than approximately 5%. On the other hand, if values of κ smaller than the simulated one are used, an important decrease of the calculated diffusion time is observed, down to 65% of the simulated value in the range of κ investigated. 3.5. Molecular Brightness. The A(D) distributions obtained with use of the FCS-CONTIN algorithm presented above are based on equal signal contributions from all diffusing species present in the measured sample. The software does not reflect possible differences in molecular or particle brightness B, defined as the number of photons emitted per second per molecule (or particle) present at the center of illumination E


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Figure 7. (a) Differential weight fraction A(M) as a function of molecular weight M for sample L119 obtained by GPC. (b) Distribution function A(Rh) of hydrodynamic radii Rh obtained on aqueous solution of polymer L119 (concentration 1 mg/mL) by dynamic light scattering using a Pearson V analytical fit (black dotted line) and CONTIN regularized fit (blue dash-dotted line) and by fluorescence correlation spectroscopy using FCS-CONTIN (red dashed line).

interest. It is also feasible to explore interactions of two or several components with different sizes, e.g., adsorption of proteins on nanoparticles, or to detect association or dissociation of supramolecular polymer structures. The components may be stained with different dyes or only one of them can be stained if the studied interaction leads to a big change in diffusion coefficient. In the case when the diffusion coefficients of the tracked components are distinguishable, their relative amplitudes and width in distribution may give qualitative assessment of the development of the explored process. For these cases it is not necessary to have the exact knowledge about components’ molecular brightness; therefore, applications discussed in this contribution assume that the knowledge of molecular brightness is not crucial for data evaluation.

volume. Molecular (or particle) brightness is an important characteristic of each fluorescent compound and may give useful information, e.g., on aggregation or association behavior of macromolecules, binding, conformational changes, etc. Therefore, several procedures were developed to determine the number of particles and their brightness in FCS measurements. By use of the method of moment analysis,23,24 the brightness of a fluorescent particle is obtained from the moments of photon counts distribution. Even though this attitude represented a significant progress in FCS data analysis, a drawback of this method was the need of higher moments’ calculations and a large data set available25 to distinguish between multiple components. Later, a more general method was introduced26 using the photon-counting histogram (PCH). The PCH method exploits the entire distribution of the detected photon counts, and it is capable of distinguishing multiple components. It gives both information about molecular brightness and concentrations, thus providing information similar to A(D) in eq 4. Nevertheless, this technique is computationally demanding and requires a large data set as well, which is significantly disadvantageous, e.g., in the case when imaging mode is used for data acquisition. Simplified methods based on histograms were also introduced imposing additional constraints and regularization conditions that eliminate nonphysical solutions and solutions contradicting empirical knowledge.27 Recently, another approach was presented based on analytical expressions of molar weight and fluorescent brightness distribution functions instead of discretizing autocorrelation functions;28 the Schulz−Zimm distribution function for the dispersity in the degree of polymerization is used in the model. Such attitude represents an elegant method of extraction size distributions from FCS data provided that the analytical form describing the composition of the polymer is known. The FCS-CONTIN can be easily used to analyze measurements of samples with neither any previous knowledge of dispersity nor large data set available. On the other hand, the software inherently works with equal molecular brightness of all species. Despite this feature, its use is not strictly limited to cases complying with such requirement; for a qualitative evaluation it is possible to neglect differences in molecular brightness of fractions. Even without any piece of information concerning molecular or particle brightness, it is possible to follow the presence or absence of one chosen component of

4. RESULTS AND DISCUSSION: EXPERIMENTAL PART The application of FCS-CONTIN is demonstrated on several experimental examples below. Samples of poly(N-(2-hydroxypropyl)methacrylamide) (HPMA) have been synthesized, fluorescently labeled, characterized according to the procedures given in the Supporting Information, and investigated in dilute aqueous solutions. Figure 7 shows the distribution of molecular weights or polymer coil sizes for the sample L119 obtained by several approaches. Figure 7a shows the GPC result indicating a weight-average molecular weight of Mw = 213 kDa and polydispersity index Mw/Mn = 4.29. Figure 7b shows the distributions of hydrodynamic radii obtained on a solution of the same polymer by dynamic light scattering and by fluorescence correlation spectroscopy. Although a relation between molecular weight and size of a polymer coil is not known for HPMA, the data can be compared quantitatively. In general, monomodal distributions of polymer sizes can be well represented by a Pearson function of type V given by12 ⎛ τ ⎞ A(τ ) = τ0pτ −p − 1 exp⎜ − 0 ⎟Γ(p) ⎝ τ⎠

This function has the remarkable property that its Laplace inversion represented by the correlation function in eq 5 has a simple form F

(

G (t ) − 1 = 1 +

t τ0

−p

)

and can be fitted


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Mw Mn

=

Γ(p)Γ p +

2 ν

1 ν

)

(

(

Γ2 p +

main component in the distribution function representing the polymer sample with average polymer coil radius Rh = 11.3 nm, a weaker smaller component is observed with characteristic size 0.41 nm. The latter very well corresponds to the size of the notpolymer-bound fluorescent label, which when measured separately in aqueous solution exhibits a hydrodynamic radius Rh = 0.48 nm (data not shown). It is therefore to be concluded that the dialysis procedure for sample L118 did not remove completely the not-polymer-bound fluorescent label during sample preparation. We note here that the low-molecularweight dye is physically a monodisperse object and should be seen as a narrow component. However, as explained in detail in ref 12, the regularization procedure in the integral inversion optimizes the distribution so as to minimize sharp changes in the distribution function in equal manner in all regions of the distribution. Therefore, if the distribution consists of a narrow and a wide component, the narrow component is recovered with a larger width. The presence of unbound dye was confirmed by analyzing the absorption signal collected at wavelength 500 nm during a GPC analysis. This absorption signal, however, cannot be quantitatively compared to the FCS distribution in Figure 8 since it is known that the molecular brightness of a fluorescent label bound to a polymer or nanoparticle is significantly smaller than for the free dye.30−32 An important reason for this effect is the red-shift of the absorption spectrum of the dye after binding. Methods for studying the amounts of free and bound dye based on the amplitudes of the corresponding fluorescent signals have been proposed,11,30 which should be applicable irrespectively of the fact whether the amplitudes have been obtained by a discrete component fit or from FCS-CONTIN. Figure 9 examines the case of a mixture of two types fluorescently labeled nanoparticles made by nanoprecipitation

)

where ν is the Flory exponent. The resulting fitted distribution is shown in Figure 7b, and assuming the value of ν = 0.6, as appropriate for a dilute solution of a linear polymer in good solvent, we obtain Mw/Mn = 6.65. This can be considered as a reasonable agreement with the GPC result given the fact that results of completely different techniques are compared. In GPC the polymer distribution is physically separated to individual components according to their hydrodynamic radii on the column, and each component is identified separately to construct in the end a distribution of molecular weights assuming a linear separation performance of the column in its nominal separation range. For wider distribution this may not be fully satisfied. In DLS and FCS the correlation function contains superposed information from all components of molecular weight, and the separation is performed mathematically by postprocessing the correlation function, which is not a simple process as documented by ∼30 years of development of this approach. In addition to the Pearson V distribution function, Figure 7b shows also distributions obtained on the same sample by integral inversion of a DLS correlation function and a FCS correlation function. Both are somewhat wider than the Pearson V distribution which is due to the influence of the regularization parameter in the integral inversion that limits the effect of noise in the data, the widths being w = 0.61 for Pearson DLS fit, w = 0.80 for DLS CONTIN fit, and w = 1.09 for FCS-CONTIN fit. In Figure 7b, the intensity-weighted distribution functions for DLS have been converted to massweighted distribution functions (assuming again the Flory exponent ν = 0.6 for a polymer in a good solvent) to be comparable with the mass-weighted distributions yielded by GPC (see e.g. ref 29) and by FCS. Indeed, for FCS it is assumed that (i) the number of fluorescent labels attached to a polymer chain is proportional to the number of monomers in the chain and (ii) the molecular brightness (see section 3.5) of a labeled polymer chain is proportional to the number of fluorescent labels attached to the chain. Figure 8 shows the distribution of hydrodynamic radii for another sample L118 of the fluorescently labeled poly[N-(2hydroxypropyl)methacrylamide-co-2-(aminopropyl)methacrylamide] (HPMA-co-APMA) copolymer. Besides the

Figure 9. Distribution functions of hydrodynamic radius Rh for an aqueous solution of mixture of PLA273-b-PEO113 nanoparticles with two radii, 53 and 204 nm, obtained by integral inversion from an FCS measurement.

from a commercial block copolymer poly(DL-lactide)-b-poly(ethylene oxide), PLA273-b-PEO113. Both types of nanoparticles are observed in the distribution function of sizes obtained by FCS-CONTIN, with hydrodynamic radii 58 and 221 nm in good agreement with the sizes measured individually for each component. The peak areas in the distribution depend on contributions from the respective fluorescent signals in the same manner as in the previous example, meaning that the square of the molecular brightness (and possible triplet state of the fluorophore) must be considered. The exact relation between the peak areas and concentrations of the species can be obtained either using additional data provided by

Figure 8. Distribution function of hydrodynamic radii Rh for a dilute aqueous solution of polymer L118 obtained by FCS. G


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independent measurements or by the analysis of photon counting histogram mentioned above in the text.

5. CONCLUSIONS We have presented a newly developed integral inversion technique for calculating the distribution of diffusion times from measured correlation functions in FCS, based on a modification of the algorithm CONTIN widely used for analysis of data in dynamic light scattering. The availability of such a method makes FCS analysis more accessible for interdisciplinary fields, since distributions of diffusion times now can be obtained with the same ease as in dynamic light scattering. The resolution power of the method depends on the experimental noise in the data that is on typical setups larger than in dynamic light scattering due to lower light intensity levels. The resolution can be improved using the parsimony principle,13 i.e., adjusting the regularization parameter α so that the resulting distribution shows the minimum of unexpected information. Finally, the importance of correct calibration of the focal volume was demonstrated. Errors not exceeding 5% in diffusion time and amplitude appear when the used value of the length to width ratio κ of the focal volume is larger than the correct calibration; however, much larger errors appear when the used value κ is smaller than the correct one. For practical applications of the method it has to be kept in mind that the molecular brightness of the fluorescent label depends on its environment and changes after binding to a polymer or nanoparticle, which has to be taken into account when considering quantitatively the amplitudes of the peaks in the distribution function.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b02158. The modified subroutine USERK in FCS-CONTIN software as well as a sample data file to FCS-CONTIN; experimental details of polymer synthesis and preparation of nanoparticles used in the main text for experimental verification of the method (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Martin Hrubý: 0000-0002-5075-261X Petr Štěpánek: 0000-0003-1433-678X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work was supported by the Ministry of Education, Youth and Sports of CR within the National Sustainability Program I (NPU I), Project POLYMAT LO1507.

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REFERENCES

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Distribution of Diffusion Times Determined by Fluorescence (Lifetime

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