Article pubs.acs.org/ac
Quantitative Analysis in Capillary Electrophoresis: Transformation of Raw Electropherograms into Continuous Distributions Joseph Chamieh,† Michel Martin,‡ and Hervé Cottet*,† †
Institut des Biomolécules Max Mousseron (IBMM, UMR 5247 CNRS-Université de Montpellier 1, Université de Montpellier 2), Place Eugène Bataillon CC 1706, 34095 Montpellier Cedex 5, France ‡ Ecole Supérieure de Physique et de Chimie Industrielles, Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH, UMR 7636 CNRS, ESPCI-ParisTech, Université Pierre et Marie Curie, Université Paris-Diderot), 10 rue Vauquelin, 75231 Paris Cedex 05, France S Supporting Information *
ABSTRACT: Quantitative analysis in capillary electrophoresis based on time-scale electropherograms generally uses timecorrected peak areas to account for the differences in apparent velocities between solutes. However, it could be convenient and much more relevant to change the time-scale electropherograms into mass relative distribution of the effective mobility or any other characteristic parameter (molar mass, chemical composition, charge density, ...). In this study, the theoretical background required to perform the variable change on the electropherogram was developed with an emphasis on the fact that both x and y axes should be changed when the time scale electropherograms are modified to get the distributions. Applications to the characterization of polymers and copolymers by different modes of capillary electrophoresis (CE) are presented, including the molar mass distribution of poly-L-lysine oligomers by capillary gel electrophoresis (CGE), molar mass distribution of end-charged poly-L-alanine by free solution CE, molar mass distribution of evenly charged polyelectrolytes by CGE, and charge density distribution of variously charged polyelectrolytes by free solution CE.
C
followed by an overlaying of the electropherograms based on a marker peak. This transformation allowed a better identification of the solutes and the direct comparison of the peak areas. Later, Schmitt-Kopplin et al.2,3 proposed the transformation of the time-scale electropherogram into a mobilityscale electropherogram. The peak integration showed an improvement of the precision in the mobility scale compared to the time or the time-corrected scales.2 The transformation in mobility scale has since been used by many research groups.4−12 Other groups made transformations from the time scale into a physicochemical characteristic scale such as the degree of polymerization,6,13 the quantity of electric charge,14 and migration time ratios.15 Most of these transformations were made for a qualitative purpose for the electropherogram to be easily interpretable by inspection in terms of variations in the chemical structure of the various analytes. Besides, there is a growing need for quantitative analysis, especially for biopolymers and synthetic polymers to determine the distribution of the parameter of interest (charge density, molar mass, chemical composition, ...).
apillary electrophoresis (CE) is a powerful separation technique allowing efficient analysis of a wide variety of (macro)molecules. CE regroups a great number of modes that can be implemented according to the nature of the analytes and the goal of the separation, e.g., free solution capillary electrophoresis, micellar capillary electrophoresis, gel capillary electrophoresis, capillary isoelectrofocusing, and capillary isotachophoresis. In capillary zone electrophoresis (CZE), where only one continuous background electrolyte is used, the separation is based on the differential migration of the analytes under the influence of a constant electric field. In contrast to chromatography where all the analytes have the same velocity in front of the detector connected to the column outlet, the apparent migration velocity of the analytes in the sensing zone of the on-column detector of the CE system is directly proportional to the apparent mobility and is thus different for different analytes. Furthermore, the apparent mobility depends not only on the electrophoretic effective mobility of the analyte, but also on the electroosmotic mobility that may affect the repeatability of the migration times (and of the peak area), due to possible electroosmotic flow (EOF) fluctuations from run to run. To remedy the EOF fluctuations and the differences in apparent velocity, Mammen et al.1 proposed the transformation of the timebased electropherogram into a 1/time-scale electropherogram © 2015 American Chemical Society
Received: September 25, 2014 Accepted: December 16, 2014 Published: January 8, 2015 1050
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Correction Due to the Difference in Velocity of the Analytes. It is assumed here that the sample contains a large number of analytes which differ by some physicochemical property P and that the analyte elution order is a monotonous function of this property. Furthermore, the dispersion of the individual analytes due to their migration along the separator is assumed negligible in comparison with the dispersion due to the distribution of the property within the sample. In these conditions, the function g(t) of the elution time, t, such that g(t) dt represents the fraction of sample amount eluting between times t and t + dt, can be expressed as
It is the objective of this work to provide a theoretical frame describing the modifications to the x- and y-scales of a raw electropherogram that should be performed to transform the time axis into another variable (mobility, μ, or any other physicochemical parameter, noted P, that is characteristic of the solute) while simultaneously keeping peak areas that are representative of the relative amounts of the analytes in the sample. In particular, this work aims at describing how one can obtain the continuous distribution in P for doing quantitative analysis in the case of continuous distributions of analytes. Similar aims have been attempted in size exclusion chromatography (SEC)16 and in field-flow fractionation.17 Apart from dealing with CZE, the specificity of the present work concerns the use of in-column rather than out-column detectors. Since CZE has proven to be a technique complementary to SEC18−25 for polymer analysis, four different examples of polymer characterization by CZE are considered in this work. Two examples deal with the free solution capillary zone electrophoresis mode with (i) the determination of the molar mass distribution of end-charged oligomers6 and (ii) the determination of the charge density distribution of evenly charged polyelectrolytes.12 The two other examples deal with the capillary gel electrophoresis mode for the sizebased separation of oligomers26 or longer polymers.27
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g (t ) d t =
dm mtot
(1)
where dm is the sample amount eluting in this infinitesimal time interval and mtot the total sample amount. The amount dm passing through the detector in the time interval dt occupies, at time t, a volume dV of the column such that dm = c(t ) dV = c(t )Ac dz
(2)
where Ac is the column cross-sectional area, dz the length of column occupied, and c(t) the analyte concentration in the sensing zone of the detector at time t. Hence, from eqs 1 and 2
THEORY
Starting from an experimental time-scale electropherogram S(t), where S is the detector signal at time t, one can transform it into a mobility-scale electropherogram f(μ) or directly into a distribution j(P) representative of a physicochemical characteristic P of the solute (e.g., molar mass, degree of polymerization, chemical composition, ...). This general scheme is presented in Figure 1. The change in variable from a time-scale to a
g (t ) =
c(t )Ac dz mtot dt
(3)
dz/dt is the migration velocity of the analyte eluting at time t. When the migration velocity of the analyte is constant all along its migration from the column inlet to the detector, it is then equal to Ld/t, where Ld is the column length from inlet to detector. Hence g (t ) =
AcLd c(t ) V c(t ) = d mtot t mtot t
(4)
where Vd is the column volume from inlet to detector. When the detector is a concentration detector, its signal S(t) is expressed as S(t ) = k(t ) c(t )
(5)
where k is the response factor of the detector for the analyte eluting at time t. Then eq 4 becomes g (t ) =
Vd S(t ) mtot k(t )t
(6)
If the response factor is the same for all analytes contained in the sample, it does not vary with time, which gives g (t ) = Figure 1. General scheme for changing a time-scale electropherogram into mobility-scale or P-scale distributions. P is the parameter of interest which is a characteristic of the polymer sample. The raw time-scale electropherogram is first corrected from the differences in analyte velocities (path 1 from S(t) to h(t)). The h(t) time-corrected electropherogram can be changed to get the P-scale distribution using a calibration curve relating P to t (path 2) or the mobility-scale distribution using the relationship between μ and t (path 3) and then from the mobility-scale to the P-scale distribution using a calibration curve relating P to μ (path 4).
Vd h(t ) h(t ) = kmtot ∫ h(t ) dt
(7)
S(t ) t
(8)
with
h(t ) =
It should be noted at this point that the hypothesis of a constant response factor for all analytes is associated with a specification of the kind of concentration that c is representing. Indeed, for polymeric samples, with typical UV photometers, if the same chromophore is present in each repeating unit of the polymer chain, the signal becomes proportional to the number of repeating units and c becomes a mass concentration. If, instead, the chromophores are located in the end groups of the chains, the
mobility-scale electropherogram, or from a time-scale to a P-scale electropherogram (or distribution) requires modifications of both the x- and y-axes. 1051
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that a non-normalized distribution can be obtained from the electropherogram S(t) by replacing the abscissa axis t by the axis μ according to eq 11 and the ordinate axis S by St.
signal is proportional to the number of (macro)molecules, irrespective of their molar mass, and c is then a molar (or number) concentration. Consequently, according to eqs 7 and 8, when the migration of the analytes occurs in steady conditions (sometimes called “uniform” conditions) with a concentration-sensitive detector giving a constant response factor for all analytes, the detector signal S(t) must be divided by the elution time to obtain an amount-based distribution g(t) of the elution times, this distribution being a number distribution when c is a molar concentration and a mass distribution when c is a mass concentration (path 1 in Figure 1). Transformation from Time-Scale to Mobility-Scale or P-Scale Distributions. The amount-based distribution of the elution times g(t) can be transformed into the amount-based distribution j(P) relative to a characteristic parameter P of the solutes (path 2 in Figure 1). Writing that the fractional sample amount eluting between times t and t + dt is the same as that eluting between P and P + dP, one gets | g ( t ) d t | = | j( P ) d P |
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APPLICATIONS TO THE CHARACTERIZATION OF POLYMER SAMPLES BY CZE As mentioned in the introduction, four different experimental cases relative to the characterization of synthetic polymer samples are considered in this work (either in free solution or in gel electrophoresis). The chemical structures of the different polymer samples studied in this work are given in Figure S1 (Supporting Information). Poly- L -lysine (PLys), poly(styrenesulfonate) (PSS), and poly(acrylamide-co-2-acrylamido2-methyl-1-propanesulfonic acid) (PAMAMPS) are evenly charged polyelectrolytes, while poly-L-alanine (PAla) is an endcharged polymer. The characteristics of the samples as well as the parameter P of interest in this work are gathered in Table S1 (Supporting Information). The experimental data (electropherograms) used in this work were derived from previously published data. The objective of this work is to apply the transformations developed in the theoretical section to get the continuous distributions j(P) which are characteristics of the polymer samples. For the aforementioned polymers considered in this work, the UV detector is a mass-sensitive concentration detector since the repeating unit is responsible for the UV absorbance. Size-Based Separation of Evenly Charged Oligomers by Capillary Gel Electrophoresis. In acidic conditions, PLys’s are fully protonated and have similar charge-to-mass ratios. Consequently, they cannot be separated in free solution CZE, and capillary gel electrophoresis should be used instead.26 The experimental electropherogram S(t) (Figure 2A) displays an oligomeric separation of the polymer sample owing to the sieving effect of the dextran solution contained in the background electrolyte. Peak identification was obtained by spiking the sample with a pure tetralysine oligomer, the identification of the other peaks being obtained by incrementing the degree of polymerization, DP, by 1 unit between two successive peaks. From this raw electropherogram, the final goal is to get the distribution in DP (Figure 2B) and to calculate the number- and weight-average DPs of the polymer sample. For that, we used a correlation between DP and the migration time according to a log-lin equation (see the calibration curve in Figure 2A):
(9)
which gives j( P ) =
g (t ) dP dt
=
S (t ) t
dP dt
Vd kmtot
(10)
Replacing P by the effective mobility of the solute, the distribution in electrophoretic mobility f(μ) can be obtained (path 3 in Figure 1) using the relationship between the EOF and solute migration times, teo and t, and the mobility: μ=
Ld ⎛ 1 1⎞ ⎟ ⎜ − E ⎝t teo ⎠
(11)
where E is the applied electric field. Thus L dμ = d2 dt Et
(12)
and, combining eqs 7, 8, 10, and 12 leads to f (μ) =
EAc tS(t ) kmtot
(13)
Similarly, the j(P) distribution can be derived from f(μ) (path 4 in Figure 1) using the following equation:
j( P ) =
ln(DP) = at + b
Setting P = DP and following the general scheme described by path 2 in Figure 1, the variable change from t to DP is given by
f (μ) dP dμ
(15)
g (t ) dt = j(DP) dDP
(14)
To achieve a polymer sample characterization, the final goal is to obtain the continuous distribution j(P) relative to the parameter of interest P from the electropherogram S(t). This requires a relationship (generally obtained by calibration using standards) between the parameter P and either (i) the migration time or (ii) the effective mobility μ. In the first case (P−t relationship), j(P) is directly obtained from the raw signal S(t) (path 2, Figure 1). In the second case (P−μ relationship), j(P) is obtained in two steps (paths 3 and 4, Figure 1). It is worth noting that the transformations of the raw electropherogram proposed in Figure 1 require changes in both the x- and the y-axes. Transforming the x-axis from time to μ or to P without changing the y-axis would lead to biased quantitative results in the μ or P distributions. When an electrophoretic mobility distribution is searched for, eq 13 shows
(16)
where g(t) is given by eqs 6 and 7. Therefore, the DP distribution is obtained using the following equation and is displayed in Figure 2B: j(DP) =
g (t ) Vd S(t ) = a DP akmtot t DP
(17)
From this weight distribution in DP, the weight- and numberaverage DPs, noted DPw and DPn, respectively, can be calculated by integration according to DP
DPw =
∫DP max j(DP)DP dDP min
DPmax
∫DP
min
1052
j(DP) dDP
=
DPmax
∫DP
min
j(DP)DP dDP (18) DOI: 10.1021/ac503789s Anal. Chem. 2015, 87, 1050−1057
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from 1 to N, eqs 18 and 19 can be rewritten in their discretized forms: N
DPw =
∑i = 1 j(DP)DP(DP i i i + 1 − DP) i N
∑i = 1 j(DP)(DP i i + 1 − DP) i
(20)
N
DPn =
∑i = 1 j(DP)(DP i i + 1 − DP) i N
∑i = 1
j(DP) i (DPi + 1 DPi
− DP) i
(21)
where DPi is associated with ti by means of eq 15. Calculations according to eqs 20 and 21 lead to DPw and DPn values of 9.5 and 7.8, respectively, which allows calculation of a polydispersity index (PI = DPw/DPn) of 1.22. As a matter of comparison, when the calculations are made using S(t) directly for the expression of j(DP) in eqs 20 and 21, DPw and DPn are overestimated to respectively 12.2 and 9.7 with a PI of 1.26. This overestimation is due to the fact that, in eq 17, the detector signal S(t) should be divided by tDP, which decreases the relative contribution of the largest oligomers compared to the situation where only S(t) is considered. The general expression of j(P), given by eq 10, and the derived particular expression j(DP) of eq 17 rely on the assumption of a continuous distribution of the parameter P or of a quasicontinuous distribution when the mean value of P is much larger than the difference between two successive values of P. However, in the particular case of the oligomeric PLys mixture exhibited in Figure 2, this is not the case and displaying finite values of j(DP) for noninteger values of DP does not have a physical meaning. Therefore, the graph of Figure 2B should be regarded not as a genuine distribution, which has to be discontinuous, but as a representation in which the area of a given peak around a given integer value of DP provides an estimate of the relative amount of the oligomer species with that DP value. The normalized distribution J(DP) of the so-estimated relative amounts of the PLys oligomers is plotted for integer values of DP in Figure 2C. It is compared with the exact distribution obtained as J(DPk ) = Figure 2. Time-scale electropherogram (A) and weight distribution of the DP (B) for an evenly charged poly-L-lysine sample obtained by capillary gel electrophoresis. The inset in (A) represents the log-lin correlation between the DP and the migration time: ln DP = 0.34t − 8.19, r = 0.9991, where t is in minutes. Experimental data adapted from ref 26. (C) Comparison of the peak surface areas obtained from the time-scale distribution and the DP distribution. For the sake of clarity, data points from the DP distribution (red dots) are shifted by 0.4 DP unit to the right. See the Supporting Information for experimental conditions.
∫ DPn =
DPmax j(DP) M DPmin
∫
DPmax
=
∫DP ∫
∫
dDP =
j(DP) dDP = dDP
DPmax j(DP) DPmin M 0 DP
∫
dDP
min
DPmax j(DP) DP DPmin
DPmax j(DP) DP DPmin M 0 DP
dDP
DPn =
dDP
∫
dDP
∑k DPk (Ak /t R, k) ∑k (Ak /t R, k)
(23)
∑k (Ak /t R, k) ∑k
Ak / t R, k DPk
(24)
From the distribution in Figure 2C and using eqs 23 and 24, the obtained values of DPw, DPn, and PI are respectively 9.3, 7.8, and 1.20, which are in accordance with the aforementioned values, showing the advantage of the proposed method by allowing the analysis of the raw data S(ti) and without having to integrate each peak separately. Separation of End-Charged Oligomers by Free Solution Capillary Zone Electrophoresis. PAla’s are end-charged
1 DPmax j(DP) DP DPmin
(22)
where DPk is the degree of polymerization of the species of the kth peak, Ak its peak area in the time-scale electropherogram, and tR,k its migration time, the sum extending to all species present in the mixture. The agreement between the two distributions in Figure 2C is excellent. From the exact distribution (eq 22) DPw, DPn, and PI can be calculated using the following equations: DPw =
DPmax j(DP) DP M DPmin
Ak /t R, k ∑k (Ak /t R, k)
(19)
where M0 is the monomer molar mass and M the molar mass of the species of degree of polymerization DP. Since, in modern CE instruments, the detector signal is recorded in digitalized form of N data points S(ti) with i varying 1053
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between DP and the migration time, as generally observed in the literature for end-charged solutes:28,29
polypeptides (one positive charge on the protonated N-terminal group) and can be separated by free solution CZE since the oligomers have different charge to mass (or charge to size) ratios. The separation of these low-solubility hydrophobic oligomers was only possible in a hexafluoro-2-propanol (HFIP)/water mixture with excellent resolution, as shown in Figure 3A.6 SEC
(25)
DP = at + b
The j(DP) distribution shown in Figure 3B is obtained using the following equation: j(DP) =
g (t ) dDP dt
=
Vd S(t ) akmtot t
(26)
DPw and DPn can be calculated by integration of the DP distribution using eqs 20 and 21. This leads to a DPw of 17.3, a DPn of 10.7, and a polydispersity index of 1.62. When j(DP) is assumed proportional to S(t), instead of proportional to S(t)/t as shown in eq 26, the average degrees of polymerization and the polydispersity index are strongly overestimated (DPw = 39.1; DPn = 15.3; PI = 2.55). For the two aforementioned oligomeric distributions, the approach used in the publications6,26 for the determination of the numerical data (DPw, DPn, PI) consisted in the integration of each peak in the raw electropherogram, the area of which was time-corrected according to eq 22. The J(DP) distribution resulting from this correct approach is shown in Figure 3C together with that obtained from the j(DP) distribution of Figure 3B by attributing the area of each peak to the closest integer DP value, as in the precedent section. Again the agreement between these two distributions is excellent. However, the approach based on the integration of the continuous distribution presented in Figure 3B was less time-consuming since the integration was not performed peak by peak. Size-Based Separation of Evenly Charged Poly(styrenesulfonates) by Capillary Gel Electrophoresis. In the case of large evenly charged polyelectrolytes, the size-based electrophoretic separation can only be obtained in the presence of a gel (or entangled polymer solution). The separation mechanism is known as biased reptation,19,30−34 for which the effective electrophoretic mobility of the solute (polyelectrolyte) depends on the molar mass M according to the simplified equation35−37 a |μ| = b + c(E) (27) M where a, b, and c(E) are constants relative to M. c(E) depends on the electric field E (which is constant during the separation process). In the nonbiased reptation regime, at infinitely low electric field, b tends to 1 and c(E) tends to 0. In the presence of an electric field, b is in the 0−1 range and c(E) increases with E. Figure 4A shows the separation of six standards of evenly charged poly(styrenesulfonate)s having the same charge to mass ratio but different molar masses.35 The separation is due to the sieving effect of the entangled polymer solution (hydroxyethyl cellulose). Experimentally, in the conditions given in the caption of Figure 4, the calibration curve obtained using these standards is displayed in the inset of Figure 4B. The molar mass M is thus related to the effective mobility μ by
Figure 3. Time-scale electropherogram (A) and weight distribution of the DP (B) for an end-charged polyalanine sample obtained by free solution capillary electrophoresis. The inset in (A) represents the linear correlation between the DP and the migration time: DP = 1.36t − 6.29, r = 0.9997, where t is in minutes. Experimental data adapted from ref 6. (C) Comparison of the peak surface areas obtained from the timescale distribution and the DP distribution. For the sake of clarity, data points from the DP distribution (red dots) are shifted by 0.4 DP unit to the right. See the Supporting Information for experimental conditions.
M=
⎛ μ − c ⎞−1/ b ⎜ ⎟ ⎝ a ⎠
(28)
a, b, and c are constants and depend on the experimental conditions. Numerical values are given in the caption of Figure 4, where M is expressed in grams per mole and μ is in 10−9 m2 V−1 s−1. To get the molar mass distribution of a polymer sample, one has first to
analysis of these polymers and in these conditions is not possible due to the poor solubility of these polypeptides because of their aggregation due to intra- and intermolecular interactions. In such electrophoretic conditions, a linear correlation is obtained 1054
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∫M
Mw =
j( M ) M d M
(30)
min
1
Mn =
M max j(M ) M M min
∫
dM
(31)
These calculations can be applied for any polymer standard (for each peak of Figure 4C). For example, the peak shown in the inset of Figure 4C represents the PSS 41000 sample, and the corresponding weight-average and number-average molar masses are respectively 39370 and 38890 g·mol−1 with a polydispersity index of 1.01. Of course, these average M values are close to the nominal value since the instrument was calibrated with the standards that are analyzed. However, it is noticeable that the methodology developed above allows the determination of PI values very close to 1. This cannot be obtained by SEC but is made possible thanks to the high resolution of CE. That polymer standards are sometimes of a much lower polydispersity than suspected from SEC analysis was already observed in thermal field-flow fractionation due to the high selectivity of this separation method.38,39 Charge-Density-Based Separation of Evenly Charged Polyelectrolytes by Free Solution Capillary Zone Electrophoresis. PAMAMPS copolymers of different charge densities (different molar fractions in charged monomers, f, ranking between 3% and 100%) have electrophoretic mobilities which are independent of the molar mass in free solution electrophoresis (free-draining behavior of long polyelectrolyte chains).12 The electrophoretic mobility of the polyelectrolytes depends only on f as shown in Figure 5A. The temporal electropherogram was next transformed into a mobility-scale electropherogram using eq 13, as displayed in Figure 5B. The relationship between f and μ shown in the inset of Figure 5B can be fitted using the following empirical law:
f = A exp(Bμ)
(32)
Equation 32 can then be used to get the f distribution using the following equation:
Figure 4. Time-scale electropherogram (A), mobility-scale distribution (B), and weight distribution of the molar mass (C) for an evenly charged poly(styrenesulfonate) mixture obtained by capillary gel electrophoresis. The inset in (B) represents the correlation between the molar mass M and the electrophoretic mobility μ: M = ((|μ| − c)/a)−1/b, a = −461.85, b = 1.17 × 10−2, c = 495.23, r = 0.9993, where μ is in 10−9 m2 V−1 s−1. The inset in (C) displays the molar mass distribution of a standard of PSS (41 × 103 g/mol). PSS molecular mass × 103 g mol−1: 1, 16; 2, 41; 3, 88; 4, 177; 5, 350, and 6, 990. Experimental data adapted from ref 35. See the Supporting Information for experimental conditions.
j(f ) =
EAc f (μ) tS(t ) = AB exp(Bμ) kmtot AB exp(Bμ)
(33)
From this transformation, a weight-average charge density can be defined and calculated according to fw =
∫f
fmax
j( f ) f d f
min
(34)
The standard deviation reflecting the width of the distribution can be calculated according to transform the time-scale electropherogram into a mobility-scale electropherogram (path 3 in Figure 1) using eq 13, as displayed in Figure 4B. Then the transformation from the mobility scale to the molar mass scale (path 4 in Figure 1) is performed according to eq 14, giving in combination with eq 28 j( M ) =
EAc ⎛ μ − c ⎞(b + 1)/ b ⎟ ab⎜ tS(t ) kmtot ⎝ a ⎠
σf 2 =
∫ j(f )(f − fw )2 df
(35)
The inset in Figure 5C represents the charge density distribution of the f = 20% PAMAMPS sample. From this distribution, it is then straightforward to get the standard deviation in f using eq 35. This standard deviation is about 10% of the weight-average value. From Table S1 (Supporting Information), it can be seen that for all the PAMAMPS samples the polydispersity in charge density remains rather low (standard deviation below 10% of the average charge density). The standard deviation value obtained by this approach is a valuable characteristic parameter giving an idea about the polydispersity in charge of the samples. In the
(29)
From the molar mass distribution presented in Figure 4C, one can then calculate the weight- and number-average molar masses, as well as the PI according to the following equations: 1055
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distribution of this parameter or one can get this distribution by passing through a mobility distribution. In any case, it should be kept in mind that both x- and y-axes should be changed when the time-scale electropherograms are modified to get the DP distributions.
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ASSOCIATED CONTENT
S Supporting Information *
Experimental details and polymer characteristics (Figure S1, Table S1). This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*Phone: +33-4-6714-3427. Fax: +33-4-6763-1046. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS H.C. gratefully acknowledges the support from the Institut Universitaire de France.
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REFERENCES
(1) Mammen, M.; Colton, I. J.; Carbeck, J. D.; Bradley, R.; Whitesides, G. M. Anal. Chem. 1997, 69, 2165−2170. (2) Schmitt-Kopplin, P.; Garmash, A.; Kudryavtsev, A. V.; Menzinger, F.; Perminova, I.; Hertkorn, N.; Freitag, D.; Petrosyan, V.; Kettrup, A. Electrophoresis 2001, 22, 77−87. (3) Schmitt-Kopplin, P.; Menzinger, F.; Freitag, D.; Kettrup, A. LC·GC Eur. 2001, 14, 384−388. (4) Welch, C. F.; Hoagland, D. A. Polymer 2001, 42, 5915−5920. (5) Petersen, N.; Hansen, S. Electrophoresis 2012, 33, 1021−1031. (6) Miramon, H.; Cavelier, F.; Martinez, J.; Cottet, H. Anal. Chem. 2010, 82, 394−399. (7) Galbusera, C.; Thachuk, M.; Lorenzi, E.; Chen, D. Anal. Chem. 2002, 74, 1903−1914. (8) Fang, N.; Zhang, H.; Li, J.; Li, H. W.; Yeung, E. Anal. Chem. 2007, 79, 6047−6054. (9) Souaïd, E.; Cottet, H. Electrophoresis 2005, 26, 3300−3306. (10) Lalwani, S.; Venditto, V.; Chouai, A.; Rivera, G.; Shaunak, S.; Simanek, E. Macromolecules 2009, 42, 3152−3161. (11) Hoagland, D. A.; Arvanitidou, E.; Welch, C. Macromolecules 1999, 32, 6180−6190. (12) Cottet, H.; Biron, J. P. Macromol. Chem. Phys. 2005, 206, 628− 634. (13) Oudhoff, K.; Schoenmakers, P.; Kok, W. J. Chromatogr., A 2003, 985, 479−491. (14) Tetsuo, I.; Jun, K.; Yasuyuki, K. J. Chromatogr., A 1998, 810, 183− 191. (15) Ju, Y.; Sahana, B.; Hage, D. S. J. Chromatogr., A 1996, 735, 209− 220. (16) Tung, L. H. J. Appl. Polym. Sci. 1966, 10, 375−385. (17) Dondi, F.; Martin, M. Physicochemical Measurements and Distributions from Field-Flow Fractionation. In Field-Flow Fractionation Handbook; Schimpf, M. E., Caldwell, K., Giddings, J. C., Eds.; WileyInterscience: New York, 2000; pp 103−132. (18) Kok, W.; Stol, R.; Tijssen, R. Anal. Chem. 2000, 72, 468A−476A. (19) Engelhardt, H.; Grosche, O. Adv. Polym. Sci. 2000, 157, 189−217. (20) Engelhardt, H.; Martin, M. Adv. Polym. Sci. 2004, 165, 211−247. (21) Cottet, H.; Simó, C.; Vayaboury, W.; Cifuentes, A. J. Chromatogr., A 2005, 1068, 59−73. (22) Maniego, A. R.; Ang, D.; Guillaneuf, Y.; Lefay, C.; Gigmes, D.; Aldrich-Wright, J. R.; Gaborieau, M.; Castignolles, P. Anal. Bioanal. Chem. 2013, 9009−9020.
Figure 5. Time-scale electropherogram (A), mobility-scale distribution (B), and distribution of the chemical charge density f (C) for an evenly charged PAMAMPS mixture obtained by free solution capillary electrophoresis. The inset in (B) represents the correlation between f and the electrophoretic mobility μ: f = 1.24 exp(0.12μ), r = 0.9986, where μ is in 10−9 m2 V−1 s−1. The inset in (C) displays the f distribution of the 20% PAMAMPS sample. Samples: 1, PAMAMPS f = 3%; 2, f = 10%; 3, f = 20%; 4, f = 30%; 5, f = 40%; 6, f = 55%; 7, f = 70%; 8, PSS Mw = 3.33 × 105 g mol−1, Mw = 1.45 × 105 g mol−1, and Mw = 2.9 × 104 g mol−1. Experimental data adapted from ref 12. See the Supporting Information for experimental conditions.
other cases (Figures 2−4), the Mw/Mn ratio was used as an estimation of the molar mass polydispersity.
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CONCLUSION Electropherogram transformation from a time scale to a mobility scale was shown to be an excellent solution to overcome the EOF fluctuations.2 In this paper, it was shown that when the transformation is fully conducted, several quantitative physicochemical parameters can be obtained from the initial temporal experimental electropherogram, namely, in the case of polymer analysis. Also, it was shown that a direct correlation between the migration time and a physicochemical parameter can lead to a 1056
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Article
Analytical Chemistry (23) Mnatsakanyan, M.; Thevarajah, J.; Roi, R.; Lauto, A.; Gaborieau, M.; Castignolles, P. Anal. Bioanal. Chem. 2013, 6873−6877. (24) Liyanage, A. U.; Ikhuoria, E. U.; Adenuga, A. A.; Remcho, V. T.; Lerner, M. M. Inorg. Chem. 2013, 52, 4603−4610. (25) Cottet, H.; Gareil, P. Separation of Synthetic (Co)polymers by Capillary Electrophoresis Techniques. In Methods in Molecular Biology; Schmitt-Kopplin, P., Ed.; Humana Press: Clifton, NJ, 2008; Vol. 384, pp 541−567. (26) Collet, H.; Souaid, E.; Cottet, H.; Deratani, A.; Boiteau, L.; Dessalces, G.; Rossi, J. C.; Commeyras, A.; Pascal, R. Chem.Eur. J. 2010, 16, 2309−2316. (27) Cottet, H.; Gareil, P.; Viovy, J. L. Electrophoresis 1998, 19, 2151− 2162. (28) Meagher, R.; Won, J. I.; McCormick, L.; Nedelcu, S.; Bertrand, M.; Bertram, J.; Drouin, G.; Barron, A.; Slater, G. Electrophoresis 2005, 26, 331−350. (29) Vreeland, W.; Desruisseaux, C.; Karger, A.; Drouin, G.; Slater, G.; Barron, A. Anal. Chem. 2001, 73, 1795−1803. (30) Slater, G.; Noolandi, J. Biopolymers 1989, 28, 1781−1791. (31) Slater, G.; Rousseau, J.; Noolandi, J. Biopolymers 1987, 26, 863− 872. (32) Slater, G.; Noolandi, J. Phys. Rev. Lett. 1985, 55, 1579−1582. (33) Mitnik, L.; Salomé, L.; Viovy, J.; Heller, C. J. Chromatogr., A 1995, 710, 309−321. (34) Sudor, J.; Novotny, M. Anal. Chem. 1994, 66, 2446−2450. (35) Cottet, H.; Gareil, P. J. Chromatogr., A 1997, 772, 369−384. (36) Grossman, P.; Soane, D. Biopolymers 1991, 31, 1221−1228. (37) Dolník, V. J. Microcolumn Sep. 1994, 6, 315−330. (38) Schimpf, M. E.; Myers, M. N.; Giddings, J. C. J. Appl. Polym. Sci. 1987, 33, 117−135. (39) Reschiglian, P.; Martin, M.; Contado, C.; Dondi, F. J. Liq. Chromatogr. Relat. Technol. 1997, 20, 2723−2739.
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