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Rapid determination of hydrodynamic radii beyond the limits of Taylor dispersion
Journal of Chromatography A, 1472 (2016) 66–73
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Rapid determination of hydrodynamic radii beyond the limits of Taylor dispersion Seyi Latunde-Dada a,∗,1 , Rachel Bott a,1 , Jack Crozier a,1 , Markos Trikeriotis a,1 , Oksana Iryna Leszczyszyn a,1 , David Goodall b,1 a b
Malvern Instruments Ltd, Grovewood Road, Malvern, Worcestershire, WR14 1XZ, UK Paraytec Ltd, York House, Outgang Lane, Osbaldwick, York, YO19 5UP, UK
a r t i c l e
i n f o
Article history: Received 31 March 2016 Received in revised form 15 August 2016 Accepted 12 October 2016 Available online 13 October 2016 Keywords: Taylor dispersion analysis Aggregation Diffusion coefficients
a b s t r a c t Taylor dispersion analysis (TDA) is an absolute method for determining the diffusion coefficients, and hence the hydrodynamic radii, of particles by measuring the dispersion in a carrier medium flowing within a capillary. It is applicable under conditions which allow the particles to radially diffuse appreciably across the cross-section of the flow before the measurement and therefore implies long measurement times are required for large particles with small diffusion coefficients. In this paper, a method has been developed by which the diffusion coefficients of large particles can be rapidly estimated from the shapes of the concentration profiles obtained at much earlier measurement times. The method relies on the fact that the shapes of the early-time concentration profiles are dependent on the diffusion coefficient, flow rate and the capillary radius through the dimensionless residence time which, theoretically, is a measure of the amount of radial diffusion undergone by the particles. The amount of radial diffusion for nanospheres of varying sizes was estimated by quantifying the relative change in the shapes of concentration profiles obtained at two points in the flow and a correlation was obtained with the variation of the dimensionless residence time to confirm the theory. This correlation was then tested by applying it to another set of measurements of solutes and solute mixtures of different sizes including a protein. The estimated diffusion coefficients were found to be in good agreement with the expected values. This demonstrates the potential for the method to extend dispersion analysis to regimes well outside the TDA limits to enable the rapid characterization of large particles. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Taylor dispersion analysis (TDA) is an absolute method for determining the diffusion coefficients, and hence the hydrodynamic radii of particles. The method, sometimes referred to as Taylor-Aris dispersion, was first described by Taylor in his classic paper [1]. In 1956, Aris developed the method further by accounting for the longitudinal diffusion of the particles [2]. This technique was first applied to the determination of gaseous [3] and then liquid diffusion coefficients [4–6]. With the use of fused silica capillaries, TDA regained interest and has been used to analyze amino acids, peptides, proteins, small molecules, macromolecules, nanoparticles and biosensors [7–27].
∗ Corresponding author. E-mail address: [email protected] (S. Latunde-Dada). 1 These authors contributed equally. http://dx.doi.org/10.1016/j.chroma.2016.10.032 0021-9673/© 2016 Elsevier B.V. All rights reserved.
Taylor dispersion within a capillary arises as a combination of the spreading due to axial convection which is regulated by molecular diffusion across the capillary radius. Hence, for TDA to be applicable, the measurement time must be long enough for radial diffusion and hence complete Taylor dispersion to occur and the characteristic Gaussian concentration profiles to develop [28]. This condition is usually expressed with a dimensionless quantity, the dimensionless residence time = Dt/rc 2 , which is the ratio of the residence time t to the characteristic time required for a molecule of diffusion coefficient D to diffuse across the capillary radius rc . is therefore a measure of the degree of radial diffusion and is typically required to be greater than 1.4 [13]. This implies that for large particles (with small values of D) long measurement times are required for TDA to be applicable. Note also that is a similarity parameter, i.e. molecules with differing diffusion coefficients but measured at points with the same values of have similar concentration profiles. Recently, a dispersion solution which is applicable at all measurement times [29] has been used to extract the diffusion coefficients from early-time concentration profiles [24]. This approach,
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however, requires the location of the transition point between convection and Taylor dispersion which can be prone to error or in some cases be obscured by the response of smaller particles which may be present and which undergo complete Taylor dispersion. Furthermore, mismatches between the concentrations of the solute buffer and run buffer may significantly alter the shape of the trace and result in a poor fit. An empirical method for the estimation of the diffusion coefficients of large particles based on the shape of their concentration profiles has been developed in the past [30–32]. The method is applicable to particles which undergo partial Taylor dispersion and correlates the relative heights of the Taylor dispersion peaks and the convection fronts with the diffusion coefficient and flow rate. In this paper, an alternative method for estimating the diffusion coefficients of large particles at times much earlier than required for TDA is proposed. The method relies on the quantification of the degree of radial diffusion (or partial Taylor dispersion) that occurs between two spatially separated points along the flow. Two measures of the degree of radial diffusion are defined and correlated. The first measure is the variation of the dimensionless residence time between two measurement points, which is a function of the diffusion coefficient, whilst the second measure is a directly measureable quantity f which is a function of the ratios of the maximum amplitudes of the convection fronts observed at the two measurement points. Using samples of known diffusion coefficients, the correlation between f and is determined so that when D (and hence ) is unknown, it may be estimated by determining f which is directly measurable from the observed concentration profiles. The paper is organized as follows. First, the dimensionless residence time is introduced as a measure of the degree of radial diffusion. Next, the early-time radial diffusion of particles and the shapes of the corresponding concentration profiles are discussed. A method for estimating the degree of radial diffusion f from these concentration profiles is then described. The correlation between f and is determined and subsequently used to make predictions for the diffusion coefficients of a wide range of particles.
2.1. The dimensionless residence time
rc2
For the Poiseuille flow of a fluid in a circular capillary of radius a, the velocity u at a distance r from the central line is
u = u0
r2 1− 2 a
(2)
where u0 is the maximum velocity at the axis. If a symmetrical distribution of solute particles is introduced into the flow, the dispersion equation for the concentration distribution is given by [1]
∂C ∂C +u =D ∂t ∂x
2
2
∂ C 1 ∂C ∂ C + + r ∂r ∂r 2 ∂x2
(3)
where C is the mean concentration of the solute particles over the cross-section of the tube, t is the time and x is the distance from the point of injection. If we denote the solute concentration at a radial distance r as Cr , the mean concentration C is defined by C=
2 a2
a
Cr rdr
(4)
0
Under pure convection, the diffusion term on the right hand side of the dispersion equation can be neglected. Therefore, a solute of initial concentration C0 injected for a time tinj under constant pressure into the capillary will occupy an initial length X = u0 tinj . If the solute injection is assumed to be stopped at time t = 0, the solution obtained for the initial average spatial concentration distribution Cc is given by: Time t = 0 (Injection): Cc = 0 : (x < 0)
Cc = C0 1 −
x X
(5)
: (0 < x < X)
Cc = 0 : (x > X) For flow-times t > 0 after the injection, there are two time domains with different concentration profiles. These are: Time t < X/u0 (Post-injection):
Cc =
Taylor dispersion is achieved when there is a balance between axial convection which tends to disperse the particles along the streamlines and the radial diffusion that arises from the resulting concentration gradients which limits the dispersion. At early times, convective transport is dominant before eventually the degree of radial diffusion becomes sufficient for the solute particles to limit the dispersion and give rise to the spatially symmetric concentration profiles attributable to Taylor dispersion. A measure of the degree of the radial diffusion is the dimensionless residence time which is defined as the ratio of the residence time tm to the characteristic time required for a molecule to diffuse across the capillary radius and is given by Dtm
2.2. Early-time dispersion and the Taylor-dispersed fraction f
Cc = 0 : (x < 0)
2. Theory
=
67
(1)
where D is the diffusion coefficient and rc is the capillary radius. The larger the value of , the greater the degree of radial diffusion that has occurred. As mentioned in the previous section, a value of greater than 1.4 is used as the condition for complete Taylor dispersion and the applicability of TDA.
C0 u0 t
x−
Cc = C0 1 + Cc =
C0 u0 t
X 2
x2 2X
: (0 < x < u0 t)
u0 t − 2x 2X
(6)
: (u0 t < x < X)
− (x − u0 t) 1 −
x − u0 t 2X
: (X < x < X + u0 t)
Cc = 0 : (x > X + u0 t) Time t > X/u0 (Post-injection): Cc = 0 : (x < 0) Cc =
C0 u0 t
Cc = C0 Cc =
x−
x2 2X
: (0 < x < X)
X : (X < x < u0 t) 2u0 t
C0 u0 t
X
2
− (x − u0 t) 1 −
(7) x − u0 t 2X
: (u0 t < x < X + u0 t)
Cc = 0 : (x > X + u0 t) Full derivations of these concentration profiles are given in the Appendix. Similar expressions for the concentration profiles that
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Fig. 1. Spatial concentration profiles for solutes undergoing pure convection at times t = 0 and t > X/u0.
the front amplitudes Cc1 and Cc2 of the convected concentration profiles is given by max =
Fig. 2. The temporal concentration profile for particles undergoing pure convection for l > X.
arise as a result of pure convection are given in Refs. [1,29] but with different injection conditions to the case under consideration. The spatial profile in Eq. (7) is relevant to this study since the measurement times are larger than X/u0 . The profiles in Eqs. (5) and (7) are illustrated in Fig. 1 and show that at times t > X/u0 a region of constant solute concentration develops within the capillary. For capillaries with fixed detection points, the evolving spatial concentration profile is observed as a function of time. If the detection point is at a distance l from the capillary inlet which is greater than the injection distance X, the profile observed is the temporal evolution of Eq. (7). This is given by: Cc = 0 : Cc =
C0 u0 t
Cc = C0
t<
X 2
l−X u0
− (l − u0 t) 1 −
l − u0 t 2X
l − X :
u0
l u0
(8)
l X :t> 2u0 t u0
This temporal concentration profile is illustrated in Fig. 2. The second term of the expression describes the steep rise in the profile when particles travelling along the central streamline at twice the average flow speed of the fluid arrive at the measurement point whilst the third term describes the long tail observed subsequent to the rise. X The temporal profile has a maximum value of C0 2l . Therefore, for two spatially separated measurement points p1 and p2 at respective distances l1 and l2 from the injection point, the ratio max of
Cc2 l1 = Cc1 l2
(9)
For a given injection length and fixed measurement points, the ratio max is a constant if the solute undergoes pure convection. Note that the two spatially separated measurement points may be realized experimentally on a capillary with two detection windows (as is done in this study), with two measurements at different run pressures on a capillary with a single detection window or with the use of two capillaries with single detection windows at different distances from the inlet. Whilst the first method requires a single measurement, the latter two require two measurements. In reality, the solute undergoes some radial diffusion at early times before becoming completely Taylor-dispersed at times for which the dimensionless residence time > 1.4. Fig. 3 shows examples of early-time concentration profiles obtained at two spatially separated measurement points along a capillary. These concentration profiles were obtained by measuring the UV absorbance (which is proportional to the concentration) as a function of time. The broad peaks observed after the initial fronts indicate that some radial diffusion has occurred behind the front and as expected, this is more pronounced at measurement point p2 which is further away from the injection point. Due to the radial diffusion, which limits the dispersion due to pure convection, the peak amplitudes A1 and A2 of the fronts are reduced in comparison to A the values expected for pure convection. Likewise, the ratio = A2 1 is less than the ratio max expected under pure convection because of the additional radial diffusion that occurs between the two measurement points. When complete Taylor dispersion is observed at the second measurement point, the front height A2 and hence is equal to 0. Therefore, a parameter f defined as f =1−
max
(10)
varies from 0 under pure convection to 1 at complete Taylor dispersion. This is therefore a measure of the degree of radial diffusion within the capillary. Furthermore, as it is defined, it may be considered an estimate of the fraction of the solute particles that undergoes Taylor dispersion between the two measurement points. As mentioned earlier, the dimensionless residence time is another measure of the degree of radial diffusion and therefore
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Fig. 3. Concentration profiles for nanospheres with hydrodynamic radii of 100 nm in 0.01 M NaCl at a run pressure of 500 mbar.
a correlation between and f is expected. Since f is an estimate of the Taylor-dispersed fraction of the particles between the two measurement points, it will be correlated with the variation in the dimensionless residence time defined as: = 2 − 1 =
D (t2 − t1 ) rc2
(11)
where t1 and t2 are the respective times it takes the solute travelling at the mean flow speed to arrive at measurement points 1 and 2. The correlation will be tested with particles with a range of hydrodynamic radii and measured at a range of run pressures. A range of run pressures and hydrodynamic radii have been chosen to establish the universality of the correlation. Note that in previous work [30–32], the ratio of the height of the broad peaks (at about twice the arrival time of the fronts) to the front height was correlated with .
Table 1 Grouped measurements to determine the correlation (m is the sum total of the measurements in each row of the table). Group
Solute Rh
Pressure/mbar
m
I
30 nm 100, 200, 250 nm 30 nm 100 nm
833, 1000 350–1000 1000–3000 1000–2000
6 36 27 16
II
Table 2 Measurements to test the correlation (m is the sum total of the measurements in each row of the table). Group
Solute Rh
Pressure/mbar
m
III
100 + 200 nm 100, 200, BSA + 100 nm, BSA + 200 nm,BSA + (100 + 200 nm)
250 250
15 15
3. Experimental section NanosphereTM 3000 Series size standards with nominal hydrodynamic radii of 30, 100, 200 and 250 nm were purchased from Fisher Scientific, (Leicestershire, UK). The standards at a concentration of 1% w/v. They were diluted in 0.01 M NaCl (Sigma Aldrich, Suffolk, UK) in a ratio of 4 drops of size standard per millilitre of NaCl solution. Concentration profiles were acquired using the Viscosizer TD instrument (Malvern Instruments Ltd., Worcestershire, UK) fitted with a standard two-window uncoated capillary (ID 75 m, OD 360 m, Malvern Instruments Ltd, Worcestershire, UK) having dimensions l1 = 0.45 m and l2 = 0.85 m for the distances from the inlet to two detection windows and a total capillary length of 1.30 m. The measurements were conducted at a wavelength of 254 nm. Delivery of narrow solute plugs was achieved by pressuredriven injection at 50 mbar for 12 s. From these injection conditions and the capillary dimensions, the injection length X is 8.1 mm and the front height ratio max for pure convection was determined from Eq. (9) to be 0.53. Elution of sample plugs was undertaken at a variety of run pressures ranging from 350 to 3000 mbar. Table 1 shows the hydrodynamic radii of the nanospheres and the corresponding run pressures used for the sum total of m measurements for each row in the table. To determine the correlation between f and , the measurements were split into two groups as shown in the table. Group I consists of 30, 100, 200 and 250 nm nanospheres run at pressures ranging from 350 to 1000 mbar. Group II consists of 30 and 100 nm radius nanospheres run at higher pressures ranging from 1000 to 3000 mbar. These measurements were split into separate groups to study the effects of the run pressure on the correlation. The mea-
surements at a run pressure of 3000 mbar have a Reynolds number of about 10. Hence laminar flow conditions are maintained up to the highest pressure used in this study. The heights, A1 and A2 , of the fronts were determined by subtracting the value of the baseline at the time of first arrival of the front from the measured peak value. The mean flow speed was determined from the times of peak arrival at the two measurement points and was determined from Eq. (9). The Stokes-Einstein equation [33] was used to determine the diffusion coefficients D from the nominal hydrodynamic radii. The correlation determined from the measurements in Groups I and II will then be used to make predictions for the hydrodynamic radii of the samples in Table 2 and the results compared with the expected values. As can be seen from the table, binary and tertiary mixtures were also prepared in 50:50 (v/v) and 50:25:25 (v/v) ratios respectively. 2.5 mg/mL Bovine Serum Albumin (BSA, Sigma Aldrich, Poole, UK; Rh ∼ 3.8 nm; prepared in 0.01 M NaCl) was used in the mixtures. Five replicates of each solute/mixture of solutes were measured giving a sum total of m = 15 measurements for each row in the table. All the measurements were conducted at a fixed run pressure of 250 mbar. This run pressure (which is lower than the run pressures used to establish the correlations) was chosen to avoid self-referencing the measurements as much as possible. Note that the Group III measurements were not used to determine the correlation.
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Fig. 4. Data from Groups I and II and a highly correlated subset of the data.
Table 3 Slopes and coefficients of determination obtained for subsets of the measurements. Group
Solute Rh /nm
Range of
Slope
R2
I
30 100 200 250 30 100 Group I: 30, 100Group II: 30
0.081–0.110 0.026–0.078 0.012–0.039 0.010–0.030 0.017–0.062 0.009–0.022 0.017–0.110
3.3 ± 0.5 3.7 ± 0.3 4.2 ± 0.9 4.4 ± 2 3.2 ± 0.4 4.6 ± 1.7 3.4 ± 0.2
0.85 0.92 0.69 0.45 0.81 0.5 0.92
II Highly correlated subset
4. Results and discussion 4.1. Correlations Fig. 4 shows plots of f against for the Groups I and II measurements with labelled data points for different hydrodynamic radii. Linear regression fits constrained to pass through the origin were applied to the data for each hydrodynamic radius and the slopes and coefficients of determination R2 obtained are shown in Table 3. The fits are fixed at the origin since this is a valid data point resulting from the definitions of f and . As can be seen from the errors in the determined slopes and the coefficients of determination, the correlation between f and is poor for the 200 nm and 250 nm measurements from Group I and the 100 nm measurements from Group II. These measurements have low values and the poor correlation may be due to the complicated interplay between convection and Taylor dispersion being more stochastic and hence less deterministic at these short residence times. The increased scatter in the Group II data may also be attributed to the uncertainties in the flow speeds at high run pressures. Typically, there is a ramp time for the pressure to attain the set value and the higher the run pressure required, the greater the ramp time and hence the greater the uncertainty in the estimated flow speed. Furthermore, at higher run pressures, the signal to noise ratio decreases which could lead to uncertainties in the determined front amplitudes. Good correlations were obtained for the 30 nm and 100 nm measurements from Group I and the 30 nm measurements from Group II with lower errors on slopes which are in good agreement. This
Table 4 Estimated hydrodynamic radii from the correlation between f and . Solute
Expected Rh /nm
Estimated Rh /nm
100 nm 200 nm 100 + 200 nm BSA + 100 nm BSA + 200 nm BSA + (100 + 200 nm)
100 200 140 100 200 140
99 ± 3 189 ± 17 140 ± 5 105 ± 5 196 ± 24 121 ± 14
suggests a lower bound of about 0.02 for is required for a high degree of confidence in the correlation. The data from this highly correlated subset are shown combined together in Fig. 4. A linear regression fit was applied to the data and the slope and coefficient of determination obtained are shown in Table 3. Hence the correlation between f and determined by this study is given by: f = 3.4 : > 0.02
(12)
Note that for a given run pressure, the lower bound on can be converted to the maximum value (upper bound) of the hydrodynamic radius measurable with a high degree of confidence using this method. Note that, according to Eq. (12), f reaches its maximum value of 1 at = 0.3 after which it becomes invariant with as expected for full Taylor dispersion. Further work would involve investigation whether this correlation holds in the region close to f = 1 (Table 4).
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Fig. 5. Concentration profiles for BSA + (100 +200) nm radius nanospheres in 0.01 M NaCl at a run pressure of 250 mbar.
4.2. Estimates for Group III measurements To test the correlation further, the equation for the regression line obtained from the highly correlated data in Fig. 5 was used to estimate for the Group III measurements and hence the hydrodynamic radii Rh from D. This implies a range of 0.045–0.1 for and hence Eq. (12) is expected to be applicable. The quantity f was estimated from the measured concentration profiles using Eq. (9) and from Eq. (12). The average results and standard deviations from the replicates of each measurement are compared to the expected values in Table 4. Note that the expected average value for the radii of the mixtures was determined by computing the ratio of the sums of the front heights measured of each individual component. The expeccted value of f and hence was then computed from which the expected value of Rh was determined to be 140 nm. Fig. 5 shows the concentration profile obtained for a tertiary mixture of BSA, 100 nm and 200 nm nanospheres. For the mixtures containing BSA, it is assumed that the BSA molecules do not contribute to the fronts since they undergo complete Taylor dispersion and their contribution is therefore negligible at the time of arrival of the front. Hence, the second peaks observed at about twice the time of arrival of the fronts are due to the BSA molecules which undergo complete Taylor dispersion. The hydrodynamic radius of BSA was determined from fits to the Taylor-dispersed peaks in the last three sets of measurements in Table 4 using the convectivedispersive fits described in the literature [24] and found to be 4.3 ± 0.5 nm which is in satisfactory agreement with the expected value of 3.8 nm.
5. Conclusions Taylor dispersion analysis of large particles (>30 nm) typically require slow flow rates and hence long measurement times. In this paper, an empirical method for the determination of the diffusion coefficients of large solutes at much shorter times than required for TDA has been demonstrated successfully. The method utilizes the correlation between two measures of the degree of early-time radial diffusion; the variation of the dimensionless residence time and an estimate of the fraction of the solute that Taylor-disperses between two measurement points. This correlation was determined from a wide range of solute sizes and run pressures and the Taylor-dispersed fraction was found to be proportional to , the difference between the dimensionless residence times at the two measurement points. This may be attributed to the fact that the dimensionless residence time is a ratio of the residence time
to the time required for diffusion across the capillary radius and hence may be expected to be proportional to the amount of solute particles that undergo diffusion before measurement. Furthermore, it was found that the correlations were relatively insensitive to changes in the shapes of the concentration profiles at times longer than those of the fronts, caused for example by the presence of rapidly diffusing components in mixtures containing the larger species. In this study, the correlation was found to be poor for values of less than 0.02. Furthermore, the maximum value of explored in this study was 0.1. Hence it is recommended that the correlation be used for values ranging from 0.02 to 0.1. On the Viscosizer TD instrument, for example, these correspond approximately to a range of run pressures from 200 to 1000 mbar for 200 nm radius nanospheres diffusing in a carrier solution with a viscosity similar to water. A value of 0.05 would be recommended for since it is close to the middle of the range. This corresponds approximately to run pressures of 3000, 900, 450 and 350 mbar for 30, 100, 200 and 250 nm radius nanospheres respectively. Further work in this field would investigate values greater than 0.1 and close to 1 to determine if the correlation holds in this regime. Though this work has been conducted entirely with nanospheres, it is expected that the results can be extended to different particles such as proteins and peptides due to the universal nature of convection and Taylor dispersion. Further work in this field would confirm this assertion and also investigate the applicability of this method to polydisperse mixtures. To conclude, these results highlight the potential for extending dispersion analysis to regimes outside the limits of TDA which would allow the rapid characterization of large particles with the use of high flow rates.
Appendix A. A. Derivations of the spatial concentration profiles resulting from pure convection (A.1) A. Derivations of the spatial concentration profiles resulting from pure convection The spatial concentration profiles in Eqs. (5)–(7) can be derived by considering the convection of particles initially confined to an infinitesimal length dx of a capillary filled with a fluid undergoing laminar flow. As derived in Ref. [1], if the initial mean concentration
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Fig. A2. Contributions of two infinitesimal strips from the concentration profile of solute injected at time t = 0. (Not to scale). Fig. A1. Convection of an infinitesimal length of solute. Cc = 0 : (x < 0)
of the solute located at the point x = 0 is C0 , then after a time t, the mean concentration Cc of the solute is:
x
Cc = 0
Cc = 0 : (x < 0) Cc =
C0 C dx = u0 t u0 t x
Cc =
C0 dx : (0 < x < u0 t) u0 t
(A.1)
x−u0 t
Cc = 0 : (x > u0 t)
X
Cc = x−u0 t
Cc = 0 : (x < 0)
Cc =
x
Cc = 0
Cc = 0 : (x > X)
C = C0 1 −
x X
x
the concentration profile is given by
X
C0 C dx = u0 t u0 t
(A.3)
X − (x − u0 t) 2
: (u0 t < x < X)
1−
x − u0 t 2X
: (X < x < X + u0 t)
x−
x2 2X
: (0 < x < X)
C X : (X < x < u0 t) dx = C0 u0 t 2u0 t
u0 t − 2x 2X
Cc = 0 : (x < 0)
(A.2)
This profile is shown in Fig. 4. Taking this as the initial concentration profile, its subsequent evolution as a function of time may again be computed by integrating the contributions from the convection of infinitesimal strips of the profile up until a distance u0 t of their starting points. The contributions from two such strips to the integrated concentration profile at a time t is illustrated in Fig. A2. The ranges of x will now have to be carefully considered as the overlaps of the contributions differ as a function of the distance along the capillary. For a time t < X/u0 , if we define the function
: (0 < x < u0 t)
Likewise for a time t > X/u0 , the concentration profile is
0
: (0 < x < X)
1+
(A.4)
C0 dx x = C0 1 − X X
x2 2X
C C0 dx = u0 t u0 t
Cc = 0 : (x > X + u0 t)
Cc =
x−
C dx = C0 u0 t
Hence the mean concentration is uniform over a length u0 t of the capillary from the starting point x = 0. This is illustrated in Fig. A1. For the continuous injection of a solute of initial concentration C0 into the capillary, the mean concentration after an injection time tinj can be computed by integrating the contributions from the convection of infinitesimal strips of the solute up until a distance X = u0 tinj from their starting points. This gives:
X
Cc = x−u0 t
X
C0 C dx = u0 t u0 t
X − (x − u0 t) 2
1−
x − u0 t 2X
: (u0 t < x < X + u0 t)
Cc = 0 : (x > X + u0 t) (A.5)
The profile in Eq. (A.5) is shown in Fig. 4. Note that in Eq. (A.5) the spatial concentration is constant in the region X < x < u0 t because it is within the distance of convection u0 t of all the injected solute particles and hence they all contribute to give a constant concentration in this region. For completeness, the profile in Eq. (A.4) for times t < X/u0 is shown in Fig. A3. Note that at x = X, the profile switches from linear to quadratic in x. For a fixed detection point at a distance l > X away from the inlet of the capillary, the temporal profile in Eq. (8) can be derived from Eq. (A.5) by making the substitution l = x.
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73
Fig. A3. The spatial concentration profile for solutes undergoing pure convection at times t < X/u0 .
References [1] G. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. Lond. A 219 (1953) 186–203. [2] R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc. R. Soc. Lond. A 235 (1956) 67–77. [3] J.C. Giddings, S.L. Seager, Rapid determination of gaseous diffusion coefficients by means of gas chromatography apparatus, J. Chem. Phys. 33 (1960) 1579–1580. [4] E. Grushka, E.J. Kikta, Extension of the chromatographic broadening method of measuring diffusion coefficients to liquid systems. I. Diffusion coefficients of some alkylbenzenes in chloroform, J. Phys. Chem. 78 (1974) 2297–2301. [5] A.C. Ouano, Diffusion in liquid systems. I. A simple and fast method of measuring diffusion constants, Ind. Eng. Chem. Fundam. 11 (1972) 268–271. [6] K.C. Pratt, W.A. Wakeham, The mutual diffusion coefficient of ethanol-water mixtures: determination by a rapid new method, Proc. R. Soc. Lond. A 336 (1974) 393–406. [7] M.S. Bello, R. Rezzonico, R.G. Righetti, Use of Taylor-Aris dispersion for measurement of a solute diffusion coefficient in thin capillaries, Science 266 (1994) 773–776. [8] E.P.C. Mes, W. Th. Kok, H. Poppe, R.J. Tijssen, Comparison of methods for the determination of diffusion coefficients of polymers in dilute solutions: the influence of polydispersity, Polym. Sci. Part B: Polym. Phys. 37 (1999) 593–603. [9] W.L. Hulse, R. Forbes, A Taylor dispersion analysis method for the sizing of therapeutic proteins and their aggregates using nanolitre sample quantities, Int. J. Pharm. 416 (2011) 394–397. [10] W.L. Hulse, R. Forbes, A nanolitre method to determine the hydrodynamic radius of proteins and small molecules by Taylor dispersion analysis, Int. J. Pharm. 411 (2011) 64–68. [11] A. Hawe, W.L. Hulse, R. Jiskoot, R. Forbes, Taylor dispersion analysis compared to dynamic light scattering for the size analysis of therapeutic peptides and proteins and their aggregates, Pharm. Res. 28 (2011) 2302–2310. [12] J. Ostergaard, J. Hensen, Simultaneous evaluation of ligand binding properties and protein size by electrophoresis and Taylor dispersion in capillaries, Anal. Chem. 81 (2009) 8644–8648. [13] U. Sharma, N.J. Gleason, J.D. Carbeck, Diffusivity of solutes measured in glass capillaries using Taylor’s analysis of dispersion and a commercial CE instrument, Anal. Chem. 77 (3) (2005) 806–813. [14] R. Niesner, A. Heintz, Diffusion coefficients of aromatics in aqueous solution, J. Chem. Eng. Data 45 (2000) 1121–1124. [15] H. Cottet, M. Martin, A. Papillaud, E. Souaid, H. Collet, A. Commeyras, Determination of dendrigraft poly-L-lysine diffusion coefficients by Taylor dispersion analysis, Biomacromolecules 8 (2007) 3235–3243. [16] H. Cottet, J.P. Biron, L. Cipelletti, M. Matmour, M. Martin, Determination of individual diffusion coefficients in evolutive binary mixtures by Taylor dispersion analysis: application to the monitoring of polymer reaction, Anal. Chem. 82 (2010) 1793–1802. [17] H. Cottet, J.P. Biron, M. Martin, Taylor dispersion analysis of mixtures, Anal. Chem. 79 (2007) 9066–9073.
[18] T. Le Saux, H. Cottet, Size-based characterization by the coupling of capillary electrophoresis to taylor dispersion analysis, Anal. Chem. 80 (2008) 1829–1832. [19] F. D’Orlye, A. Varenne, P. Gareil, Determination of nanoparticle diffusion coefficients by Taylor dispersion analysis using a capillary electrophoresis instrument, J. Chromatogr. A 1204 (2008) 226–232. [20] U. Franzen, C. Vermehren, H. Jensen, J. Ostergaard, Physicochemical characterization of a PEGylated liposomal drug formulation using capillary electrophoresis, J. Electrophoresis 32 (2011) 738–748. [21] W.A. Boyle, R.F. Buchholz, J.A. Neal, J.L. McCarthy, Flow injection analysis estimation of diffusion coefficients of pauci- and polydisperse polymers such as polystyrene sulfonates, J. Appl. Polym. Sci. 42 (1991) 1969–1977. [22] J.G. Quinn, Modeling Taylor dispersion injections: determination of kinetic/affinity interaction constants and diffusion coefficients in label-free biosensing, Anal. Biochem. 421 (2012) 391–400. [23] J.G. Quinn, Evaluation of Taylor dispersion injections: determining kinetic/affinity interaction constants and diffusion coefficients in label-free biosensing, Anal. Biochem. 421 (2012) 401–410. [24] S. Latunde-Dada, R. Bott, K. Hampton, O. Leszczyszyn, Application of the exact dispersion solution to the analysis of solutes beyond the limits of Taylor dispersion, Anal. Chem. 87 (2015) 8021–8025. [25] S. Latunde-Dada, R. Bott, K. Hampton, O. Leszczyszyn, Analytical mitigation of solute-capillary interactions in double detection Taylor dispersion analysis, J. Chrom. A. 1408 (2015) 255–260. [26] S. Latunde-Dada, R. Bott, K. Hampton, J. Patel, O. Leszczyszyn, Methodologies for the Taylor Dispersion Analysis for mixtures, aggregates and the mitigation of buffer mismatch effects, Anal. Methods 7 (2015) 10312–10321. [27] S. Latunde-Dada, R. Bott, D. Barker, O. Leszczyszyn, Methodologies for the rapid determination of the diffusion interaction parameter using Taylor dispersion analysis, Anal. Methods 8 (2016) 386–392. [28] G. Taylor, Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion, Proc. R. Soc. Lond. A 225 (1954) 473–477. [29] W.N. Gill, R. Sankarasubramanian, Exact analysis of unsteady convective diffusion, Proc. R. Soc. Lond. A 316 (1970) 341–350. [30] F.M. Kelleher, C.N. Trumbore, Dispersion of protein solutions in short, open capillary tubes as a method for rapid molecular weight determination, Anal. Biochem. 137 (1984) 20–24. [31] C.N. Trumbore, M. Grehlinger, M. Stowe, F.M. Kelleher, Further experiments on a new, fast method for determining molecular weights of diffusing species in a liquid phase, J. Chrom. A 322 (1985) 443–454. [32] C.N. Trumbore, L.M. Jackson, S. Bennett, A. Thompson, High-speed analytical sensor for in-line monitoring of dissolved analytes flowing in a tube employing a combination of limited diffusion, laminar flow and plug solvent injections, J. Chromatogr. A 481 (1989) 111–120. [33] A. Einstein, On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat, Ann. Phys. 322 (1905) 549–560.
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Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Rapid determination of hydrodynamic radii beyond the limits of Taylor dispersion Seyi Latunde-Dada a,∗,1 , Rachel Bott a,1 , Jack Crozier a,1 , Markos Trikeriotis a,1 , Oksana Iryna Leszczyszyn a,1 , David Goodall b,1 a b
Malvern Instruments Ltd, Grovewood Road, Malvern, Worcestershire, WR14 1XZ, UK Paraytec Ltd, York House, Outgang Lane, Osbaldwick, York, YO19 5UP, UK
a r t i c l e
i n f o
Article history: Received 31 March 2016 Received in revised form 15 August 2016 Accepted 12 October 2016 Available online 13 October 2016 Keywords: Taylor dispersion analysis Aggregation Diffusion coefficients
a b s t r a c t Taylor dispersion analysis (TDA) is an absolute method for determining the diffusion coefficients, and hence the hydrodynamic radii, of particles by measuring the dispersion in a carrier medium flowing within a capillary. It is applicable under conditions which allow the particles to radially diffuse appreciably across the cross-section of the flow before the measurement and therefore implies long measurement times are required for large particles with small diffusion coefficients. In this paper, a method has been developed by which the diffusion coefficients of large particles can be rapidly estimated from the shapes of the concentration profiles obtained at much earlier measurement times. The method relies on the fact that the shapes of the early-time concentration profiles are dependent on the diffusion coefficient, flow rate and the capillary radius through the dimensionless residence time which, theoretically, is a measure of the amount of radial diffusion undergone by the particles. The amount of radial diffusion for nanospheres of varying sizes was estimated by quantifying the relative change in the shapes of concentration profiles obtained at two points in the flow and a correlation was obtained with the variation of the dimensionless residence time to confirm the theory. This correlation was then tested by applying it to another set of measurements of solutes and solute mixtures of different sizes including a protein. The estimated diffusion coefficients were found to be in good agreement with the expected values. This demonstrates the potential for the method to extend dispersion analysis to regimes well outside the TDA limits to enable the rapid characterization of large particles. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Taylor dispersion analysis (TDA) is an absolute method for determining the diffusion coefficients, and hence the hydrodynamic radii of particles. The method, sometimes referred to as Taylor-Aris dispersion, was first described by Taylor in his classic paper [1]. In 1956, Aris developed the method further by accounting for the longitudinal diffusion of the particles [2]. This technique was first applied to the determination of gaseous [3] and then liquid diffusion coefficients [4–6]. With the use of fused silica capillaries, TDA regained interest and has been used to analyze amino acids, peptides, proteins, small molecules, macromolecules, nanoparticles and biosensors [7–27].
∗ Corresponding author. E-mail address: [email protected] (S. Latunde-Dada). 1 These authors contributed equally. http://dx.doi.org/10.1016/j.chroma.2016.10.032 0021-9673/© 2016 Elsevier B.V. All rights reserved.
Taylor dispersion within a capillary arises as a combination of the spreading due to axial convection which is regulated by molecular diffusion across the capillary radius. Hence, for TDA to be applicable, the measurement time must be long enough for radial diffusion and hence complete Taylor dispersion to occur and the characteristic Gaussian concentration profiles to develop [28]. This condition is usually expressed with a dimensionless quantity, the dimensionless residence time = Dt/rc 2 , which is the ratio of the residence time t to the characteristic time required for a molecule of diffusion coefficient D to diffuse across the capillary radius rc . is therefore a measure of the degree of radial diffusion and is typically required to be greater than 1.4 [13]. This implies that for large particles (with small values of D) long measurement times are required for TDA to be applicable. Note also that is a similarity parameter, i.e. molecules with differing diffusion coefficients but measured at points with the same values of have similar concentration profiles. Recently, a dispersion solution which is applicable at all measurement times [29] has been used to extract the diffusion coefficients from early-time concentration profiles [24]. This approach,
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however, requires the location of the transition point between convection and Taylor dispersion which can be prone to error or in some cases be obscured by the response of smaller particles which may be present and which undergo complete Taylor dispersion. Furthermore, mismatches between the concentrations of the solute buffer and run buffer may significantly alter the shape of the trace and result in a poor fit. An empirical method for the estimation of the diffusion coefficients of large particles based on the shape of their concentration profiles has been developed in the past [30–32]. The method is applicable to particles which undergo partial Taylor dispersion and correlates the relative heights of the Taylor dispersion peaks and the convection fronts with the diffusion coefficient and flow rate. In this paper, an alternative method for estimating the diffusion coefficients of large particles at times much earlier than required for TDA is proposed. The method relies on the quantification of the degree of radial diffusion (or partial Taylor dispersion) that occurs between two spatially separated points along the flow. Two measures of the degree of radial diffusion are defined and correlated. The first measure is the variation of the dimensionless residence time between two measurement points, which is a function of the diffusion coefficient, whilst the second measure is a directly measureable quantity f which is a function of the ratios of the maximum amplitudes of the convection fronts observed at the two measurement points. Using samples of known diffusion coefficients, the correlation between f and is determined so that when D (and hence ) is unknown, it may be estimated by determining f which is directly measurable from the observed concentration profiles. The paper is organized as follows. First, the dimensionless residence time is introduced as a measure of the degree of radial diffusion. Next, the early-time radial diffusion of particles and the shapes of the corresponding concentration profiles are discussed. A method for estimating the degree of radial diffusion f from these concentration profiles is then described. The correlation between f and is determined and subsequently used to make predictions for the diffusion coefficients of a wide range of particles.
2.1. The dimensionless residence time
rc2
For the Poiseuille flow of a fluid in a circular capillary of radius a, the velocity u at a distance r from the central line is
u = u0
r2 1− 2 a
(2)
where u0 is the maximum velocity at the axis. If a symmetrical distribution of solute particles is introduced into the flow, the dispersion equation for the concentration distribution is given by [1]
∂C ∂C +u =D ∂t ∂x
2
2
∂ C 1 ∂C ∂ C + + r ∂r ∂r 2 ∂x2
(3)
where C is the mean concentration of the solute particles over the cross-section of the tube, t is the time and x is the distance from the point of injection. If we denote the solute concentration at a radial distance r as Cr , the mean concentration C is defined by C=
2 a2
a
Cr rdr
(4)
0
Under pure convection, the diffusion term on the right hand side of the dispersion equation can be neglected. Therefore, a solute of initial concentration C0 injected for a time tinj under constant pressure into the capillary will occupy an initial length X = u0 tinj . If the solute injection is assumed to be stopped at time t = 0, the solution obtained for the initial average spatial concentration distribution Cc is given by: Time t = 0 (Injection): Cc = 0 : (x < 0)
Cc = C0 1 −
x X
(5)
: (0 < x < X)
Cc = 0 : (x > X) For flow-times t > 0 after the injection, there are two time domains with different concentration profiles. These are: Time t < X/u0 (Post-injection):
Cc =
Taylor dispersion is achieved when there is a balance between axial convection which tends to disperse the particles along the streamlines and the radial diffusion that arises from the resulting concentration gradients which limits the dispersion. At early times, convective transport is dominant before eventually the degree of radial diffusion becomes sufficient for the solute particles to limit the dispersion and give rise to the spatially symmetric concentration profiles attributable to Taylor dispersion. A measure of the degree of the radial diffusion is the dimensionless residence time which is defined as the ratio of the residence time tm to the characteristic time required for a molecule to diffuse across the capillary radius and is given by Dtm
2.2. Early-time dispersion and the Taylor-dispersed fraction f
Cc = 0 : (x < 0)
2. Theory
=
67
(1)
where D is the diffusion coefficient and rc is the capillary radius. The larger the value of , the greater the degree of radial diffusion that has occurred. As mentioned in the previous section, a value of greater than 1.4 is used as the condition for complete Taylor dispersion and the applicability of TDA.
C0 u0 t
x−
Cc = C0 1 + Cc =
C0 u0 t
X 2
x2 2X
: (0 < x < u0 t)
u0 t − 2x 2X
(6)
: (u0 t < x < X)
− (x − u0 t) 1 −
x − u0 t 2X
: (X < x < X + u0 t)
Cc = 0 : (x > X + u0 t) Time t > X/u0 (Post-injection): Cc = 0 : (x < 0) Cc =
C0 u0 t
Cc = C0 Cc =
x−
x2 2X
: (0 < x < X)
X : (X < x < u0 t) 2u0 t
C0 u0 t
X
2
− (x − u0 t) 1 −
(7) x − u0 t 2X
: (u0 t < x < X + u0 t)
Cc = 0 : (x > X + u0 t) Full derivations of these concentration profiles are given in the Appendix. Similar expressions for the concentration profiles that
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Fig. 1. Spatial concentration profiles for solutes undergoing pure convection at times t = 0 and t > X/u0.
the front amplitudes Cc1 and Cc2 of the convected concentration profiles is given by max =
Fig. 2. The temporal concentration profile for particles undergoing pure convection for l > X.
arise as a result of pure convection are given in Refs. [1,29] but with different injection conditions to the case under consideration. The spatial profile in Eq. (7) is relevant to this study since the measurement times are larger than X/u0 . The profiles in Eqs. (5) and (7) are illustrated in Fig. 1 and show that at times t > X/u0 a region of constant solute concentration develops within the capillary. For capillaries with fixed detection points, the evolving spatial concentration profile is observed as a function of time. If the detection point is at a distance l from the capillary inlet which is greater than the injection distance X, the profile observed is the temporal evolution of Eq. (7). This is given by: Cc = 0 : Cc =
C0 u0 t
Cc = C0
t<
X 2
l−X u0
− (l − u0 t) 1 −
l − u0 t 2X
l − X :
u0
l u0
(8)
l X :t> 2u0 t u0
This temporal concentration profile is illustrated in Fig. 2. The second term of the expression describes the steep rise in the profile when particles travelling along the central streamline at twice the average flow speed of the fluid arrive at the measurement point whilst the third term describes the long tail observed subsequent to the rise. X The temporal profile has a maximum value of C0 2l . Therefore, for two spatially separated measurement points p1 and p2 at respective distances l1 and l2 from the injection point, the ratio max of
Cc2 l1 = Cc1 l2
(9)
For a given injection length and fixed measurement points, the ratio max is a constant if the solute undergoes pure convection. Note that the two spatially separated measurement points may be realized experimentally on a capillary with two detection windows (as is done in this study), with two measurements at different run pressures on a capillary with a single detection window or with the use of two capillaries with single detection windows at different distances from the inlet. Whilst the first method requires a single measurement, the latter two require two measurements. In reality, the solute undergoes some radial diffusion at early times before becoming completely Taylor-dispersed at times for which the dimensionless residence time > 1.4. Fig. 3 shows examples of early-time concentration profiles obtained at two spatially separated measurement points along a capillary. These concentration profiles were obtained by measuring the UV absorbance (which is proportional to the concentration) as a function of time. The broad peaks observed after the initial fronts indicate that some radial diffusion has occurred behind the front and as expected, this is more pronounced at measurement point p2 which is further away from the injection point. Due to the radial diffusion, which limits the dispersion due to pure convection, the peak amplitudes A1 and A2 of the fronts are reduced in comparison to A the values expected for pure convection. Likewise, the ratio = A2 1 is less than the ratio max expected under pure convection because of the additional radial diffusion that occurs between the two measurement points. When complete Taylor dispersion is observed at the second measurement point, the front height A2 and hence is equal to 0. Therefore, a parameter f defined as f =1−
max
(10)
varies from 0 under pure convection to 1 at complete Taylor dispersion. This is therefore a measure of the degree of radial diffusion within the capillary. Furthermore, as it is defined, it may be considered an estimate of the fraction of the solute particles that undergoes Taylor dispersion between the two measurement points. As mentioned earlier, the dimensionless residence time is another measure of the degree of radial diffusion and therefore
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Fig. 3. Concentration profiles for nanospheres with hydrodynamic radii of 100 nm in 0.01 M NaCl at a run pressure of 500 mbar.
a correlation between and f is expected. Since f is an estimate of the Taylor-dispersed fraction of the particles between the two measurement points, it will be correlated with the variation in the dimensionless residence time defined as: = 2 − 1 =
D (t2 − t1 ) rc2
(11)
where t1 and t2 are the respective times it takes the solute travelling at the mean flow speed to arrive at measurement points 1 and 2. The correlation will be tested with particles with a range of hydrodynamic radii and measured at a range of run pressures. A range of run pressures and hydrodynamic radii have been chosen to establish the universality of the correlation. Note that in previous work [30–32], the ratio of the height of the broad peaks (at about twice the arrival time of the fronts) to the front height was correlated with .
Table 1 Grouped measurements to determine the correlation (m is the sum total of the measurements in each row of the table). Group
Solute Rh
Pressure/mbar
m
I
30 nm 100, 200, 250 nm 30 nm 100 nm
833, 1000 350–1000 1000–3000 1000–2000
6 36 27 16
II
Table 2 Measurements to test the correlation (m is the sum total of the measurements in each row of the table). Group
Solute Rh
Pressure/mbar
m
III
100 + 200 nm 100, 200, BSA + 100 nm, BSA + 200 nm,BSA + (100 + 200 nm)
250 250
15 15
3. Experimental section NanosphereTM 3000 Series size standards with nominal hydrodynamic radii of 30, 100, 200 and 250 nm were purchased from Fisher Scientific, (Leicestershire, UK). The standards at a concentration of 1% w/v. They were diluted in 0.01 M NaCl (Sigma Aldrich, Suffolk, UK) in a ratio of 4 drops of size standard per millilitre of NaCl solution. Concentration profiles were acquired using the Viscosizer TD instrument (Malvern Instruments Ltd., Worcestershire, UK) fitted with a standard two-window uncoated capillary (ID 75 m, OD 360 m, Malvern Instruments Ltd, Worcestershire, UK) having dimensions l1 = 0.45 m and l2 = 0.85 m for the distances from the inlet to two detection windows and a total capillary length of 1.30 m. The measurements were conducted at a wavelength of 254 nm. Delivery of narrow solute plugs was achieved by pressuredriven injection at 50 mbar for 12 s. From these injection conditions and the capillary dimensions, the injection length X is 8.1 mm and the front height ratio max for pure convection was determined from Eq. (9) to be 0.53. Elution of sample plugs was undertaken at a variety of run pressures ranging from 350 to 3000 mbar. Table 1 shows the hydrodynamic radii of the nanospheres and the corresponding run pressures used for the sum total of m measurements for each row in the table. To determine the correlation between f and , the measurements were split into two groups as shown in the table. Group I consists of 30, 100, 200 and 250 nm nanospheres run at pressures ranging from 350 to 1000 mbar. Group II consists of 30 and 100 nm radius nanospheres run at higher pressures ranging from 1000 to 3000 mbar. These measurements were split into separate groups to study the effects of the run pressure on the correlation. The mea-
surements at a run pressure of 3000 mbar have a Reynolds number of about 10. Hence laminar flow conditions are maintained up to the highest pressure used in this study. The heights, A1 and A2 , of the fronts were determined by subtracting the value of the baseline at the time of first arrival of the front from the measured peak value. The mean flow speed was determined from the times of peak arrival at the two measurement points and was determined from Eq. (9). The Stokes-Einstein equation [33] was used to determine the diffusion coefficients D from the nominal hydrodynamic radii. The correlation determined from the measurements in Groups I and II will then be used to make predictions for the hydrodynamic radii of the samples in Table 2 and the results compared with the expected values. As can be seen from the table, binary and tertiary mixtures were also prepared in 50:50 (v/v) and 50:25:25 (v/v) ratios respectively. 2.5 mg/mL Bovine Serum Albumin (BSA, Sigma Aldrich, Poole, UK; Rh ∼ 3.8 nm; prepared in 0.01 M NaCl) was used in the mixtures. Five replicates of each solute/mixture of solutes were measured giving a sum total of m = 15 measurements for each row in the table. All the measurements were conducted at a fixed run pressure of 250 mbar. This run pressure (which is lower than the run pressures used to establish the correlations) was chosen to avoid self-referencing the measurements as much as possible. Note that the Group III measurements were not used to determine the correlation.
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Fig. 4. Data from Groups I and II and a highly correlated subset of the data.
Table 3 Slopes and coefficients of determination obtained for subsets of the measurements. Group
Solute Rh /nm
Range of
Slope
R2
I
30 100 200 250 30 100 Group I: 30, 100Group II: 30
0.081–0.110 0.026–0.078 0.012–0.039 0.010–0.030 0.017–0.062 0.009–0.022 0.017–0.110
3.3 ± 0.5 3.7 ± 0.3 4.2 ± 0.9 4.4 ± 2 3.2 ± 0.4 4.6 ± 1.7 3.4 ± 0.2
0.85 0.92 0.69 0.45 0.81 0.5 0.92
II Highly correlated subset
4. Results and discussion 4.1. Correlations Fig. 4 shows plots of f against for the Groups I and II measurements with labelled data points for different hydrodynamic radii. Linear regression fits constrained to pass through the origin were applied to the data for each hydrodynamic radius and the slopes and coefficients of determination R2 obtained are shown in Table 3. The fits are fixed at the origin since this is a valid data point resulting from the definitions of f and . As can be seen from the errors in the determined slopes and the coefficients of determination, the correlation between f and is poor for the 200 nm and 250 nm measurements from Group I and the 100 nm measurements from Group II. These measurements have low values and the poor correlation may be due to the complicated interplay between convection and Taylor dispersion being more stochastic and hence less deterministic at these short residence times. The increased scatter in the Group II data may also be attributed to the uncertainties in the flow speeds at high run pressures. Typically, there is a ramp time for the pressure to attain the set value and the higher the run pressure required, the greater the ramp time and hence the greater the uncertainty in the estimated flow speed. Furthermore, at higher run pressures, the signal to noise ratio decreases which could lead to uncertainties in the determined front amplitudes. Good correlations were obtained for the 30 nm and 100 nm measurements from Group I and the 30 nm measurements from Group II with lower errors on slopes which are in good agreement. This
Table 4 Estimated hydrodynamic radii from the correlation between f and . Solute
Expected Rh /nm
Estimated Rh /nm
100 nm 200 nm 100 + 200 nm BSA + 100 nm BSA + 200 nm BSA + (100 + 200 nm)
100 200 140 100 200 140
99 ± 3 189 ± 17 140 ± 5 105 ± 5 196 ± 24 121 ± 14
suggests a lower bound of about 0.02 for is required for a high degree of confidence in the correlation. The data from this highly correlated subset are shown combined together in Fig. 4. A linear regression fit was applied to the data and the slope and coefficient of determination obtained are shown in Table 3. Hence the correlation between f and determined by this study is given by: f = 3.4 : > 0.02
(12)
Note that for a given run pressure, the lower bound on can be converted to the maximum value (upper bound) of the hydrodynamic radius measurable with a high degree of confidence using this method. Note that, according to Eq. (12), f reaches its maximum value of 1 at = 0.3 after which it becomes invariant with as expected for full Taylor dispersion. Further work would involve investigation whether this correlation holds in the region close to f = 1 (Table 4).
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71
Fig. 5. Concentration profiles for BSA + (100 +200) nm radius nanospheres in 0.01 M NaCl at a run pressure of 250 mbar.
4.2. Estimates for Group III measurements To test the correlation further, the equation for the regression line obtained from the highly correlated data in Fig. 5 was used to estimate for the Group III measurements and hence the hydrodynamic radii Rh from D. This implies a range of 0.045–0.1 for and hence Eq. (12) is expected to be applicable. The quantity f was estimated from the measured concentration profiles using Eq. (9) and from Eq. (12). The average results and standard deviations from the replicates of each measurement are compared to the expected values in Table 4. Note that the expected average value for the radii of the mixtures was determined by computing the ratio of the sums of the front heights measured of each individual component. The expeccted value of f and hence was then computed from which the expected value of Rh was determined to be 140 nm. Fig. 5 shows the concentration profile obtained for a tertiary mixture of BSA, 100 nm and 200 nm nanospheres. For the mixtures containing BSA, it is assumed that the BSA molecules do not contribute to the fronts since they undergo complete Taylor dispersion and their contribution is therefore negligible at the time of arrival of the front. Hence, the second peaks observed at about twice the time of arrival of the fronts are due to the BSA molecules which undergo complete Taylor dispersion. The hydrodynamic radius of BSA was determined from fits to the Taylor-dispersed peaks in the last three sets of measurements in Table 4 using the convectivedispersive fits described in the literature [24] and found to be 4.3 ± 0.5 nm which is in satisfactory agreement with the expected value of 3.8 nm.
5. Conclusions Taylor dispersion analysis of large particles (>30 nm) typically require slow flow rates and hence long measurement times. In this paper, an empirical method for the determination of the diffusion coefficients of large solutes at much shorter times than required for TDA has been demonstrated successfully. The method utilizes the correlation between two measures of the degree of early-time radial diffusion; the variation of the dimensionless residence time and an estimate of the fraction of the solute that Taylor-disperses between two measurement points. This correlation was determined from a wide range of solute sizes and run pressures and the Taylor-dispersed fraction was found to be proportional to , the difference between the dimensionless residence times at the two measurement points. This may be attributed to the fact that the dimensionless residence time is a ratio of the residence time
to the time required for diffusion across the capillary radius and hence may be expected to be proportional to the amount of solute particles that undergo diffusion before measurement. Furthermore, it was found that the correlations were relatively insensitive to changes in the shapes of the concentration profiles at times longer than those of the fronts, caused for example by the presence of rapidly diffusing components in mixtures containing the larger species. In this study, the correlation was found to be poor for values of less than 0.02. Furthermore, the maximum value of explored in this study was 0.1. Hence it is recommended that the correlation be used for values ranging from 0.02 to 0.1. On the Viscosizer TD instrument, for example, these correspond approximately to a range of run pressures from 200 to 1000 mbar for 200 nm radius nanospheres diffusing in a carrier solution with a viscosity similar to water. A value of 0.05 would be recommended for since it is close to the middle of the range. This corresponds approximately to run pressures of 3000, 900, 450 and 350 mbar for 30, 100, 200 and 250 nm radius nanospheres respectively. Further work in this field would investigate values greater than 0.1 and close to 1 to determine if the correlation holds in this regime. Though this work has been conducted entirely with nanospheres, it is expected that the results can be extended to different particles such as proteins and peptides due to the universal nature of convection and Taylor dispersion. Further work in this field would confirm this assertion and also investigate the applicability of this method to polydisperse mixtures. To conclude, these results highlight the potential for extending dispersion analysis to regimes outside the limits of TDA which would allow the rapid characterization of large particles with the use of high flow rates.
Appendix A. A. Derivations of the spatial concentration profiles resulting from pure convection (A.1) A. Derivations of the spatial concentration profiles resulting from pure convection The spatial concentration profiles in Eqs. (5)–(7) can be derived by considering the convection of particles initially confined to an infinitesimal length dx of a capillary filled with a fluid undergoing laminar flow. As derived in Ref. [1], if the initial mean concentration
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Fig. A2. Contributions of two infinitesimal strips from the concentration profile of solute injected at time t = 0. (Not to scale). Fig. A1. Convection of an infinitesimal length of solute. Cc = 0 : (x < 0)
of the solute located at the point x = 0 is C0 , then after a time t, the mean concentration Cc of the solute is:
x
Cc = 0
Cc = 0 : (x < 0) Cc =
C0 C dx = u0 t u0 t x
Cc =
C0 dx : (0 < x < u0 t) u0 t
(A.1)
x−u0 t
Cc = 0 : (x > u0 t)
X
Cc = x−u0 t
Cc = 0 : (x < 0)
Cc =
x
Cc = 0
Cc = 0 : (x > X)
C = C0 1 −
x X
x
the concentration profile is given by
X
C0 C dx = u0 t u0 t
(A.3)
X − (x − u0 t) 2
: (u0 t < x < X)
1−
x − u0 t 2X
: (X < x < X + u0 t)
x−
x2 2X
: (0 < x < X)
C X : (X < x < u0 t) dx = C0 u0 t 2u0 t
u0 t − 2x 2X
Cc = 0 : (x < 0)
(A.2)
This profile is shown in Fig. 4. Taking this as the initial concentration profile, its subsequent evolution as a function of time may again be computed by integrating the contributions from the convection of infinitesimal strips of the profile up until a distance u0 t of their starting points. The contributions from two such strips to the integrated concentration profile at a time t is illustrated in Fig. A2. The ranges of x will now have to be carefully considered as the overlaps of the contributions differ as a function of the distance along the capillary. For a time t < X/u0 , if we define the function
: (0 < x < u0 t)
Likewise for a time t > X/u0 , the concentration profile is
0
: (0 < x < X)
1+
(A.4)
C0 dx x = C0 1 − X X
x2 2X
C C0 dx = u0 t u0 t
Cc = 0 : (x > X + u0 t)
Cc =
x−
C dx = C0 u0 t
Hence the mean concentration is uniform over a length u0 t of the capillary from the starting point x = 0. This is illustrated in Fig. A1. For the continuous injection of a solute of initial concentration C0 into the capillary, the mean concentration after an injection time tinj can be computed by integrating the contributions from the convection of infinitesimal strips of the solute up until a distance X = u0 tinj from their starting points. This gives:
X
Cc = x−u0 t
X
C0 C dx = u0 t u0 t
X − (x − u0 t) 2
1−
x − u0 t 2X
: (u0 t < x < X + u0 t)
Cc = 0 : (x > X + u0 t) (A.5)
The profile in Eq. (A.5) is shown in Fig. 4. Note that in Eq. (A.5) the spatial concentration is constant in the region X < x < u0 t because it is within the distance of convection u0 t of all the injected solute particles and hence they all contribute to give a constant concentration in this region. For completeness, the profile in Eq. (A.4) for times t < X/u0 is shown in Fig. A3. Note that at x = X, the profile switches from linear to quadratic in x. For a fixed detection point at a distance l > X away from the inlet of the capillary, the temporal profile in Eq. (8) can be derived from Eq. (A.5) by making the substitution l = x.
S. Latunde-Dada et al. / J. Chromatogr. A 1472 (2016) 66–73
73
Fig. A3. The spatial concentration profile for solutes undergoing pure convection at times t < X/u0 .
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