Precision in differential field-flow fractionation: A chemometric study

$Page 1: Precision in differential field-flow fractionation: A chemometric study$
Letizia Bregola1

Catia Contado1

Michel Martin2

Luisa Pasti1

Francesco Dondi1

1Department of Chemistry,Ferrara, Italy

2Ecole Sup�rieure de Physique etde Chimie Industrielles,Laboratoire de Physique etM�canique des MilieuxH�t�rog�nes (PMMH – UMR7636 CNRS – ESPCI – Universit�Paris 6 – Universit� Paris 7),France

Original Paper

Precision in differential field-flow fractionation:A chemometric study

In the present paper, the capabilities of differential field-flow fractionation, i. e., thedetermination of an incremental quantity of a colloidal species, e. g., an uptakeadsorbed mass, determined by the joint use of two independent FFF measurements,over a species and the same modified species respectively, are considered. The differ-ent error types, those related to the retention time determinations and those com-ing from the operating parameter fluctuations were considered. The different com-ponents were computed with reference to SdFFF determinations of bare polystyrene(PS) submicronic particles and the same PS particles covered by IgG. Comparisonwas made between theoretically computed precision and experiments. The errorcoming from the experimental measurement of retention times was identified to bethe main source of errors. Accordingly, it was possible to make explicit the detec-tion limits and the confidence intervals of the adsorbed mass uptake, as a functionof experimental quantities such as the retention ratio, the detector calibrationratio, the injected quantity, the baseline noise, and the void time relative error. Anexperimentally determined and theoretically foreseen dependence of both theexperimental detection and confidence limits (L l 10 – 17 g) on the square root of theinjected concentration, for constant injected volume, was found.

Keywords: Differential field-flow fractionation / Error on retention time / Immunoglobulin G /Mass uptake / Precision /

Received: May 7, 2007; revised: June 21, 2007; accepted: July 4, 2007

DOI 10.1002/jssc.200700200

1 Introduction

Field-flow fractionation (FFF) is generally used as a sep-aration and characterization method of analytes and, inparticular, it is a methodology able to measure physicalparameters of particulate matter. The characterizationrelies on a retention model, which establishes a linkbetween the analyte parameter that one wishes to deter-mine (for instance, particle size, d) [1] and the experimen-tally measured retention time tR. In this approach, thetrueness and precision, which together define the accu-racy [2] of the absolute value of the retention time orretention parameter (k, see in the following) directlyaffect the accuracy of the searched analyte parameter.Another possibility of exploiting FFF consists in deter-mining parameters, which rely, not on a single FFF reten-tion experiment, but on retention in, at least, two FFF

runs performed not with a single identical analyte, buton an analyte, which has undergone a modification. Thecomparison between the retention data in the two runsallows to get information on that modification. This isthe basis of the concept of differential FFF.

Example of this concept is provided by the case of barecolloidal particles on which some chemical species isphysically or chemically adsorbed [3]. Information on theequilibrium or kinetic parameters of the coating can beobtained. The difference between retention times of thebare particles and of the modified particles allows accessto information linked to this modification [4, 5]. Most ofthe referenced studies in this topic have been performedby means of sedimentation FFF (SdFFF). This is mainlybecause the size and mass selectivities of this techniqueare larger than those of other FFF techniques. Neverthe-less, flow FFF [6, 7] and electrical FFF [8] have also beenused for this purpose.

In this differential FFF approach, the precision of thefinal information is heavily dependent on the precisionof the individual retention measurements, because ofthe random character of the experimental errors, whichaffects the precision.

Correspondence: Professor Francesco Dondi, Department ofChemistry, Via L. Borsari, 46, I-44100, Ferrara, ItalyE-mail: [email protected]: +39-0532-240709

Abbreviations: FFF, field-flow fractionation; PS, polystyrene;SdFFF, sedimentation FFF

i 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.jss-journal.com

2760 L. Bregola et al. J. Sep. Sci. 2007, 30, 2760 – 2779

J. Sep. Sci. 2007, 30, 2760 –2779 Other Techniques 2761

In this paper, the capabilities of differential FFF areinvestigated from a chemometric point of view. The keyparameter determined is the detection limit, i. e., theminimum amount of the quantity adsorbed by a particlewhich can be determined at a given probability value(e. g., 95%). Other related quantities such as decision andquantification limits which are related to the precisionare obtained too. This study will focus on the SdFFF tech-nique. The different error sources related to the measure-ment of the retention time and to the fluctuations of theoperating parameters will be investigated. The results ofthe theoretical analysis will be compared to experiment.This will allow us to both evaluate the capabilities of thedifferential FFF and to better define the performance ofthe FFF techniques as methods of determination of rele-vant physicochemical parameters of colloidal systems.Since it is known that SdFFF is an absolute techniquebeing able to measure the mass of a particle in the submi-cronic range, the study will allow to define the mini-mum measurable mass uptake by a particle in an adsorp-tion process. In this paper, the main point investigatedconcerns the aspect of precision. The global accuracyconcerns in fact many additional aspects not related tothe FFF operating procedure, but linked to the reliabilityof some physical constant employed for obtaining thefinal measurement. This will be considered in a subse-quent study.

2 Theory

The theory and the versatility of FFF for particle and poly-mer fractionation and characterization are well docu-mented in the literature [9]. FFF encompasses varioussubtechniques characterized by the specific applied forcefield, and, consequently by the type of particle propertysingled out by a subtechnique. SdFFF is considered here.The force field and the responsive particle property arethe sedimentation field and the effective mass, respec-tively.

The retention mechanism of SdFFF has been widelydescribed [10, 11]. Therefore, for the sake of brevity, onlythe theoretical aspects which are relevant to the presentstudy are reported below.

Particles of a sample suspension injected into the FFFchannel are forced to migrate to one of the channel walls(accumulation wall). When the sedimentation equili-brium has been established, the sample components areconfined to layers whose concentration c decays expo-nentially with distance x from the accumulation wall [9,12]

cðxÞ ¼ cð0Þ expð�x=lÞ ð1Þwhere c(0) is the concentration at the accumulation walland l is the mean particle distance from the wall, relatedto the sedimentation force F by [9]

l ¼ kBTFj j ð2Þ

where kB is the Boltzmann's constant and T is the absolutetemperature during elution. The mean particle elevationl is usually normalized with respect to the channel thick-ness w [1, 9, 12] as

k ¼ lw¼ kBT

Fj jw ð3Þ

In the case of a sedimentation field created by an accel-eration G on a particle of effective mass meff, the dimen-sionless layer thickness is given by

k ¼ kBTmeff Gw

ð4Þ

where meff is the effective mass of the particle suspendedin the carrier given by

meff ¼ mqp � ql

qp¼ m

Dq

qpð5Þ

where m is the particle mass, qp, ql are the particle andliquid carrier densities, respectively, and Dq = (qp –ql). IfDq is positive, particles accumulate at the outer wall ofthe SdFFF channel, whereas, if it is negative, they accu-mulate at the inner wall.

By considering a compact sphere particle of diameterd, one has

m ¼ qppd3

6ð6Þ

By combining Eqs. (4–6) one has

k ¼ 6kBTpd3DqGw

ð7Þ

k can be measured through the general equation of theretention [9, 13, 14]

R ¼ t0

tR¼ 6kðcoth

12k� 2kÞ ð8Þ

where t0 is the void time and tR is the retention time, i. e.,the time required to elute the sample. R is the retentionratio. Because of the curvature of the SdFFF channel, thet0/tR ratio is not rigorously equal to the k functiondescribed by the RHS term of Eq. (8). Nevertheless, whenthe ratio of the channel thickness to the curvature radiusis as low as that of the channel used in the present study,the correction to be brought to Eq. (8) can be safelyneglected [15]. Experimentally, R allows to calculate thevalue of the k parameter, which gives information onparticle effective mass or size using Eqs. (4) and (7),respectively.

If the particle is coated by adsorption onto the surface(see Fig. 1), a retention time increment is observed, sincein SdFFF the experimental retention time is a function ofthe effective particle masses (see Eqs. 4 and 5). The effec-



tive masses of the bare particle (p) and of the coating(adsorbed mass c), meff,p and meff,c, respectively, are

meff ;p ¼ mpqp � ql

qp

!ð9Þ

meff ;c ¼ mcqc � ql

qc

� �ð10Þ

where qc is the density of the coating. The total effectivemass of the coated particle (p + c) is

meff ;pþc ¼ meff ;p þmeff ;c ð11Þ

From Eqs. (4, 9, 10), kp and kp + c are

kp ¼kBT

Gw mp 1� ql

qp

!#" ð12Þ

kpþc ¼kBT

Gw mc 1� ql

qc

� �þmp 1� ql

qp

!" # ð13Þ

respectively, with kp and kp + c being the k parametersexperimentally determined for the bare and coated par-ticles, respectively. By combining Eqs. (12) and (13), it ispossible to calculate mc [3, 4, 6, 16–18] as follows

mc ¼kBT

Gw 1� ql

qc

� � 1kpþc

� 1kp

� �ð14Þ

This equation could be simplified, for high retention,i. e., for low k values, by using the approximation [14]

R ¼ t0

tRL 6k ð15Þ

and the following approximation can be put forward

1kpþc

� 1kp

� �L

6DtR

t0ð16Þ

with

DtR ¼ tR;pþc � tR;p ð17Þ

where tR,p and tR,p + c are the retention times of the bareand the coated particles.

In the derivation of Eqs. (9–17), it has been assumedthat the densities of the bare particles and of the coatingare both larger than the carrier fluid density. When theyare both samller than the latter, the modification of Eqs.(9–17) is straightforward and requires taking the abso-lute values of the density difference terms rather thantheir algebraic values. When the carrier liquid density isintermediate between the densities of the bare particlesand of the coating, the situation becomes somewhatmore complex, but its handling is relatively easy. In thepresent study, bare particles and the coating are denserthan the carrier liquid and the equations presented inthis study do not require modification. By putting

6kBT

Gw 1� ql

qp

! ¼ constant ¼ Ap ð18aÞ

6kBT

Gw 1� ql

qc

� � ¼ constant ¼ Ac ð18bÞ

and by combining Eqs. (12, 13, 15, 18a and 18b), the fol-lowing expressions for the particle and adsorbed massesrespectively are obtained

mp ¼ AptR;p

t0ð19Þ

mc ¼ AcDtR

t0ð20Þ

By combining Eqs. (18–20), one gets

mc

mp¼ DtR

tR;p

qp � ql

qc � ql

qc

qp

!ð21Þ

From Eqs. (20) and (21), it can be seen that the absoluteadsorbed mass and its relative value, mc and mc/mp respec-tively, are proportional to DtR or to (DtR/tR,p) respectively.The sensitivity of differential FFF, i.e., the lowest detect-able values of either mc or (mc/mp) depends thus on thelowest detectable values of either DtR or (DtR/tR,p) respec-tively, provided that the other quantities in Eqs. (18a andb) ql, qp, qc, G, kB, T are assumed to be constant: their inac-curacy will simply affect the exactness of either mc or(mc/mp) respectively [19]. The point is not for the momentspecifically considered here.

The sensitivity of a given method is expressed by well-known quantities [2], which are discussed below. The crit-ical decision limit, LC, for mc is that retention time incre-


Figure 1. Schematic representation of the bare and of thecoated particle. mc is the mass of adsorbed protein and theqp and qc are the particle and protein density, respectively.


ment which permits to state (at a given probability level)that the tested particle is different from the bare one, i. e.,a certain amount of adsorbent was taken up. This casehappens when the measured DtR is greater than

L DtRÞC ¼ kC

ffiffiffi2p

rtR

�ð22Þ

where rtR is the random error on tR andffiffiffi2p

rtR is the erroron DtR. The

ffiffiffi2p

factor comes from the fact that two inde-pendent measurements, tR;p and tR,p + c, are made. More-over, it is assumed that the measurement errors of bothtR,p and tR,p + c are equal. We observe that, in Eq. (22),LðDtRÞC is expressed in time units. This is because thedecision regarding whether there is adsorbed materialon the particles or not relies on knowing whether DtR isstatistically different from 0 or not. The error a related tothis hypothesis test is a type I error [2]. It represents theprobability that, in a particular experiment, the bare par-ticles lead to a retention time increment over the meanof many experiments greater than LðDtRÞC [2]. As for thekC value, see below.

Errors on tR and t0 affect the mass determination (seeEqs. 19 and 20). Regarding the mass uptake, the detec-tion limit, LD, expresses the lowest value of mc incrementwhich can be detected at a given probability level. If oneconsiders Eq. (20), one has [2]

LðmcÞD ¼ kDAcrDtR

t0

� �DtR!0

ð23aÞ

where rDtR

t0

� �DtR!0

is the SD of the quantity contained

in the brackets, computed for DtR fi 0, i. e., when tR, p + c fitR, p (see Eq. 17). The detailed expression for Eq. (23a) canbe obtained by applying the error transmission lawwhich establishes that, for a function of type z = x/y underthe hypothesis of independence of x and y, the squaredrelative error of z is equal to the sum of the squared rela-tive errors of x and y [20]. Thus, under the assumption ofindependence of the errors on the quantities t0, tR, p + c andtR, p under the above defined limit conditions, and takinginto account Eqs. (15) and (17), one has from Eq. (23a)

LðmcÞD ¼kDAc

ffiffiffi2p

R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirtR;p

tR

� �2

þ rt0

t0

� �2s

ð23bÞ

Again we observe that the factorffiffiffi2p

appearing in thisequation comes from the assumption that the errors onthe retention times of the bare and of the coated par-ticles are the same under conditions of a vanishinglysmall mc value. The assumptions of independence oferrors will be discussed in the following. The quantifica-tion limit LQ will be likewise given by a similar expressionby substituting the quantity kD in Eqs. (23a and b) withthe quantity kQ [2]. kC and kD appearing in Eqs. (22) and(23) respectively, are kC = 1.645, kD = 3.29, for a = b = 0.05(b is a type II error). kQ is 10 or 20, for an assumed relativepercent SD, 100rmc=mc, of 10 or 5%, respectively [2]. In

the case where the estimates stR of rtR have to beemployed, specific kC, kD and kQ values have to be com-puted from the Student distribution with pertinentdegrees of freedom.

Equation (23b) can be the basis for obtaining the confi-dence intervals (CI) at a given 1–a probability value forthe average determination of mass uptake, mc based on nrepeated determinations (this is further discussed in Sec-tion 6) [20]

CI ¼ mc lt1�an�1Ac

ffiffiffi2pffiffiffi

np

R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirtR;p

tR

� �2

þ rt0

t0

� �2s

ð24Þ

where t1�an�1 is the Student’s t-value for the 1–a probability

(usually 0.95) and n–1 degrees of freedom. One can seethat the quantities

rtR;p

tR

� �and

rt0t0

� �determine the preci-

sion of the methods. Sources of errors on tR and t0 will bediscussed in the following sections.

3 Sources of error on retention time in FFF

Goedert and Guiochon [21] in their study on the sourcesof error in measurement of gas chromatographic reten-tion times, have distinguished two kinds of randomerrors: (i) those arising from the detector noise, rtR ;N, and(ii) those arising from the random fluctuations of operat-ing parameters affecting retention times, rtR ;P. This holdstrue for FFF also since these random errors are independ-ent of each other. Thus one has [20]

rtR ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

tR ;Nþ r2

tR;P

qð25Þ

The errors on retention time tR, arising from the sec-ond kind of random errors can be calculated by means ofthe error propagation coefficients derived from the equa-tions relating the retention time and the fluctuatingoperating parameters. They are discussed in Section 3.4with reference to SdFFF.

The following three Sections (3.1–3.3) address theeffect of the first kind of random errors on retentiontime, the one arising from the detector noise. For thedetermination of this error, we develop below two prag-matic approaches which are later compared in Section 5.The first one is based on an estimate of the noise ampli-tude and is discussed in Sections 3.1 and 3.2. The secondone is based on a fitting of the experimental data pointsaround the peak maximum and is discussed in Section3.3. Their discussion is quite general and applies to anyzonal separation method, such as chromatography, FFF,or CE.

3.1 Precision on tR arising from detector noise

The retention time tR in zonal separation methods isdefined as the first moment of the sojourn time distribu-tion in the separator. For determining this first moment,



integration algorithms or peak fitting algorithms arerequired. Frequently, for simplifying the determination,and because the integration and peak fitting procedureshave their own sources of error, the retention time is con-sidered to be equal to the time of elution of the peak max-imum. This is exact for symmetrical peaks, but not cor-rect for asymmetrical peaks. Nevertheless, the approachfollowed below for the determination of the peak maxi-mum can be extended also to the latter case.

Because of the noise of the detector signal, there is afinite lapse of time around the peak maximum duringwhich it is not possible to determine whether or not thesignal increases, is constant, or decreases. This gives riseto an error in the determination of the time of peak max-imum.

Let j be the detector random noise. rtR ;N can be esti-mated as that time random span, after (or before) thetime of elution of the peak maximum, during which thesignal is smaller than the peak height, hm, and largerthan (hm – j), as shown in Fig. 2. j is here taken as equal tothe SD of the noise amplitude. As stated by Goedert andGuiochon [21], the actual value of j could be calculatedby combining spectral analysis of the noise and peak pro-file. This approach would require a specific handlingwhich lies beyond the aims of the present study.

Let us assume that the peak is Gaussian, with an SD, rt.The detector signal, h, is given by

hðtÞ ¼ hm exp � ðt� tRÞ2

2r2t

" #ð26Þ

The error rtR ;N corresponds to the value of t– tR forwhich h is equal to hm – j. Thus, one gets

exp �r2

tR ;N

2r2t

" #¼ 1� j

hm

� �ð27Þ

i. e.,

rtR ;N

rt¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 ln 1� j

hm

� �sð28Þ

Equation (28) shows that whatever the separationmethod at hand, rtR ;N arising from detector signal ran-dom error is scaled to the peak SD and depends only onthe relative value, j/hm. The variation of rtR ;N=rt as a func-tion of j/hm is shown in Fig. 3. The curve has been limitedto j/hm values lower than 0.25, as it is generally admittedthat the LOD corresponds to an S/N of 4. When j/hm issmall, Eq. (28) can be approximated, by using Mac Laurinseries properties, by

rtR ;N

rt¼

ffiffiffiffiffiffiffiffiffiffiffiffi2

jhm

rð29Þ

Equation (29) represents the approximation of thepeak shape with a parabolic function near the peak max-imum (j/hm f 1). The corresponding approximation isalso shown in Fig. 3. In a small interval around the maxi-mum, the difference between the exact curve and theapproximated one is small. This approximation holdstrue even when j reaches 10% of the peak height, a rela-tively large value. The difference in the rtR ;N=rt valuesobtained from Eqs. (28) and (29) is then lower than 2.6%.Hence, Eq. (29) can be safely used for estimating theretention time error arising from the detector signal ran-dom error. Figure 3 shows that the retention time errorhas a square root dependence on the j/hm ratio. For j/hm

equal to 1, 0.1, and 0.01%, rtR ;N amounts to 14.1, 4.47,and 1.41%, respectively, of the peak SD.

The relative error in retention time rtR ;N=tR isexpressed as a function of hm/j as

rtR ;N

tR¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2N

jhm

rð30Þ

where N is the plate number


Figure 2. Schematic representa-tion of the error DtR on the peakmaximum due to the detectornoise, j. tR is the retention time(here, for a symmetrical peak,time of elution of the peak maxi-mum), rt is the peak SD, and hm,the peak height.


N ¼ tR

rt

� �2

ð31Þ

Hence, this relative error depends not only on the j/hm

ratio, but also on the plate number, N, and, for highlyefficient separation methods, this relative error is small.

3.2 Effect of peak height on the retention timeerror arising from detector noise

Let us assume that the detector delivers a signal propor-tional to the mass concentration, c, of the analyte in thedetection cell

h ¼ kdc ð32Þwhere kd is the response factor of the detector for the ana-lyte in the linear range. Let q be the analyte mass injectedin the separator, cq the analyte maximum concentrationat the separator outlet, and rV the SD of the peak at theseparator outlet in unit of volume of eluent. As the distri-bution of analyte outlet concentration is assumed to beGaussian, one has, according to Eq. (32)

hm ¼ kdcq ¼ kdq

rVffiffiffiffiffiffi2pp ¼ kd

cinjVinj

rVffiffiffiffiffiffi2pp ð33Þ

where Vinj and cinj are the injected sample volume andanalyte concentration in the injected sample, respec-tively [22].

The SD in volume unit is related to the retention vol-ume, VR, as

rv ¼VRffiffiffiffi

Np ¼ V0

RffiffiffiffiNp ð34Þ

where V0 is the void volume of the separator.Combining Eqs. (30), (33), and (34), one gets

rtR ;N

tR¼

ffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffi2p

pp

2:24

ffiffiffiffiffiffiffiffiffiffiffiV0

Vinj

s

f1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffij

kdcinj

s

f2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

RN3=2

rf3

ð35Þ

The relative error on tR arising from the detector noiseappears to depend on three groups of parameters.

The first factor, f1 in Eq. (35), is a geometric factor,depending on the ratio of the separator volume to thesample injection volume. Clearly, for reducing the rela-tive error on tR, a large sample volume should be injected.There is however a limit in Vinj because a too large injec-tion volume will provide a significant contribution tothe peak broadening and a reduction in the plate num-ber, or an increase in 1/N. In optimized separations, themaximum sample volume should be selected in such away that the relative increase in 1/N due to the sampleprocess is kept equal to the tolerated fraction h2 [23].Then, the optimum V0/Vinj ratio appearing in Eq. (35) isequal to

V0

Vinj¼ R

ffiffiffiffiNp

hcð36Þ

where c is a constant which depends on the injectiondevice. For ideal devices delivering a rectangular sampleprofile to the separator, c is equal to

ffiffiffiffiffiffi12p

L 3.5. Forsyringe injection through a septum, c was found to beequal to about 2 [24]. For poorly designed injectors, itmay be even smaller. By introducing Eq. (36) into Eq. (35)one has

rtR ;N

tR¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffi2pp

hc

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffij

kdcinj

s ffiffiffiffiffiffiffi1N

rð37Þ

The second factor, f2, in Eq. (35) represents the noise-to-signal ratio that would be obtained by injecting directlythe sample in the detection cell. In fact, it should beremembered that j is here defined as the SD of the noise.Obviously, the larger the sample concentration, thesmaller the factor f2 and the smaller the relative error ontR.

The third factor, f3, in Eq. (35) reflects the influence ofthe operating parameters on the retention and band


Figure 3. Relative error on reten-tion time, rtR ;N=rt , arising fromdetector noise, as a function of thenoise-to-signal ratio, j/hm. Theplain curve corresponds to theexact expression (Eq. 28). Thedashed curve corresponds to theapproximated expression (Eq. 29).


broadening characteristics of the peak. It is observed thatwhen the sample injection volume is kept constant, thisinfluence is exerted through both R and N, while, whenthe sample injection volume is expressed by Eq. (36), thisinfluence of operating parameters occurs only through N(see Eq. 37).

3.3 Experimental determination of the retentiontime error arising from detector noise

In this section, a fitting method for the determination oftR allowing to determine rtR ;N appearing in Eqs. (27–30),is introduced. In the present study, we assume that tR isdetermined from the peak maximum and not from thefirst moment. This preference is justified by consideringthat most of the softwares employed in the current FFFpractice determine the peak maximum. This method issuitable for the purposes of this study devoted to differ-ential FFF, where it is also assumed that the peak shapesof the two compared fractograms are equal.

Let us assume that the peak is Gaussian, i. e., can be rep-resented according to Eq. (26). The retention time tR canbe computed either as the time of elution of the peakmaximum, tM, or a peak first moment, l1, which in thecase of a Gaussian peak is coincident, i. e., tM = tR. tR can bedetermined by interpolating the experimental data by aquadratic regression, since, around tM, the peak shape is

essentially parabolic, i. e., one has

hðtÞ ¼ aþ b1tþ b2t2 ð38Þ

where a and bi (i = 1, 2) are fitting coefficients and tM isthe maximum of the function h(t) of Eq. (38), i. e.,

b1 þ 2b2t ¼ 0 ð39Þ

and then tM is from Eq. (39)

tM ¼ tR ¼ �b1

2b2ð40Þ

In this case, the error in the retention time, due to thedetector noise, rtR ;N , can be determined by the errors ofregression coefficients. In fact, by considering Eq. (40)and by applying the error propagation law [20] one has

rtR ;N ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2b2

� �2

r2b1þ b1

ð2b2Þ2

!2

r2b2

vuut ð41Þ

This quantity can be compared with that appearing inEq. (29) which was derived under the same approxima-tion hypothesis, i. e., that the peak near the maximum

can be approximated by a parabola. We observe, how-ever, that, in this handling, the error in the determina-tion of tR due to baseline fluctuations is not considered.This effect introduces a systematic error which can be,nonetheless, estimated through the equations compara-ble to Eqs. (38–41). In fact, if the baseline shift is of para-bolic type, i. e., it can be represented by an equation simi-lar to Eq. (38), the peak signal and the baseline shift addtogether. The corresponding coefficients of the peak andof the perturbation will add too and b1 and b2 will bebiased: this will affect the determination of tR as either apositive or negative error, in agreement with Eq. (40).This effect will result in observed experimental errorgreater than that expressed by Eq. (25).

3.4 Precision on tR and fluctuations of theoperating parameters

The operating parameters affecting the retention timeprecision in SdFFF are: the temperature T, the fieldstrength (i. e., the number of revolutions per minute,rpm), and the carrier flow rate Q.

The absolute error ra;tR and the relative errors rr;tR ofthe retention time resulting from the random fluctua-tions of these operating parameters are derived inAppendix A. Here, the final ra;tR expression is reported

where ra;t, ra;rpm and ra;Q are the absolute fluctuations oftemperature, rotation velocity, and flow rate, respec-tively. In Eq. (42), the coefficient for the temperatureparameter can be defined as

qa;T ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid lnRd lnk

� �2 1Tþ

qpap � qlal

qp � ql

!2vuut ð43Þ

where ap and al are the thermal expansion coefficients ofthe particles and of the carrier, respectively (see Appen-dix A). The coefficient for the field strength (i. e., rotationvelocity) parameter is defined as

qa;G ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid lnRd lnk

� �2 1rpm2

sð44Þ

At last, the flow rate parameter coefficient is

qa;Q ¼1Q

ð45Þ

By combining Eqs. (42–45) one has

ra;tR ;P ¼ tR

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2

a;Tr2a;T þ q2

a;Gr2a;rpm þ q2

a;Q r2a;Q

qð46Þ


ra;tR ¼ tR

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidlnRdlnk

� �2 1Tþ

qpap � qlal

qp � ql

!2

r2a;T þ 4

dlnRdlnk

� �2 r2a;rpm

rpm2þ

r2a;Q

Q 2

vuut ð42Þ


4 Experimental

4.1 SdFFF equipment

The SdFFF system used in this study is Model S-101 Par-ticles/colloid fractionator (Postnova Analytics, Germany).The channel has nominal dimensions of 89.5 cm inlength, 2 cm in breadth, and 0.0254 cm in thickness. TheFFF system was equipped with an HPLC Pump PN1121(Postnova Analytics) to pump the carrier liquid into thechannel. The eluted samples were detected by a UV detec-tor Spectra SERIES UV 100 (Thermo Separation Products,USA) with a wavelength set at 254 nm.

4.2 Data handling

SPIN130 is a Windows compatible program whichacquires data which can be read by FFF's data analysissoftware. In general, the software emulates a controlpanel for setting the desired operating conditions, suchas the data rate. The data rate is the number of datapoints recorded per minute; in this work 20 points perminute was used. The determination of the channel voidvolume was made by measuring the elution time, t0, ofan unretained analyte (sodium benzoate) at various flowrates, after correction of the extrachannel contribution(computed as the ratio of the extrachannel geometric vol-ume, 0.24 mL, and of the carrier flow rate). A least squarelinear regression of the plot of t0 versus 1/Q resulted in anintercept of 0.0058 l 0.03 (statistically equal to zero) anda slope of 4.30 l 0.12 mL, which is according to Eq. (B-1)in Appendix B, the V0 value of the channel.

All computations were performed by means of home-made programs. The confidence intervals refer to Stu-

dentized interval, i. e., l ¼ x l t0:05n�1

sn

, where x is the mean

of the n measurement, s the SD of the measurements,and t0:05

n�1 the Student coefficient with n–1 degrees of free-dom and a probability level of 0.05 [2].

4.3 SEM

SEM has been used to monitor particle shape and size(model STEREOSCAN 360, Cambridge Instruments, UK).

4.4 Materials

Polystyrene (PS) latex standard particles with diameter of204 and 299 nm (Seradyn, Indianapolis, USA) were usedas adsorption surfaces. According to the manufacturer,the density of the PS particles, qp, is 1.040 g/mL at 258C.The protein adsorbed was lyophilized polyclonal humanIgG (Sigma–Aldrich, Milano, Italy), of a nominal density,qc, given as 1.353 g/mL at 258C. Carrier liquid for FFF sys-tem was a solution of 0.1% v/v FL-70 detergent (Fisher Sci-entific) with density 0.997 g/mL (258C).

4.5 Preparation of coated particles

The immunoglobulin sample, used to coat the PS latex,was dissolved in water from Milli-Q at a concentration of2% w/v and heated for 30 min at 588C. In an Eppendorftube an aliquot of 825 lL of a glycine buffer solution(pH 9.4), containing 0.15% w/v of glycine and 0.2% w/v ofNaCl was mixed with 75 lL of the IgG previous solution.Hundred microliters of the PS latex suspension wereadded to this solution. The tube was vortexed for 5 minand then incubated for 4 h with constant end-over-endshaking at room temperature. The coated latex particleswere centrifuged for 1 h at 8000 rpm and 850 lL of thesupernatant was replaced by the buffered solution. Thiscleaning process was repeated six times. Before injectinga sample, the tube containing the PS coated particles wasultrasonicated for 3 min and then vortexed for about30 s [3].

4.6 SdFFF conditions

The samples were injected through a 50 lL Rheodyneloop. The used rotation rates were: 1500 rpm (372 G) forthe 204 nm particles and 1000 rpm (165 G) for 299 nmparticles. After the injection, the carrier was stopped for12 min (relaxation period) to allow the sample to equili-brate under the applied field. The flow rates were 1.5, 2and 2.5 mL/min. The exact flow rate was determined bymonitoring the effluent volume as a function of time. Allwork was done at room temperature (23 l 18C). Differentconcentrations of suspensions for the uncoated andcoated particles were 0.0025, 0.005, 0.01, 0.015, and0.025% w/v.

5 Results and discussion

Two standard PS samples (204 and 299 nm) were consid-ered and both were coated with IgG according to the pro-cedure described in Section 4. The SEM technique hasbeen used to verify the nominal size of the PS particles.The images confirm that the suspensions are free fromaggregates and the two standards are homogeneous anduniform in size (see Figs. 4a and b). For each dimensionand sample concentration, a series of fractograms werecollected at various flow rates. In Figs. 5a and b fracto-grams at different injected concentrations of the two PSbare standards are reported. The analyses results arereproducible under all experimental conditions and thedetector response was linear (see discussion below). Thecorresponding series of analyses were performed byinjecting PS coated with IgG. In Figs. 6a and b, fracto-grams of the 204 and 299 nm PS latex samples with andwithout added IgG are compared, respectively.

Precision of the computed mass-uptake depends onthe precision of each single retention time and void time



determination (see Eqs. 23a, b and 24). Several are thecontributions which can affect both the tR and the t0

determinations: (i) “noise” in the detector signal, (ii) var-iation of the flow rate, (iii) variation of sample concentra-tion, and (iv) fluctuations in operational parameterssuch as field strength, temperature, and flow rate (seeEqs. 42 and 46). tR determination will be first considered.

5.1 Experimental determination of the tR error

5.1.1 Experimental determination of tR noise error

Equation (41) is the basic one for the experimental deter-mination of rtR ;N, i. e., the contribution to retention timeprecision derived from noise (see Eq. 25). All the quanti-ties appearing in Eq. (41) were obtained by fitting thepeak profile near the maximum by Eq. (38) (seconddegree polynomial). An example of peak fitting isreported in Fig. 7a. A third degree polynomial was alsotested and the relative fitting is reported in Fig. 7b forthe sake of comparison. One can see that the fitting

degrees obtained by the two equations are comparable toeach other. However, fitting with the third degree poly-nomial is meaningless due to overfitting. In fact, in thethird degree polynomial equation y = a + b1x + b2x2 + b3x3,the coefficients are: a = (–32.1 l 7.5)102, b1 =(4.0 l 1.6)102, b2 = 5 l 11, and b3 = 0.91 l 0.26. One can seethat, in the case of b2, the error is greater than its value.Instead, the fitting with the quadratic regression equa-tion, y = a + b1x + b2x2, gives coefficients significant in allcases: a = –5858 l 52; b1 = 961.4 l 7.4; b2 = –33.64 l 0.26.This behavior was also observed even with peaks at thehighest injected concentrations, where the S/N is high-est. Quadratic regression is thus employed for the calcu-lations, and tR is taken as the point at which the first deri-vate is zero (see Eqs. 39 and 40). By using Eq. (41), rtR ;N wascomputed. Note that, since a number of points greaterthan 30 is employed, this last quantity is reported as r

and not as s [2]. A series of runs were performed by chang-ing the flow rate and the concentration of the injectedsample, for both the 204 and 299 nm particles. The rtR ;N

values are reported in Tables 1 and 2.The first question that arises is about the correctness

of the rtR ;N values obtained from parabolic fitting (seeEqs. 38–41). By rearranging Eq. (35) one obtains


Figure 4. SEM of PS particles: (a) 204 nm PS, marked inter-distance: 202 nm; (b) 299 nm PS, marked interdistance299 nm.

Figure 5. SdFFF fractograms of bare PS standards: (a)204 nm PS. Experimental conditions: 1500 rpm (372G); stopflow 12 min; carrier flow rate 2 mL/min; concentrations:0.0025, 0.005, 0.01, and 0.025% w/v. (b) 299 nm PS. Exper-imental conditions: 1000 rpm (165G); stop flow 12 min; car-rier flow rate 2 mL/min; concentrations: 0.0025, 0.005, 0.01,0.025% w/v.


rtR ;N

tR

� �2 1

2ffiffiffiffiffiffi2pp Ving

V0RN3=2� �

kdcinjÞ ¼ jcalc�

ð47Þ

From Eq. (47), one can obtain an estimation of the jvalue, jcalc, by measuring rtR ;N and by evaluating the kd

value, all the other quantities being accessible in theexperiment. By combining Eqs. (33 and 34), one gets

hm ¼ kdVinjR

ffiffiffiffiNpffiffiffiffiffiffi

2pp

V0cinj ð48Þ

then

hm ¼ k9dcinj ð49Þ

where

kd ¼ k9dV0

ffiffiffiffiffiffi2pp

RffiffiffiffiNp

Vinjð50Þ

Consequently, the kd constant appearing in Eq. (47) canbe obtained from Eq. (50). In fact, the other quantitiesappearing in Eq. (50), V0, Vinj, N, and R can be obtainedfrom the FFF run and k9d from the detector calibrationplot according to Eq. (49). In Table 3, the k9d, kd, R, and Nvalues for the two PS particle diameters are reported. Byusing the data of Table 3 and Eq. (47), jcalc of Tables 1 and

2 were calculated. These jcalc can be compared to the cor-responding experimental j values, i. e., the data obtainedby calculating the SD of the baseline signal in a timeinterval equal to the peak width in the same fractogram(jbaseline). The data are reported in Tables 1 and 2 for thesake of comparison. It can be seen that the agreement issatisfactory, and the method of obtaining rtR ;N from para-


Figure 6. SdFFF fractograms of bare versus Ig-coated PSstandards: (a) SdFFF fractograms of bare and Ig-coated PS204 nm. Experimental conditions: 1500 rpm (372G); stopflow 12 min; carrier flow rate 1.5 mL/min; concentrations0.005 and 0.01% w/v. (b) SdFFF fractograms of bare PS andIg-coated 299 nm particles. Experimental conditions:1000 rpm; stop flow 12 min; carrier flow rate 2.5 mL/min;concentrations 0.01 and 0.025% w/v

Figure 7. Polynomial linear least square fitting for tR and rtR ;Ndetermination for bare PS 204 nm. (a) Quadratic approxima-tion, (b) third degree polynomial approximation.

Table 1. Noise errors rtR ;N and comparison between calcu-lated jcalc (from Eq. 47) and experimentally measured jbaseline

for PS 204 nm (rpm = 1500)

cinj

(%w/v)Q(mL/min)

rtR ;N

(min)jcal (mV)(Eq. 47)

jbaseline

(mV)

0.005 1.5 0.51 5.74 5.102 0.37 5.49 4.542.5 0.26 4.12 3.80

0.01 1.5 0.40 4.90 4.502 0.21 3.47 3.602.5 0.18 4.15 4.34

0.025 1.5 0.10 4.40 4.302 0.08 4.05 3.902.5 0.08 4.05 3.60

j 4.50 l 0.74 4.20 l 0.50


bolic fitting (Eqs. 38 –41) results is validated. The depend-ence of rtR ;N values on flow rate (i. e., on tR for the sameparticle size) is further exploited in Appendix C.

Up to now only the noise component of the error wasconsidered. Instead the error in the tR determination ofrepeated runs also contains the component derivingfrom the fluctuation of all the experimental parameters,as expressed by Eq. (25) for rtR ;P. This second componentof the error is investigated in the following.

5.1.2 tR precision and operating parameterfluctuation

On the basis of Eqs. (42–46), rtR ;P was estimated underdifferent values of the assumed fluctuations of the mainexperimental parameters T, rpm, and Q. Two conditionsare considered: the first one is a long term condition (typ-ically 3 months) and the second a short term condition(1 day). The following parameter variations were foundto hold true in our experimental practice (see under Sec-tion 4).

(i) Long term. Fluctuation of temperature T: ra,T = 28C;fluctuations of number of rpm: ra,rpm = 0.5 and 0.04 rpm,for PS 204 nm (rpm = 1500) and PS 299 nm (rpm = 1000)respectively; fluctuation of Q: rs,Q = 0.01 mL/min.

(ii) Short term. Fluctuation of temperature T: ra,T =0.28C; fluctuations of number of rpm: ra,rpm = 0.2 and0.02 rpm for PS 204 nm (rpm = 1000) and PS 299 nm(rpm = 1500) respectively; fluctuation of Q: ra,Q = 0.005mL/min.

We observe that the ra,rpm is significantly dependent onthe rpm value. On the basis of Eqs. (42) and (46), and byusing experimental retention data (tR, R and k) obtainedin selected experimental runs, rtR ;P values for the twocases were calculated and reported in Tables 4 and 5. Inpractice, one estimates a rtR ;P component ranging from0.10 to 0.30 min for the long term fluctuation and 0.03to 0.10 min for the short term fluctuation. One can seethat even in the worse condition (long term fluctuation),the expected rtR ;P component is lower or near to the rtR ;N

component (c.f.r. rtR ;P in Tables 4 and 5 with correspond-ing rtR ;N data in Tables 1 and 2). This underlines the goodstability of the SdFFF instrumentation.


Table 2. Noise errors rtR ;N and comparison between calcu-lated jcalc (from Eq. 47) and experimentally measured jbaseline

for PS 299 nm (rpm = 1000)

cinj

(%w/v)Q(mL/min)

rtR ;N

(min)jcalc (mV)(Eq. 47)

jbaseline

(mV)

0.005 1.5 0.48 4.16 4.502.0 0.31 4.03 4.102.5 0.23 2.94 3.55

0.01 1.5 0.31 3.92 4.102.0 0.20 3.43 4.702.5 0.15 2.50 3.60

0.025 1.5 0.22 4.80 4.212.0 0.12 3.15 3.802.5 0.11 3.40 3.46

j 3.59 l 0.70 4.00 l 0.40

Table 3. Detector calibration constants (V0 = 4.3 ml;Vinj = 50 lL; concentration range 0.0025 f cinj f 0.025% w/v)

Size (nm) k9d (102) R N kd (107)

204 2000 € 51 0.15 9.11 9.96 € 0.02299 2037 € 37 0.11 13.7 11.1 € 0.02

Table 4. Experimental parameter error propagators (qa,T, qa,G, qa,Q), and retention time errors rtR;P . Long term conditions. PS bareparticles 204 and 299 nm (see Eqs. 42 and 46)

Size(nm)

cinj

(%w/v)Q(mL/min)

rpm tR

(min)qa,T

(K – 1)qa,G

(rpm – 1)qa,Q

(min/mL)rtR;P

(min)

204 0.005 1.5 1500 20.46 3.34610 – 3 1.26610 – 3 0.66 1.94610 – 1

0.01 1.5 1500 21.51 3.35610 – 3 1.27610 – 3 0.66 2.04610 – 1

0.025 1.5 1500 20.52 3.34610 – 3 1.26610 – 3 0.66 1.94610 – 1

0.005 2.0 1500 14.35 3.33610 – 3 1.26610 – 3 0.50 1.20610 – 1

0.01 2.0 1500 14.66 3.33610 – 3 1.26610 – 3 0.50 1.22610 – 1

0.025 2.0 1500 15.00 3.33610 – 3 1.26610 – 3 0.50 1.25610 – 1

0.005 2.5 1500 12.32 3.34610 – 3 1.26610 – 3 0.40 0.96610 – 1

0.01 2.5 1500 11.99 3.33610 – 3 1.26610 – 3 0.40 0.94610 – 1

0.025 2.5 1500 12.21 3.34610 – 3 1.26610 – 3 0.40 0.95610 – 1

299 0.005 1.5 1000 28.80 3.39610 – 3 1.93610 – 3 0.66 2.74610 – 1

0.01 1.5 1000 27.11 3.39610 – 3 1.92610 – 3 0.66 2.58610 – 1

0.025 1.5 1000 26.76 3.38610 – 3 1.92610 – 3 0.66 2.54610 – 1

0.005 2.0 1000 18.87 3.38610 – 3 1.92610 – 3 0.50 1.58610 – 1

0.01 2.0 1000 18.69 3.38610 – 3 1.92610 – 3 0.50 1.57610 – 1

0.025 2.0 1000 18.50 3.37610 – 3 1.91610 – 3 0.50 1.55610 – 1

0.005 2.5 1000 16.43 3.39610 – 3 1.92610 – 3 0.40 1.29610 – 1

0.01 2.5 1000 16.42 3.39610 – 3 1.92610 – 3 0.40 1.29610 – 1

0.025 2.5 1000 16.33 3.39610 – 3 1.92610 – 3 0.40 1.28610 – 1


Several cinj and Q values were considered for both PS204 nm and PS 299 nm sizes (see Tables 6 and 7) and, foreach set of conditions, three repeated injections wereperformed. tR and ðstR Þexp, where ðstR Þexp is the SD of tR,were determined. In the same Tables 6 and 7, the valuesof rtR ;N for each of the three runs (a, b, c), for the givenconditions of cinj and Q are reported. The values of rtR ;N ofthe same tables were estimated as described above (Eq.46). From rtR ;N and rtR ;P, the values of ðrtR Þcalc, i. e., thetotal calculated error on tR, were computed by using the

following equation derived according to the additiverule expressed by Eq. (25)

ðrtR Þcalc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrtR ;NðaÞÞ

2 þ ðrtR ;NðbÞÞ2 þ ðrtR ;NðcÞÞ

2

3þ ðrtR ;NÞ

2

sð51Þ

One can see that only in selected cases ðstR Þexp is signifi-cantly greater than ðrtR Þcalc (see rows no. 6 and nos. 15–


Table 5. Experimental parameter error propagators (qa,T, qa,G, qa,Q), and retention time error rtR;P . Short term conditions. PS bareparticles 204 and 299 nm (see Eqs. 42 and 46)

Size(nm)

cinj

(%w/v)Q(mL/min)

rpm tR

(min)qa,T

(K – 1)qa,G

(rpm – 1)qa,Q

(min/mL)rtR;P

(min)

204 0.005 1.5 1500 20.46 3.34610 – 3 1.26610 – 3 0.66 6.97610 – 2

0.01 1.5 1500 21.51 3.35610 – 3 1.27610 – 3 0.66 7.33610 – 2

0.025 1.5 1500 20.52 3.34610 – 3 1.26610 – 3 0.66 6.99610 – 2

0.005 2.0 1500 14.35 3.33610 – 3 1.26610 – 3 0.50 3.73610 – 2

0.01 2.0 1500 14.66 3.33610 – 3 1.26610 – 3 0.50 3.81610 – 2

0.025 2.0 1500 15.00 3.33610 – 3 1.26610 – 3 0.50 3.90610 – 2

0.005 2.5 1500 12.32 3.34610 – 3 1.26610 – 3 0.40 2.61610 – 2

0.001 2.5 1500 11.99 3.33610 – 3 1.26610 – 3 0.40 2.54610 – 2

0.025 2.5 1500 12.21 3.34610 – 3 1.26610 – 3 0.40 2.59610 – 2

299 0.005 1.5 1000 28.80 3.39610 – 3 1.93610 – 3 0.66 9.80610 – 2

0.01 1.5 1000 27.11 3.39610 – 3 1.92610 – 3 0.66 9.22610 – 2

0.025 1.5 1000 26.76 3.38610 – 3 1.92610 – 3 0.66 9.10610 – 2

0.005 2.0 1000 18.87 3.38610 – 3 1.92610 – 3 0.50 4.88610 – 2

0.01 2.0 1000 18.69 3.38610 – 3 1.92610 – 3 0.50 4.84610 – 2

0.025 2.0 1000 18.50 3.37610 – 3 1.91610 – 3 0.50 4.79610 – 2

0.005 2.5 1000 16.43 3.39610 – 3 1.92610 – 3 0.40 3.47610 – 2

0.01 2.5 1000 16.42 3.39610 – 3 1.92610 – 3 0.40 3.47610 – 2

0.025 2.5 1000 16.33 3.39610 – 3 1.92610 – 3 0.40 3.44610 – 2

Table 6. Comparison with ðrtR Þcal calculated and ðstR Þexp obtained from three repeated FFF analyses (a, b, and c) for PS 204 nm

Terms of r components Eq. (51)

No. cinj

(%w/v)Q(mL/min)

tR

(min)ðstR Þexp(min)

ðrtR Þcalc (min)Eq. (51)

rtR ;N ðaÞ(min)

rtR ;N ðbÞ(min)

rtR ;N ðcÞ(min)

rtR ;P

(min)

1 0.0025 2.0 13.69 0.42 0.39 0.39 0.39 0.38 0.042 0.005 2.0 14.35 0.16 0.37 0.38 0.36 0.37 0.043 0.01 2.0 14.66 0.37 0.21 0.23 0.19 0.21 0.044 0.015 2.0 14.88 0.36 0.26 0.29 0.23 0.24 0.045 0.02 2.0 13.67 0.12 0.10 0.1 0.08 0.1 0.046 0.025 2.0 15.00 0.55 0.08 0.06 0.08 0.08 0.047 0.0025 1.5 20.98 0.10 0.63 0.58 0.7 0.6 0.078 0.005 1.5 20.46 0.50 0.42 0.31 0.5 0.4 0.079 0.01 1.5 21.51 0.30 0.29 0.23 0.34 0.25 0.07

10 0.015 1.5 20.93 0.35 0.38 0.38 0.37 0.37 0.0711 0.02 1.5 20.65 0.10 0.22 0.2 0.21 0.21 0.0712 0.025 1.5 20.52 0.18 0.25 0.16 0.3 0.25 0.0713 0.0025 2.5 12.61 0.04 0.38 0.36 0.4 0.37 0.0314 0.005 2.5 12.32 0.50 0.27 0.1 0.3 0.35 0.0315 0.01 2.5 11.99 0.03 0.19 0.2 0.25 0.09 0.0316 0.015 2.5 12.86 0.03 0.21 0.21 0.21 0.22 0.0317 0.02 2.5 12.70 0.03 0.21 0.19 0.18 0.17 0.0318 0.025 2.5 12.21 0.16 0.09 0.08 0.09 0.08 0.03


17 in Tables 6 and 7, respectively), but these seem to benot statistically significant if one considers that for eachparticle type one has, in total, 18 cases. Consequently,one can conclude that the error on tR, under standardworking conditions, can be explained on the basis ofbaseline noise and the parameter fluctuations, rtR ;N

being the most important component (c.f.r. rtR ;P and rtR ;N

in Tables 6 and 7).

5.2 Application to the studied case

Retention time data and the component of the errors forPS bare and IgG-coated particles are reported in Tables 8and 9. One can see that the agreement for both the bareand Ig-coated particles between ðrtR Þcalc and ðstR Þexp isgood. These data can be the basis for determining thedetection limits.

First of all one can see that the rtR values, for a giveninjected concentration, for both the bare and the coated

particles are statistically equal (see Table 8 for PS 204 nmand Table 9 for PS 299 nm). Consequently, we can useEqs. (23a and b) with the

ffiffiffi2p

constant for the determina-tion of the detection limits as discussed under Section 2.Moreover, we observe that noise is the major error sourcefor tR, as one can see by comparing ðrtR Þcalc to rtR ;N inTables 6 and 7 for PS 204 nm and PS 299 nm, respectively.Consequently, one can assume

rtRð Þcalc L rtR ;N ð52aÞ

for the detection limits calculation by Eq. (23b). Then wemust remember that for a given injected concentrationvalue one proved that

rtRð Þcalc L rtR ¼ r9tR6tR ð52bÞ

where r9tR is a constant for a given injected concentration(Fig. C-1 in Appendix C). r9tR can be computed from theslopes of the experimental rtR ;N versus tR plots at a givencinj value, (see Appendix C). Consequently by combining


Table 7. Comparison with ðrtR Þcal calculated and ðstR Þexp obtained from three repeated FFF analyses (a, b, and c) for PS 299 nm

Terms of r components Eq. (51)

No. cinj

(%w/v)Q(mL/min)

tR

(min)ðstR Þexp(min)

ðrtR Þcalc (min)Eq. (51)

rtR ;N ðaÞ(min)

rtR ;N ðbÞ(min)

rtR ;N ðcÞ(min)

rtR ;P

(min)

1 0.0025 2.0 17.68 0.13 0.50 0.46 0.51 0.53 0.052 0.005 2.0 18.87 0.04 0.31 0.32 0.32 0.29 0.053 0.01 2.0 18.69 0.11 0.20 0.19 0.20 0.20 0.054 0.015 2.0 18.99 0.17 0.16 0.14 0.16 0.16 0.055 0.02 2.0 18.63 0.11 0.13 0.12 0.11 0.12 0.056 0.025 2.0 18.50 0.05 0.13 0.12 0.11 0.12 0.057 0.0025 1.5 28.56 0.02 0.91 0.90 1.00 0.80 0.098 0.005 1.5 28.80 0.22 0.50 0.48 0.48 0.50 0.099 0.01 1.5 27.11 0.11 0.32 0.33 0.30 0.30 0.09

10 0.015 1.5 27.52 0.04 0.35 0.34 0.33 0.34 0.0911 0.02 1.5 26.65 0.36 0.27 0.28 0.23 0.25 0.0912 0.025 1.5 26.76 0.30 0.23 0.21 0.22 0.22 0.0913 0.0025 2.5 17.14 0.02 0.97 0.90 1.00 1.00 0.0914 0.005 2.5 16.43 0.40 0.23 0.22 0.23 0.23 0.0315 0.01 2.5 16.42 0.50 0.15 0.15 0.16 0.14 0.0316 0.015 2.5 16.07 0.28 0.18 0.17 0.18 0.18 0.0317 0.02 2.5 15.83 0.45 0.14 0.14 0.15 0.13 0.0318 0.025 2.5 16.33 0.08 0.12 0.11 0.12 0.11 0.03

Table 8. Mean retention times and component of the errors for PS bare particles and IgG-coated PS (PS-IgG) computed onthree repeated injections (core particle PS 204 nm, flow rate 2 mL/min, 1500 rpm)

cinj (%w/v) PS PS-IgG

tR (min) ðrtR Þcalc (min)Eq. (51)

ðstR Þexp (min) tR (min) ðrtR Þcalc (min)Eq. (51)

ðstR Þexp (min)

0.0025 13.69 0.39 0.42 18.46 0.41 0.210.005 14.35 0.37 0.16 19.91 0.30 0.190.01 14.66 0.21 0.37 20.62 0.23 0.200.015 14.88 0.26 0.36 20.84 0.29 0.300.02 13.66 0.10 0.12 19.68 0.10 0.320.025 15.00 0.08 0.55 20.86 0.10 0.05


Eqs. (52a and b), with Eq. (23b) and by consideringj = jbaseline, one obtains

LðmcÞD LkDAc

ffiffiffi2p

R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffi2p

pV0

VinjkdcinjRN3=2jbaseline þ

rt0

t0

� �2s

ð53aÞ

The L(mc)D for IgG adsorption for both PS 204 nm and PS299 nm, computed according to Eq. (53a), are reported

in Table 10. Here, and in the followingrt0

to

� �¼ 0:01 was

employed as specified in Appendix B.One can see that for the same cinjvalue, the L(mc)D values

are slightly greater for PS 299 nm than for PS 204 nm.This is mainly due to the Ac values: in fact, for PS 299 nm,Ac = 1.53610–16 g (with G = 165.59 (980 cm/s2)) which isroughly double that for PS 204 nm (Ac = 6.82610–17 gwith G = 372.58 (980 cm/s2)), the difference in Ac thusovercomes the differences due to the other parameters(R = 0.15, N = 9.11, and kd = 9.966107 for PS 204 nm versus

R = 0.11, N = 13.7, and kd = 11.1 107 for PS 299 nm). In con-trast, the other quantities jbaseline, Vinj, and V0 are constant.

In Figs. 8a and b the IgG L(mc)D values with respect to cinj

are reported for PS 204 nm and PS 299 nm, respectively.One can see that the detection limits decrease with theamount of the injected quantity, as expected from Eq.(53a).

When the conditionrt0

to

� �f

13

rtR ;P

tR

� �in Eq. (23b)

holds true, we can neglect the termrt0

to

� �2

with a maxi-

mum error of about 10% under square root in Eq. (53a)(which was derived from Eq. 23b) and we obtain the fol-lowing approximated expression


Table 9. Retention times and component of the errors for PS bare particles and IgG-coated PS (PS-IgG) computed on threerepeated injections (core particle PS 299 nm, flow rate 2 mL/min, 1000 rpm)

cinj (%w/v) PS PS-IgG

tR (min) ðrtR Þcalc (min)Eq. (51)

ðstR Þexp (min) tR (min) ðrtR Þcalc (min)Eq. (51)

ðstR Þexp (min)

0.0025 17.68 0.50 0.13 25.56 0.34 0.400.005 18.87 0.31 0.04 26.46 0.25 0.360.01 18.69 0.20 0.11 27.05 0.19 0.200.015 18.99 0.16 0.17 27.32 0.35 0.180.02 18.63 0.13 0.11 26.16 0.11 0.100.025 18.50 0.13 0.05 26.97 0.15 0.09

Table 10. Comparison with L(mc)D for deterministic t0 (Eq.53b) and L(mc)D for experimental t0 (Eq. 53a) in both the par-ticles size (PS 204 nm and PS 299 nm)

Size(nm)

cinj L(mc)D Eq. (53a)a)

(10 – 16 g)L(mc)D Eq. (53b)(10 – 16 g)

204 0.0025 1.40 1.380.005 1.00 0.980.01 0.72 0.690.015 0.60 0.570.02 0.53 0.490.025 0.48 0.44

299 0.0025 1.89 1.860.005 1.37 1.320.01 1.00 0.930.015 0.84 0.760.02 0.75 0.660.025 0.69 0.59

a) Consideringrt0

t0

� �= 0.01.

Figure 8. Detection limits L(mc)D (Eq. 53a) [f] and mass-uptake averaged with respect to flow rate [g] versus injectedconcentration. (a) PS 204 nm; (b) PS 299 nm.


LðmcÞD L kDAc2:24

R3=2N3=4k1=2d

( ) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV0

Vinjcinjjbaseline

sð53bÞ

The IgG L(mc)D values computed according to this sim-plified expression are reported in Table 10, last column.One can see that the difference in L(mc)D values is signifi-cant, as expected, only at high values of cinj when the con-tribution to precision due to the tR determinationdecreases.

In Tables 11 and 12 three repeated experimental IgGmc values computed by using Eq. (14) for both PS 204 nmand PS 299 nm particles, together with their statistics,are reported. In Fig. 8 the mc values versus cinj are reported.One can see that the mc data are almost constant andindependent from cinj, proving the robustness of the mc

determination with respect to cinj in the exploited condi-tions. Moreover, one can see that only in the case of PS204 nm for cinj = 0.0025%, mc is lower than L(mc)D.

The CI values can be likewise computed by using Eq.(24), under the above referred conditions holding truefor Eq. (53a) (see Eq. 52a, b with j = jbaseline) and by consider-ing the effective number of replicated measurements in

the different steps of the adsorbed quantity determina-tion

CIcalc L mc lt1�an�1Ac

ffiffiffi2p

Rffiffiffiffiffiffiffiffiffiffin1n2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffi2pp V0

VinjkdcinjRN3=2jbaseline þ

rt0

t0

� �2s

ð54aÞ

where mc value is assumed to be the average of the mc val-ues obtained at different cinj, and n1 is the number ofrepeated kp and kp + c determinations performed to com-pute each mc for a given cinj according to Eq. (14), and n2 isthe number of mc used to determine the mc, over thethree Q, for each cinj (thus n1 and n2 stand for n of Eq. 24).Likewise the experimental CI values were computed as:

CIexp L mc lt1�an�1smcffiffiffiffiffi

n2p ¼ mc l t1�a

n�1smc ð54bÞ

where smc is the SD of the n2 repeated mc determinations(each based on given n1 k determinations, as specifiedabove). In Fig. 9, CIexp and CIcalc are reported versus cinj. Onecan see that the agreement between CIcalc and CIexp is


Table 11. Summary of experimental mass-uptake together with the SD values for PS 204 nm bare particles. Conditions forSdFFF were: rpm 1000 and temperature (23 l 1)8C

cinj (%w/v) Q 1.5 mL/min Q 2 mL/min Q 2.5 mL/min Average mc smc experimental CIcalc Eq. (54a)a) CIexp Eq. (54b)

Mass-uptake (10 – 16 g)(10 – 16 g) (10 – 16 g) (10 – 16 g) (10 – 16 g)

0.0025 1.17 1.15 1.15 1.16 0,01 0.51 0.030.005 1.17 1.34 1.21 1.24 0,09 0.37 0.220.01 1.38 1.42 1.32 1.37 0,05 0.26 0.120.015 1.27 1.43 1.28 1.33 0,09 0.22 0.220.02 1.33 1.44 1.35 1.37 0,06 0.19 0.150.025 1.27 1.40 1.30 1.32 0,07 0.17 0.17Mean 1.26 1.36 1.27 mc 1.30 smc

0.06SD 0.08 0.11 0.07

a) Consideringrt0

t0

� �= 0.01.

Table 11. Summary of experimental mass-uptake together with the SD values for PS 299 nm bare particles. Conditions forSdFFF were: rpm 1000 and temperature (23 l 1)8C

cinj (%w/v) Q 1.5 mL/min Q 2 mL/min Q 2.5 mL/min Average mc smc experimental CIcalc Eq. (54a)a) CIexp Eq. (54b)

Mass-uptake (10 – 16 g)(10 – 16 g) (10 – 16 g) (10 – 16 g) (10 – 16 g)

0.0025 4.35 4.30 4.02 4.22 0.18 0.70 0.440.005 4.15 4.10 4.10 4.12 0.03 0.50 0.070.01 4.55 4.48 4.13 4.39 0.22 0.37 0.560.015 4.48 4.50 4.15 4.38 0.19 0.31 0.490.02 4.43 4.10 4.30 4.27 0.17 0.28 0.410.025 4.31 4.48 4.20 4.36 0.10 0.25 0.25Mean 4.38 4.31 4.15 mc 4.28 s

mc0.12

S D 0.14 0.18 0.09

a) rt0

t0

� �= 0.01 was assumed.


good. Moreover, the single mc values are randomly dis-tributed around the common average mc and no depend-ence is observed with respect to cinj. Consequently, theability of Eqs. (53a) and (54a) in predicting the sensitivityof the differential FFF on the sole basis of the operativeparameters was proved.

The success of this handling is based on the theoreticaltractability of the FFF technique. Moreover, the characterof the SdFFF, being an absolute analytical technique isthus extended also to its precision features, which seemsquite unique in the scenario on the analytical separationtechniques. At this point, in order to fully define theaccuracy of the procedure, one should also face the true-ness, which, together with precision, define the accuracyof a measurement [2]. This substantially requires todefine the trueness and the precision of the Ac and Ap

quantities appearing in the various expression (see Eqs.23, 24, 57, 58). This, in turn, calls for exploiting the true-ness of the quantities ql, qp, and qc appearing in theseexpressions. The point calls for a specific handling whichgoes beyond the aims of this first study. Besides, this wayof fully exploiting the method accuracy follows the gen-eral guidelines for a method validation which consists in

firstly defining the precision and then the trueness andthe whole accuracy.

6 Concluding remarks

This study proved that it is possible to model the preci-sion of retention time determination in SdFFF. Two mainsources of errors were identified, that coming from thenoise signal and that from the system fluctuations, i. e.,from the variation of the operative parameters. Systemfluctuations give generally a lower contribution withrespect to the noise in terms of signal variance. More-over, from the sole knowledge of the baseline noise, theretention time noise component of the error can be eval-uated Eq. (35). The general equations expressing thedetection error are furnished. By using Eqs. (46) and (25),the retention time error can be a priori estimated.

The detection and quantification limits in differentialFFF were obtained. The determined uncertainty demon-strated the reliability of SdFFF in mass uptake measure-ment. The present general chemometric approach wasthus experimentally validated by establishing practicalrules valid for diagnostics of the current SdFFF measure-ment. The result obtained with reference to SdFFF can beeasily extended to other FFF subtechniques. This simplyrequires to define a constant appearing, i. e., Ac in Eqs.(23) and (24) for the specific FFF subtechnique which, forSdFFF, is given by Eq. (18b). The present study can thus bethe starting point for exploiting fully the accuracy of dif-ferential FFF.

This work was financially supported by Italian Ministry of Univer-sity and Scientific Technological Research COFIN 2005(2005037725_002)

Appendix A

Sources of error on retention time in FFF:Fluctuations of operating parameters

General treatment for zonal elution methods

As noted by Goedert and Guiochon [21], if the operatingparameters x, y, and z are varying respectively byamounts dx, dy, and dz, the variation, dtR, of the reten-tion time, tR, is expressed by

dtR ¼qtR

qxdxþ qtR

qydyþ qtR

qzdz ðA1Þ

or

dtR ¼ qa;xdxþ qa;ydyþ qa;zdz ðA2Þwith

qa;s ¼qtR

qs; s ¼ x; y; z ðA3Þ


Figure 9. Average mass-uptake values mcof protein [f] andthe associated intervals CIexp (Eq. 54b) together mc [dottedline]. The continuous lines are the CIcalc given by Eq. 54a. (a)PS 204 nm; (b) PS 299 nm.


The relative variation of tR is

dtR

tR¼ q ln tR

q ln xdxxþ q ln tR

q ln ydyyþ q ln tR

q ln zdzz

ðA4Þ

or

dtR

tR¼ qr;x

dxxþ qr;y

dyyþ qr;z

dzz

ðA5Þ

with:

qr;s ¼q ln tR

q ln s; s ¼ x; y; z ðA6Þ

The coefficients qa,s and qr,s are called error propagationcoefficients. The subscripts a and r refer to absolute andrelative errors, respectively.

The SD, rtR ;p, of the retention time arising from thefluctuations of operating parameters:

rtR ;p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2

xr2x þ q2

yr2y þ q2

zr2z

qðA7Þ

and the corresponding RSD, rtR ;P=tR are such that

r tR ;PtR

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2

r;xrx

x

� �2þq2

r;y

ry

y

� �2

þq2;z

rz

z

� �2

sðA8Þ

where rs are the absolute SDs of the fluctuations of theoperating parameter s (s = x, y, z). In putting Eqs. (A7) and(A8) it is assumed that the operative parameter fluctua-tions are independent of each other.

The retention time is related to the retention ratio, R(or relative analyte average velocity), as

tR ¼t0

R¼ L

Ru¼ V0

RQðA9Þ

where t0 is the void time (or elution time of an unre-tained analyte), L the separator length, u the cross-sec-tional average carrier velocity, V0 the separator geometri-cal void volume, and Q the carrier flow rate. One assumesin the following that the void volume does not dependon the operating parameters. This may not be exactly so,especially when considering the effect of temperaturefluctuations, but this influence on V0 and on channeldimensions is likely to be of second order compared tothat on R. Hence, one gets for the error propagation coef-ficients

qa;s ¼ � V0

R2Q

� �qRqs� V0

RQ 2

� �qQqs

¼ �tR1R

qRqsþ 1

QqQqs

� �ðA10Þ

and

q;s ¼ �q ln Rq ln s

� q ln Qq ln s

ðA11Þ

Combining Eqs. (A10) and (A11) gives

qa;s ¼tR

sqr;s ðA12Þ

Due to the simplicity of this relationship, one focusesin the following on the expression of qr,s.

In fact, by combining Eqs. (A12) and (A7) one has

rtR ;P ¼ tR

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2

r;xrx

x

� �2þq2

r;y

ry

y

� �2

þq2r;z

rz

z

� �2

sðA13Þ

For the particular case where s = Q, as R generally doesnot depend on Q, one gets

qr;Q ¼ �1 ðA14Þ

SdFFF

In SdFFF, the retention ratio is related to the analyte–field interaction parameter. Equation (7) is recalled

k ¼ 6kBTpd3DqGw

¼ 6kBTpd3j qp � q1 jGw

ðA15Þ

where kB is the Boltzmann constant, T the absolute tem-perature, d the analyte particle diameter, G the centrifu-gal acceleration, w the channel thickness, and Dq the dif-ference between the particle density, qp, and the carrierdensity, ql. As in the Brownian retention mode, Rdepends solely on k (see Eq. 8) and one can express Eq.(A11) as

qr;s ¼d ln Rd ln k

d ln k

d ln s� d ln Q

d ln sðA16Þ

The first derivative in the RHS of Eq. (A16) dependsonly on the retention level. From Eq. (8) one has

d ln Rd ln k

¼ 1þ 12R

e�1=k

ð1� e�1=kÞ � k2

¼ 3þ R

12k2 �3R

ðA17Þ

and becomes equal to 1 in the high retention domain[14]. Figure A-1 shows the variations of dln R/d ln k withR. As mentioned above, the main operating parameterslikely to fluctuate are the temperature, the centrifugalacceleration, and the flow rate.

Effect of temperature fluctuations

Regarding the influence of the fluctuations of the tem-perature on the retention time (i. e., s = T), Eq. (A15) showsthat k is directly proportional to T. However, the temper-ature may also influence the density difference, Dq.Whether qp is larger than ql or not, one gets from Eq.(A15)

q ln k

q ln T¼ 1þ T

qpap � q1a1

qp � q1ðA18Þ



where ap and al are, respectively, the thermal expansioncoefficients of the particles and of the carrier, defined as

ai ¼ �1qi

qqi

qT; i ¼ p; l ðA19Þ

The Dq contribution to this coefficient vanishes if thetemperature dependences of the particle and carrier den-sities are the same.

The influence of temperature on the carrier flow ratedepends on the flow delivery system. If this system deliv-ers a constant carrier volumetric flow rate, there is noinfluence of temperature on Q. We will consider that this

is the case. But it should be noted that, if, instead, thissystem delivers a constant carrier mass flow rate, _mm = q1Q,then one would get

q ln Qq ln

¼ q ln ð _mm=q1Þq ln T

¼ aiT ðat constant _mmÞ ðA20Þ

Effect of fluctuations of centrifugal acceleration

From Eq. (A15), one gets, for the fluctuations of the cen-trifugal acceleration, G

q ln k

q ln G¼ �1 ðA21Þ

When measuring the fluctuations of the angular rota-tion velocity, x, or the number of revolutions perminute, rpm, instead of those of G, since

G ¼ RCx2 ¼ RCp

30rpm

� �2ðA22Þ

where RC is the radius of curvature of the centrifuge, onegets

q ln k

q ln rpm¼ �2 ðA23Þ

Effect of the fluctuations of the carrier flow rate

The azimuthal (or circumferential) carrier flow rate inthe gap of two infinitely long cylinders gives rise to a sec-ondary flow in the channel cross-section due to the Cori-olis and centrifugal forces above a certain flow ratethreshold [25]. Generally, this threshold is much abovetypical flow rate values in SdFFF channels. Nevertheless,in practice SdFFF channels have a rectangular cross-sec-tion and the presence of the two small edges leads to agenerally weak, but noticeable, flow recirculation whichinfluences the retention ratio [26]. We assume in the fol-lowing that the cross-sectional aspect ratio of the FFFchannel is large enough for this effect to be negligible. Inthese conditions, the error propagation coefficient forthe flow rate fluctuations is given by Eq. (A14).

Overall retention time error in SdFFF arising fromthe fluctuations of operating parameters

Combining Eqs. (A7, 14, 16, 18, 21 and 23), one gets therelative error, rr;tR , of the retention time arising from therelative fluctuations of temperature, rotation velocity,and flow rate in SdFFF

where the dlnR/d lnk term is given by Eq. (A17).The absolute error, ra;tR , of the retention time arising

from the absolute fluctuations of temperature, rotationvelocity, and flow rate is, with help of Eqs. (A12) and(A24), given by

Appendix B

Sources of error on void time

In this study, the void time t0 is derived from the channelvoid volume V0 corrected for extrachannel contribution(see Section 4), as


rr;tR ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid ln Rd ln k

� �2

1þ Tqpap � qlal

qp � ql

!2

r2r;T þ 4

d ln Rd lnk

� �2

r2r;rpm þ r2

r;Q

vuut ðA24Þ

ra;tR ¼ tR

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid ln Rd ln k

� �2 1Tþ

qpap � qlal

qp � ql

!2

r2a;T þ 4

d ln Rd lnk

� �2 r2a;rpm

rpm2þ

r2a;Q

Q 2

vuut ðA25Þ

Figure A-1. Variations of d ln R/dln k versus R in the classi-cal Brownian retention model.


t0 ¼V0

QðB1Þ

By applying the error transmission law [20] at Eq. (B1),one obtains the absolute error contribution to t0, rt0;P,coming from the absolute V0 and Q parameter fluctua-tions, rV0 and rQ, respectively

rt0;P

t0¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirV0

V0

� �2

þ rQ

Q

� �2s

ðB2Þ

However, this error is not specifically exploited as doneabove for the tR, since t0 was determined according tothe Eq. (B1), under different conditions of the assumedindependent variable 1/Q. For the hypothesis of the ordi-nary regression, 1/Q is assumed to be not affected byerrors, or better, its errors are reversed into those of thedependent variable t0.

According to the least square theory Massart et al. [2]applied to regression line, the residual SD computed isan estimate of the pure error onto rt0

rt0 L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

ðt0;exp � t0;calcÞ2

n� 2

vuutðB3Þ

where n is the number of t0 determinations and t0,exp andt0,calc are the experimental and calculated void times overthe regression line. It must be pointed out that if a differ-ent strategy of experimental measurement of the quanti-ties DtR and t0 is followed, different conditions as for theerrors hold true. For example, if DtR and t0 are measuredon the same fractogram, their errors are not independ-ent and the covariance between these two quantitiesmust be considered.

t0 Error: Experimental determination

Several t0 measurements in the range of 1.5 f Q f 2.5 mL/min were performed (see under Section 4), and (t0;1/Q)data were processed according to Eq. (B-1) regression. TheV0 value of 4.30 l 0.12 mL, statistically equal to the nomi-nal value V0 = 4.30 mL, given by the manufacturer, wasobtained. Also st0 = 0.02 min according to Eq. (B-3) wasobtained. Consequently, for 2 f t0 f 3 min, the t0 relativeerror was thus in the range: 0.007 f (rt0 /t0) f 0.01 min.This error contains two contributions, that arising fromthe experimental determination ðrt0;NÞ and that fromparameter fluctuation ðrt0;PÞ. In turn this latter one,according to Eq. (B2), is built up by two components. Thefirst component is rV0=V0, arising from the thermalexpansion due to temperature fluctuation of l28C, isapproximately equal to 2610–5. The second componentrQ =Q is f 0.005 (see above under “fluctuation of theexperimental parameters”, for 2 f Q f 3 mL/min). Conse-quently, the above referred value of st0 = 0.02 min isroughly determined half by the noise component rt0;N,i. e., by the experimental determination of direct meas-urement of t0, and half by Q fluctations. st0 is significantly

lower than rtR because the void time is significantlylower than tR of a retained component.

In practice, we can consider two possibilities for the t0

determination and the rt0=t0 value:(i) the t0 value is directly measured; its experimental

error (Eq. B3) is: rt0 L st0 = 0.02 min, and the relative erroris thus: rt0=t0 L 0.01 for 2 f t0 f 3 min.

(ii) the t0 value is evaluated from V0 = 4.30 mL, assumedas the deterministic value (its fluctuation from tempera-ture fluctuation is negligible), and from the Q value, byusing Eq. (B1). In this case, the relative error contributionis only that coming from Q fluctuation, i. e.,rt0=t0 L rQ =Q f 0.005 in the worst case (long term fluctu-ations).

For the subsequent determinations, the first proce-dure was followed, i.e., t0 was determined directly byunretained compound injection. Nonetheless, we can seethat the second procedure gives better precision in t0

determination than the first one.

Appendix C

Dependence of the noise error rtR ;N on tr

Referring to the discussion presented in Section 5.1.1,another way is exploited here to present the global coher-ence of the data, i. e., the correctness of the rtR ;N estima-tion method (Eqs. 39 –41) and of the interpretationmodel (Eq. 35). In fact, in Fig. C-1 the rtR ;N values for thesame cinj and Vinj quantities are reported versus tR for thecase of PS 204 nm. One can see that the trends of rtR ;N ver-sus tR, for constant cinj values, are linear. This can be

explained on the basis of the following equation

rtR ;N ¼ tR

ffiffij

p ffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffi2ppq ffiffiffiffiffiffiffiffiffiffiffi

V0

Vinj

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

kdcinj

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

RN3=2

r !ðC1Þ

obtained by rearranging Eq. (35), where the expression inbrackets is an almost constant quantity. In fact in allthese experiments, the quantities V0, Vinj, kd, cinj, and R areconstant (same injected species). Moreover, both the aver-age j values are almost constant for the different cinj val-


Figure C-1. Dependence of the noise error rtR ;N on tR forbare PS 204 nm: constant injected concentration plots.


ues (see j values in Table 1), and the N values change onlywithin 5%, for the experimental conditions exploitedhere. Consequently, ðrtR ;N

ffiffiffiffiffiffifficinjp Þ versus tR is expected to be

linear, as confirmed in Fig. C-2. Similarly, by deriving anestimated j value from the slope on the basis of Eq. (C1),an average value of j is estimated, jcal;slope = (4.7 l 0.1) mVwhich can be compared to the experimental ones derivedfrom the data of Table 1 ðjcal = (4.5 l 0.1) mV versus jbaseline

= (4.2 l 0.5) mV).

References

[1] Dondi, F., Martin, M., in: Schimpf, M., Caldwell, K. D., Giddings,J. C. (Eds.), Field-Flow Fractionation Handbook, Wiley-Interscience,New York 2000, pp. 103 – 132.

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Figure C-2. Dependence of the noise error rtR ;N on tR forbare PS 204 nm: normalized concentration plot.

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Precision in differential field-flow fractionation: A chemometric study