Turbidimetric Detection Method in Flow-Assisted Separation of

Characterization of dispersed samples is an outstanding trend in analytical science. Among flow-assisted separation techniques for dispersed samples, ...
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Anal. Chem. 2003, 75, 6469-6477

Turbidimetric Detection Method in Flow-Assisted Separation of Dispersed Samples Andrea Zattoni,† Elena Loli Piccolomini,‡ Giancarlo Torsi,† and Pierluigi Reschiglian*,†

Department of Chemistry “G. Ciamician”, University of Bologna, Via Selmi 2, I-40126 Bologna, Italy, and Department of Mathematics, University of Bologna, Piazza di Porta S. Donato 5, I-40127 Bologna, Italy

Characterization of dispersed samples is an outstanding trend in analytical science. Among flow-assisted separation techniques for dispersed samples, size exclusion chromatography, hydrodynamic chromatography, and field-flow fractionation are the most widely applied. With dispersed analytes separated by these techniques, the UV/ vis spectrophotometric detectors work as turbidimeters. To directly convert the analytical signal for quantitative analysis, the extinction properties of the dispersed analyte must be known. A new method is proposed to experimentally obtainsby single-run, flow-assisted separation with UV/vis diode-array detectorssthe mass-size (or number-size) distribution function of the analytes when a retention-to-size relationship is either theoretically or empirically available for the chosen separation technique. This approach needs neither standards nor reliance on a method to predict the optical properties of the analytes. Theory and original algorithms are presented. Algorithms are then tested to optimize the numerical routines. Accuracy and robustness of the method are evaluated by simulation, and limitations for the application to experimental data are described. Finally, first application to field-flow fractionation shows validity of the method when applied to a few real cases. Among methods for the analysis and characterization of dispersed samples, flow-assisted techniques such as size exclusion chromatography, hydrodynamic chromatography, and field-flow fractionation (FFF) are suitable to separate these samples on the basis of differences in physical indexes of the analytes. The detectors most often coupled to these techniques still are the UV/ vis spectrophotometers which, for dispersed analytes, work as turbidimeters. To obtain quantitative response, the relationship between the turbidimetric signal and analyte mass must be theoretically or empirically worked out. A systematic approach to the conversion of the turbidimetric signal into mass of dispersed particles separated by flow-assisted techniques has been initiated in previous work.1,2 Therein we showed that an expression similar to the Beer-Lambert (B-L) law holds true for UV/vis, flow* Corresponding author. Fax: +39 (0)51 209 9456. E-mail: resky@ ciam.unibo.it. † Department of Chemistry “G. Ciamician”. ‡ Department of Mathematics. (1) Reschiglian, P.; Zattoni, A.; Melucci, D.; Locatelli, C.; Torsi, G. In Recent Research Developments in Applied Spectroscopy; Pandalai, S. G., Ed.; Research Signpost: Trivandrum, India, 2000; Vol. 3, pp 61-80. 10.1021/ac034729c CCC: $25.00 Published on Web 11/01/2003

© 2003 American Chemical Society

through turbidity measurements.3 Through the use of this B-Llike expression, flow-through, quantitative analysis of dispersed analytes can be thus performed once the spectroscopic constant (i.e. the analyte extinction coefficient) is obtained. We then developed previous methods to obtain the conversion of the FFF analytical response (the fractogram) into sample particle mass(or number-) size distribution (what we called the particle size and sample amount distribution, PSAD)4,5 through evaluation of the analyte optical properties either by experimental measurements on standard particles (calibration)3,4 or by prediction with a model.5,6 Because the extinction properties of dispersed samples are size-dependent, calibration-based methods should require monodispersed standards, the availability of which is often limited. The extinction coefficient of dispersed samples can generally be predicted by applying either the Mie scattering theory, which involves methods with highly demanding numerical complexity or other simpler, approximate approaches that can only be applied within limited domains of sample features.7 However, any modelbased prediction of the optical properties of dispersed analytes requires either the exact knowledge of sample specifications (i.e. size, shape, refractive index, density) or restriction of the range of experimental and instrumental conditions within which the optical properties can be considered relatively constant.6 For complex samples the above requisites are rarely met. Because of the above limitations of our previously developed methods, we present here a totally new approach for quantitative analysis of dispersed samples (from nanometer-sized to micrometersized analytes) through flow-assisted separation techniques with UV/vis turbidimetric detection. The new method experimentally obtains the optical properties of the analyte without requiring standards for calibration or a model that needs knowledge of sample specifications, and it then allows to independently obtain the PSAD of the analyte from a single separation experiment. The method is based on the fundamental property of the extinction efficiency to be a function of the ratio between the diameter of (2) Reschiglian, P.; Zattoni, A.; Torsi, G.; Melucci, D. Rev. Anal. Chem. 2001, 3, 239-269. (3) Reschiglian, P.; Melucci, D.; Torsi, G. Chromatographia 1997, 44, 172178. (4) Reschiglian, P.; Melucci, D.; Zattoni, A.; Torsi G. J. Microcolumn Sep. 1997, 9, 545-556. (5) Reschiglian, P.; Melucci, D.; Torsi, G.; Zattoni, A. Chromatographia 2000, 51, 87-94. (6) Zattoni, A.; Melucci, D.; Torsi, G.; Reschiglian, P. J. Chromatogr. Sci. 2000, 38, 122-128. (7) van de Hulst, H. C. Light Scattering by Small Particles; Dover Publications: New York, 1981.

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dispersed, spherical particles and the incident wavelength, when the particle relative refractive index (i.e. the ratio between particle refractive index and refractive index of the dispersing medium) is constant.7 The use of a UV/vis diode-array detector (UV/vis DAD) is thus the first requisite since the method requires to register turbidity as a function of the incident wavelength. It must be noted that the assumption of constant particle relative refractive index is respected only if absorption is independent of the incident wavelength. The method cannot be thus applied within wavelength ranges in which specific absorption is present. If particles are not spherical, the diameter of a sphere of equivalent cross section can be used as an average value since nonspherical particles are known to behave as though their average cross section were onefourth of their surface area.8 From the fundamental property of the extinction efficiency upon which the method is based, it also derives the second requisite to apply the method: a relationship between retention and size of the dispersed analyte separated by the chosen flow-assisted technique. The need of a retention-tosize relationship could be considered the most critical point of the method. For instance, a retention-to-size relationship for nonspherical particles can be difficult to define in flow-assisted separation techniques. Moreover, with these separation techniques some complex mechanisms affecting the analyte migration along the separation device can come into play like, for instance, interaction of the migrating analytes with the separation device. If not available from first principles, a retention-to-size correlation can be, in most cases, empirically obtained with standards or by means of orthogonal methods. A systematic analysis of the possible approaches (and relevant accuracy issues) for the conversion of retention to size in flow-assisted separation techniques for dispersed samples lies, however, outside the scope of this work. The final problem in independently obtaining the extinction efficiency and mass-size distribution functions is solved by solving a nonlinear equation system. Theory and original algorithms developed for the numerical treatment from data handling to the nonlinear equation system solution are presented. Tests on simulated data are then used as fundamental tools to check the efficiency of the numerical treatment, as well as its precision and robustness in the presence of noise. The good results obtained by simulating realistic, noisy data finally made it possible to test the method in a few experimental cases. Method limitations to be taken into account for the very first application to experimental data are discussed, and within these limitations few particulate samples of different particle sizes and optical properties were considered. Samples were either narrowly dispersed particles like nanometer-sized, polystyrene (PS) latex beads, broadly dispersed, micrometer-sized particles like chromatographic silica beads, or complex particulate matter of biological origin like human red blood cells (HRBCs). Flow-assisted, size separation of these samples was obtained with two different FFF systems: gravitational FFF (GrFFF); hollow-fiber flow FFF (HF FlFFF). Comparison of the optical properties and PSAD analysis of samples obtained with the presented method and by either simulation, modeling, or uncorrelated methods of size analysis provided generally good results. (8) Cauchy, A. Complete Works of Augustin Cauchy; Gauthier-Villars: Paris, 1908; 1st Series, Vol. II, pp 167-477 (in French).

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THEORY The theoretical derivation of the new method is based on the following development of the theory upon which our previous methods for quantitative analysis of dispersed samples by flowthrough turbidimetric measurements with UV/vis spectrophotometric detectors were based (see refs 1 and 2 and references therein). The turbidity of the sample τ (cm-1) is an experimental quantity that can be obtained from the analytical signal of the UV/ vis detector that operates as a turbidimeter:

τ)

()

P0 1 ln b P

(1)

Here P0 and P (erg s-1 cm-2) are the radiant powers exiting the detector cell when respectively the blank and the sample fill it and b (cm) is the optical cell path length. In previous work we have shown that

τ ) ln(10) Kc

(2)

where c (g cm-3) is the sample mass concentration and K (cm2 g-1) the sample extinction coefficient, which we proved to be independent of c in a broad range. In a sufficiently dilute sample of spherical particles turbidity can be expressed as a function of particle diameter d (cm), number concentration of the particles N (particles cm-3), and dimensionless quantity Q (-), the extinction efficiency7

π τ ) d 2NQ 4

(3)

A fundamental property of Q states that, at constant particle shape, Q is a function of the type Q(x, m), where x is the size parameter defined as x ) πd/λ, in which λ (cm) is the incident wavelength in the dispersing medium, which is λ ) λ0/n, where λ0 (cm) is the wavelength in the vacuum and n (-) the refractive index of the dispersing medium.7 The parameter m (-) is the particle relative refractive index, defined as m ) np/n, where np (-) is the particle refractive index. At constant m, Q ) Q(x). Then, by combining eqs 2 and 3, one gets

K)

3Q(x) 2 ln(10) Rd

(4)

where R (g cm-3) is the particle density. When in flow-assisted, size separation of dispersed samples it is assumed that the retention time axis can be converted into size/ molecular weight values of the sample via a model or an empirical relationship, the particle size distribution (PSD) of the sample can be obtained by transformation of the analytical signal recorded as a function of time, as shown in a previous report on FFF of nanometer-sized latex beads.9 If the sample extinction coefficient (K) can be obtained from the extinction efficiency Q(x) (eq 4), also the particle mass-size distribution (what we called particle size-amount distribution, PSAD4,5) can be obtained as follows. (9) Blanda, M.; Reschiglian, P.; Dondi, F.; Beckett, R. Polym. Int. 1994, 33, 61-69.

The expression of turbidity in eq 2 can be rewritten as a function of the retention volume Vr (cm3) and of the mass m (g) of eluting particles:

∂m τ ) ln(10) K ∂Vr(d)

(5)

The mass-size distribution f(d) (g cm-1) is by definition

f(d) )

∂m ∂d

(6)

It must be pointed out, however, that since f(d) is not a normalized function, the mass-size distribution (PSAD) obtained from a flowassisted separation experiment actually is f(d, m0):

∫ f(d) dd ) m ∞

0

0

(7)

Here m0 (g) is the total mass of the eluted sample. Equation 6 can be written as

f(d) )

∂m ∂m ∂Vr(d) ) ∂d ∂Vr(d) ∂d

(8)

By combining eqs 5 and 8, we get

f(d) )

∂Vr(d) τ ln(10) K ∂d

(9)

By substituting eq 4 in eq 9, we finally obtain

f(d) )

2 τRd ∂Vr(d) 3 Q(x) ∂d

(10)

It must be pointed out that the above expression is valid in the case of constant relative refractive index (m), that is for particles which neither absorb light at a specific wavelength nor show different absorbance values at different values of incident wavelength. In the case of UV/vis DAD measurements with a flow-assisted separation technique, τ can be recorded as a function of Vr in addition to λ. If we assume that retention volume values can be converted into particle size values, we have τ ) τ(d, λ). Hence

f(d)Q(x) ) f(d)Q

(πdλ ) ) y(d, λ)

∂Vr(d) 2 y(d, λ) ) τ(d, λ)Rd 3 ∂d

(11)

(12)

Equation 11 indicates that the y(d, λ) values are the experimental, input values from which the values for f(d) and Q(x) can be computed. Equation 12 shows that the y(d, λ) values can be obtained, once particle density (R) is known and constant, by first measuring the turbidity values τ(Vr, λ). Conversion from retention to size then gives τ(d(Vr), λ) and ∂Vr(d)/∂d. One must finally point

Table 1. Nominal Specifications of the Silica and PS Samples sample

d (nm)

sa (nm)

silica PS 50 nm PS 100 nm PS 150 nm PS 200 nm

50 102 155 204

2 3 4 6

a

d10 (µm)

d50 (µm)

d90 (µm)

density (g cm-3)

3.7

5.0

6.8

2.3 1.05 1.05 1.05 1.05

s ) standard deviation.

out that, because the mass-size distribution of the size-separated sample is not a normalized frequency function (see eq 7), y(d, λ) is also a function of the eluted sample mass (m0). Details on the numerical procedure to handle the experimental values of τ(Vr, λ) and to solve the system in eq 11 are given below in the Computational Section. EXPERIMENTAL SECTION FFF Systems. The instrumental system used for the FFF experiments was a modified apparatus for liquid chromatography, in which the GrFFF or HF FlFFF channel replaced the column. The channels were built and controlled as described elsewhere.10-12 In all cases, the channel outlet was connected to a Spectra System UV6000LP photodiode array UV/vis spectrophotometer (Thermo Finnigan, Austin, TX) operating within a range of wavelengths from 250 to 800 nm. Detection parameters and signal acquisition were controlled by the ChromQuest Chromatography Data System software (version 2.51, Thermo Finnigan). Samples. Silica particles were porous, spherical silica for HPLC column packing (LiChrospher Si-60 5 µm, Merck, Darmstad, Germany). PS latex beads were nanosphere size standards (Duke Scientific Co., Palo Alto, CA). Sample size and density specifications are given in Table 1. Fresh human blood was drawn from donors under clinical control. Samples were collected in tubes containing EDTA to inhibit coagulation. Certified determination of average HRBC concentrations were obtained for all blood samples by standard clinical analysis methods. COMPUTATIONAL SECTION The algorithm to compute the good, discrete solution of the system was numerically implemented using Matlab 6.0 (The Mathworks, Natick, MA), with the Optimization Toolbox 2.0 (The Mathworks) and applied to either simulated or experimental data. The routines ran on a PC Pentium III 1 GHz with 512 MB of RAM. Pseudocode and detailed description of the applied numerical methods are separately reported as Supporting Information. In practice, from flow-assisted separation of the dispersed sample with UV/vis DAD, once retention time values are converted to particle size values, we handle the discrete turbidity values τij ) τ(di, λj) obtained with the diameter d spanning the interval [dmin, dmax] and the wavelength λ spanning the interval [λmin, λmax]. (10) Reschiglian, P.; Zattoni, A.; Roda, B.; Casolari, S.; Moon, M. H.; Lee, J.; Jung, J.; Rodmalm, K.; Cenacchi, G. Anal. Chem. 2002, 74, 4895-4904. (11) Reschiglian, P.; Roda, B.; Zattoni, A.; Min, B. R.; Moon, M. H. J. Sep. Sci. 2002, 25, 490-498. (12) Reschiglian, P.; Zattoni, A.; Roda, B.; Cinque, L.; Melucci, D.; Min, B. R.; Moon, M. H. J. Chromatogr. A 2003, 985, 519-529.

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From eq 12 we then compute the values yij ) y(di, λj). By discretizing eq 11 in the points di, i ) 1, ..., m, and λj, j ) 1, ..., n, we obtain

f(di)Q

( )

πdi ) y(di, λj) λj

(13)

From this relationship we propose an efficient algorithm to compute the good, discrete approximations of the functions f and Q. The developed algorithm is split into two parts. In the first part, we give a suitable formulation for the nonlinear system (eq 13). By calling ξij ) πdi/λj, i ) 1, ..., m, and j ) 1, ..., n, the function Q can be approximated at the points ξij. Then, each equation of the system (eq 13) can be written as hfiqjij ) yij. With this formulation, the system can have no solution because the number of unknowns (m + (n × m)) is greater than the number of equations (n × m). Then the function Q must be approximated on a reduced set of equally spaced points xk, k ) 1, ..., r (r e n), in the interval [xmin, xmax], where xmin ) minij(ξij) and xmax ) maxij(ξij). The right-hand side values yˆik, i ) 1, ..., m, and k ) 1, ..., r, must be computed on a new grid (di, xk), i ) 1, ..., m, and k ) 1, ..., r, through interpolation of the initial values yij, i ) 1, ..., m, and j ) 1, ..., n. Cubic Hermite or cubic spline functions are used for the interpolation because they guarantee good approximation even for a large data set and in the presence of noise. In the second part of the algorithm, the nonlinear equations system, with m + r unknowns and m × r equations, which is expressed as

hf iqjk ) yˆik i ) 1, ..., m; k ) 1, ..., r

(14)

is solved as a nonlinear least-squares problem. The solution components (fh1, ..., hfm; qj1, ... qjr) are the approximating values of the function f at the points d1, ..., dm and of the function Q at the points x1, ..., xr, except for a constant factor C. In fact, if hfi and qjk are solutions of eq 14, then Cfhi and (1/C)qjk are solutions as well. The constant C is determined as the ratio between the value m0 of the total eluted sample mass (see eq 7) and the value m1 of a numerical integration on hf1, ..., hfm. Application of the method thus requires that, to compute C, the analyte mass (m0) after the separation-detection process is known. The algorithm output values ˜f1, ..., ˜fm and ˜q1, ..., ˜qr are eventually obtained as C(fh1, ..., hfm) and (qj1, ..., qjr)/C, respectively. RESULTS AND DISCUSSION 1. Tests by Simulation. To fully understand how simulations can test our method, it is necessary to first describe how simulated data is generated. The exact solutions of the method are generally represented by analytical expressions of functions f(d) and Q(x). For simulations, these expressions are assumed as known and used to build the simulated yij data. To reproduce as close as possible an experimental case, noise was also added to the simulated yij values, so that the ˜yij values were obtained. The numerical routine that implements the method took the simulated yij (or, in case of added noise, ˜yij) values as input values and computed the output values ˜f1, ..., ˜fm and ˜q1, ..., ˜qr, which approximate f1 ) f(d1), ..., fm ) f(dm) and q1 ) Q(x1), ..., qr ) Q(xr). Since the exact, analytical expressions of both f and Q were 6472

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assumed to be known, it was eventually possible to evaluate the error introduced by the numerical method, that is the error values between the output ˜f1, ..., ˜fm and ˜q1, ..., ˜qr and the input, exactly known values for f and Q. For this evaluation the graphical outputs and the dependence of the percent coefficient of variation (% CV) on different parameters were analyzed. % CV on f and Q are respectively defined as

x

m

∑(f - ˜f )

% CVf )

i

i

i)1

2

100

m-1

〈fi〉

and

x

r

∑(q

% CVQ )

k

- ˜qk)2

k)1

r-1

100 〈qk〉

where 〈fi〉 and 〈qk〉 are the average values for f1, ..., fm and q1, ..., qk, respectively. It is evident that, through this approach, we can only estimate the approximation degree introduced by the numerical methods, as well as their efficiency, stability, and precision for further application to real cases. The more representative functions f and Q are of real cases, and the more realistic the added noise, the more reliable the results of such tests. Otherwise, the broader the exploration of f, Q, and noise characteristics, the better the evaluation of the efficiency of the numerical algorithm for the application to real cases. 1.1. Numerical Optimization and Accuracy Tests. The first testing step consisted of some executions on test problems. This first step aimed at establishing the optimal values of some fundamental parameters and the relevant tolerances required by the method. This made it possible to use such optimal parameter values for further data processing. Specifically, we evaluated (a) the best interpolating functions for the data, chosen between cubic Hermite and cubic spline functions, and (b) the best method for solving the nonlinear system, chosen between the trust region (TR) and the Levenberg-Marquardt (LM) method. Test problems were generated as separately described in Supporting Information. Detailed numerical results are also therein reported. It generally appears that cubic spline interpolation is preferable and that the TR method is more efficient for solving the nonlinear system. As a consequence, for all further tests on simulated data and for all the runs on experimental data described in the next sections, the cubic spline interpolation and the TR method were employed. The data for the first test problem were then added with white noise or with noise intensity proportional to the values of the simulated functions at each point. Results separately reported as Supporting Information show that the method is robust with respect to the presence of noise in the processed data. This is a key feature for the application to experimental data. Further tests were performed to study how the method reacts to the increased problem dimension, that is by increasing input data points. Results separately reported as Supporting Information show that the increase in input data points produces better results, especially

on noised data, at the expense of computational time, which in fact increased quite rapidly. This finding suggests that when the method is applied to experimental data, the precision value for solving the system through the TR method must be properly chosen with respect to the required accuracy and computational time. 1.2. Simulated PSAD Analysis. Realistic simulations can indicate benefits and limits of the method. Although the method can, in principle, be applied to dispersed analytes of any size and shape, we restrict our simulations to expressions for f and Q that can be representative of the real cases of micrometer- and nanometer-sized, spherical particles that have been size-separated by FFF for the experimental validation of the method. Two different expressions for f were considered: the Gaussian and the log-normal distribution. These expressions are among the most frequent size distribution profiles of particulate samples.13 In the first case, bimodal distributions for micrometer- or nanometer-sized particles were considered to check the feasibility of the method to the analysis of multimodal size distribution samples. In the second case, monomodal log-normal size distribution functions of nanometer- or micrometer-sized particles were considered. For the micrometer-sized particle distributions, the Q functions were simulated using the van de Hulst-Walstra (vdH-W) model since this model best describes the UV/vis turbidimetric behavior of micrometer-sized particles.2,6 The vdH-W equations for Q are reported in Supporting Information (eq 1a-e). In the cases of nanometer-sized particle distributions, the Q functions were simulated using the expression derived from Mie theory,14 as reported in the Supporting Information. Final expressions for the functions f and Q results in terms of % CV, and graphical outputs are given in the Supporting Information. Figure 1a,b shows the solution for f and Q for a simulated sample of micrometer-sized silica particles of monomodal, Gaussian size distribution. In this case Q was calculated for a value of the detector acceptance angle θ ) 0.08 rad. The acceptance angle is defined as the sum of half the angle of divergence and convergence of the incident beam and half the angle subtended by the detector. In previous work we have described how to directly or indirectly measure the acceptance angle of UV/vis spectrophotometric detectors.6 The value here chosen (θ ) 0.08 rad) lies within the range for common UV/vis detectors.2,6 For the simulation of Q it was also taken into account the correction term we have elsewhere employed for low-refractive particles such as porous silica beads (m ) 1.04).5 The f function was simulated as it had been obtained from the conversion of a fractogram of a simulated FFF experiment in steric/hyperlayer elution mode, in which di values are inversely proportional to Vr,i.15 Proportional noise was added. In Figure 2a,b it is shown the simulated case of a mixture of four nanometer-sized PS latex beads. The size distribution profile of each PS sample in the mixture was assumed to be Gaussian. The four size distribution functions were respectively centered in d ) 50, 100, 150, and 200 nm, with variance equal to (13) Allen, T. Particle Size Measurement, Vol. 1-Powder sampling and particle size measurement, 5th ed.; Chapman & Hall: London, U.K., 1997. (14) van de Hulst, H. C. Light Scattering by Small Particles; Dover Publications: New York, 1981; Chapter 9. (15) Caldwell, K. D. In Field-Flow Fractionation Handbook; Schimpf, M. E., Caldwell, K. D., Giddings, J. C., Eds.; Wiley-Interscience: New York, 2000; Chapter 5.

Figure 1. Simulation for micrometer-sized, spherical silica. Test parameters: f ) Gaussian; Q ) vdH-W model with correction for low-refracting particles; µ ) 5.0 µm; d ) 3-7 µm; x ) 40-100; m × n ) 80 × 80. Proportional noise is equal to 4% of the signal intensity. % CV on input data: 2.28%. Key: (a) solution (×) for the simulated f function (s), % CVf ) 0.58%; (b) solution (×) for the simulated Q function (s), % CVQ ) 0.61%.

50 nm2 and maximum height equal to 0.01. The expression for such a simulated size distribution f then becomes 4

f(d) ) 0.01

∑e

-(d-50i)2/50

i)1

The f function was simulated as it had been obtained from the conversion of a fractogram of a simulated FFF experiment in normal mode,16 in which di values are directly proportional to Vr,i. The function Q was simulated using the expression derived from Mie theory.14 Proportional noise was added. All the simulated cases prove that the method is able to accurately rebuild the f and Q functions used to generate the simulated input values and, when noise is added, the error values between output and input values (% CVf, % CVQ) are comparable to the level of noise introduced on input data (% CV on input data). The latter finding made it possible the application to experimental data without specific signal denoising. Incidentally, according to good laboratory practice rules, this is key feature for possible, further implementation of the new method to routine analysis. (16) Schure, M. R.; Schmipf, M. E.; Schettler, P. D In Field-Flow Fractionation Handbook; Schimpf, M. E., Caldwell, K. D., Giddings, J. C., Eds.; WileyInterscience: New York, 2000; Chapter 2.

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Figure 2. Simulation for multimodal, nanometer-sized PS latex beads. Test parameters: f ) sum of Gaussian functions; µ ) 50, 100, 150, 200 nm; σ ) 10 nm; Q ) Mie model; m × n ) 160 × 80. Proportional noise is equal to 4% of the signal intensity. % CV on input data: 3.47%. Key: (a) solution (×) for the simulated f function (s), % CVf ) 5.78%; (b) solution (×) for the simulated Q function (s), % CVQ ) 5.07%.

However, if specific data smoothing is required, specific noise filtering on input data can be independently performed without affecting the numerical method performance. 2. FFF-UV/Vis PSAD of Real Samples. When the method is applied to experimental data, five assumptions are required. First, the sample polydispersity should be sufficiently broader than the apparent polydispersity due to sample broadening in the fractionation instrument. Second, the sample polydispersity present in the UV/vis DAD cell must be assumed to be sufficiently low to give accurate, instantaneous estimates of d and Q at each Vr. Third, if sizing is performed through measurements taken throughout the fractionation process, the accuracy of the method chosen for the conversion from retention to size determines the accuracy of the calculated f and Q functions. In other words, the method was derived by assuming an exact evaluation of sample size at each retention time value. It must be noted that the three above assumptions concern any flow-assisted technique in which separation itself causes band broadening and for which sizing is obtained from conversion of the retention volume scale to the sample size scale by means of a model orsfor most casess empirical calibration. Fourth, it is assumed that at each Vr must correspond a single Q value: if two analytes coelute and they have different Q range values, the method fails. Fifth, there must exist a range of incident wavelength values at which the sample does 6474 Analytical Chemistry, Vol. 75, No. 23, December 1, 2003

Figure 3. GrFFF-UV/vis DAD of micrometer-sized, spherical silica. Mobile phase: 0.05% w/v SDS; 0.01% w/v NaN3. Injected sample mass: 20 µg. Injection time: 20 s. Injection flow rate: 0.2 mL/min. Stop-flow time: 3 min. Elution flow rate: 1.0 mL/min. Key: (a) output f function; (b) output Q function.

not absorb the incident radiation or in which the absorption is independent of the incident wavelength. 2.1. GrFFF of Micrometer-Sized Silica. Figure 3a,b shows the solution for f and Q obtained after processing a real GrFFFUV/vis DAD fractogram of silica beads used for chromatographic support. The GrFFF/PSAD analysis of the same sample had been obtained5 using our previous method for PSAD in FFF. The previous method nonetheless required a model to describe the optical behavior of the particulate sample, while the new method requires only the same approach therein used to convert retention to particle size, which is based on the evaluation of the mean particle elevation during GrFFF elution. The required value of m0 (see Computational Section) was obtained by evaluation of sample recovery. Sample recovery was determined by off-channel injection of the same sample amount injected to obtain the fractogram, as the ratio between in-channel and off-channel peak areas. Table 2 reports the percentile values obtained by the integration of f in Figure 3a, the corresponding values obtained by means of our previous method, and the nominal values given by the manufacturer, obtained by laser light diffraction. If we compare the values, good agreement is generally found. Comparison between the Q function profile obtained here (Figure 3b) and the experimental values for Q obtained by calibration in the previous study (Figure 2 of ref 5) indicates relatively good agreement if the different value of the detector acceptance angles is taken into account.

Table 2. Sizing Results for Silica Particles: Comparison between Different Methods size distribution percentiles

nominal previous worka this work a

d10 (µm)

d50 (µm)

d90 (µm)

3.7 3.3 4.1

5.0 5.0 4.9

6.8 6.4 5.9

Reference 5.

Figure 4. HF FlFFF-UV/vis DAD of nanometer-sized, spherical PS mixture. Mobile phase: 0.01% v/v FL-70; 0.02% w/v NaN3. Inlet flow rate: 1.52 mL/min. Radial flow rate: 0.085 mL/min. Key: (a) output f function (s), raw fractogram, λ ) 254 nm (---); (b) output Q function.

2.2. HF FlFFF of Nanometer-Sized PS Latex Bead Mixtures. Figure 4 reports the solution for f (Figure 4a, full line) and Q (Figure 4b) obtained after processing a real HF FlFFFUV/vis DAD fractogram of a mixture of nanometer-sized PS latex beads (Figure 4a, dashed line). Since the sample showed specific absorption in the UV region, only data collected in the nonabsorbing region were processed (λmin ) 350 nm; λmax ) 800 nm). Conversion between retention and particle size was obtained from the evaluation of particle diffusion coefficient by applying the theoretical expression derived by other authors for the retention in HF FlFFF (eq 2 of ref 17). The required value of m0 (see (17) van Bruijnsvoort, M.; Kok, W. Th.; Tijssen, R. Anal. Chem. 2001, 73, 47364742.

Computational Section) was obtained by the evaluation of sample recovery as described in section 2.1. Good agreement is found between the diameter values obtained from the calculated f (Figure 4a, full line) and the nominal diameters given by the manufacturer and obtained by laser light scattering (Table 1). Good agreement with the nominal specifications is also found for the f obtained for the simulated case of the same mixture of nanometer-sized PS latex beads discussed in section 1.3. However, one may argue that, for narrowly dispersed samples such as the nanometer-sized PS latex beads analyzed here, the polydispersity contribution to the total band broadening could be comparable to the instrumental band broadening (the so-called apparent polydispersity18). Figure 4a indicates, in fact, that the total band broadening for each peak (∼10-13 nm in standard deviation) is mainly caused by band broadening in the fractionation instrument, since nominal values of size polydispersity of the analytes (Table 1) are lower (3-4 nm in standard deviation). This finding does not match the first assumption above-described for the application of the new method to real cases, and the method in fact gives incorrect results for such narrow-dispersed populations if the apparent polydispersity contribution is not deconvolved from the fractogram before application of the method to obtain f and Q. The main value of the case discussed in this section concerns, however, the observed good agreement between the simulated Q (Figure 2b), the Q function obtained by the new method from the experimental fractogram (Figure 4b), and the Q profile derived from Mie theory (Figure 2b). This is particularly noteworthy since it experimentally confirms that the new method can be effectively applied to nanometer-sized other than micrometer-sized analytes. It must in fact be recalled that in UV/vis turbidity detection the only approach to date possible for the detector signal conversion to the mass of nanometer-sized beads has been optical extinction property evaluation on the basis of either calibrations or models derived by the Mie theory. Application of such models not only is computationally demanding but also requires exact knowledge of particle features and detector optics, which is not the case for the new method here presented. Figure 4a,b indirectly suggests the error one would indeed make if the fractogram in Figure 4a were converted to PSAD by assuming a uniform detector response with size. Incidentally, this assumption is, still, quite common in FFF practice. 2.3. HF FlFFF of HRBCs. Morphology and size distribution of HRBCs are known to depend on medium osmolarity.19,20 It is known that in 170 mOsm media at pH 7.4 (e.g. phosphate-buffered saline, PBS, 85 mM) the HRBC is perfectly spherical, its hydrodynamic diameter corresponding to 1.24 AGV1/3, where AGV is the average HRBC volume. Previous HF FlFFF experiments on spherical HRBCs have shown good agreement between the HRBC size obtained from the AGV measured by uncorrelated methods of clinical analysis and the size determined from HF FlFFF retention time.12 However, for such a comparison between average size values obtained by uncorrelated methods, it was therein noted that the standard method of clinical analysis (18) Schure, M. R.; Barman, B.; Giddings, J. C. Anal. Chem. 1989, 61, 27352743. (19) Weinstein, R. S. In The Red Blood Cell, 2nd ed.; Surgenor, D. N., Ed.; Academic Press: New York, 1974; Vol. 1. (20) Assidjo, N. E.; Chiane´a, T.; Clarot, I.; Dreyfuss, M. F.; Cardot, Ph. J. P. J. Chromatogr. Sci. 1999, 37, 229-236.

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respectively. It must be recalled that differences as low as 10% in size are quite often obtained by independent, uncorrelated methods for particle size analysis.21 Figure 5b reports the solution for Q. The Q values result relatively constant and close to unity. This finding is in agreement with the scattering theory for nonabsorbing particles.7

Figure 5. HF FlFFF-UV/vis DAD of spherical HRBCs. Key: (a) output f (---) and fn (s) functions; (b) output Q function.

employed for the AGV measurements on the blood sample (i.e. the Coulter counter) actually gave a number-average value, while the value obtained from the HF FlFFF retention time is a crosssection distribution average. The new method makes it possible not only to obtain mass-size distribution but also number-size distribution analysis. In the case of spherical particles, we can write f(d) ) ∂m/∂d ) (π/6)d 3Rfn(d), where fn(d) is the numbersize distribution. Equation 11 then becomes fn(di)Q(di/λj) ) (4τij/πdi2)(∂Vr,i/∂di), which represents the nonlinear system whose solution, in this case, is obtained in terms of fn and Q. The required value of m0 (see Computational Section) was obtained by evaluation of sample recovery as described in section 2.1. Figure 5 reports the solutions for f (Figure 5a, dashed line), fn (Figure 5a, full line), and Q (Figure 5b) obtained after processing a HF FlFFF-UV/vis DAD fractogram of HRBCs. As in the case of nanometer-sized PS beads discussed in section 2.2, only data collected in the region were HRBCs did not show specific absorption were processed (λmin ) 450 nm; λmax ) 800 nm). Injected HRBCs were 7.0 × 104, the channel flow rate at the HF inlet was 3.00 mL/min, and the radial flow rate was 0.30 mL/ min. To have spherical HRBCs the suspending medium and the mobile phase were 170 mOsm PBS. Because of the independence of HF FlFFF retention on particle density, this made the necessary conversion from retention to spherical HRBC size possible by calibration with micrometer-sized, PS spheres, as described in ref 12. If the mean value of the number-average size distribution (fn) reported in Figure 5a (full line) is compared to the mean value given by the reference method employed for sizing the HRBCs sample,12 reasonably good agreement is found: 7.2 µm vs 6.6 µm, 6476 Analytical Chemistry, Vol. 75, No. 23, December 1, 2003

CONCLUSIONS An original method for the independent evaluation of the optical properties and mass- (or number-) size distribution of dispersed analytes in flow-assisted techniques with UV/vis turbidimetric detection is described. With respect to our previously developed approaches, the key, original feature of the new method is that, to describe the analyte optical properties, it requires neither standards nor to rely on models. When applied for the first time to few FFF-UV/vis DAD fractograms of particulate analytes of different origin, size, and optical features, the method gives results whose accuracy is comparable to that obtained by simulation, by independent analysis methods or by our previous methods based on calibration or models to determine the extinction efficiency. In these real cases, however, the method accuracy has been just qualitatively estimated. Unlike simulated cases, in fact, in real cases f and Q cannot be known a priori, and thus, an estimation of the error introduced in the output f and Q values cannot be quantitatively estimated via % CVf or % CVQ. On the other hand, accurate evaluation of the extent to which accuracy of the sizing method for the employed separation technique is reflected in the output f and Q values also stands beyond the aims of this paper. Also band-broadening effects must be generally taken into account for accurate application of the method to PSAD analysis of real samples through any flow-assisted technique. The method has been up to now applied only to nanometerand micrometer-sized particles of spherical shape. In principle, however, the method can be applied to dispersed analytes of any size and shape, including macromolecules. In FFF a suitable correction for nonspherical particles is needed if size wants to be obtained directly through retention time measurements. In general, the conversion from retention to size/molecular weight may require calibration with standards or the use of orthogonal methods for sizing. The use of high-sensitivity detectors may be another limiting aspect, since the amount of light extinguished by macromolecules (in absence of absorption) decreases with decreasing molecular weight/size. Further developments may include an optimized numerical routine that could be used as an add-on software for commercial packages for UV/vis DAD data acquisition and processing. This implementation could permit easier, optimized application of the method. LIST OF SYMBOLS A

absorbance (-)

b

optical cell path length (cm)

c

mass concentration (g cm-3)

C

normalization factor (-)

% CV

percent coefficient of variation

(21) Barth, H. G., Ed. Modern Methods of Particle Size Analysis; Wiley-Interscience: New York, 1984.

d

particle diameter (cm)

τ

turbidity (cm-1)

f, f(d)

particle mass-size distribution (PSAD) (g cm-1)

ξ

size parameter before interpolation (-)

fn, fn(d)

particle number-size distribution (particles cm-1)

˜f

algorithm output values for f

hf

system solution values for f before normalization

K

extinction coefficient (cm2 g-1)

m

particle mass (g)

m

particle relative refractive index (-)

m

number of input data points for d

m0

total mass of the eluted sample (g)

m1

value of the numerical integral on hf

N

particle number concentration (particles cm-3)

n

refractive index of the medium (-)

n

number of input data points for λ

np

particle refractive index (-)

˜q

algorithm output values for Q

qj

system solution values for Q before normalization

Q, Q(x)

extinction efficiency (-)

s

standard deviation

Vr

retention volume (cm3)

x

size parameter (-)

y

system input values (fQ)

ACKNOWLEDGMENT Hollow fibers were kindly supplied by Prof. B.-R. Min, Department of Chemical Engineering, Yonsei University, Korea. Fresh blood samples were supplied and clinically tested by the Department of Internal Medicine and Gastroenterology, University of Bologna. The authors wish to thank P.S. Williams, The Cleveland Clinic Foundation, and M. Schimpf, Boise State University, for manuscript revision and the helpful discussions. Thanks also goes to E. Cartoon for English revision and to M. Massari for part of the experimental work here enclosed. Work presented at the Ninth (Golden, CO) and the Tenth (Amsterdam, The Netherlands) International Symposium on Field-Flow Fractionation. This work was financially supported by the European Commission (4th Framework Program, INCO-Copernicus, Contract N ERB IC15CT98-0909) and by the Italian Ministry of the Education, University and Research (MIUR, COFIN 2000 (MM03247343-005)). SUPPORTING INFORMATION AVAILABLE Computational details (pseudocode and description of the algorithms) and additional results and discussion (details on method tests by simulation). This material is available free of charge via the Internet at http://pubs.acs.org.

yˆ

interpolated fQ values

˜y

y with added noise

R

particle density (g cm-3)

θ

detector acceptance angle (rad)

Received for review July 2, 2003. Accepted September 24, 2003.

λ

incident wavelength in the dispersing medium (cm)

AC034729C

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