Pertti
Vastamäki
*^{ad},
P. Stephen
Williams
^{b},
Matti
Jussila
^{a},
Michel
Martin
^{c} and
Marja-Liisa
Riekkola
^{a}
^{a}Laboratory of Analytical Chemistry, Department of Chemistry, University of Helsinki, P.O. Box 55, FIN-00014, Finland. E-mail: pertti.vastamaki@vtt.fi; Fax: +358 20 722 4374
^{b}Cambrian Technologies Inc., 1772 Saratoga Avenue, Cleveland, OH 44109, USA
^{c}École 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
^{d}VTT Technical Research Centre of Finland, P.O.Box 1000, FIN-02044 VTT, Finland

Received
29th May 2013
, Accepted 8th October 2013

First published on 8th October 2013

A theoretical and experimental study of continuous two-dimensional thermal field-flow fractionation (2D-ThFFF) is presented. Separation takes place in radial flow between two closely spaced discs, one of which is heated and the other cooled in order to maintain a temperature gradient across the channel. The cooled disc, which serves as the accumulation wall, is rotated relative to the other to create a shear component to the fluid flow. Under the influence of the thermal gradient and flow components, the sample components spiral outwards along different paths to the outer rim of the channel to be collected. The general principle of operation is described and an approximate theoretical model formulated for predicting the outlet position for the path of each sample component. The influence of the principal operational parameters, such as radial and angular flow rates and thermal gradient, on the deflection angle of the sample trajectory is investigated. Fractionation is demonstrated for polystyrene polymer standards in a binary solvent consisting of cyclohexane and ethylbenzene. Experimental results are compared with theoretical predictions.

The common fields employed in FFF are gravitation and centrifugation (together being referred to as sedimentation), displacement by a cross-channel flow component, thermal gradient, and electrical and magnetic fields. FFF, as it was conceived^{1} and generally still implemented, is a batchwise analytical and characterization technique. The mechanism does not lend itself to larger samples (greater than, say, 50 or 100 μL), and FFF in its conventional form is not suitable for preparative separations. The same may be said of other separation techniques, such as chromatography and electrophoresis. There has been some success in simply up-scaling liquid chromatographic separations, of course, but approaches have also been developed to adapt these techniques for continuous operation, as described below. In the present work we present an adaptation of thermal FFF for continuous operation.

Different kinds of continuous separation methods have been developed. Examples of continuous electrophoresis methods are continuous two-dimensional electrophoresis,^{10} and continuous free-flow electrophoresis.^{11} Continuous separation is achieved in the dipole magnetic fractionator (DMF)^{12–14} using a field-flow configuration identical to that of continuous free-flow electrophoresis. Turina et al.^{15} described a continuous two-dimensional gas–liquid chromatographic apparatus, but only preliminary experiments were carried out. Giddings^{16} described in theory a continuous annular gas chromatograph. Sussman and co-workers^{17–19} have developed an approach to continuous two-dimensional gas chromatography using radial flow between two closely spaced glass plates. The method was referred to as continuous disc chromatography or continuous surface chromatography. Continuous two-dimensional liquid chromatographic separations have been carried out in annular packed columns.^{20–22} Another important implementation of continuous two-dimensional separation is that of split-flow thin channel (SPLITT) fractionation, invented by Giddings,^{23} and developed further with several designs.^{24–26}

Giddings^{27} also pointed out that continuous two-dimensional FFF could be achieved by combining FFF with either a bulk displacement or a flow displacement perpendicular to the FFF displacement. In order to realize selective deflection for continuous fractionation using a flow at right angles to the FFF displacement, it would be required that the flow profiles in the two directions are different (for example, Poiseuille and electro-osmotic flows). He suggested that if FFF was carried out in an annular channel (with a radial field), then continuous operation could be achieved by rotation of the channel as in continuous annular chromatography. He also suggested that continuous FFF could be carried out using radial flow between rotating discs. This is the same configuration as rotating disc chromatography. A rotating disc gravitational FFF was constructed in the Giddings laboratory with separation to be carried out in the steric mode. The operation of the system was not successful. It is possible that secondary flow patterns may have interfered with separation in this device also. It would probably be advantageous to have a stationary channel, and rotate the sample inlet port and the outlet ports around the channel.

The system is similar in some aspects to those of continuous disc chromatography^{17–19} and rotating disc gravitational FFF.^{27} The carrier stream flows from a central inlet in the heated upper wall to 23 outlets spaced at equal intervals around the perimeter of the disc-shaped channel. Collection syringes are installed at the channel outlets and the flow maintained by suction at the syringes. The carrier reservoir supplying the central inlet is pressurized to reduce the tendency for bubble formation. In addition to a parabolic flow profile established in the radial direction, a shear-driven flow component is generated in the angular direction by slowly rotating the cooled lower channel wall. A continuous sample stream is introduced at a second inlet in the upper wall at a point close to the center of the channel. The sample components are subjected to two orthogonal displacement mechanisms: selective field-flow fractionation in the radial direction, and a flow displacement in the angular direction that has a relatively small selectivity, provided it is the accumulation wall that rotates. The combination of these simultaneously acting mechanisms underlies the two-dimensional continuous fractionation.^{8,27,33} The sample components are separated by these two displacements into continuous filaments that strike off at different angles over the 2D surface. The separated components are then collected at different locations around the perimeter of the channel. The superposition of shear flow onto the radial flow may help to suppress secondary flow disturbances, but this aspect is yet to be explored.

Theoretical equations have been formulated to examine the capability of the new method for analytical and preparative applications. These will aid in the selection of experimental conditions for optimal performance. A full theoretical treatment would be capable of predicting the deflection of a sample component trajectory and its lateral dispersion to each side of its path. Such a full theoretical characterization of the 2D-ThFFF operation will be required for further development of instrument configuration, channel structure, and operative practice. Our previous experimental work,^{33} where the 2D-ThFFF system was shown capable of almost baseline continuous fractionation of two standard polymers of different molar masses, will support these further developments. The starting point for the theoretical model is presented in this paper and predicts the paths and outlet positions for sample components. The theory is based on that of thermal field-flow fractionation,^{34–37} and classical fluid dynamics assumptions. The objective of this study was to formulate these basic theoretical equations and to examine the agreement between theory and experiment by introducing polystyrene standards of different molar masses and of narrow molar mass distribution.

(1) The radial fluid velocity profile is assumed to be parabolic. The small departure from the parabolic profile, caused by variation of viscosity with temperature across the channel thickness, can easily be taken into account by using the corrected equation for retention ratio already developed.^{38}

(2) It is assumed that there is a simple shear flow in the angular direction. (Variation in viscosity will also influence the flow profile in the angular direction.)

(3) The concentration profile for each sample component is assumed to be exponential across the channel thickness.

(4) The sample flow rate is assumed to be negligibly small in comparison to the radial carrier flow rate.

A schematic of the 2D-ThFFF system, along with the flow components and geometrical parameters, is illustrated in Fig. 1.

(1) |

(2) |

R_{r} = 6λ(coth(1/2λ) − 2λ) ≈ 6λ(1 − 2λ) | (3) |

For λ of 0.170, the accurate expression for R_{r} yields a value of 0.679, corresponding to rather poor retention, yet the approximate expression on the right of eqn (3) gives a result of 0.673. The error is less than 0.9% at this point, and falls rapidly with reduction of λ.

Similarly, the rate at which a retained zone changes its angular position is

(4) |

(5) |

(6) |

The angular velocity profile is given as

Ω = Ω_{0}(1 − ξ) | (7) |

Ω = Ω_{0}ξ | (8) |

(9) |

The respective equations for Ω (eqn (7) and (8)) are substituted into the equation

(10) |

(11) |

(12) |

At the retention level considered in relation to eqn (3), where λ = 0.170, the accurate R_{θ} given by eqn (11) is 1.666, and the approximate form of the equation is in error by less than 0.4%. At the same level of retention, the approximate form of eqn (12) is in error by almost 1.7%.

Note that R_{θ} approaches a constant value of 2 at high retention, if the accumulation wall rotates. This is consistent with the requirement that orthogonal selectivities differ. The angular selectivity in this case approaches zero, and rotational migration is non-selective. This is not the case for a static accumulation wall. Angular selectivity for a static accumulation wall approaches that for radial migration, so that overall selectivity approaches zero and no separation is possible.

Dividing eqn (2) by eqn (4) gives, with eqn (9)

(13) |

The zone trajectory is given by rearranging eqn (13) and integrating from the initial radial distance r_{i} and zero angle to radial distance r at angle θ:

(14) |

(15) |

The final angle θ_{r} at the time of elution t_{r} is given by

(16) |

We can replace π(r_{o}^{2} – r_{i}^{2})w by the effective void volume of the channel V^{0}, and write eqn (16) in the form

(17) |

(18) |

Hence, the θ_{r}/θ^{0} ratio is equal to R_{θ}/R_{r}, and using the high retention limits for these retention ratios we obtain for a rotating accumulation wall:

(19) |

(20) |

At λ = 0.170, eqn (19) is in error by just over 0.5%; it predicts a ratio of 2.466 as opposed to the true ratio of 2.453. At the same level of retention, eqn (20) predicts a ratio of 0.505 while the true ratio is 0.493; the error is 2.5%.

It is interesting to note that the elution angle is larger than the non-retained elution angle if accumulation occurs at the rotating wall while it is smaller for a static accumulation wall. This provides a means to determine the status of this accumulation wall, and hence the sign of the Soret coefficient (at least, when θ_{r} is smaller than 2π). Furthermore, for a rotating accumulation wall, the selectivity with respect to λ is given by

(21) |

(22) |

The selectivity for a rotating accumulation wall approaches unity for strong retention, whereas selectivity approaches zero for a stationary accumulation wall. It is therefore obvious that the accumulation wall must rotate with respect to the sample introduction port and the collection ports if a separation is to be achieved.

After the continuous 2D-ThFFF run, the collected samples from the syringes, containing equal amounts of the binary carrier, were transferred to test tubes and evaporated to dryness. A small amount of tetrahydrofuran (100 μL) was added to each test tube to dissolve the polymer fractions. The same solvent was used as a carrier in the analytical ThFFF instrument. Relative concentrations of the polymer in each fraction were calculated from the peak areas of the fractograms.

Runs were carried out using two different temperature gradients and also without a temperature gradient, and the outlet positions determined for the different samples. In addition, four rotation rates of the accumulation wall and two radial flow rates of the carrier were applied. Other run conditions were kept constant. The first series of the runs was performed without an applied temperature gradient for the determination of the outlet positions of a non-retained sample. The theoretical non-retained elution angles were calculated using eqn (18) and these were compared with the experimental results. The runs were repeated using an applied temperature gradient, and as previously, the elution angles for retained samples were determined. The predicted elution angles were calculated using eqn (17) with retention parameters λ predicted from the retention data obtained in our previous work using the analytical ThFFF instrument.^{32} The λ values were predicted using the plots of λ vs. 1/ΔT (see Fig. 2), and it was assumed that the thicknesses of the analytical and continuous FFF channels were identical.

It was shown in our previous work that the 2D-ThFFF method is suitable for continuous fractionation of lipophilic polymers under the influence of a thermal gradient.^{31,32} In this study, new experimental results, together with some from our previous work, have been compared with predictions of the theoretical retention model described in the previous section. The method is rather complex due to the number of relevant parameters. In this work we have examined the influence of the molar mass of the sample, the angular rotation rate and radial flow rate of the carrier on the deflection of the sample zone. The parameters affecting deviations from the theory, such as angular broadening of the deflected sample zone, the effect of the temperature of the lower wall, and the lack of relaxation time, have also been discussed.

Fig. 3 shows the theoretically predicted (by eqn (3), (11) and (17)) dependence of elution angle (θ_{r}) on the dimensionless retention parameter λ, for (A) different rotation rates Ω_{0} of the accumulation wall at a fixed radial flow rate, and for (B) different radial flow rates at a fixed rotation rate. It was assumed that the rotating cold wall was the accumulation wall, and therefore eqn (3), (11), and (17) were assumed to apply. From these curves it is clear that the elution angle is predicted to increase with decrease of λ or radial flow rate, or with increase of angular velocity of the cold wall. These findings are, in fact, in good agreement with the experimental results described below.

It was generally the case that a polymer sample was eluted over several collection ports at the channel periphery. The result of a given run can thus be reported by plotting the fractional recovery of the polymer collected at each port. Results obtained for a set of experiments performed using the three individual polystyrene samples under conditions of five different temperature gradients (0, 10, 14, 17 and 21 K) are shown in Fig. 4. A fixed carrier flow rate of 221 μL min^{−1}, and a fixed angular velocity of the accumulation wall Ω_{0} of 0.024 rpm were used in these experiments. In Fig. 4A–C the fractional recoveries of the polymers are plotted as functions of the different outlet port numbers. For each run, the mean elution angle is computed according to (360°/23)(Σ(i − 1) φ_{i}), where φ_{i} is the fractional recovery of the polymer collected at port i, port number 1 being aligned with the sample introduction port, the sum extending from i = 1 to i = 23, the number of ports. These mean angles are expressed as elution angles θ_{r} if a temperature gradient is applied and as θ^{0} if no gradient is applied. The theoretically predicted and the experimentally observed mean elution angles together with their percentage errors are listed in Table 1. The theoretically predicted elution angles were calculated by means of eqn (16) using the accurate forms of eqn (3) and (11) in all cases.

Sample | PS 51k | PS 520k | PS 1000k | |
---|---|---|---|---|

ΔT = 0 K |
θ
^{0} (exp) |
38° | 36° | 33° |

θ
^{0} (theory) |
32.8° | 32.8° | 32.8° | |

% error | 15% | 11% | −0.3% | |

θ
^{0}/Ω_{0} min (exp) |
4.4 | 4.2 | 3.8 | |

ΔT = 10 K |
θ
_{r} (exp) |
44° | 57° | 69° |

θ
_{r} (theory) |
42.0° | 54.1° | 87.9° | |

% error | 5.6% | 5.3% | −22% | |

θ
_{r}/θ^{0} (exp) |
1.2 | 1.6 | 2.1 | |

θ
_{r}/θ^{0} (theory) |
1.3 | 1.6 | 2.7 | |

ΔT = 14 K |
θ
_{r} (exp) |
43° | 67° | 90° |

θ
_{r} (theory) |
46.3° | 65.5° | 119° | |

% error | −7.5% | 2.5% | −25% | |

θ
_{r}/θ^{0} (exp) |
1.1 | 1.8 | 2.7 | |

θ
_{r}/θ^{0} (theory) |
1.4 | 2.0 | 3.6 | |

ΔT = 17 K |
θ
_{r} (exp) |
47° | 81° | 100° |

θ
_{r} (theory) |
49.8° | 75.0° | 145° | |

% error | −4.9% | 8.1% | −30% | |

θ
_{r}/θ^{0} (exp) |
1.3 | 2.2 | 3.1 | |

θ
_{r}/θ^{0} (theory) |
1.5 | 2.3 | 4.4 | |

ΔT = 21 K |
θ
_{r} (exp) |
57° | — | 129° |

θ
_{r} (theory) |
54.6° | 88.8° | 182° | |

% error | 4.0% | — | −29% | |

θ
_{r}/θ^{0} (exp) |
1.5 | — | 3.9 | |

θ
_{r}/θ^{0} (theory) |
1.7 | 2.7 | 5.5 |

Experimental results obtained for the sample PS 1000k carried out at four different rotation rates (0.016, 0.024, 0.032, and 0.040 rpm) and three different temperature gradients (ΔT = 0, 12, and 20 K), and a fixed carrier flow rate ( = 276 μL min^{−1}) are shown in Fig. 5. The results for the mean elution angles are reported in Table 2, again compared with theoretically predicted values.

Ω
_{0} (rpm) |
0.016 | 0.024 | 0.032 | 0.040 | |
---|---|---|---|---|---|

ΔT = 0 K |
θ
^{0} (exp) |
29° | 38° | 42° | 46° |

θ
^{0} (theory) |
17.5° | 26.3° | 35.1° | 43.8° | |

% error | 67% | 46% | 19% | 4.4% | |

θ
^{0}/Ω_{0} min (exp) |
5.1 | 4.4 | 3.6 | 3.2 | |

ΔT = 12 K |
θ
_{r} (exp) |
58° | 66° | 78° | 97° |

θ
_{r} (theory) |
55.0° | 82.5° | 110° | 138° | |

% error | 6% | −20% | −29% | −29% | |

θ
_{r}/θ^{0} (exp) |
2.0 | 1.7 | 1.9 | 2.1 | |

θ
_{r}/θ^{0} (theory) |
3.1 | 3.1 | 3.1 | 3.1 | |

ΔT = 20 K |
θ
_{r} (exp) |
55° | 90° | 144° | — |

θ
_{r} (theory) |
92.0° | 138° | 184° | 230° | |

% error | −41% | −35% | −22% | — | |

θ
_{r}/θ^{0} (exp) |
1.9 | 2.3 | 3.4 | — | |

θ
_{r}/θ^{0} (theory) |
5.2 | 5.2 | 5.2 | 5.2 |

Similarly, elution profiles for PS 1000k obtained at a fixed rotation rate (0.016 rpm), but at two different carrier flow rates (276 and 221 μL min^{−1}) and two temperature gradients (ΔT = 0 and 20 K) are shown in Fig. 6. Experimentally observed mean elution angles are compared with theoretical predictions in Table 3.

(μL min^{−1}) |
276 | 221 | |
---|---|---|---|

ΔT = 0 K |
θ
^{0} (exp) |
29° | 59° |

θ
^{0} (theory) |
17.5° | 21.9° | |

% error | 67% | 170% | |

ΔT = 20 K |
θ
_{r} (exp) |
55° | 124° |

θ
_{r} (theory) |
92.0° | 115° | |

% error | −41% | 8.0% | |

θ
_{r}/θ^{0} (exp) |
1.9 | 2.1 | |

θ
_{r}/θ^{0} (theory) |
5.2 | 5.2 |

The identity of the accumulation wall may be obtained from Fig. 4–6. Fig. 4, for example, shows that θ_{r} increases with increase of either ΔT or the molecular mass of the sample, and therefore, with decrease of λ. This is consistent with eqn (19) and inconsistent with eqn (20). It follows that the cold wall must serve as the accumulation wall. The increase of θ_{r} for PS 1000k with ΔT in Fig. 5 and 6 also indicates that the cold wall is the accumulation wall. In fact, if the accumulation wall was the stationary hot wall eqn (20) would predict θ_{r} to be smaller than θ^{0}, which is not the case for any of the results shown in Fig. 4–6.

This observation indicates that the thermal diffusion-induced migration of polystyrene dissolved in a binary ethylbenzene–cyclohexane solvent mixture acts to drive molecules from warmer to colder regions, as has been observed for polystyrene in either pure ethylbenzene^{38} or pure tetrahydrofuran.^{39} A molecular theory of thermal diffusion in liquids has not yet been developed. Neither the values nor the signs of thermal diffusion coefficients may be predicted. Experimental determination of the coefficients is required, and continuous 2D-ThFFF appears to be a convenient alternative to existing methods for the investigation of the phenomenon.

Effect of molar mass and thermal gradient on deflection angle.
According to Table 1 and Fig. 4, the experimental mean elution angle θ^{0}(exp) obtained without an applied field, for a constant radial flow rate and rotation rate, for each polymer sample, agrees rather well with the predicted non-retained elution angle θ^{0}(theory), calculated from eqn (18). When the thermal gradients are applied, the experimental deflection angles θ_{r}(exp) for PS 51k agree very well with the predicted values of θ_{r}(theory). The experimental peak elution angles for the moderately retained sample PS 520k also agree very well with the predicted values at ΔT of 10 to 17 K. However, for the more retained sample PS 1000k the experimental elution angles are significantly smaller than the predicted mean values when the field is applied. However, the results show clearly that the deflection angles increase as the molar mass of the sample or the thermal gradient increases, as predicted by the theory.

Effect of rotation rate of the lower wall and thermal fields on deflection angle.
According to Table 2 and Fig. 5, the experimental mean elution angle θ^{0}(exp) for the sample PS 1000k is consistently larger than the theoretically predicted angle θ^{0}(theory) without the field, but the difference decreases as rotation rate increases. The elution angle θ^{0} obtained with theory increases gradually, when the rotation speed increases. The θ^{0}/Ω_{0} ratio should not depend on the rotation rate and, according to eqn (18), be equal to t^{0}/2, i.e., to 3.0 min in the flow conditions of Fig. 5 and Table 2. In fact, it is seen that, experimentally, θ^{0}/Ω_{0} is, in all cases, larger than its theoretical value, but it decreases with increasing rotation rate.

When the field is applied, the experimental mean elution angles, θ_{r}(exp), are seen to be smaller than the theoretical values θ_{r} (theory) in almost all cases, the exception being for ΔT = 12 K and Ω_{0} = 0.016 rpm. As the temperature drop is increased from 0 K to 12 K, then to 20 K, whatever the rotation rate, the distribution of the polymer along the exit ports becomes wider and shifted toward larger port numbers, as expected for samples accumulating at the rotating wall. For ΔT = 12 K, the relative error between experimental and theoretical values of θ_{r} tends to increase with the rotation rate. For ΔT = 20 K, the relative error decreases with increasing rotation rate. The experimental θ_{r}/θ^{0} ratios are all smaller than the theoretical values. According to eqn (19), these values should only depend on λ and be equal to 3.1 and 5.2 for ΔT = 12 K and 20 K, respectively.

Effect of radial flow rate on deflection, with and without the field.
The influence of the carrier flow rate on the elution behavior of the polymer in unretained and retained conditions is shown in Fig. 6 and Table 3 for ΔT = 0 K and 20 K, respectively, at the lowest rotation rate of 0.016 rpm. As the carrier flow rate is lowered, both θ^{0} and θ_{r} increase. This is expected from eqn (17) and (18) since a reduction in leads to an increase in the time spent in the channel and thus to an increased deflection induced by the rotation of one plate. Whatever the carrier flow rate, the experimental value of θ^{0} (exp) is larger than expected, the more so for the lower flow rate. However, while the experimental elution angle θ_{r} (exp) is lower than the theoretical angle when = 276 μL min^{−1}, it is observed to be slightly larger than theoretically expected when = 221 μL min^{−1}. Still, for both flow rates, the experimental θ_{r}/θ^{0} ratio is seen to be smaller than theoretically expected.

Solvent partitioning effect.
In order to fulfill some practical constraints, we choose to use a mixture of cyclohexane and ethylbenzene as the carrier liquid. Indeed, cyclohexane was preferentially selected because its relatively low boiling point facilitates evaporation of the collected effluents of the various exit ports for redissolution of the polymeric residues in THF before analysis by conventional thermal FFF. Ethylbenzene was added for increasing the solubility of the polymers with the highest molar masses and for increasing the retention with the limited temperature gradient obtainable in the 2-D device. Under a temperature gradient, this binary mixture is subject to the Soret effect. A weak gradient in the composition of the binary mixture is then formed across the channel thickness. Under this composition gradient, the polystyrene molecules tend to concentrate more in the better solvent, in that case, ethylbenzene. As it was observed that the cold regions are slightly enriched in cyclohexane and the hot regions in ethylbenzene,^{40} this solvent partitioning effect is somehow moderating the proper Soret process of the polystyrene molecules which lead them to concentrate at the cold wall. As a result, the steady-state concentration profile of the polymer in the channel thickness is no longer exponential and the value of λ determined from Fig. 2 is an apparent value. This should not affect the correctness of the determination of R_{r} from eqn (3) using this apparent λ value because the λ values reported in Fig. 2 are obtained from experimental measurements of R_{r} using eqn (3) in a conventional thermal FFF system using the same binary solvent mixture. However, this will lead to erroneous calculations of R_{θ} from eqn (11) because this equation is based on the hypothesis of an exponential distribution profile in the channel thickness. This might explain why the elution angles of the PS 1000k sample are lower than expected, the amplitude of the solvent partitioning effect increasing with the molar mass of the polymer.^{40} A more detailed study is required to clarify the impact of this effect of the mean elution angle.

Flow velocity profile.
In thermal FFF, the carrier flow velocity profile is not parabolic because of the temperature dependences of the carrier viscosity and thermal conductivity. Furthermore, these physico-chemical parameters might be affected by the partitioning effect of the binary solvent mixture discussed above. As a consequence, the relationships between R_{r} and λ and between R_{θ} and λ are not given exactly by eqn (3) and by eqn (11) and (12), respectively. Still, for the reason described above, this does not affect the R_{r} values used in the computations since these were deduced from the calibration regressions given in Fig. 2, but this should affect the value of R_{θ}. However, because of the relatively low ΔT applied in the experiments, the corresponding deviations are likely to be small and a minor effect.

Effective channel outlet radius.
The theoretical model assumes that the sample trajectory can intersect the periphery of the channel at any location. In practice, it is not so because the number of outlet collection ports is finite (and equal to 23 in the present system). If there was no rotation, the flow streamlines, which would be radially distributed near the central injection port, would not follow a radial line as they approach the edge of the channel, but would be bent to focus to the closest outlet port. When a plate is rotating, the flow streamlines follow curved lines which similarly are modified as they approach the edge to converge to an outlet port. There is therefore a perturbation region near the channel periphery where the fractionation effect does not operate because all nearby flow streamlines converge to the same outlet port. This may be thought of in terms of an effective channel radius, r_{eff}, within which the fractionation takes place, that is smaller than the true channel radius. Clearly, the difference, r_{o} − r_{eff}, is of the order of the distance between two consecutive ports. The radius r_{eff} can be approximated by the radial distance of the third vertex of an equilateral triangle for which the two other vertices are consecutive outlet ports. Simple geometrical consideration shows that, for a channel with n outlet ports

(23) |

For n = 23, this gives r_{eff}/r_{o} = 0.755. The effective void volume, V^{0}_{eff}, of the 2D-ThFFF system to be used in eqn (17) and (18) instead of V^{0} is then equal to 0.91 mL instead of 1.68 mL. The corresponding reductions in θ^{0} and θ_{r} are thus significant (46%). This is because the radial flow decreases with increasing distances from the channel center, thus the polymer spends more time in traveling a given radial distance near the channel edge than near the center. This however should not affect the θ_{r}/θ^{0} ratio. If the effective channel radius must be correctly determined in order to obtain accurate retention parameters, the fact that it is smaller than the channel radius can clearly not explain why experimental θ^{0} are larger than expected.

Furthermore, for the computation of the mean deflection angle, whatever the angular position at which a streamline reaches the periphery of the channel, or even the effective radius, it is set to the value of the angular position of the nearest exit port. This introduces a bias which limits the accuracy of the determination of the mean elution angles θ^{0} and θ_{r}.

Sample flow rate.
The model described in the Theory section assumes that the sample flow rate is negligibly small compared to the carrier flow rate. In the present system, the sample to carrier flow rate ratio was in the range between 0.0045 and 0.0113, which is indeed small. However, the flow of the sample into the channel could slightly reduce the void time and, consequently, the elution angle according to eqn (17), although it has been previously found that the sample flow rate has a negligible influence on θ^{0}.^{32} Computational fluid dynamics could help in understanding the effect and possibly estimating its magnitude.

The above estimations of the effects of the number of ports and of the sample flow rate are based on highly simplified assumptions. A detailed fluid mechanical study of the system would help to get a more precise description of the effects. Additional hydrodynamic effects may be encountered if the sample viscosity is initially significantly different from that of the carrier liquid.

Sample relaxation.
The theory assumes that the samples migrate according to eqn (3) and (11) from the point of entry. This is not the case, however. The sample components must approach their equilibrium distributions across the channel thickness under the mechanisms of thermophoresis and molecular diffusion under flowing conditions. The thermal diffusion coefficient and therefore the rate of transport across the channel thickness are independent, or nearly independent, of molecular weight. It is therefore expected that the relaxation toward equilibrium distribution should result in reduced retention times and reduced θ_{r}, with absolute decrements not being strongly dependent on molecular weight. This is not observed by experiment (see Table 1). The experimental values for θ_{r} are in most cases lower than the predicted values, and relaxation effects could contribute to these discrepancies but do not fully account for the discrepancies.

Non-uniformity of ΔT.
It is assumed that the temperature of each disc is constant across its surface and that ΔT is constant throughout the 2D channel. This may not be the case, and any variation in ΔT that falls in the path of an eluting sample would influence its θ_{r}.

Geometrical imperfections.
Finally, we should consider geometrical imperfections of the 2D channel. The theory assumes that the discs are perfectly flat and the spacing between them is uniform throughout, from the axis to the circumference. The slightest variation in disc spacing across the faces would have a very strong effect on the radial flow velocity as a function of angle and also result in angular components to flow that vary with radius and angle. In addition, this pattern could rotate relative to either disc, or to both, as they rotate relative to one another. Small channel imperfections could account for remaining discrepancies between theory and experiment.

The instrumental limitations, such as rather low maximum temperature gradient, the limited number (23 at present) of collection ports, and non-continuous fraction collection due to the limited volume of the syringes, restrict the full-scale operation and extension of the technique to lower molar mass applications. However, the temperature gradients applied in this work resulted in the retention of PS 1000k with R_{r} < 0.5 for ΔT ≥ 14 K, which allowed the effect of field strength on the continuous fractionation to be demonstrated.

In spite of the fact that there were significant discrepancies between predicted and experimental θ_{r}, it may be seen from Table 1 that with the current system PS 51k and PS 1000k would be quite well separated from one another on a continuous basis at ΔT of 17 K. With ΔT of 20 K the separation would be even better. The theoretical model presented here will provide guidelines for future interpretation and optimization of separations in continuous 2D-ThFFF. It constitutes a first step in the modeling of the system and it suggests that there may be flow disturbances caused by non-uniform w and/or ΔT in the current system. These are design and construction issues that can be addressed in future instruments. With improved instrumentation, the extension of the model to account for relaxation and dispersion phenomena will be the natural next step.

c | Concentration |

c
_{0}
| Concentration at the accumulation wall |

r | Radial distance |

r
_{i}
| Radial distance of the sample inlet |

r
_{o}
| Outlet radius |

r
_{eff}
| Effective channel radius |

R | Retention ratio |

R
_{r}
| Retention ratio in the radial direction |

R
_{
θ
}
| Angular retention ratio |

t | Time |

t
^{0}
| Void time (V^{0}/) |

T | Absolute temperature |

T
_{c}
| Cold wall temperature |

ΔT | Thermal difference between channel walls |

V
^{0}
| Effective void volume of the channel |

Carrier fluid flow rate (total) | |

〈v_{r}〉 | Mean radial fluid velocity |

w | Channel thickness |

x | Distance from the accumulation wall |

λ | Dimensionless retention parameter |

ξ | Reduced distance from accumulation wall (=x/w) |

θ | Angular displacement |

θ
_{r}
| Final elution angle |

θ
^{0}
| Non-retained elution angle |

φ
_{i}
| Fractional recovery of the polymer collected at port i |

Ω | Angular fluid velocity |

〈Ω〉 | Mean angular fluid velocity |

Ω
_{0}
| Angular velocity of the rotating wall |

- J. C. Giddings, Sep. Sci., 1966, 1, 123 CrossRef.
- J. C. Giddings, Science, 1993, 260, 1456 CAS.
- M. E. Schimpf, Trends Polym. Sci., 1996, 4, 114 CAS.
- L. J. Gimbert, K. N. Andrew, P. M. Haygarth and P. J. Worsfold, Trends Anal. Chem., 2003, 22, 615 CrossRef CAS.
- J. C. Giddings, M. N. Myers, G.-C. Lin and M. Martin, J. Chromatogr., 1977, 142, 23 CrossRef CAS.
- P. C. Wankat, AIChE J., 1977, 23, 859 CrossRef CAS.
- P. C. Wankat, Sep. Sci. Technol., 1984, 19, 801 CrossRef CAS.
- J. C. Giddings, Anal. Chem., 1984, 56, 1258A CrossRef CAS.
- J. C. Giddings, in Unified Separation Science, John Wiley & Sons, Inc., New York, NY, 1991, pp. 112–140 Search PubMed.
- H. H. Strain and J. C. Sullivan, Anal. Chem., 1951, 23, 816 CrossRef CAS.
- A. C. Arcus, A. E. McKinnon, J. N. Livesey, W. S. Metcalf, S. Vaughan and R. B. Keey, J. Chromatogr., A, 1980, 202, 157 CrossRef CAS.
- M. Zborowski, L. R. Moore, S. Reddy, G.-H. Chen, L. Sun and J. J. Chalmers, ASAIO J., 1996, 42, 666 CrossRef PubMed.
- L. R. Moore, M. Zborowski, L. Sun and J. J. Chalmers, J. Biochem. Biophys. Methods, 1998, 37, 11 CrossRef CAS.
- T. Schneider, L. R. Moore, Y. Jing, S. Haam, P. S. Williams, A. J. Fleischman, S. Roy, J. J. Chalmers and M. Zborowski, J. Biochem. Biophys. Methods, 2006, 68, 1 CrossRef CAS PubMed.
- S. Turina, V. Krajovan and T. Kostomaj, Fresenius' J. Anal. Chem., 1962, 189, 100 CrossRef CAS.
- J. C. Giddings, Anal. Chem., 1962, 34, 37 CrossRef CAS.
- M. V. Sussman and C. C. Huang, Science, 1967, 156, 974 CAS.
- M. V. Sussman, K. N. Astill, R. Rombach, A. Cerullo and S. S. Chen, Ind. Eng. Chem. Fundam., 1972, 11, 181 CAS.
- M. V. Sussman, K. N. Astill and R. N. S. Rathore, J. Chromatogr. Sci., 1974, 12, 91 CrossRef CAS PubMed.
- C. D. Scott, R. D. Spence and W. G. Sisson, J. Chromatogr., 1976, 126, 381 CrossRef CAS.
- M. Goto and S. Goto, J. Chem. Eng. Jpn., 1987, 20, 598 CrossRef CAS.
- A. Uretschläger and A. Jungbauer, Bioprocess Biosyst. Eng., 2002, 25, 129 CrossRef PubMed.
- J. C. Giddings, Sep. Sci. Technol., 1985, 20, 749 CrossRef CAS.
- Y. Gao, M. N. Myers, B. N. Barman and J. C. Giddings, Part. Sci. Technol., 1991, 9, 105 CrossRef CAS.
- S. R. Springston, M. N. Myers and J. C. Giddings, Anal. Chem., 1987, 59, 344 CrossRef CAS.
- C. B. Fuh, M. N. Myers and J. C. Giddings, Ind. Eng. Chem. Res., 1994, 33, 355 CrossRef CAS.
- J. C. Giddings, J. Chromatogr., 1990, 504, 247 CrossRef CAS.
- M. N. Myers and J. C. Giddings, Powder Technol., 1979, 23, 15 CrossRef CAS.
- M. R. Schure, M. N. Myers, K. D. Caldwell, C. Byron, K. P. Chan and J. C. Giddings, Environ. Sci. Technol., 1985, 19, 686 CrossRef CAS PubMed.
- C. F. Ivory, M. Gilmartin, W. A. Gobie, C. A. McDonald and R. L. Zollars, Biotechnol. Prog., 1995, 11, 21 CrossRef CAS.
- P. Vastamäki, M. Jussila and M.-L. Riekkola, Sep. Sci. Technol., 2001, 36, 2535 CrossRef PubMed.
- P. Vastamäki, M. Jussila and M.-L. Riekkola, Analyst, 2003, 128, 1243 RSC.
- P. Vastamäki, M. Jussila and M.-L. Riekkola, Analyst, 2005, 130, 427 RSC.
- G. E. Thompson, M. N. Myers and J. C. Giddings, Anal. Chem., 1969, 41, 1219 CrossRef CAS.
- M. E. Hovingh, G. H. Thompson and J. C. Giddings, Anal. Chem., 1970, 42, 195 CrossRef CAS.
- M. Martin and R. Reynaud, Anal. Chem., 1980, 52, 2293 CrossRef CAS.
- J. C. Giddings, M. Martin and M. N. Myers, J. Chromatogr., 1978, 158, 419 CrossRef CAS.
- J. C. Giddings, M. Martin and M. N. Myers, Sep. Sci. Technol., 1979, 14, 611 CrossRef CAS.
- M. Martin, B.-R. Min and M. H. Moon, J. Chromatogr., A, 1997, 788, 121 CrossRef CAS.
- C. A. Rue and M. E. Schimpf, Anal. Chem., 1994, 66, 4054 CrossRef CAS.

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