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Measuring Arbitrary Diffusion Coefficient Distributions of NanoObjects by Taylor Dispersion Analysis Luca Cipelletti,*,† Jean-Philippe Biron,‡ Michel Martin,§ and Hervé Cottet*,‡ †

Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS, Université de Montpellier, Montpellier, France Institut des Biomolécules Max Mousseron (IBMM, UMR 5247 CNRS, Université de Montpellier, Ecole Nationale Supérieure de Chimie de Montpellier), Campus Triolet, Place Eugène Bataillon, CC 1706, 34095 Montpellier Cedex 5, France § Ecole Supérieure de Physique et de Chimie Industrielles, Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH, UMR 7636 CNRS, ESPCI-ParisTech, Université Pierre et Marie Curie, Université Paris-Diderot), 10 rue Vauquelin, 75231 Paris Cedex 05, France ‡

S Supporting Information *

ABSTRACT: Taylor dispersion analysis is an absolute and straightforward characterization method that allows determining the diffusion coefficient, or equivalently the hydrodynamic radius, from angstroms to submicron size range. In this work, we investigated the use of the Constrained Regularized Linear Inversion approach as a new data processing method to extract the probability density functions of the diffusion coefficient (or hydrodynamic radius) from experimental taylorgrams. This new approach can be applied to arbitrary polydisperse samples and gives access to the whole diffusion coefficient distributions, thereby significantly enhancing the potentiality of Taylor dispersion analysis. The method was successfully applied to both simulated and real experimental data for solutions of moderately polydisperse polymers and their binary and ternary mixtures. Distributions of diffusion coefficients obtained by this method were favorably compared with those derived from size exclusion chromatography. The influence of the noise of the simulated taylorgrams on the data processing is discussed. Finally, we discuss the ability of the method to correctly resolve bimodal distributions as a function of the relative separation between the two constituent species.

T

Typically, only a few nL of sample are injected, which makes modern TDA suitable for the analysis of biological or other low-abundant samples. Since TDA is an absolute method, no calibration is required and the exact knowledge of the sample concentration is not a prerequisite. It is applicable to samples of different nature in both aqueous or nonaqueous liquid phases (small molecules and drugs,10,11 polymers,12,13 dendrimers,14 proteins,10 nanoparticles,15,16 clusters,17 liposomes,18 drug delivery systems19), with sizes from angstroms9,11 to about 300 nm,15 as far as the Taylor dispersion conditions are met.4 Recently, TDA (or flow-induced dispersion) was also used to study interactions between molecules or for research on biomarkers.20,21 In the case of polydisperse samples, the taylorgram is the sum of individual Gaussian peaks, each of the Gaussian contributions pertaining to a given diffusion coefficient. From the overall temporal variance of the elution peak, an average diffusion coefficient can be determined by TDA. For a mass concentration (respectively molar concentration) sensitive

aylor dispersion analysis (TDA) is an absolute method that allows determining the diffusion coefficient, D, and hydrodynamic radius, Rh.1,2 TDA is based on the dispersion of a narrow sample band injected in an open tube under laminar flow.1 The combination of molecular diffusion and the dispersive parabolic velocity profile under flow leads to a specific dispersion, known as the Taylor dispersion. For a monodisperse (or monomolecular) sample, the elution profile recorded as a function of time at a given point in the dispersion tube, is a Gaussian peak,1 provided that the conditions reviewed below in the Theory section are met.1,3 When these conditions are verified, the temporal variance of the elution peak is directly related to the molecular diffusion coefficient, D, of the sample component, independently of the chosen experimental conditions (capillary diameter and length, mobilizing pressure).4 Therefore, it is possible and straightforward to experimentally determine the solute D value from the temporal variance of the elution profile (taylorgram). TDA was first implemented in long open tubes in gaseous phase5 and then in liquid phase.6−8 More recently, capillary electrophoresis instruments, which allow an inline recording of the elution profile at a given point in a narrow capillary (∼50 μm), were found to be particularly well adapted for TDA.9 © 2015 American Chemical Society

Received: May 22, 2015 Accepted: July 19, 2015 Published: August 5, 2015 8489

DOI: 10.1021/acs.analchem.5b02053 Anal. Chem. 2015, 87, 8489−8496

Article

Analytical Chemistry detector, TDA leads to a harmonic weight-averaged (respectively, harmonic number-averaged) diffusion coefficient and, therefore, to the weight-averaged (respectively, numberaveraged) hydrodynamic radius.22 Some attempts were done to deconvolve the taylorgram in order to get the sample size distribution for specific cases, such as bimodal mixtures,23 or three-component systems.24 The latter approach was relatively complex, as it involves diffusion cross-terms, which however vanish when the sample is highly diluted in the carrier liquid. Fitting the taylorgram by a sum of Gaussian functions should, in principle, allow the determination of the distribution of the diffusion coefficients of the mixture. This approach has been applied to synthetic mixtures of up to six-mers of ethylene glycol oligomers.25 However, it requires the knowledge of the exact number of components and becomes hardly applicable for more complex mixtures/ distributions. Recently, we proposed a new approach for the data processing of taylorgrams, based on a cumulant development similar to that used in dynamic light scattering.26 Both simulated and real experimental data for solutions of moderately polydisperse polymers and polymer mixtures were analyzed leading to the quantification of the sample polydispersity and to the determination of an “equivalent” log-normal distribution having the same polydispersity as the real sample distribution.26 In this work, we present a new approach for the data analysis of polydisperse samples based on the Constrained Regularized Linear Inversion (CRLI) method, which we implement using the freely available CONTIN software package.30−32 The paper is organized as follows. In the next section, the theoretical bases of the CRLI analysis are presented. The CRLI method is then applied to simulated taylorgrams generated from size exclusion chromatography (SEC) distributions, to demonstrate the validity and interest of this approach. Finally, the CRLI method is applied to experimental taylorgrams obtained for polystyrenesulfonate standards and their bimodal or trimodal mixtures.

S(t ) =

∞

⎡ (t − t )2 12D ⎤ 0 ⎥d D CM(D)ρ(D) D exp⎢ − R c2t0 ⎦ ⎣ (2)

where the concentration is assumed to be low enough for cross terms to be dropped and where M(D) and ρ(D) are the mass and molar concentration in the injected sample of the objects with diffusion coefficient D. It is convenient to consider a normalized expression of the taylorgram: s (t ) ≡

S(t ) =c S(t0)

∫0

∞

⎡ (t − t )2 12D ⎤ 0 ⎥d D PD(D) D exp⎢ − 2 R ⎦ ⎣ c t0 (3) 1/2 −1

−1 with c = [∫ ∞ = [ D ] a normalization 0 PD(D)√DdD] factor and PD(D) = M(D)ρ(D)[∫ 0∞M(D)ρ(D)dD]−1 the (mass-weighted) probability distribution function (PDF) of the diffusion coefficient. Note that by definition s(t) is dimensionless and varies between 0 and 1. The various moments of the distribution of D are obtained from D β = β ∫∞ 0 PD(D)D dD, where β needs not to be a positive integer. In particular, β = 1 corresponds to the mass-weighted average diffusion coefficient D̅ , which however is not directly accessible by TDA. By contrast, the temporal variance of the taylorgram directly yields the harmonic average of D (β = −1),22 which we shall denote as the “Taylor average” of D, ⟨D⟩T:

∞

∫ s(t )dt R 2t ⟨D⟩T = c 0 ∞ 0 = [ D−1]−1 24 ∫ s(t )(t − t0)2 dt 0

(4)

Cumulant Analysis. In ref 26 we have introduced the cumulant analysis of Taylor dispersion data. When combined with the usual measurement of ⟨D⟩T, the cumulant analysis provides a means to quantify the sample polydispersity, by introducing a suitable polydispersity index (PI). In the following, we will use the PI as a way to quantify any differences between the measured PD(D) and the theoretical ones; we thus briefly recall the main features of the cumulant analysis and the definition of the PI. In analogy to the analysis of dynamic light scattering data,31 the taylorgram is plotted using variables such that the signal is a straight line for a perfectly monodisperse sample, that is, by plotting ln[s(t)] versus (t − t0)2, as seen from eq 1. Deviations from a straight line are indicative of polydispersity and can be modeled by expanding ln[s(t)] in powers of (t − t0)2. Keeping only terms up to (t − t0)4, one finds26

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THEORY Taylor Dispersion Analysis on Polydisperse Samples. The temporal variation of the signal recorded in a Taylor dispersion experiment for a solution of identical objects of molar mass M has a Gaussian shape: ⎡ (t − t )2 12D ⎤ 0 ⎥ S(t ) = CMρ D exp⎢ − R c2t0 ⎣ ⎦

∫0

(1)

with C an instrumental constant, ρ the molar concentration of the objects in the injected sample, t0 the peak time, and Rc the capillary radius. The factor M has been introduced to account for the response of the detector, which is usually proportional to the mass concentration. Note that eq 1 only holds under the following assumptions that can be easily met in practice: (i) the capillary radius is small enough (to ensure that molecular diffusion allows thorough mixing across its cross-section);1,4 (ii) axial diffusion is negligible compared to the contribution of Taylor dispersion;3,4 (iii) the injected volume is small enough, so that corrections due to the finite injection time are negligible;30 and (iv) the peak standard deviation is sufficiently smaller than t0 for the peak shape to be reasonably assumed to be Gaussian.4 For solutions of objects with a finite size distribution, eq 1 is replaced by an integral over the contributions of all sizes and, hence, diffusion coefficients:

ln[s(t )] = −

Γ 12 ⟨D⟩Γ (t − t0)2 + 2 (t − t0)4 + ... 2 2 R c t0

(5)

where we have introduced the second cumulant Γ2 and the “Gamma average” of D: ⟨D⟩Γ =

D3/2 D1/2

(6)

Although Γ2 is directly related to the width of the PDF of D, we have shown in ref 26 that a more robust indicator of the polydispersity is obtained by comparing the Gamma and Taylor averages through the polydispersity index PI, defined by PI = 8490

2 ⟨D⟩Γ ln 3 ⟨D⟩T

(7) DOI: 10.1021/acs.analchem.5b02053 Anal. Chem. 2015, 87, 8489−8496

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Analytical Chemistry

searches for a set of non-negative Pj that minimize the following quantity:34

While eqs 6 and 7 hold whatever the shape of PD(D), the Gamma average and the PI have a particularly straightforward interpretation if the PDF of the diffusion coefficient is lognormal. In this case, ⟨D⟩Γ and the PI are directly related to the parameters μ and σ governing the position and width of the peak of the log-normal distribution: PD(D) =

⎡ (ln D − μ)2 ⎤ exp⎢ − ⎥ Dσ 2π ⎦ ⎣ 2σ 2

Nt

1 2 ln⟨D⟩Γ + ln⟨D⟩T , σ = PI 3 3

(8)

(9)

Constrained Regularized Linear Inversion of TDA Data. We now aim to go beyond the cumulant method and retrieve the full PDF of D from the taylorgram. This requires solving eq 3 for the unknown PD(D), given a taylorgram s(t) and the experimental parameters Rc and t0. Note that in experiments the taylorgram is unavoidably affected by measurement uncertainties and s(t) is sampled for a discrete set of times t1, ..., tNt. It is convenient to discretize also D and its PDF, so that the unknown are ND values Pj (j = 1, ..., ND) such that Pj = PD(Dj). Taking into account the discretization of t and D and the experimental uncertainties, eq 3 is rewritten as a set of Nt linear equations in the ND unknowns Pj: ND

sk =

∑ AkjPj + εk ,

k = 1, ..., Nt (10)

j=1

with sk the experimental taylorgram at time tk and εk the corresponding experimental error. The matrix A has elements Akj =

⎡ 12D (t − t )2 ⎤ j k 0 ⎥cj Dj exp⎢ − ⎢⎣ ⎥⎦ R c2t0

∑ AkjPj)2 + α

k=1

j=1

ND − 1

∑

(Pj − 1 − 2Pj + Pj + 1)2 (12)

j=2

The first term of eq 12 corresponds to the sum of the squared deviations between the data and the fit, as in usual least-squares fitting. The second term is the so-called regularizator, which penalizes physically unsmooth PDFs. For details on the way the regularization parameter α is chosen, see the Supporting Information and ref 27. A final point of practical importance is the choice of the set of {Dj}. To avoid increasing the computational load and because PD(D) is a smooth function, the number ND of solution points needs not and should not be too large: we typically use ND ≤ 200 values of D, linearly or logarithmically spaced between a lower and an upper bound, Dmin and Dmax, respectively. The interval [Dmin, Dmax] should be sufficiently wide to encompass the range of diffusion coefficients of the species that compose the sample. On the other hand, it should be kept as small as possible in order not to lose too many details of the actual PDF due to discretization. One possible approach is to start with a conservatively large D interval and reduce it by running the CRLI algorithm iteratively. However, this has a non-negligible computational cost. We propose two alternative approaches, based on the cumulant analysis described above. The first approach relies on the observation that in a cumulant plot the slope of ln(s), ∂ ln s/∂(t − t0)2, is proportional to D, to leading order in (t − t0)2 (see eq 5). For polydisperse samples, the cumulant plot is not a straight line and one expects the range spanned by the local slope of ln s to be directly related to the range of values of the diffusion coefficient. In particular, Dmin (respectively, Dmax) should be related to the minimum (respectively, the maximum) of ∂ ln s/ ∂(t − t0)2. Based on the analysis of a wide range of simulated taylorgrams, for both monomodal and multimodal size distributions, we find that a convenient choice of Dmin, Dmax is

1

with μ=

ND

∑ (sk −

(11)

where cj incorporates the prefactor c and the quadrature coefficients32 required to discretize the integral of eq 3. Equation 10, together with the physical constraint that the sought PDF must be non-negative, Pj ≥ 0 for j = 1, ..., ND, can be solved by standard least-squares fitting.33 Unfortunately, the fit is unstable, reflecting the fact that inversion problems such as that of eq 10 are ill-posed. In other words, the solution {Pj} is not unique; rather, there is an infinite set of vastly different PDFs that yield the same s(t), to within experimental uncertainty. To overcome this difficulty, one may add additional constraints to the fit, in order to reject unphysical solutions and finally converge to a “regularized” (though not unique) solution, a procedure known as Constrained Regularized Linear Inversion method. We shall not discuss the details of CRLI, which can be found, for example, in refs 27 and 28. Here, we simply outline the principles behind CRLI and discuss some practical issues when applying it to TDA, for which we adapt the popular and freely available CONTIN package,29 originally designed to analyze dynamic light scattering data. In CRLI, one constrains the solution using all available a priori knowledge. In our case, in addition to the constraint {Pj ≥ 0}j=1, ..., ND, we expect the PDF to be a smooth function of D, since large changes of the molar concentration with respect to a small change in size are unlike to occur in real samples. This constraint is implemented by penalizing the solutions for which ∂2PD/∂D2 is, on average, too large. In practice, the algorithm

Dmin(max) = 0.1(100) ×

t0R c2 ∂ ln s min(max) 12 ∂(t − t0)2

(13)

The second approach is based on the notion of “equivalent” log-normal PDF of D.26 Regardless of the actual shape of PD(D), one can use ⟨D⟩T and ⟨D⟩Γ to define an equivalent lognormal PDF of D through eqs 8 and 9. We have shown that the equivalent log-normal PDF represents well the range of D values of the sample;26 one can thus use it to guess Dmin and Dmax. One convenient choice validated by analyzing several simulated taylorgram is

D min = exp(μ ∓ 2kσ ) max

(14)

where we typically use k = 2, corresponding to a range of D that covers 99.53% of the area under the log-normal PDF (see the Supporting Information for details). So far, we have discussed how to retrieve PD(D) from the taylorgram. If one instead seeks the PDF of the hydrodynamic radius or of the molar mass, PR(Rh) and PM(M), respectively, the most robust approach is to rewrite eq 3 in terms of the variable of interest (Rh or M) and apply the CRLI method to this new equation. Practical formulas for searching PR(Rh) and 8491

DOI: 10.1021/acs.analchem.5b02053 Anal. Chem. 2015, 87, 8489−8496

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Analytical Chemistry PM(M), including the choice of the corresponding lower and upper bounds, are given in the Supporting Information. Simulated Taylorgrams. In order to test the CRLI method, it is desirable to apply it to taylorgrams for samples whose PD(D) is known. To this end, we simulate taylorgrams using PM(M) determined by SEC for PSS polymers of various average molar mass. The PDF of M is transformed into PD(D) using standard probability transformation laws and assuming a power-law dependence of D on M: 1/3 k T ⎛ 5NA ⎞ ⎟ D= B ⎜ M −1 + a /3 η ⎝ 324π 2K ⎠

Sample injection was performed hydrodynamically on the inlet side of the capillary (0.3 psi, 9 s; ∼1% of the capillary volume). Mobilization pressures of 2 psi were applied with buffer vials at both ends of the capillary. Pressure ramp time was 15 s. The elution time was systematically corrected for the delay in the application of the pressure by subtracting 7.5 s (half-time of the pressure ramp) to the observed (recorded) elution time. The temperature of the capillary cartridge was set at 25 °C.

■

RESULTS AND DISCUSSION Constrained Regularized Linear Inversion Analysis of Simulated Taylorgrams. Figure 1a shows the raising slope of

(15)

where kB is the Boltzmann’s constant, T is the absolute temperature, η is the solvent viscosity, NA is Avogadro’s number, and where K and a are the Mark−Houwink parameters relating to the intrinsic viscosity to M through [η] = KMa. Details on the choice of K and a can be found in ref 26. To simulate multimodal equimass mixtures, we add the PDFs of monomodal samples and normalize the resulting distribution. Once PD(D) is determined, eqs 10 and 11 are used with realistic parameters (Rc = 25 μm, t0 ≈ 77 s, and tk+1 − tk = 0.25 s) to generate the simulated taylorgram, that is, a set of data {tk,sk}k=1,...,Nt. The exact value of t0 is generated randomly; it is stored for later comparison but it is not known at the time the data are analyzed. This mimics TDA in experiments, where the peak time is not known exactly. Finally, for each data point we add a noise term (εk in the r.h.s. of eq 10), drawn randomly from a Gaussian distribution, with zero mean and standard deviation εrms typically in the range 10−4 to 10−2.

■

EXPERIMENTAL SECTION Chemicals and Polymers. Borax (disodium tetraborate decahydrate) was purchased from Prolabo (Paris, France). The water used to prepare all buffers was further purified with a Milli-Q-system from Millipore (Molsheim, France). The borate buffers were directly prepared by dissolving the appropriate amount of borax in water. Standards of poly(styrenesulfonate) (PSS, weight-average molar masses Mw 5.29 × 103, 29 × 103, 333 × 103 g/mol, corresponding to Rh = 1.9, 4.5, and 15.8 nm, respectively) were purchased from Polymer Standards Service (Mainz, Germany). The polydispersity index of the PSS is below 1.2. The degree of sulfonation of the PSS is higher than 90%. All PSS standards were provided with the PDF of M (numerical data, derived from SEC data, were obtained from Polymer Standards Services on simple request). The Mark− Houwink parameters (K, a) of polystyrenesulfonate were determined in 80 mM sodium borate buffer at 25 °C (a = 0.5083, K = 0.11 (c.g.s. units)).26 Taylor Dispersion Analysis. TDA experiments were performed on a PACE MDQ Beckman Coulter (Fullerton, CA) apparatus. Capillaries were prepared from bare silica tubing purchased from Composite Metal Services (Worcester, U.K.). Capillary dimensions were 40 cm (30 cm to the detector) × 50 μm I.D. New capillaries were conditioned with the following flushes: 1 M NaOH for 30 min, 0.1 M NaOH for 30 min, and water for 10 min. Before sample injection, the capillary was filled with the buffer (80 mM borate buffer, pH 9.2, 8.9 × 10−4 Pa·s viscosity). PSS samples were dissolved in the buffer at 0.5 g/L. Between two TDA analyses, the capillary was successively flushed with (i) water (50 psi, 1 min), (ii) 1 M NaOH (50 psi, 2 min), and (iii) buffer (50 psi, 3 min). Solutes were monitored by UV absorbance at a wavelength of 200 nm.

Figure 1. (a) Raising slope of the simulated taylorgram for a PSS5k sample. Here and in (b) the symbols are the data and the line a CRLI fit. (b) Cumulant plot of the same data as in (a). (c) Probability distribution function of D used to simulate the taylorgram (obtained from PM(M) given by SEC) and PD(D) obtained from the CRLI analysis using various guesses for the peak time t0 (in s), as shown by the labels. The best guess obtained from a cumulant-style analysis of the data is t0 = 76.804 s, see text for more details. The actual peak value used to generate the data (unknown at the time the data are analyzed, but stored for further comparison) is t0 = 76.808 s.

the simulated taylorgram using PM(M) determined by SEC and Mark−Houwink coefficients (see simulated taylorgram in the Theory section) for sample PSS5k, with a noise level εrms = 10−4. The same data are shown in a cumulant plot in Figure 1b. We recall that in this representation the taylorgram of a perfectly monodisperse sample appears as a straight line. The data for the PSS5k are close to a straight line, as expected due to the relatively small polydispersity of this sample (polydisersity index below 1.2). In Figure 1a,b, the CRLI fit is shown as a red line: an excellent agreement is observed over the whole range of variation of s(t). The fit was obtained by imposing the peak time t0 = 76.804 s, as obtained from the cumulant fit procedure detailed in ref 26. Without describing the details of the procedure, we simply recall that this procedure was 8492

DOI: 10.1021/acs.analchem.5b02053 Anal. Chem. 2015, 87, 8489−8496

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Analytical Chemistry introduced because the cumulant fit was observed to be very sensitive to the peak time value: errors on t0 smaller than the typical experimental uncertainty led to significant changes in the fit parameters. Motivated by this observation, we investigate whether the CRLI fit is as sensitive to the choice of the peak time as the cumulant one. Figure 1c shows the SEC PDF of D used to simulate the data, together with the result of the CRLI analysis for various choices of t0. We find no significant variation of PD(D) if the error on t0 is smaller than about 0.2 s, which is typically achievable by a simple analysis of the taylorgram shape, for example, by a parabolic fit of the peak summit. For larger deviations, artifacts do appear in the fitted PDF, as exemplified by the bimodal shape of PD(D) for t0 = 76.5 s. When the peak time is correctly input, the CRLI method yields a diffusion coefficient distribution in good agreement with the one used to generate the data, although the CONTIN algorithm slightly overestimates the PDF width (see Figure SI-3 in the Supporting Information and the related text for a short discussion on why the CRLI method tends in general to smooth the PDF). As a way to quantify the deviation of the inverted PD(D) from the SEC one, we compare the Taylor and Γ averages (eqs 4 and 6, respectively) and the polydispersity index PI (eq 7), calculated from the SEC distribution to those obtained using the PDF issued from the CRLI fit. As shown in Table SI-1, the CRLI fit captures extremely well both the Taylor and Γ averages (to within less than 1%) and provides a very good estimate of the PDF (with less than 6% error on the PI for εrms = 10−4). We test the ability of the CRLI to capture complex PDFs by simulating the taylorgrams of equimass binary and ternary mixtures (PSS5k/PSS29k and PSS5k/PSS29k/PSS333k, respectively), with a noise level εrms = 10−4. Figure 2 shows the

diffusion coefficient distributions used to generate the data (SEC, solid thin lines), together with those retrieved using the CRLI method (thick dashed lines). For both bi- and trimodal solutions, we find that the quality of the CRLI fit is excellent, as shown by the small values of the fit residues, s(t) − sFIT(t) (insets of Figure 2a,b). Note that the residues fluctuate randomly around zero, showing no overall trend, which would be the signature of a poor fit. Overall, the probability distribution function of the diffusion coefficient obtained from the CRLI is in reasonably good agreement with the theoretical one; moreover, the fit does correctly capture the bimodal or trimodal nature of the sample. As for the data of Figure 1c, we note that the CONTIN algorithm used to implement the CRLI tends to smooth the PDF. We quantify in Table SI-1 the deviations between the input PDF and that retrieved by CRLI, by inspecting the corresponding values of ⟨D⟩T, ⟨D⟩Γ, and PI. For the bimodal sample, we find a good agreement: all values obtained from the CRLI PDF differ by less than 10% from the theoretical ones. For the trimodal sample, the agreement is excellent for ⟨D⟩T (less than 1% deviation), although it is somehow poorer for ⟨D⟩Γ and PI (18 and 15% deviation, respectively). Influence of the Noise on Constrained Regularized Linear Inversion Analysis of Simulated Taylorgrams. Inversion methods are notoriously sensitive to data noise:27,28 we investigate this issue by running the CONTIN software on taylorgrams generated from the same PDF, to which different amounts of noise are added, in the range 10−4 ≤ εrms ≤ 10−2. For comparison, the typical experimental noise is at most of the order of a few 10−3, as we shall discuss later. Figure 3a,c shows the results for the monomodal PSS5k sample. For all noise levels, the CRLI fits are excellent, as shown by the residues, which exhibit unbiased, random fluctuations around zero (Figure 3a). We furthermore find that the root-mean-square (rms) average of the residues always coincides, to within 5%, with the standard deviation of the noise added to the simulated data. This demonstrates that deviations from the fit are due only to data noise, and not to poor fitting. As also observed in Figures 1 and 2, the PDF obtained running the CONTIN software is less sharp than the one used to generate the data. Moreover, PD(D) becomes increasingly wide as the noise level increases. These changes have small impact on the average value of the diffusion coefficient: ⟨D⟩T and ⟨D⟩Γ differ in all cases by less than 8% from the theoretical value obtained from the noise-free SEC distribution (see Table SI-1). The impact of noise is larger on the PI, where deviations up to a few 10% are observed for εrms ≥ 5 × 10−3. This is consistent with the intuitive notion that the polydispersity index is more sensitive to the details of the probability distribution function of D as compared to the average values ⟨D⟩T and ⟨D⟩Γ. The impact of noise on the analysis run on the bidisperse equimass mixture PSS5k/PSS29k is shown in Figure 3b,d. The fit quality remains excellent: as for the monodisperse case, the residues oscillate randomly around zero with no overall trend (see Figure 3b) and their rms average coincides to within 4% with the imposed noise level, except for the lowest noise level, εrms = 10−4, for which the rms residues exceed εrms by 10%. While the agreement between the SEC and CRLI ⟨D⟩T, ⟨D⟩Γ, and PI values is comparable to that for the bimodal mixture (see Table SI-1), a qualitative change in the shape of PD(D) is observed according to the noise level. Indeed, only for noise levels εrms ≤ 5 × 10−4 is the bimodal nature of PD(D) unambiguously captured by the CONTIN fit. Beyond εrms = 5

Figure 2. Probability distribution function of D used to simulate a taylorgram (obtained from PM(M) given by SEC, solid black line) and issued from a CRLI fit (dashed red line), for (a) an equimass binary mixture of PSS5k and PSS29k and (b) and equimass ternary mixture of PSS5k, PSS29k, and PSS333k. The insets show the CRLI fit residuals on the rising slope of the taylorgram. 8493

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Analytical Chemistry

Figure 3. Influence of data noise on the CRLI fits to simulated taylorgrams, for PSS5k (a) and (c) and an equimass mixture of PSS5k and 29k (b) and (d). (a, b) Fit residues, from top to bottom, the noise level is 10−4, 10−3, and 10−2. For the sake of clarity, the residues have been offset and magnified, as indicated by the dashed lines and the labels. (c, d) Probability distribution function of D used to simulate the taylorgram (obtained from PM(M) given by SEC) and obtained from the CRLI fits, for various levels of noise, as indicated by the labels.

Figure 4. Raising slope of the experimental taylorgram for a PSS5k sample, plotted in a conventional (a) or cumulant-style (b) representation. Open circles: data; red line: CRLI fit. (c) Probability distribution function of D, as measured by SEC (dashed line) and as obtained from a CRLI fit (solid lines), for various choices of the peak time t0 (in s), as shown by the labels. The best guess obtained from a cumulant-style analysis of the data is t0 = 76.522 s, see text for more details. Data acquisition frequency is 4 Hz.

× 10−3, one single broad peak is seen, whose shape does not further change with the noise level. These results illustrate the importance of reducing as much as possible the measurement noise when deconvolving taylorgrams issued from samples with complex particle size distributions. Constrained Regularized Linear Inversion Analysis of Experimental Taylorgrams. We now turn to the CRLI analysis of experimental taylorgrams. Figure 4a shows the raising slope of the experimental taylorgram obtained for a PSS5k sample (open circles), together with the CRLI fit (red line). In this representation, the fit quality appears excellent. However, in a cumulant plot of the same data some deviations are seen for s(t) close to zero (Figure 4b). These deviations are most likely due to data noise close to the baseline; they are representative of the experimental errors that are often (but not systematically) encountered in Taylor dispersion analysis. Figure 4c compares PD(D) measured by SEC to the probability distribution function obtained from the CRLI analysis. As for the simulated data of Figure 1c, we test the impact of the choice of the peak time by varying t0 around the optimum value t0 = 76.522 s, determined using the procedure described in ref 26. We find that the retrieved PDF is stable with respect to variations of t0 of the order of 0.1 s, confirming that the choice of t0 is not a critical parameter in the fitting procedure. For the optimum value t0 = 76.522 s, the PDF obtained from the taylorgram by CRLI is close to that measured by SEC: this demonstrates the ability of the Taylor dispersion analysis to provide detailed information on not only on the average

particle size, but also on the size distribution, to a degree of finesse similar to that of other separation methods, such as SEC. Both ⟨D⟩T and ⟨D⟩Γ calculated using the CRLI and SEC distributions of D agree to within about 10%, which is remarkable given that the PDFs are obtained from two completely different experimental methods. By contrast, the PI associated with the CRLI PDF is about 1 order of magnitude smaller than that obtained from the SEC data, indicating a sharper distribution of sizes, as seen in Figure 4c (compare the purple thick line to the dashed line). One should be cautious in commenting this discrepancy: contrary to the case of the simulated taylorgrams, we do not know the true PD(D) of the sample and it is thus impossible to assess whether the SEC or the CRLI PDF deviate more from the actual distribution. It is well-known that SEC analysis is also much influenced by column peak broadening that may artificially increase the apparent sample polydispersity, especially in the case of polymer standards.35 Indeed, for some polymer standards, much lower polydispersities than estimated by SEC were determined by means of field-flow fractionation36,37 and capillary electrophoresis,38 thanks to their larger selectivities. We present in Figure 5a,b the results of the CRLI analysis on the experimental taylorgrams measured for equimass binary and ternary mixtures. We find that the CRLI fits are of excellent quality, as shown by the residues (insets of Figure 5a and b) that exhibit no systematic deviations from zero. The magnitude of the residues may be used to estimate the typical uncertainty of the experimental data. Note that this is actually an upper 8494

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Supporting Information and the related text for a short discussion on why the CRLI method tends to smooth the PDF). Overall, the data shown in Figure 5 demonstrate that under typical experimental conditions it is possible to retrieve the salient features of the size distribution by inverting the taylorgram by CRLI, even when the various species are relatively close in size. Interestingly, we note that for the experimental data this procedure seems to be more robust than what expected from the simulations, in the sense that the multimodal nature of the PDF is clearly captured even if the experimental noise level is on the order of a few 10−3, a value for which the various peaks of the PDF could not always be resolved in the simulations. Resolution of Constrained Regularized Linear Inversion Analysis of Simulated Taylorgrams from Equimass Bimodal Distributions. To get a better insight in the resolution that may be reached by a CRLI fit, we have performed a systematic investigation of the deconvolution of simulated taylorgrams of equimass bimodal mixtures. For that, taking PD(D) of the PSS5k as a template having D5k as a weight-average diffusion coefficient and σ25k as variance, we build bimodal equimass distributions composed of the sum of two distributions PD1(D) and PD2(D) that differs one each other according to the following characteristics (see Supporting Information for more details):

Figure 5. Probability distribution function of D as obtained from SEC data and from a CRLI fit to the experimental taylorgram, for (a) an equimass binary mixture of PSS5k and PSS29k, and (b) equimass ternary mixture of PSS5k, PSS29k, and PSS333k. The insets show the CRLI fit residuals on the rising slope of the taylorgram. Data acquisition frequency is 4 Hz (a) and 32 Hz (b).

D1 = αD D5k

D2 =β D1

bound, since one implicitly assumes that the fit is “perfect” and that nonzero residues are only due to experimental uncertainty. For the binary mixture, the rms average of the residues is 6 × 10−3, while it is sensibly smaller, 10−3, for the ternary mixture. The residues for the ternary PSS5k/PSS29k/PSS333k mixture oscillate around zero randomly and rapidly, similarly to the case of the simulated taylorgrams (see, e.g., Figures 2a,b or 3a,b), where the noise term was assumed to be temporally uncorrelated. For the binary mixture of Figure 5a, by contrast, the residues exhibit relatively large, slower oscillations, which might be due to small fluctuations of the mobilizing pressure. We conclude that typical experimental uncertainties on the normalized taylorgram are of order 10−3. Moreover, their magnitude is the same regardless of t, that is, the absolute uncertainty rather than the relative one is constant. The PDFs obtained by CRLI are overall close to those measured by SEC. For the binary mixture, the agreement is comparable to that for the simulated data, compare Figure 5a to Figure 2a and see Table SI-1. Again, we emphasize that for the experiments no a priori knowledge of the true PDF of D is available, so that the difference between the SEC and CRLI should be regarded as being representative of the typical variations when probing a sample with different techniques, rather than an indicator of the goodness of either method per se. Furthermore, we recall that the SEC distributions for the mixtures have been built from experimental data on monomodal samples, as explained in Simulated Taylorgrams. Accordingly, they represent the “best” PDFs that may be obtained in principle by SEC, rather than the outcome of an actual measurement on a binary or ternary sample. For the ternary mixture of Figure 5b, we note that the CRLI fit is able to capture the trimodal nature of the sample, although the amplitude of the PSS29k contribution appears to be somehow underestimated (again, see Figure SI-3 in the

and

and

σD2 σD1

D2 = αDβ D5k

(16)

=β (17)

where σD1 and σD2 are the standard deviations of the distributions PD1 (D) and PD2 (D), respectively. For various values of αD in the range 0.1 to 6, β was systematically varied between 2.0 and 2.5 to search for βcrit, the minimum β value for which CONTIN can still distinguish the two modes of the distribution (see the last section of SI and Figure SI-4 for more details). We find that a βcrit value of ∼2.2−2.3 is required for CONTIN to distinguish the two modes for different αD values (0.5, 1.0, 6.0, see Table SI-2). It is interesting to compare this value to the relative width of the individual distributions, σD/D̅ ≈ 0.3: average hydrodynamic radii of the two polymer constituents should differ by at least a factor of about 7× the relative width for CONTIN to be able to resolve the bimodal distribution.

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CONCLUSION In this work, it has been demonstrated that a Constrained Regularized Linear Inversion method can be successfully used for retrieving sample polydispersity from Taylor dispersion data. This approach was applied to the analysis of moderately polydisperse polymer samples and bimodal/trimodal mixtures of these samples. We find that in the CRLI analysis the determination of the exact taylorgram peak time t0 is not a critical requirement, contrary to what was observed for the cumulant approach. No significant variation was observed on the PDF of D if the error on t0 is smaller than about 0.2 s, which can be easily obtained by a simple analysis of the taylorgram. For simulated data, the CRLI method yields a diffusion coefficient distribution in good agreement with the one used to generate the data. This also applies to the case of equimass binary and ternary mixtures, for which the fit does 8495

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(7) Pratt, K. C.; Wakeham, W. A. Proc. R. Soc. London, Ser. A 1974, 336, 393−406. (8) Grushka, E.; Kikta, E. J. J. Phys. Chem. 1974, 78, 2297−2301. (9) Bello, M. S.; Rezzonico, R.; Righetti, P. G. Science 1994, 266, 773−776. (10) Hulse, W. L.; Forbes, R. T. Int. J. Pharm. 2011, 411, 64−68. (11) Ye, F.; Jensen, H.; Larsen, S. W.; Yaghmur, A.; Larsen, C.; Østergaard, J. J. Pharm. Biomed. Anal. 2012, 61, 176−183. (12) Barooah, A.; Chen, S. H. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 2457−2468. (13) Mes, E. P. C.; Kok, W.; Th; Poppe, H.; Tijssen, R. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 593−603. (14) Liu, T.; Oukacine, F.; Collet, H.; Commeyras, A.; Vial, L.; Cottet, H. J. Chromatogr. A 2013, 1273, 111−116. (15) Belongia, B. M.; Baygents, J. C. J. Colloid Interface Sci. 1997, 195, 19−31. (16) d’Orlyé, F.; Varenne, A.; Gareil, P. J. Chromatogr. A 2008, 1204, 226−232. (17) Wuelfing, W. P.; Templeton, A. C.; Hicks, J. F.; Murray, R. W. Anal. Chem. 1999, 71, 4069−4074. (18) Franzen, U.; Vermehren, C.; Jensen, H.; Østergaard, J. Electrophoresis 2011, 32, 738−748. (19) Oukacine, F.; Bernard, S.; Bobe, I.; Cottet, H. J. Controlled Release 2014, 196, 139−145. (20) Jensen, H.; Østergaard, J. J. Am. Chem. Soc. 2010, 132, 4070. (21) Bielejewska, A.; Bylina, A.; Duszczyk, K.; Fialkowski, M.; Holyst, R. Anal. Chem. 2010, 82, 5463−5469. (22) Cottet, H.; Biron, J.-P.; Martin, M. Anal. Chem. 2007, 79, 9066− 9073. (23) Cottet, H.; Biron, J. P.; Cipelletti, L.; Matmour, R.; Martin, M. Anal. Chem. 2010, 82, 1793−1802. (24) Price, W. E. J. Chem. Soc., Faraday Trans. 1 1988, 84, 2431− 2439. (25) Kelly, B.; Leaist, D. G. Phys. Chem. Chem. Phys. 2004, 6, 5523− 5530. (26) Cipelletti, L.; Biron, J.-P.; Martin, M.; Cottet, H. Anal. Chem. 2014, 86, 6471−6478. (27) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 213−227. (28) Provencher, S. Comput. Phys. Commun. 1982, 27, 229−242. (29) CONTIN package, available at http://s-provencher.com/pub/ contin/. (30) Chamieh, J.; Cottet, H. J. Chromatogr. A 2012, 1241, 123−127. (31) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976. (32) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes: The Art of Scientific Computing, 3rd ed.; Cambridge University Press, Cambridge, U.K., 2007. (33) See, e.g., the routine scipy.optimize.nnls of the Python-based, open-source software SciPy at www.scipy.org. (34) For simplicity and with no loss of generality, in writing eq 12, we assume that the Dj are equally spaced and we approximate the second derivative by a central finite difference to second order accuracy. (35) Popovici, S. T.; Kok, W. T.; Schoenmakers, P. J. J. Chromatogr A 2004, 1060, 237−252. (36) Schimpf, M. E.; Myers, M. N.; Giddings, J. C. J. Appl. Polym. Sci. 1987, 33, 117−135. (37) Reschiglian, P.; Martin, M.; Contado, C.; Dondi, F. J. Liq. Chromatogr. Relat. Technol. 1997, 20, 2723−2739. (38) Chamieh, J.; Martin, M.; Cottet, H. Anal. Chem. 2015, 87, 1050−1057.

correctly capture the bimodal or trimodal nature of the sample for the data noise levels up to εrms = 10−3, a level comparable to that of experimental data. For both experimental and simulated data, the CRLI fit was generally of excellent quality, with residues exhibiting unbiased random fluctuations around zero. The Taylor and Gamma diffusion coefficient averages were correctly captured by the CRLI fits, generally to within 10% of their expected value. Deviations on the PI were somehow larger up to a few 10%, especially for the larger noise levels. Finally, the resolutive power of the CRLI fit was shown to be preserved in the case of an equimass bimodal mixture as long as the ratio of the average hydrodynamic radii of the two polymer constituents exceeds by about 7× the relative width of the individual distributions. Given the simplicity of implementation of TDA and its wide applicability, we believe that the CRLI approach will establish itself as a powerful, routine tool for the characterization of a large variety of solutions and suspensions, with constituents in the size range from a few angstroms to a fraction of micron.

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ASSOCIATED CONTENT

S Supporting Information *

Table SI-1: Taylor and Gamma averages of D and polydispersity index PI calculated from the PDF of D obtained from SEC and by inversion of the simulated and experimental taylorgrams; the derivation of the general expression of the Taylorgram signal and its particularization as a function of the probability distribution of the diffusion coefficient D, hydrodynamic radius Rh, and molar mass M; the generalization of the CRLI method when one wishes to retrieve the PDF of Rh or M, rather than that of D (Figure SI-1); the empirical determination of convenient bound values for the sought PDF, for Rh, M, or D; a synthetic description of how the regularization parameter α in eq 12 of the main manuscript is chosen by the CONTIN algorithm (Figures SI-2 and SI-3); estimation of the resolution of Constrained Regularized Linear Inversion analysis of simulated taylorgrams from equimass bimodal distributions (Figure SI-4 and Table SI-2). The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.analchem.5b02053.

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AUTHOR INFORMATION

Corresponding Authors

*Tel.: +33 4 6714 3589. Fax: +33 4 67 14 34 98. E-mail: luca. [email protected]. *Tel.: +33 4 6714 3427. Fax: +33 4 6763 1046. E-mail: herve. [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS H.C. gratefully acknowledges the support from the Institut Universitaire de France (2011−2016).

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REFERENCES

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DOI: 10.1021/acs.analchem.5b02053 Anal. Chem. 2015, 87, 8489−8496